complex analysis and applications ’13complex analysis and applications ’13 proceedings of...

Complex Analysis and Applications ’13

Proceedings of International ConferenceSofia, October 31-November 2, 2013

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CAA‘13

CAA ’13

BOOK OF ABSTRACTS

Sofia, 2013

Institute of Mathematics and Informatics

Bulgarian Academy of Sciences

This memorial conference is one of the events by which the Institute ofMathematics and Informatics - Bulgarian Academy of Sciences and the co-organizing institutions mark the 100 Anniversary of Academician LjubomirIliev (1913-2000), a long-term Director of the Institute, President of Unionof Scientists in Bulgaria and Union of Bulgarian Mathematicians, Vice-Rector of Sofia University, General Secretary and Vice-President of Bul-garian Academy of Sciences, leading figure in several international organi-zations related to Mathematics, Informatics, Science and Education.

The main organizer is Department “Analysis, Geometry and Topology”,formerly Section “Complex Analysis” – whose founder and head (1962-1988) Acad. L. Iliev was – that included at his time members from bothAcademy’s Institute and Sofia University’s Faculty.

Cite to this book as:

“Complex Analysis and Applications ’13”(Proc. Intern. Conf., Sofia, 2013)

– Book of Abstracts, 92 pp. + i-iv

Editor: Virginia Kiryakova

c© Institute of Mathematics and Informatics, Bulg. Acad. Sci.Sofia, 2013

ISBN 978-954-8986-37-3(Book of Abstracts and CD with Full Length Papers)


Proceedings of International ConferenceSofia, October 31-November 2, 2013

S O F I A 2 0 1 3

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CAA ’13

BOOK OF ABSTRACTS

Sofia, 2013

Institute of Mathematics and Informatics


• International Steering Committee: Blagovest Sendov (Bulgaria),Bogoljub Stankovic (Serbia), Blagoj Popov (Macedonia), Wolfgang Tutschke(Austria), Jozef Siciak (Poland), Hans-Juergen Glaeske (Germany), MouradE.H. Ismail (USA)• International Program Committee: Virginia Kiryakova (Chair),Armen Sergeev (Russia), Ivan Pierre Ramadanoff (France), Stevan Pilipovic(Serbia), Julian Lawrynowicz (Poland), Peter Pflug (Germany), Lyubomir Boy-adjiev (Kuwait), Arpad Takaci (Serbia), Toma Tonev (USA); Bulgaria: PetarPopivanov, Peter Rusev, Ivan Dimovski, Oleg Muskarov, Ralitsa Kovacheva,Johann Davidov, Georgi Ganchev• Local Organizing Committee: Georgi Dimkov (Chair), Donka Pash-kouleva, Valentin Hristov, Rumyan Lazov, Jordanka Paneva-Konovska, GeorgiIliev, Vanja Hadzijski, Nikolay Ikonomov, Yulian Tsankov, Emilia Bazhlekova,Nikolaj Nikolov, Stoyu Barov, Lilia Apostolova, Velichka Milusheva

• The Organizers Acknowledge with Thanks the Support by:

Institute of Mathematics and Informatics as organizing institutionhosting the Conference;

Bulgarian Academy of Sciences and its International Affairs De-partment for supporting the visits of some foreign participants in frames ofthe bilateral agreements for scientific cooperation with corresponding institu-tions from Belgium, Hungary, Russia, Serbia. Specially, we mention the partialsupport under Project “Mathematical Modelling by means of ...” (in which work-ing program the Conference is included) within a Bilateral agreement betweenBulgarian Academy of Sciences and Serbian Academy of Sciences and Arts;

Union of Scientists in Bulgaria as organizing institution supporting andsponsoring the Conference;

Faculty of Mathematics and Informatics – Sofia University “St.Kliment Ohridski” and Union of Bulgarian Mathematicians for theirsupport as co-organizing institutions;

Colleagues who supported the Conference as private persons.

Academician Ljubomir Iliev

(1913 – 2000)

“Complex Analysis and Applications ’13”

International Memorial Conference for the 100th Anniversaryof Birth of Academician Ljubomir Iliev

IMI – Sofia, 31 October - 2 November 2013e-mail: [email protected]

Website: http://www.math.bas.bg/complan/caa13

organized by

Institute of Mathematics and Informatics (IMI)– Bulgarian Academy of Sciences (BAS)

and Union of Scientists in Bulgaria (USB)

in collaboration with:

Faculty of Mathematics and Informatics (FMI)– Sofia University (SU) ”St. Kliment Ohridski”

and Union of Bulgarian Mathematicians (UBM)

This conference is one of the events by which the Institute of Mathematicsand Informatics - Bulgarian Academy of Sciences the co-organizing institutionsmark the 100 Anniversary of Academician Ljubomir Iliev (1913-2000), a long-term Director of the Institute (IMI), President of USB and UBM, Vice-Rectorof SU, Vice-President of BAS, etc.

The main organizer is Section ”Analysis, Geometry and Topology”, the for-mer ”Complex Analysis” that included then members from both IMI-BAS andFMI-SU, and whose founder and head (1962-1988) Acad. L. Iliev was.

International Steering Committee:

Blagovest Sendov (Bulgaria), Bogoljub Stankovic (Serbia), Blagoj Popov(Macedonia), Wolfgang Tutschke (Austria), Jozef Siciak (Poland),

Hans-Juergen Glaeske (Germany)

Topics:

Functions of One Complex Variable; Several Complex Variables and ComplexGeometry; Special Functions and Integral Transforms; Fractional and Opera-tional Calculi; Real and Functional Analysis; Geometry and Topology; Metho-dology of Science and Education; Varia in Analysis, Differential Equations,Applications


(Proc. of International Conference, Sofia, 31 Oct.-2 Nov. 2013)

Preface

ACADEMICIAN LJUBOMIR ILIEV AND

THE DAY OF NATIONAL LEADERS

Virginia Kiryakova

Academician Ljubomir Iliev was born on April 20, 1913. The centenary ofhis birthday was celebrated partly by the Bulgarian mathematical communityyet in April, in the frames of the traditional Spring Conference of the Unionof Bulgarian Mathematicians. However, the section “Analysis, Geometry andTopology” as a successor of the “Complex Analysis” section founded by him,chose to host this memorial conference at the Institute (whose building of 1972 isconsidered to a great extent as his personal achievement) and to open it exactlyon November 1, the Bulgarian Day of National Leaders.

The Day of National Leaders, called also National Revival Day, is a Bul-garian national holiday celebrated each year on November 1. It is to honor theleaders of the National Revival period, the Bulgarian educators, revolutionaries,spiritual mentors and scholars. A ritual raising of the national flag and officialchange of guards happen in front of the main entrance of the Presidential Ad-ministration, together with festival events, parades and torchlight processionsorganized by the universities, scientific institutions, religious and spiritual cen-ters over the country. It is a festival of the historical memory and of our nationalself-confidence standing for year after year during the centuries of slavery, vio-lence and people’s suffering under foreign oppression, led and supported morallyby these great men and women.

The Day of National Leaders arose in the difficult time of spiritual ruin afterthe First World War. The Bulgarian society collapsed the Renaissance ideals.For many, it was clear the real threat of disintegration of our national values.At that time Bulgarians chose the experience of their society, and stared at thegreat leaders of the Bulgarian spiritual past to find the way back to their equilib-rium and stability as a nation. For the first time, this holiday was celebrated inthe town of Plovdiv in 1909. Until then and nowadays, the date of November 1is celebrated (in calendar’s old style) as the Day of St. Ivan (John) Rilski (also,a commemoration of All Saints, or known as All Hallows’ Day in many countries

c© 2013 IMI – BAS, Sofia

throughout the world). He is honored as the patron-saint of the Bulgarian peopleand as one of the most important saints in the Bulgarian Orthodox Church.In 1922, the National Assembly declared this holiday for all “deserving Bul-garians”: “Let the day of St. John of Rila be a Day of National Leaders incelebration of the greatest Bulgarians, to awake in young people the good senseof existence and interest towards the figures of our past...”. Since 1945 thesecelebrations have been temporarily interrupted and afterwards, revived by anact adopted by the 36th National Assembly in 1992, to resume the tradition ofthe feast. But yet since 1991 the Union of Scientists in Bulgaria has adoptedthe National Leaders Day also as a Day of Bulgarian Science ! This Union isa co-organizer of our memorial Conference and we like to acknowledge theirsponsorship.

Among the most popular Bulgarian national leaders are St. Ivan (John) ofRila, St. Paisius of Hilendar (Paisij Hilendarski), Sofronij Vrachanski, GregoryTsamblak, Konstantin Kostenechki, Vladislav the Grammarian, Matthew theGrammarian, Neophyte Bozveli, brothers Dimitar and Konstantin Miladinovi,Georgi Sava Rakovsky, Vasil Levski, Hristo Botev, Stefan Karadza, Hadji Dim-itar, Ljuben Karavelov, Dobri Chintulov, Ivan Vazov, and many others. Andamong them, the name of Academician Ljubomir Iliev finds place without anydoubt !

He was born in the town of Veliko Tarnovo (with a meaning “Great” Tarnovo)referred also as the “City of the Tsars” (Emperors) and being the historical cap-ital of the Second Bulgarian Empire (1185 – 1396). In the Middle Ages, the citywas among the main European centres of culture and gave its name to thearchitecture of the Tarnovo Artistic School, painting of the Tarnovo ArtisticSchool and literature; a quasi-cosmopolitan city, with many foreign merchantsand envoys (incl. Armenians, Jewishes and Roman Catholics) besides a domi-nant Bulgarian population. May be, this town’s origin was one of the reasons forIliev to feel himself not only as a Bulgarian (proving by all his life and contribu-tions to be a “deserving” one), but also a citizen of the World. He graduated themale secondary school there, then finished his mathematical education in 1936at the Physics-Mathematics Department of Sofia University. L. Iliev had been ateacher in a Sofia secondary school, worked as Assistant Prof. (1941), AssociateProf. (1947), Full Prof. (1952) and head of “Advanced Analysis” Dept. (since1952) at the University. Obtained his PhD degree in 1938, and Dr.Sc. - in 1958,became a Corresponding Member of Bulgarian Academy of Sciences (BAS) in1958, and Member of Academy since 1967. He was the Director of our Institute(1964-1988); General scientific secretary for a long time and Vice-president ofBAS (1968-1973); Vice-Rector of the Sofia University; long-term President of theUnion of Scientists in Bulgaria and of the Union of Bulgarian Mathematicians;etc.

Among the topics of his scientific interests were: analytic and entire func-tions theory, zeroes of polynomials, univalent functions, analytical non-expenda-bility of series, methodology of science and education, development of computerscience, etc. He is author of a great number of mathematical papers, mono-graphs and university textbooks, see List of his publications in this volume.

The scientific contributions and the impressive international activities ofAcad. Iliev gave him a series of worldwide recognitions, among them – foreignmemberships of Soviet (now Russian) Academy of Sciences, of the German Acad-emy (then of German Democratic Republic), of the Hungarian Academy; DoctorHonoris Causa of Technical University - Drezden; President of the Council of theInternational Mathematical Center “St. Banach” - Warsaw (1974-1977); Chairof Balkan Mathematical Union (now inherited by MASSEE), Vice-President ofInternational Federation for Information Processing; etc, etc.

But let us stress in this note on some of his contributions to the Bulgarianculture, education and science as a human consciously being Bulgarian and be-longing to Bulgaria. Enormous are his activities and achievements in favor ofthe modern education in Mathematics and Informatics in our country, both inschools and universities. Just to mention the creation of the specialized mathe-matical high schools, the special attention arranged to the talented pupils, theintroduction of the 3-cycles of higher education qualifications (Bachelor - Mag-ister - Ph.D.) in Sofia University yet in 1970, long time before other Europeanuniversities introduced it (as the so-called Bologna process, since 1999), etc.

It is common to speak about “Iliev’s era” in the development of Bulgarianmathematics. Among many other ideas and achievements, he devoted muchefforts to his goal to develop a wide range of research topics in the Institute, aswide as to cover almost all items in Mathematics Subject Classification. It wasan era when this Institute (named either Mathematical Institute with Comput-ing Centre, or Institute of Mathematics and Mechanics, or enlarged as UnitedCentre for Mathematics and Mechanics including also the corresponding Fac-ulty of Sofia University) incorporated several departments with more than 500scholars in all areas of pure and applied mathematics, mechanics and computersciences.

Iliev had his leading role in introducing the Computer Science and Compu-tational Technique in Bulgaria, developed in frames of Mathematical Science.The first Bulgarian Computing Centre (1961), the first Bulgarian Computer“Vitosha” (1963), the Bulgarian calculator “Elka” (1965) - the 4th electroniccalculator in the world (after the British, Italian and Japanese, and the firstone (!) executing square root function) – are all projects initiated and estab-lished under his close guidance, sometimes with the risque he took on himselfto argue with the political officials, as “Cybernetics” has been considered thenas a wrong Western influence. Along with construction of the hardware, Acad.

Iliev’s project included a well planned care to create the necessary scholars forthe Informatics era, introducing courses in numerical mathematics and math-ematical programming, arranging scholarships and PhD studies abroad for the“new people” necessary for the “new” era of Information Technologies.

Most of Iliev’s power was hidden in his organizational abilities and activities.He was a real leader of the mathematical life in Bulgaria, and did this with agift from nature and endless enthusiasm. Just to mention his policy to organizein Bulgaria a series of big international mathematical congresses, followed bymore specialized (topical) international conferences, where in the time of theIron curtain, mathematicians from both Eastern and Western countries couldcome to Bulgaria to meet and exchange ideas and experience. This was a realphenomenon, not commonly possible in that era. Thus, the Bulgarian mathe-matics, the country’s culture, the old history and the state itself, were not onlyopened to the World but could serve as a bridge between its two parts.

Among the hobbies of Acad. Iliev related to Bulgarian history and culture,it is worth to mention that he was a devoted numismatist. This was not only tocollect coins, but to touch sources of authentic information on our history andculture, a possibility to develop his own theories and hypotheses. Another trendof dedication to the Bulgarian history and culture, was his initiative to namemathematical journals and books series founded by him, after purely Bulgarianwords from history, having nothing in common (at first sight) with their mathe-matical contents. Such are the Bulgarian Mathematical Journal “Serdica”, theProceedings “Pliska” (both after old Bulgarian capitals), the series of mathe-matical monographs “Az Buki” (with the meaning: the first 2 letters of theBulgarian alphabet), etc.

Finally, let us say few words of acknowledgements to Acad. L. Iliev of behalfof our section, as he founded it and was its first Head (during 1962-1988). The“Complex Analysis” (CA) section was one of the first departments to form thestructure of Institute of Mathematics and Informatics (IMI) at Bulgarian Acad-emy of Sciences (BAS) with clearly specialized subjects and serious scientific po-tential. It was among the departments that inherited the department “AdvancedAnalysis”, directed by Acad. L. Tchakalov until 1962. In 1962 the departments“Complex Analysis” (with a head Corr. Member of BAS (then) L. Iliev), “Realand Functional Analysis” and “Differential Equations” were formed. From thebeginning of its independent existence, the department “CA” achieved signifi-cant development both thematically and staff wise. Along with the traditionaltopics from the classical function theory - geometric function theory, distributionof zeroes of entire and meromorphic functions - new trends as several complexvariables functions, complex geometry, special functions, integral transforms,fractional and operational calculi found place. Since 2010, Section “CA” joinedwith the former IMI sections “Real and Functional Analysis” and “Geometry

and Topology”, and is presently named as a new section “Analysis, Geometryand Topology” (AGT) – the Organizer of this memorial Conference. Severalscientific groups work now in the following directions: Functions of One Com-plex Variable; Functions of Several Complex Variables and Complex Geometry;Transform Methods, Special Functions, Fractional and Operational Calculi; Ge-ometry and Topology, etc. Currently, the section “AGT” consists of more than20 members, being one of the most numerous departments at IMI.

Under the initiation and guidance of Academician L. Iliev, our departmentorganized the series of international conferences “Complex Analysis and Appli-cations” in the town of Varna (on Black Sea), held in 1981, 1983, 1985, 1987,1991, where a great number of foreign and Bulgarian mathematicians took part.Thus the name of the present meeting has been chosen as a re-make of theseconferences and to commemorate the role of their chairman. Here is to mentionthat Acad. Iliev had remarkable skills in organization of scientific events in localand international aspects. But he also left traditions and devotedly taught usand our colleagues how such congresses and conferences should be organized.

And if you have been satisfied by the organization of this “CAA ’13” meeting,our aim has been to make it a worthy analytical continuation (although insmaller scale of participants) of the previous “CAA” conferences ...

Chair of International Program Committee of “CAA ’13”and on behalf of the “AGT” section –

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences“Acad. G. Bontchev” Str., Block 8, Sofia – 1113, BULGARIA

Haut-relief of Acad. Ljubomir Ilievopened Nov. 1, 2013 on the entrance facade of Institute



ACADEMICIAN LJUBOMIR ILIEV – LEADER

OF THE BULGARIAN MATHEMATICAL COMMUNITY

(ON THE OCCASION OF HIS CENTENARY)

Blagovest Sendov

Abstract

As his student and close collaborator for many years, I have had the opportu-nity on many occasions to speak and write about the rich and fruitful activitiesof Academician Ljubomir Iliev, a leading Bulgarian mathematician and a leaderof the Bulgarian mathematical community, for more details see for example [1]- [4].

On the occasion of his centenary, it is natural to try to evaluate his achieve-ments in perspective. Among the Bulgarian mathematicians, active during themiddle of the last century, Ljubomir Iliev takes the place after Nikola Obreshkoffand Ljubomir Tschakaloff. His results in complex analysis, namely on schlicht(univalent) functions, analytic inextensibility and overconvergence of series, dis-tributions of the zeros of polynomials and entire functions, having integral rep-resentation, the inequality of Poul Turan and others, are part of the contempo-rary mathematics and are cited by today’s researchers. All this is enough to callAcademician Ljubomir Iliev one of the leading Bulgarian mathematicians in allhistory.

MSC 2010 : 01A60, 01A70, 97-XX, 68-03, 30-XXKey Words and Phrases: history of mathematics - 20th century; functions of

a complex variable; secondary schools and university education in mathematicsand computer science; development of mathematics, computer science, infor-mation processing, electronics and computer industry in Bulgaria; electroniccalculator; first Bulgarian computer

*This is really true and a generally accepted fact, but Ljubomir Iliev was not

only a leading Bulgarian mathematician, we may name among many others.


What is very important and specific is, that Academician Ljubomir Iliev wasan outstanding leader of the Bulgarian mathematical community. He had thevision and the ambition to work and organize, during his long active live, allavenues of the mathematical developments in Bulgaria. Today, after the radicalpolitical and economic changes in our country, not everything achieved in thepast is evaluated without personal emotions. We shall try to present severalthings connected with the name of Ljubomir Iliev, which are not disputable.

His first important contribution is in the development of Bulgarian math-ematical education at all levels. As a Vice Rector of the University of Sofia“St. Kliment Ohridski”, Ljubomir Iliev made a very important step for improv-ing the quality of higher education in mathematics for preparing professionalmathematicians and researchers. In 1950, in the Physics and Mathematics Fac-ulty of the University were formed special groups for professional mathemati-cians, which were in fact a magistrature. In addition, with the decisive help ofLjubomir Iliev, 30 secondary schools specializing in mathematics were openedall over the country and one National mathematical gymnasium was openedin Sofia. Up to now, these specialized mathematical schools have played anextremely important role in maintaining the quality of secondary education.There have been many attempts to close these elite schools, or transform thembut, they always manage to survive.

A very fruitful activity of Ljubomir Iliev is associated with the BulgarianMathematical Union. He helped, together with many other mathematicians andphysicists, with the re-establishment of the Bulgarian Physics and MathematicsSociety and the separation, afterwards into two unions. A unique character-istic of the Bulgarian Mathematical Union, inspired by the tradition and theleadership of Ljubomir Iliev is the unity of all Bulgarian mathematicians in asingle professional organization, combining teachers in secondary schools andthose working in the universities and in the Bulgarian Academy of Sciences.A demonstration of this unity is shown in the traditional Spring MathematicalConferences, where students, teachers and researchers meet together every yearin April.

Academician Ljubomir Iliev was the initiator and supporter of the develop-ment of informatics and computer science in Bulgaria. He started by proposingin 1959 the creation of the first course in numerical analysis, and after 1961,many other courses, which prepared the first specialists in applied mathematicsand programming in Bulgaria.

One of the main projects of Ljubomir Iliev was the establishment of theFirst Bulgarian computer center, created in 1962 jointly under the BulgarianAcademy of Sciences and the University of Sofia “St. Kliment Ohridski”. It

took a tremendous efforts and organizational talent to select a group of engi-neers, mathematicians an technicians to build the first Bulgarian digital elec-tronic computer, which became operational in 1963. These pioneering effortsturned out to be the basis for the development of the Bulgarian electronics andcomputer industry. It is mostly forgotten today, that in the First Bulgariancomputer center was designed and built one of the first electronic calculators inthe world called “Elka”. This electronic calculator was advertised in The Finan-cial Times and sold in Great Britain in 1968. “Elka” was the step to the popularin the former communist countries electronic computer “Pravetz”, produced inthe electronic factories in the Bulgarian town Pravetz.

As a pioneer in the development of information processing, Ljubomir Ilievbelieved that eventually, Informatics will become a natural part of Mathematics.He was very active in international cooperation and became Vice President ofthe International Federation for Information Processing (IFIP). The Bulgarianmembership in IFIP was a very good opportunity for many Bulgarian specialiststo be in contact with the world leaders in the field.

As the General scientific secretary of the Bulgarian Academy of Sciencesand for long time the Vice President of the Academy and Director of the Math-ematical Institute (nowadays, Institute of Mathematics and Informatics - Bulg.Acad. Sci.), Ljubomir Iliev used all his influence for the benefit of the Bulgar-ian mathematical community. First of all, he introduced a concrete plan for thestructure of the Mathematical Institute based on a theory for the structure ofmathematical science itself. His ambition was to open opportunities in our coun-try for the development of all branches of contemporary mathematics. LjubomirIliev defended the necessity to build a separate building for a big Institute ofMathematics and Informatics and succeeded in finishing this building. It is justa recognition of these efforts, that on the occasion of his centenary, a memorialrelief of Ljubomir Iliev is on the front wall of this building.

Academician Ljubomir Iliev was a devoted patriot. In the center of themotivation for every one of his projects was the benefit for his country. He usedevery opportunity to show that Bulgaria is a country with a rich culture, longhistory, talented people and a prosperous future. He cared especially for theyoung mathematicians, who show capacity for research and leadership. One ofhis popular formulas was: “For the new fields of research - new young people”.Even in the period of the Cold War, Ljubomir Iliev was trying to fulfill hisprinciple about the specialization of young scientists in Bulgaria: “Every youngscientist has to have at least one specialization in the East and one in the West”.A leader is a leader, not because he is on top, but because he cares for the peoplehe leads.

All the activities of Ljubomir Iliev as the leader of the Bulgarian mathe-matical community took place during the so-called totalitarian period. The big

political and economic changes and the complete democratization of the countryis accompanied by emotional criticism of almost everything created during thetotalitarian regime. Nevertheless, everything done by Ljubomir Iliev is for thebenefit of Bulgaria and its value is invariant under every political transforma-tion.

References

[1] Bl. Sendov, The 60-th Anniversary of Professor L. Iliev, In: Math. Struc-tures, Comp. Math., Math Modeling, Bulg. Acad. Sci., Sofia, 1975, 17–27.

[2] Bl. Sendov, Academician Ljubomir Iliev in the occasion of his 70 years,Fiz.-Math. Spisanie, Sofia 29, No 3 (1973), 169–171 (in Bulgarian).

[3] Bl. Sendov, The 80th Anniversary of Academician Ljubomir Iliev, Serdica(Bulg. Math. Publ.), 19, No 2-3 (1993), 91–97.

[4] Bl. Sendov, Acad. Ljubomir Iliev 1913 – 2000, Serdica Math. J. 26, No 3(2000), i–xxviii.

Institute of Information and Communication TechnologiesBulgarian Academy of SciencesSofia – 1113, BULGARIAe-mail: [email protected] Received: August 27, 2013



ACADEMICIAN LJUBOMIR ILIEV

AND THE CLASSICAL COMPLEX ANALYSIS

Peter Rusev

Abstract

Academician Ljubomir Georgiev Iliev was born in 1913, 25 years after theestablishment of the first Bulgarian institution of higher education and 10 yearsafter its renaming as University. Next year the future academicians Kyril Popov,Ivan Tzenov and Ljubomir Tchakalov attained academic ranks as associate pro-fessors at the Faculty of Physics and Mathematics (of the Sofia University).They were successors and followers of the work of the pioneers of the higher ed-ucation in mathematics in Bulgaria Emanuil Ivanov, Marin Batchevarov, AtanasTinterov, Spiridon Ganev and Anton Shourek. Eight years later the future aca-demician Nikola Obrechkov became also their colleague.

Popov and Tchakalov, who had defended their theses at European Univer-sities, transferred the spirit of cultivating the mathematical science from themost prestigious scientific centers. The seminars under their guidance with theactive participation of Obrechkov, being still a student, became ”incubators” ofyoung enthusiasts – future teachers at the Sofia University and at the arisingInstitutions of higher technical education. Among them, was Ljubomir Iliev,one of the most talented of their followers, one of the most brilliant from thethird generation of Bulgarian mathematicians.

On the occasion of the 100th anniversary of the birth of Ljubomir Iliev andthis current remake of the international conferences “Complex Analysis andApplications” (held in Varna, 1981, 1983, 1985, 1987) that were initiated andorganized under his guidance, we try to present a short survey of some of hiscontributions to topics of the classical complex analysis.

MSC 2010 : 30-XX; 30-03; 30B40, 30B50; 30C15; 30C45

Key Words and Phrases: functions of one complex variable; zeros of en-tire functions; analytical non-continuable power and Dirichlet series; classes ofunivalent functions


1. Zeros of entire Fourier transforms

In 1737, L. Euler defined the function ζ by means of the equality

ζ(σ) =∞∑

n=1

1nσ

, σ > 1,

and pointed out the validity of the representation

ζ(σ) =∏

p∈P

(1− 1

pσ

)−1

,

where P is the set of prime numbers.In his memoir Uber die Anzahl der Primzahlen unter einer gegebenen Grosse,

Monatsber. der Konigl. Preuss. Akad. der Wiss. zu Berlin aus dem Jahr 1859(1860), 671–680, B. Riemann extended Euler’s definition by the equality

ζ(s) =∞∑

n=1

exp (−s log n), s = σ + it, σ > 1, t ∈ R.

The functional equation

π−s/2 Γ(s)ζ(s) = π(1−s) Γ(1− s)ζ(1− s), (1)

bearing his name, realizes analytical continuation of this function in the wholecomplex plane as a meromorphic function with a single pole at the point s = 1.From (1) it follows, in particular, that the points −2k, k ∈ N are its simple zeros.

In the same memoir, Riemann stated the famous hypothesis which is neitherproved nor rejected till now, namely that the function ζ, except this zeros namedtrivial, has infinitely many others and that all they are on the line Re s = 1/2.It is equivalent to the hypothesis that the introduced by him entire function

ξ(z) = s(s− 1)π−s/2Γ(s/2)ζ(s), s = 1/2 + z,

has only real zeros.At the end of the 19th century but mostly in the first decades of 20th one,

the efforts of many mathematicians including first-class ones as Jensen, Polya,Hardy and Titchmarsh turn to the problem of zero-distribution of entire func-tions defined as Fourier transforms of the kind

b∫

a

F (t) exp(izt) dt, −∞ ≤ a < b ≤ ∞.

Indisputable motive for their investigations is Riemann’s representation ofthe function ξ in this form with an even function F on the interval (−∞,∞).The first Bulgarian publications in this field, due to Tchakalov and Obrechkov,are influenced by the results of Polya about the zero-distribution of the entirefunctions of the kind

a∫

−a

f(t) exp(izt) dt, 0 < a < ∞ (2)

and of their particular casesa∫

0

f(t) cos zt dt (3)

and a∫

0

f(t) sin zt dt. (4)

This direction becomes a field of intensive studies of academician Iliev.

An essential role in Polya’s investigations plays an algebraic statement, mostfrequently called theorem of Kakeya, saying that if a0 < a1 < · · · < an, n ∈ N,then the zeros of the polynomials

n∑

k=0

akzk

are in the unit disk D := z ∈ C : |z| < 1. Using it and applying the methodof variation of the argument, Polya obtains his famous result for reality andmutually interlacing of the zeros of the entire functions (3) and (4), providedthe function f is positive and increasing in the interval (0, a).

Another approach to the problem of zero-distribution of entire functions ofthe kind (3) and (4) is due to academician Iliev. It is based on his result thatif the zeros of the algebraic polynomial P of degree n ∈ N are in the regionz ∈ C : |z| > 1 and P ∗ is the polynomial defined by P ∗(z) = znP (1/z), thenthe zeros of the polynomial

P (z) + γzkP ∗(z), |γ| = 1, k ∈ N0,

are on the unit circle. This assertion, as well as the successful use of an alge-braic result of N. Obrechkov, lead to one of the most essential achievements ofacademician Iliev. It says that if the function f is positive and increasing in theinterval (0, a), 0 < a < ∞, and the zeros of the algebraic polynomial p are inthe strip z ∈ C : λ ≤ Re z ≤ µ, then the zeros of the polynomial

a∫

0

f(t) p(z + t) + γp(z − t) dt, |γ| = 1, (5)

are in the same strip too. The classical results of Polya can be obtained bysetting p(z) = zn, γ = ±1, and letting n to go to infinity. Indeed, then thepolynomials

Pn(f ; z) =

a∫

0

f(t)(

1 +izt

n

)n

+ γ

(1− izt

n

)ndt, n ∈ N,

have only real zeros and, moreover,

limn→∞Pn(f ; z) =

a∫

0

f(t) exp(izt) + γ exp(−izt) dt

uniformly on each bounded subset of C.

A brilliant realization of one of the most fruitful ideas of academician Ilievconcerns the class E(a) of entire functions (3) having only real zeros. If A(a), 0 <a < ∞, denotes the set of the real functions x(t), t ∈ R, such that x(a) = 0 and,moreover, x′(it), t ∈ R, is a restriction to the real axes of a function of theLaguerre-Polya class, i.e. it is either a real polynomial with only real zeros oran uniform limit of such polynomials. A witty algorithm ensures ”reproduction”of this class. Its first application is that if x(t) ∈ A(a), x(0) > 0, and λ > −1,then the entire function

a∫

0

xλ(t) cos zt dt

has only real zeros. The particular case when x(t) = 1 − t2q, q ∈ N, leads to aresult of Polya saying that the entire function

1∫

0

(1− t2q) cos zt dt

has only real zeros.The next application is one of the most significant achievements of aca-

demician Iliev which states that if ϕ(t), t ∈ R, is a real, nonnegative, and evenfunction, such that ϕ′(it) is a restriction to the real axes of a function from theLaguerre-Polya class, then the entire function

∞∫

0

exp(−ϕ(t)) cos zt dt

has only real zeros. The particular case when ϕ(t) = a cosh t, a > 0, is thewell-known result of Polya for the reality of the zeros of the entire function

∞∫

0

exp(−a cosh t) cos zt dt.

2. Analytically non-continuable power and Dirichlet series

To Weierstrass is due the first example of a convergent power series, namely∞∑

n=0

anzbn, a > 0, b ∈ N, b > 1,

which is non-continuable outside its circle of convergence, i.e. this circle isthe domain of existence for the analytic function defined by its sum. Thisexample became a starting point of a great number of studies on the singularpoints of functions defined by convergent power series and their analytical non-continuability.

The contributions of academician Iliev in this field are obtained mainly un-der the influence of works of such experts in the classical complex analysis, asHadamard, Ostrowski, Fabry and Szego.

Due to Szego is the result that if each of the coefficients of the power series∞∑

n=0

anzn (6)

is equal to one for finitely many complex numbers d1, d2, . . . , ds, dj 6= dk, j 6=k, then it is either analytically non-continuable outside the unit disk, or theMaclorain series of a rational function of the kind

P (z)1− zm

, m ∈ N,

where P is an algebraic polynomial, and this is possible if and only if the se-quence of its coefficients is periodic after some subscript. Essential general-izations, extensions and various modifications of this result are obtained byacademician Iliev. Typical one is the assertion for the series (6) with coeffi-cients of the kind an = γncn, n ∈ N0. If the members of the sequence γn∞0accept finitely many values and for some α ∈ R the sequence cnnα has a finitenumber of limit points all different from zero, then the requirement for non-periodicity after each subscript of the sequence γn is sufficient for analyticalnon-continuability of the series (6).

It seems that academician Iliev was the first who obtained also Szego’s typetheorems for Dirichlet series of the kind

∞∑

n=0

γncn exp(−λns).

3. Univalent functions

In the first decades of the past century, mainly after the works of P. Kobeand L. Bieberbach, a new branch of Geometric Function came into being. Itis known now as Theory of the Univalent Functions. Its main object is the

class S of functions f which are holomorphic and univalent in the unit disk andare normalized by the conditions f(0) = 0, f ′(0) = 1, i.e. the functions withMaclorain expansion of the kind

f(z) = z + a2z2 + a3z

3 + . . .

in the unit disk, and such that f(z1) 6= f(z2) whenever z1 6= z2. Differentsubclasses of S, e.g. defined by additional requirements for convexity of theimage f(D) of the unit disk by means of a function f ∈ S, or by starlikeness ofthis image with respect to the zero point, are also studied. The central object ofthe efforts of a great number of investigators are theorems of deformation andcoefficients estimates. The famous Bieberbach’s conjecture that |an| ≤ n, n ≥ 2,which remained open till its confirmation by Lui de Brange, was one of thestimuli for publishing a great number of papers in prestigious journals and otherpublications. In this field, except Kobe and Bieberbach, many mathematiciansas Littlewood, Hayman, Lewner, Szego, Golusin and others have remarkablecontributions.

Academician Iliev did not remain indifferent to this direction of studies. Theresults, published in his papers devoted to the univalent functions, weer createduring a very short period. The main attention of their author was directed tothe class Sk of the k-symmetric functions fk from the class S, i.e. to those ofthem having Maclorain expansion of the kind

fk(z) = z + a(k)1 zk+1 + a

(k)2 z2k+1 + . . . . (7)

One of the first results of academician Iliev is influenced by a theorem ofSzego about the divided difference of the functions in the class S. The suc-cessful use of a similar theorem of Goluzin for the class Σ of the functions fmeromorphic and univalent in the region C \ D ∪ ∞ and normalized byf(∞) = ∞, f ′(∞) = 1, leads him to the inequalities

1− r2

(1 + r2)4/k≤

∣∣∣∣fk(z1)− fk(z2)

z1 − z2

∣∣∣∣ ≤1

(1− r2)(1− rk)4/k(8)

for each function fk ∈ Sk provided that 0 < |zj | ≤ r < 1, j = 1, 2, z1 6= z2. Itsapplication leads to the result that the exact radius of univalence of the partialsums

σ(2)n = z + a

(2)3 z3 + · · ·+ a

(2)2n+1z

2n+1, n = 1, 2, 3, . . . (9)

of the function from the class S2, i.e. the class of odd functions in S, is equalto 1/

√3.

Another application is that the partial sum

σn(z) = z + a2z2 + · · ·+ anzn

of a function from the class S is univalent in the circle |z| < 1 − 4 log n/n foreach n ≥ 15 which improves a result of V. Levin. Similar result for the partialsums (9) of the functions from the class S2 is that they are univalent in thecircle |z| <

√1− 3 log n/n for n ≥ 12.

The problem for the radius of univalence of the partial sums

σ(3)n (z) = z + a

(3)1 z4 + · · ·+ a(3)

n z3n+1

of the 3-symmetric functions is also treated by of academician Iliev. As a resultit is obtained that it is 3

√3/2 for n 6= 2. The proof is based on coefficients

estimates for the functions from the class S3 as well as on the left inequality in(9). For n = 2 its exactness is proved directly by the method of Lowner.

The inequality∣∣∣∣fk(z1)− fk(z2)

z1 − z2

∣∣∣∣ ≥1− r2

(1 + r2)2/k, |zj | ≤ r, 0 < r < 1, z1 6= z2

is obtained under the additional assumption that the function fk is convex. Itis exact for k = 1, 2, i.e. for the functions from the class S as well as for theodd functions in this class. By its help the circle defined by the inequality|z| < 1− (1 + 2/k) log(n + 1)/(n + 1)1/k is found, where the partial sum

z + a(k)1 zk+1 + · · ·+ a(k)

n znk+1

of a function fk ∈ Sk is univalent for n > exp(k√

2k/(2 + k))− 1.An inequality for the divided difference of bounded functions in the class Sk

is obtained, i.e. for the functions fk ∈ Sk such that fk(D) is a bounded domain.

The already mentioned contributions of academician Iliev and many others,e.g. for the inequality of Hamburger and Turan, for the problem of Pompeiuas well as for the numerical method based on the Newton iterations assignedhim a merited position of one of the distinguished experts in actual areas ofmathematical analysis where his effort has been directed during several decadesin the past century.

Institute of Mathematics and InformaticsBulgarian Academy of Sciences”Acad. G. Bontchev” Str., Block 8Sofia – 1113, BULGARIA

e-mail: [email protected] Received: September 5, 2013

Complex Analysis and Applications '13(Proc. of International Conference, Soa, 31 Oct.-2 Nov. 2013)

LIST OF PUBLICATIONSOF ACADEMICIAN LJUBOMIR ILIEV

Composed by Donka PashkoulevaInstitute of Mathematics and Informatics


Application 1 List of Scientic Publications of Acad. L. Iliev

1938[1] Uber die Nullstellen gewisser Integralausdrucke. Jahresber. Dtsch. Math.-

Ver., 48 (1938) (9/12), 169-172 (German).1939

[2] Several theorems on the distribution of zeros of polynomials. Jubileeproc. of physical-mathematical society, part II, 1939, 60-64 (Bulgarian).

[3] Uber die Nullstellen einiger Klassen von Polynomen. Tohoku Math. J.,45 (1939), 259-264 (German).

1940[4] On the zeros of some classes of polynomials and entire functions. Soa,

1940, 28 p. (Bulgarian). 1942[5] Some elementary criteria for indecomposability of one polynomial of 3rd

degree. J. Phys.-Math. Soc, 27 (1942), No 9-10, 294-298 (Bulgarian).[6] Uber trigonometrische Polynome mit monotoner Koezientenfolge. An-

nuaire Univ. Soa, Fac. Sci., 38 (1942), No 1, 87-102 (Bulgarian, Germansummary).

1943[7] Einige Probleme uber nichtgleichmaessig gespannte ebene Membranen.

Annuaire Univ. Soa, Fac. Sci., 39 (1943), No 1, 251-286 (Bulgarian,German summary).

[8] Uber das Gleichgewicht von elliptischen Membranen. Annuaire Univ.Soa, Fac. Sci., 39 (1943), No 1, 409-426 (Bulgarian, German summary).

[9] Book of mathematical problems (with A. Mateev) Soa, 1943 (Bulgar-ian).

[10] Uber trigonometrische Polynome mit monotoner Koezientenfolge. Jah-resber. Dtsch. Math.-Ver., 53 (1943), 12-23 (German).

c© 2013 IMI BAS, Soa

1945[11] Algebra. Textbook for 6th class of secondary school (with L. Chakalov

and A. Mateev). 1st edition, Soa, 1945 (Bulgarian).[12] An integral property of functions of two variables (with H. Ya. Hristov)

Phys.-Math. J., (1945), No 5-6, 37-40. (Bulgarian).[13] On some problems in the education in mathematics in Bulgaria. Phys.-

Math. J., 29 (1945), No 1-2, 48-52 (Bulgarian).[14] Uber die Verteilung der Nullstellen einer Klasse ganzer Funktionen. An-

nuaire Univ. Soa, Fac. Sci., 41 (1945), 31-42 (Bulgarian).1946

[15] Algebra. TeXtbook for 5th class of secondary school (with N. Obrechkovand I. Nedyalkov). Soa, 1946 (Bulgarian).

[16] Analytisch nichtfortsetzbare Potenzreihen. Annuaire Univ. Soa, Fac.Sci., 42 (1946), No 1, 67-81 (Bulgarian. German summary).

[17] Uber ein Problem von D. Pompeiu. Annuaire Univ. Soa, Fac. Sci., 42(1946), No 1, 83-96. (Bulgarian, German summary).

[18] Book of mathematical problems with their solutions (with A. Mateev).Soa, 1946 (Bulgarian).

1947[19] On a boundary problem. Proceedings of National Assembly of Culture,

3 (1947) , No 1, 81-88 (Bulgarian).[20] Uber die in der Umgebung der Abzisse der absoluten Konvergenz einer

Klasse Dirichletscher Reihen zugehorige singulare Stellen. Annuaire Univ.Soa, Fac. Sci., 43 (1947), No 1, 239-267 (Bulgarian).

[21] Book of mathematical problems with their solutions, part 1 algebra,part 2 geometry (with L. Chakalov and A. Mateev). 1st edition, Soa,1947 (Bulgarian).

1948[22] Uber die Verteilung der Nullstellen einer Klasse ganzer Funktionen. An-

nuaire Univ. Soa, Fac. Sci., 44 (1948), No 1, 143-174 (Bulgarian).[23] Examples of the development of math. analysis which show the nature

of mathematical creativity. Annuaire Univ. Soa, Fac. Sci., 44 (1948),No 1, 83-103 (Bulgarian).

[24] Beitrag zum Problem von D. Pompeiu. Annuaire Univ. Soa, Fac. Sci.,44 (1948), No 1, 309-316 (Bulgarian).

[25] Analytisch nichtfortsetzbare Potenzreihen. C. R. Acad. Bulg. Sci., 1(1948), No 1, 25-28 (German).

[26] Beitrag zum Problem von D. Pompeiu. Bull. Sect. Sci. Acad. Repub.Pop. Roum. 30, (1948). 613-617 (German).

[27] Sur une classe de fonctions a zeros reels. C. R. Acad. Bulg. Sci., 1(1948), No. 2-3, 15-18 (French).

[28] Uber die Verteilung der singularen Stellen einer Klasse DirichletscherReihen in der Umgebung der Konvergenzgeraden. C. R. Acad. Bulg.Sci. , 1 (1948), No. 2-3, 19-22 (German).

1949[29] Zur Theorie der schlichten Funktionen. Annuaire Univ. Soa, Fac. Sci.,

45 (1949), No 1, 115-135 (Bulgarian).[30] Application of a theorem of G. M. Goluzin on univalent functions. Dok-

lady Akad. Nauk SSSR (N.S.), 69 (1949), 491-494 (Russian).[31] Anwendung eines Satzes von G. M. Golusin uber die schlichten Funktio-

nen. C. R. Acad. Bulg. Sci., 2 (1949), No 1, 21-24 (German).[32] Ganze Funktionen mit lauter reellen Nullstellen. C. R. Acad. Bulg. Sci.,

2 (1949) , No 1, 17-20 (German).[33] Sur un probleme de M. D. Pompeiu. Annuaire Univ. Soa, Fac. Sci., 45

(1949), No 1, 111-114 (French).[34] Uber die Nullstellen einer Klasse von ganzen Funktionen. C. R. Acad.

Bulg. Sci., 2 (1949), No 2/3, 9-11 (German).1950

[35] Uber die Abschnitte der schlichten Funktionen, die den Kreis |z| < 1konvex abbilden. Annuaire Univ. Soa, Fac. Sci., 46 (1950), No 1,153-159 (Bulgarian, German summary).

[36] Uber die Newtonschen Naherungswerte. Annuaire Univ. Soa, Fac. Sci.,46 (1950), No 1, 167-171 (Bulgarian, German summary).

[37] Lectures on elementary algebra (textbook for correspondent students atthe University). Soa, 1950 (Bulgarian).

[38] On nite sums of univalent functions. Doklady Akad. Nauk SSSR (N.S.),70, (1950), 9-11 (Russian).

[39] Satze uber die Abschnitte der schlichten Funktionen. Annuaire Univ.Soa, Fac. Sci., 46 (1950), No 1, 147-151 (Bulgarian).

[40] Uber die 3-symmetrischen schlichten Funktionen. Annuaire Univ. Soa,Fac. Sci., 46 (1950), No 1, 161-165 (Bulgarian, German summary).

[41] Uber die Abschnitte der 3-symmetrischen schlichten Funktionen. C. R.Acad. Bulg. Sci., 3 (1950) (1951), No 1, 9-12 (German).

1951[42] Algebra. Textbook for 10th classes of secondary school (with L. Buneva

and D. Shopova). 1st edition, Soa, 1951 (Bulgarian).[43] On three fold symmetric univalent functions. Dokl. Akad. Nauk SSSR,

N.S., 79 (1951), 9-11 (Russian).[44] Uber die Abschnitte der 3-symmetrischen schlichten Funktionen. C. R.

Acad. Bulg. Sci., 3 (1951), No 1, 9-12 (German, Russian summary).[45] Uber die Abschnitte der schlichten Funktionen. Acta Math. Acad. Sci.

Hung., 2 (1951), 109-112 (German, Russian summary).

1952[46] Theorems on triply symmetric univalent functions. Doklady Akad. Nauk

SSSR (N.S.), 84 (1952), No 1, 9-12 (Russian).[47] Schlichte Funktionen, die den Einheitskreis konvex abbilden. C. R. Acad.

Bulg. Sci., 5 (1952)(1953), No 2-3, 1-4 (German).1953

[48] Analytically noncontinuable series of Faber polynomials. Bulg. Akad.Nauk., Izv. Mat. Inst., 1 (1953), No 1, 35-56 (Bulgarian).

[49] On triply symmetric univalent functions. Bulg. Akad. Nauk., Izv. Mat.Inst., 1 (1953), No 1, 27-34 (Bulgarian).

[50] Series of Faber polynomials whose coecients assume a nite number ofvalues. Doklady Akad. Nauk SSSR (N.S.), 90, (1953), No 4, 499-502(Russian).

[51] Book of problems in algebra, part 1, for 4th and 5th classes of secondaryschool (with A. Mateev and P. Stambolov), 1st edition, Soa, 1953 (Bul-garian).

1954[52] Textbook of elementary mathematics algebra (with Sp. Manolov), 1st

edition, Soa, 1954, (Bulgarian).[53] Trigonometrische Integrale, die ganze Funktionen mit nur reellen Null-

stellen darstellen. Bulg. Akad. Nauk., Izv. Mat. Inst., 1 (1954), No 2,147-153 (Bulgarian).

1955[54] Theorem on the univalence of nite sums of triply symmetric univalent

functions. Dokl. Akad. Nauk SSSR (N.S.), 100 (1955), No 4, 621-622(Russian).

[55] Book of problems in elementary mathematics (with L. Chakalov, A. Ma-teev and Sp. Manolov), Soa, 1955 (Bulgarian).

[56] On the dierence quotient for bounded univalent functions. Dokl. Akad.Nauk SSSR (N.S.), 100 (1955), No 5, 861-862 (Russian).

[57] Uber trigonometrische Integrale, welche ganze Funktionen mit nur reellenNullstellen darstellen. Acta Math. Acad. Sci. Hung., 6 (1955) 191-194(German, Russian summary).

1956[58] Some results on the investigations of the distributions of zeros of entire

functions. Uspekhi Mat. Nauk, 11 (1956), No 5, 76 (Russian).1957

[59] On the analytic noncontinuability of power series. Thesis, Soa, 1957(Bulgarian).

[60] Ein Satz uber analytische Nichtfortsetzbarkeit von Potenzreihen. C. R.Acad. Bulg. Sci., 10 (1957), 447-450 (German).

1958[61] Textbook of elementary mathematics arithmetic. 1st edition, Soa,

1958 (Bulgarian).[62] Textbook of elementary mathematics trigonometry (with Sp. Manolov).

1st edition, Soa, 1958 (Bulgarian).1959

[63] On the analytic noncontinuability of power series. Annuaire Univ. Soa,Fac. Sci., 52 (1959), No 1, 1-22 (Bulgarian).

[64] On the analytic noncontinuability of power series. Dokl. Akad. NaukSSSR, 126 (1959), 13-14 (Russian).

[65] International congress of Mathematics in Edinburgh. Phys.-Math. J., 2(1959), No 1, 38-45 (Bulgarian).

[66] Eine Bedingung fur die Nichtfortsetzbarkeit der Potenzreihen. Bulg.Akad. Nauk., Izv. Mat. Inst., 3 (1959), No 2, 205-211 (Bulgarian).

1960[67] Eine Klasse von analytisch nichtfortsetzbaren Potenzreihen. Bulg. Akad.

Nauk., Izv. Mat. Inst., 4 (1960), No 2, 153-159 (Bulgarian).[68] On the state and needs of mathematical science in Bulgaria. J. Bulg.

Acad. Sci., (1960), No 3, 3-23 (Bulgarian).[69] Colloquium on Theory of series in Budapest. Phys.-Math. J., 3 (1960),

No 1, 71 (Bulgarian).[70] Analytische Nichtfortsetzbarkeit und Ueberkonvergenz einiger Klassen

von Potenzreihen. Mathematische Forschungsberichte. XII VEB Deut-scher Verlag der Wissenschaften, Berlin, 1960, 61 p. (German).

1961[71] Academician M. V. Keldysh elected for President of the Academy. J.

Bulg. Acad. Sci., 6 (1961), No 2, 84-86. (Bulgarian)[72] Analytic noncontinuability and overconvergence of certain classes of po-

wer series. Soa, Bulg. Acad. Sci., 1961., 70 p. (Bulgarian).[73] Fifth congress of Austrian mathematicians. Phys.-Math. J., 4 (1961),

No 1, 71 (Bulgarian).1962

[74] Penetrating of mathematics into other sciences. J. Bulg. Acad. Sci., 7(1962), No 1-2, 43-52 (Bulgarian).

[75] Colloquium on Theory of functions and functional analysis in Bucharest.Phys.-Math. J., 5 (1962), No 4, 316 (Bulgarian). [76] Congress of IFIP.Phys.-Math. J., 5 (1962), No 4, 315 (Bulgarian).

[76] The research activity of Bulg. Acad. Sci. for 1961 J. Bulg. Acad. Sci.,7 (1962), No 1-2, 18-30 (Bulgarian).

1963[77] Academician L. Chakalov life and work. Phys.-Math. J., 6 (1963), No

2, 123-129 (Bulgarian).[78] On the signicance of the computer techniques and on its application in

Bulgaria. J. Bulg. Acad. Sci., 8 (1963), No 1, 65-74 (Bulgarian).[79] Konvergente Abschnittsfolgen C-summierbarer Reihen. Rev. Math.

Pures Appl. (Bucarest), 8 (1963), 349-351 (German).[80] Uber Newton'sche Iterationen (with K. Dochev). Wiss. Z. Tech. Univ.

Dresden, 12 (1963), 117-118 (German).1964

[81] Meeting on Methods on automatic programming and computer languages(with Bl. Sendov). Phys.-Math. J., 7 (1964), No 1, 71 (Bulgarian).

[82] Turan'sche Ungleichungen C. R. Acad. Bulg. Sci., 17 (1964), 693-696(German).

[83] Uber einige Klassen von Polynomfolgen. C. R. Acad. Bulg. Sci., 17(1964), 7970-800 (German).

1965[84] Integraldarstellung einer Klasse von Polynomfolgen. C. R. Acad. Bulg.

Sci., 18 (1965), 7-9 (German).[85] Orthogonale Systeme in einigen Klassen von Polynomfolgen C. R. Acad.

Bulg. Sci., 18 (1965), 295-298 (German).[86] Orthogonale Systeme in einigen Klassen von Polynomenfolgen (with B.

Sendov). Wiss. Z. Hochsch. Archit. Bauwes. Weimar, 12 (1965), 517-519 (German).

1966[87] Certain classes of entire functions and systems of polynomials generated

by them. In: Contemporary Problems in Theory Anal. Functions (In-ternat. Conf., Erevan, 1965), Nauka, Moscow, 150-155 (Russian).

[88] The International Congress of mathematicians Moscow 1966 and fthGeneral Assembly of International Mathematical Union in Dubna. Phys.-Math. J., 9 (1966), No 4, 317 (Bulgarian).

[89] Funktionen, die eine Turan'sche Ungleichung befriedigen. C. R. Acad.Bulg. Sci., 19 (1966), 93-96 (German).

[90] Uber einige Klassen von ganzen Funktionen. C. R. Acad. Bulg. Sci., 19(1966), 575-577 (German).

1967[91] On some achievement of Bulgarian Academy of Sciences in 1966. J. Bulg.

Acad. Sci., 12 (1967), No 2, 3-28 (Bulgarian).[92] Speech of the Chairman of the Org. Committee of the Second congress of

Union of Bulgarian Mathematicians Academician L. Iliev. Phys. Math.J. , 10 (1967), No 4, 258-265 (Bulgarian).

[93] Third Balkan congress of mathematicians. Phys. Math. J., 10 (1967),No 1, 62 (Bulgarian).

[94] Uber die Laguerre'schen ganzen Funktionen. Atti dell' VIII Congressodell'unione Matematica Italiana, Trieste, 2-7 Ott., 1967 (German).

[95] Zur Theorie einer Klasse von speziellen ganzen Funktionen IV. Int. Kongr.Anwend. Math. Ingenieurwiss. Weimar 1967, Ber. 2 (1968), 99-104(German).

1968[96] Introductory speech of the Second National Conference on Application

of mathematical methods in economics, mathematics and cybernetics.Varna, 1969, 15-19 (Bulgarian).

[97] On the development of mathematical Science in Bulgaria. J. Bulg. Acad.Sci., 13 (1968), 3-33 (Bulgarian).

1969[98] Certain classes of entire functions and systems of polynomials generated

by them. Bulgar. Akad. Nauk., Izv. Mat. Inst., 10 (1969), 273-281(Bulgarian).

[99] Institute of Mathematics with Computer Center, Bulg. Acad. Sci.Phys.-Math. J., 12 (1969), No 4, 265-274 (Bulgarian).

[100] Most general trends and forecasts in contemporary mathematics. Phys.-Math. J., 12 (1969), No 1, 27-34 (Bulgarian).

[101] Allgemeine Tendenzen und Prognosen der Mathematik der Gegenwart.Wiss. Z. Hochsch. Architektur u. Bauwesen. Weimar, 16 (1969), No 5,459-463 (German).

[102] O niektorych zagadnieniah poznania naukowego i wykorzystania jegowynik. Zagadinienia Naukoznawstwa, 1 (1969), 27-38 (Polish).

1970[103] Mathematics as the science of models. Phys.-Math. J., 13 (1970), No 4,

287-296 (Bulgarian).[104] Some problems on the development of Bulgarian Academy of Sciences

and natural-mathematical science. J. Bulg. Acad. Sci., 16 (1970), No2-3, 35-45 (Bulgarian).

[105] Some problems on the theory and application related to the problems ofknowledge. Philosophical thought, (1970), No 3, 3-15.

[106] On some problems of the scientic knowledge and the use of its results.In: Control, managing, planning and organization of scientic and tech-nical investigations, 1, M., 1970, 264-279 (Russian).

[107] Most general trends and forecasts of current mathematics. Bulg. Akad.Nauk., Izv. Mat. Inst., 11 (1970), 285-294 (Russian).

[108] Quelques questions concernant la theorie et l'application en liaison avecles problemes de la connaissance des sciences. Problems of the Science

of Science. Zagadnienia naukoznawstwa (Special issue), 1970, 104-119(French).

1971[109] Academician Nikola Obreskov (on the occasion of his seventy-fth birth-

day). Phys.-Math. J., 14(47) (1971), 32-33 (Bulgarian).[110] Talk at the 10th congress of Bulgarian Communist Party. J. Bulg. Acad.

Sci., 17 (1971), No 3-4, 37-40. (Bulgarian).[111] ICM in Nice and the meeting of the General Assembly of IMU in Manton.

Phys.-Math. J., 14 (1971), No 1, 93 (Bulgarian).[112] Extremale Probleme der schlichten Funktionen. IX Congresso dell'Unione

Matematica Italiana Sunti delle Communicazioni, Bari, 1971, 57-58 (Ger-man).

1972[113] Mathematics as the science of models. Izv. Akad. Nauk SSSR, Ser.

Math., 27 (1972), No 2(164), 203-211 (Russian).[114] Introductory speech of the Chairman of the Org. Committee of the

Third congress of Bulgarian Mathematicians Academician L. Iliev. Phys.Math. J., 15 (1972), No 4, 257-260 (Bulgarian).

[115] Sur les problemes de formation et de recherche en informatique. ActaCybernetica, Szeged, 1972, No 4, 263-271 (French).

[116] Uber die Laguerre'schen ganzen Funktionen. In: Construktive FunctionTheory (Proc. Internat. Conf., Varna, 1970), Soa, 1972, 193-200 (Ger-man).

1973[117] Mathematics in the contemporary society. Vselena (1972), (1973), 17-24

(Bulgarian).[118] On mathematical creativity. Vselena (1972), (1973), 24-35 (Bulgarian).

1974[119] Mathematics in the contemporary society. J. Bulg. Acad. Sci., 20 (1974),

No 1, 14-26 (Bulgarian).[120] Joint Centre for Research and Training in Mathematics and Mechanics.

J. Bulg. Acad. Sci., 20 (1974), No 1, 27-37 (Bulgarian).[121] Joint Centre for Research and Training in Mathematics and Mechanics.

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1975[124] On the development of mathematics in the People's Republic of Bulgaria.

Publishing House of the Bulgarian Academy of Sciences, Soa, 1975, 72p. (Bulgarian).

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1983[172] Academician Blagovest Khristov Sendov (on his 50th birthday). Pliska

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[175] Splines with Laguerre functions. International conference on analyticalmethods in number theory and analysis (Moscow, 1981). Trudy Mat.Inst. Steklov, 163 (1984), 90-94 (Russian).

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[192] 70th anniversary of Prof. Alipi Mateev 1914-1979. Phys.-Math. J., 26(1984), 29-30 (Bulgarian).

[193] Mαθηµατικα Eπαγγελµατα διασταση. 1984, 3-4, 1-11.1985

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1986[200] Academician Lyubomir Chakalov (on the occasion of the century of his

birth). In: Mathematics and mathematical education (Proc. 15th SpringConf. Union Bulg. Math., 1986), 1986, 83-92 (Bulgarian).

[201] 90th anniversary of Academician Nikola Obreshkov. Scientic life (1986),No 3, 30 (Bulgarian).

[202] General theory of knowledge and articial intelligence. Voenna tehnika,(1986) No 12, 4-5 (Bulgarian).

[203] Panorama living in my memory ... Soa, 1986 (Bulgarian).[204] Theory of modelling. Soa, 1986 (Bulgarian).

1987[205] Classical extremal problems in the theory of univalent functions. In:

Mathematics and Education in Mathematics, (Proc. 16th Spring Conf.Union Bulg. Math., 1987), 1987, 9-34.

[206] A problem of Szego for univalent functions. Serdica Bulg. Math. Publ.,13 (1987), No 1, 3-17 (Russian).

[207] The dierence quotient for univalent functions. Serdica Bulg. Math.Publ., 13 (1987), No 1, 18-20 (Russian).

[208] First Computer Center in People's Republic of Bulgaria. 25 year of itsfoundation. Phys.-Math. J., 29 (1987), 86-92 (Bulgarian).

[209] Talk at the Second International Conference on Articial intelligence.Phys.-Math. J., 29 (1987), 129-131. (Bulgarian).

[210] Speech of Academician Iliev at the First spring conference of the Bulgar-ian mathematical association, Soa, 1972, 11-14 (Bulgarian).

[211] On Newton approximation for Laguerre entire functions. In: ComplexAnalysis and Applications'85. (Proc. Int. Conf., Varna 1985), 1987,276280.

[212] Store for new upswing. Phys.-Math. J., 29 (1987), 181-188 (Bulgarian).[213] Chakalov's method. Collected papers on the occasion of the century of

his birth.[214] On some problems in Bulgarian Academy of Sciences. Soa, 1987 (Bul-

garian).[215] Bulgarian people, don't forget your tribe and language. Soa, 1987 (Bul-

garian).[216] Laguerre entire functions. Second Edition, Soa, 1987, 188 p.

1988[217] Analytisch nichtfortsetzbare Reinen. Second Edition, Soa, 1988, 158 p.[218] To my Greek friends. Soa, 1987 (Greek).[219] Salonika, 1986, 1-238.[220] On Theory of Knowledge. Soa, UBM, 1988 (Bulgarian).[221] Union of Scientists in Bulgaria and reorganization. In: Union of Scien-

tists in Bulgaria and reorganization of intellectual scope. Soa, 1988, 7-1(Bulgarian).

[222] Looking forward. Soa, UBM, 14 (Bulgarian).[223] On Theory of Knowledge. In: Dimitar Haralampiev Dimitrov jubilee

volume. Soa, 1988, 97-103 (Bulgarian).[224] On Theory of Knowledge. In: Mathematics and Education in Mathemat-

ics, (Proc. 17th Spring Conf. Union Bulg. Math., 1988), 1988, 781-721(Bulgarian).

1989[225] On Theory of Knowledge. In: UNESCO, New Information Technologies

in higher Education, CEPPES, Bucharest, 1989, 57-62.[226] Looking forward. In: Mathematics and Education in Mathematics,

(Proc. 18th Spring Conf. Union Bulg. Math., 1989), 1989, 781-721(Bulgarian).

1990[227] 100 years teaching mathematics in Soa University St. Kliment Ohridski

... and mathematicians in this world. Soa, UBM (Bulgarian).[228] When the University teaching in Bulgaria gathered strength. In: Mathe-

matics and Physics 100 years. UBM, section BAS-US, Soa, 1009, 18-42(Bulgarian).

1991[229] Problems in the theory of univalent Functions, Constantin Caratheodory:

An International Tribute, Vol. 1, 1991. World Scientic Publishing Co.Ptc. Ltd, Printed in Singapore by Utopica Press, 495-500.

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[231] Mathematical methods in the Theory of Knowledge. Soa, 1991, PreprintNo 2.

1994[232] Axiomatical methods in the theory of cognition. C. R. Acad. Bulg. Sci.,

47, (1994), No. 9, 9-11.

Application 2 List of Acad. L. Iliev's Publicationson Education in Mathematics

Published Textbooks (In Bulgarian,on Elementary Mathematics and for Secondary Schools)

[1] Ëåêöèè ïî åëåìåíòàðíà àëãåáðà, ó÷åáíèê çà ñòóäåíòèòå-çàäî÷íèöè âÓíèâåðñèòåòà, Ñîôèÿ, 1950, èçä. ½Íàóêà è èçêóñòâî.

[2] Åëåìåíòàðíà ìàòåìàòèêà àëãåáðà (çàåäíî ñ äîö. Ñï. Ìàíîëîâ), I-âîèçäàíèå Ñîôèÿ, 1954 ã., II-ðî èçäàíèå Ñîôèÿ, 1956 ã., èçä. ½Íàóêàè èçêóñòâî.

[3] Ñáîðíèê îò çàäà÷è ïî åëåìåíòàðíà ìàòåìàòèêà (çàåäíî ñ àêàä. ×à-êàëîâ, äîö. Ìàòååâ, äîö. Ìàíîëîâ), Ñîôèÿ, 1955 ã., èçä. ½Íàóêà èèçêóñòâî.

[4] Ñáîðíèê îò ðåøåíè çàäà÷è ïî ìàòåìàòèêà, ÷àñò I àëãåáðà, ÷àñò II ãåîìåòðèÿ (çàåäíî ñ àêàä. ×àêàëîâ è äîö. Ìàòååâ), I-âî èçäàíèå Ñîôèÿ, 1947 ã., II-ðî èçäàíèå (â åäèí òîì), Ñîôèÿ, èçä. ½Íàóêà èèçêóñòâî, 1950 ã.

[5] Ñáîðíèê îò ðåøåíè çàäà÷è ïî ìàòåìàòèêà (çàåäíî ñ äîö. Ìàòååâ),Ñîôèÿ, 1946 ã.

[6] Àëãåáðà çà VI êëàñ íà ãèìíàçèèòå (çàåäíî ñ àêàä. ×àêàëîâ è äîö.Ìàòååâ), Ñîôèÿ, èçä. ½Íàðîäíà ïðîñâåòà, èçëÿçëà â íÿêîëêî èçäà-íèÿ.

[7] Àëãåáðà çà Õ êëàñ íà åäèííèòå ó÷èëèùà (çàåäíî ñ Ë. Áóíåâà è Ä.Øîïîâà), Ñîôèÿ, èçä. ½Íàðîäíà ïðîñâåòà, èçëÿçëà â íÿêîëêî èçäà-íèÿ.

[8] Àëãåáðà çà V êëàñ íà ãèìíàçèèòå (çàåäíî ñ àêàä. Îáðåøêîâ è Í.Íåäÿëêîâ), Ñîôèÿ, èçä. ½Íàðîäíà ïðîñâåòà.

[9] Ñáîðíèê îò çàäà÷è ïî àëãåáðà, ÷àñò I çà IV è V êëàñ íà ñðåäíèòå ó÷è-ëèùà (çàåäíî ñ äîö. Ìàòååâ è Ï. Ñòàìáîëîâ), Ñîôèÿ, èçä. ½Íàðîäíàïðîñâåòà, èçëÿçúë â 2 èçäàíèÿ.

Translated Textbooks (from Russian into Bulgarian,on Education in Mathematics, for Teachers)

[1] Ìåòîäèêà íà ãåîìåòðèÿòà îò Í. Ì. Áåñêèí (ïðåâîä îò ðóñêè, çàåäíîñ àêàä. ×àêàëîâ è äîö. Ìàòååâ), Ñîôèÿ, èçä. ½Íàðîäíà ïðîñâåòà.

[2] Ãåîìåòðèÿ (çà ó÷èòåëñêèòå èíñòèòóòè) îò Á. Â. Êóòóçîâ (ïðåâîä îòðóñêè, çàåäíî ñ ïðîô. Ïåòêàí÷èí è äîö. Ìàòååâ), Ñîôèÿ, èçä. ½Íà-ðîäíà ïðîñâåòà.

Other Popular Readings

Proposed new problems with their solutions in the magazine of the BulgarianPhysics-Mathematics Society, and in Jahresbericht der Deutschen Math. Vereini-gung; also many articles on methodology of education in Mathematics publishedin the same magazine of the Bulgarian Physics-Mathematics Society.



CERTAIN PROPERTIES OF THE GENERALIZED GAUSS

HYPERGEOMETRIC FUNCTIONS

Praveen Agarwal

Department of MathematicsAnand International College of Engineering

Jaipur – 303012, INDIAe-mail: [email protected]

The integral transforms and the operators of fractional calculus play animportant role in various sub-fields of applicable mathematical analysis. Herewe aim at establishing certain integral transform and fractional integral formu-las for the generalized Gauss hypergeometric functions introduced by Ozergin.Moreover, results for some particular values of the parameters are also pointedout.

MSC 2010 : Primary 26A33, 33B15, 33B20, 33C05, 33C15, 33C20; Sec-ondary: 33B99, 33C99, 60B99

Key Words and Phrases: Beta function; generalized Beta functions; gener-alized Gauss hypergeometric functions; fractional calculus operators



HYPERBOLIC BICOMPLEX VARIABLES

Lilia Apostolova

Institute of Mathematics and InformaticsBulgarian Academy of Sciences

”Acad. G. Bontchev” Str., Block 8Sofia – 1113, BULGARIA

e-mail: [email protected]

Bicomplex numbers appear in the work of Corrado Segre (Le rappresen-tazioni reali delle forme complesse e gli enti iperalgebrici, Math. Annalen 40(1892), 597-665) for the goals of algebraic geometry. Hyperbolic bicomplexnumbers are introduced and used later mainly for the goals of physics.

Hyperbolic bicomplex numbers are the elements of the algebra

R(j1, j2; 1) = x0 + j1x1 + j2x2 + j1j2x3 : j21 = j2

2 = 1, j1j2 = j2j1,

where x0, x1, x2, x3 are real numbers.The units 1, j1, j2 and j1j2 form cartesian base of this 4-dimensional commu-

tative non-division algebra. Matrix representation of the hyperbolic bicomplexnumbers is given here. The determinant of the matrix representation is found.Invertible elements and idempotent elements are described. Representation ofthe hyperbolic bicomplex variables by four idempotent elements is found.

Holomorphic functions of four-dimensional hyperbolic bicomplex variableare considered. Analogous results for the algebra of bicomplex numbers arelisted.

MSC 2010 : 32A30; 30G35Key Words and Phrases: hyperbolic bicomplex number; bicomplex number;

invertible element; idempotent element; holomorphic function



ON CLOSED AND CONVEX SETS IN `2 WITH EMPTY

GEOMETRIC INTERIORS

Stoyu T. Barov 1,§, Jan J. Dijkstra 2

1 Institute of Mathematics and Informatics8 Acad. G. Bonchev Str., 1113 Sofia, BULGARIA

e-mail: [email protected] Afdeling Wiskunde, Vrije Universiteit

De Boelelaan 1081a, 1081 HV AmsterdamThe NETHERLANDS


If A ⊂ `2, then the geometric interior A of A is the interior of A relativeto its closed affine hull. We consider closed and convex subsets B of `2 withB = ∅. We discuss the Exposed Point Theorem for such sets, namely, if k ∈ Nand B is a closed convex subset of `2 with B = ∅ then for every x ∈ Bthere is a k-dimensional linear subspace L of `2 such that (x + L) ∩ B = x.As a consequence, we get the Imitation Theorem: Let k ∈ N and let B be aclosed convex subset of `2 with B = ∅. If C is a closed subset of `2 such thatC + L = B + L for every k-subspace L of `2 then B = C, i.e. B has only oneclosed k-imitation.

MSC 2010 : 52A20, 46A55, 57N15Key Words and Phrases: Hilbert space, convex projection, imitation



PROPERTIES OF THE FUNDAMENTALAND THE IMPULSE-RESPONSE SOLUTIONS

OF MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS

Emilia Bazhlekova

Institute of Mathematics and Informatics – Bulgarian Academy of Sciences,Acad. G. Bonchev Str., Bl. 8, Sofia 1113, BULGARIA


We study the multi-term fractional differential equation

(Dα∗ u)(t) +

m∑

j=1

λj(Dαj∗ u)(t) + λu(t) = f(t), t > 0; u(0) = c0;

where Dα∗ is the fractional derivative operator in the Caputo sense,

0 < αm < ... < α1 < α ≤ 1, λ, λj > 0, j = 1, ..., m, m ∈ N ∪ 0.

This equation is a generalization of the classical relaxation equation, obtainedfor m = 0, α = 1, and governs some fractional relaxation processes. Ap-plying Laplace transform method, we find the fundamental and the impulse-response solutions of the equation, corresponding to f(t) ≡ 0, c0 = 1, andf(t) ≡ δ(t), c0 = 0, respectively, where δ(t) is the Dirac delta function. Theproperties of the solutions are derived directly from their representations asLaplace inverse integrals. We prove that the fundamental and the impulse-response solutions are completely monotone functions and find their asymptoticexpressions for small and large times. It appears that the asymptotic behaviourof the solutions for t → 0 is determined by the largest order of fractional differ-entiation α and for t → ∞ by the smallest order αm. In all cases an algebraicdecay is observed for t → ∞. Some useful estimates for the solutions are alsoobtained. In the limiting case m = 0, in which the solutions can be expressedin terms of the Mittag-Leffler functions, some well-known properties of thesefunctions are recovered from our results.

MSC 2010 : 26A33, 33E12, 34A08, 34C11, 34D05, 44A10Key Words and Phrases: fractional calculus; fractional relaxation; Caputo

derivative; Laplace transform; Mittag-Leffler function; completely monotonefunction



ORTHOGONAL POLYNOMIALS APPROACH TO

THE HANKEL TRANSFORM OF SEQUENCES BASED

ON MOTZKIN NUMBERS

Radica Bojicic 1, Marko D. Petkovic 2,§

1 University of Pristina, Faculty of Economy, SERBIAe-mail: [email protected]

2 University of Nis, Faculty of Sciences and Mathematics, SERBIAe-mail: [email protected]

In this paper we use the method based on orthogonal polynomials to givea closed form evaluations of the Hankel transform of sequences based on theMotzkin numbers. It includes linear combinations of consecutive two, three andfour Motzkin numbers. In some cases we were able to derive the closed-formevaluation of the Hankel transform, while in the others we showed that Hankeltransform satisfies certain difference equation. As the corollary, we reobtainknown and show some new results regarding the Hankel transform of Motzkinand shifted Motzkin numbers. Those evaluations also gives an idea how to applymethod based on orthogonal polynomials on the sequences having zero entriesin their Hankel transform.

MSC 2010 : 33C45, 42C05, 05E35Key Words and Phrases: Hankel transform, Motzkin numbers, shifted Motz-

kin numbers



CONVOLUTIONS AND COMMUTANTS CONNECTED WITH

STURM-LIOUVILLE OPERATOR ON THE HALF-LINE

Nikolai Bozhinov 1, Ivan Dimovski 2

1 Department of Mathematics, University of National and World EconomyStudentski Grad, 1700 – Sofia, BULGARIA

e-mail: niki−[email protected] Institute of Mathematics and Informatics

Bulgarian Academy of Sciences“Acad. G. Bontchev” Str. Block 8, Sofia - 1113, BULGARIA


The linear operators L : C[0, +∞) → C[0,+∞) with an invariant subspaceC1

h = f ∈ C1[0, +∞) : f ′(0)− hf(0) = 0 commuting with a Sturm-Liouvilleoperator D = d2

dx2 − q(x) in C 2[0,+∞) are characterized as the operators of theform

Lf(x) = ΦξT ξf(x) ,where Φ is a linear functional on C1[0,+∞) and T ξ is the generalized translationoperator of B.M. Levitan [1] with C1

h as an invariant subspace.It is proven that the subspace of these operators with an invariant subspace

C1h,Φ = f ∈ C1

h : Φ(f) = 0,which commute with D are the operators of the form

Lf(x) = λf(x) + m ∗ f ,

where λ is a constant,m ∈ C[0,+∞) , and ∗ is the non-classical convolutionfound by the authors [2] in 1978.

MSC 2010 : 44A35, 44A40, 34B24Key Words and Phrases: operational calculus, Sturm-Liouville operator,

translation operators, commutant

References: [1] B.M. Levitan, Theory of Generalized Translation Operators.Nauka, Moscow, 1973 (In Russian); [2] N. Bozhinov, I. Dimovski, Boundaryvalue operational calculi for linear differential operator of second order. Compt.Rend. Acad. Bulg. Sci., 31, No 7 (1978), 815-818.



LOCAL BEHAVIOR OF SOLUTIONS TO THE BELTRAMI

EQUATION WITH DEGENERATION

Melkana A. Brakalova

Department of Mathematics – Fordham University411 East Fordham Rd, JMH 407, Bronx, NY - 10458, USA


We apply extremal length techniques to study weak conformality of home-omorphic solutions to the Beltrami equation, fz = µfz, with degeneration, ina neighborhood of a point. Here µ, the complex dilatation of the solution f, isa complex-valued measurable function, defined in a neighborhood of that pointsuch that |µ| < 1 a.e.

Properties related to weak conformality at a point, a notion weaker than thewell-known conformality at a point, have been studied by Belinskii, Gutlyanskii,Jenkins, Lehto, Martio, Ryazanov, Reich, Teichmuller, Wittich, the author andothers. A homeomorphisms f defined in a neighborhood of the origin, f(0) = 0,is weakly conformal at 0, if it preserves circles, namely

limr→0

max|z|=r

|f(z)|min|z|=r

|f(z)| = 1,

and angles between rays emanating from the origin, i.e. for an appropriatechoice of a branch of the argument lim

r→0

[arg f(reiθ2)− arg f(reiθ1)

]= θ2 − θ1,

uniformly in θ1 and θ2.We also address the stronger notion of asymptotic homogeneity. A homeo-

morphisms f defined in a neighborhood of the origin, f(0) = 0, is asymptoticallyhomogeneous, if lim

z→0

f(ζz)f(z) = ζ.

Using analytic estimates, involving the complex dilatation, for the extremallength of families of curves we obtain results that are related to the author’sPhD thesis from 1988.

MSC 2010 : 30C20, 30C25, 30C62Key Words and Phrases: solutions to the Beltrami Equation with degenera-

tion, local behavior, extremal length, weak conformality, asymptotic homogene-ity, conformality



ABOUT CLOSED MAPPINGS AND ONE THEOREMOF ARHANGEL’SKII-BELLA

Mitrofan Choban ∗,1, Ekaterina Mihaylova ∗∗,2

∗ Department of Mathematics, Tiraspol State UniversityIablochikin 5, MD 2069, Kishinev, Republic of MOLDOVA

e-mail: [email protected]∗∗ University of Architecture Civil Engineering and Geodesy

1 Hr. Smirnenski Blvd., Sofia 1046, BULGARIAe-mail: katiamih [email protected]

A subset L of a topological space X is a bounded subset of X if for everylocally finite family γ of X the set V ∈ γ; V ∩ L 6= ∅ is finite. A space Xis called feebly compact if X is bounded in X. For the definition of a q-space,see [2]. The network weight of a space X is the smallest cardinal number of theform |S|, where S is a network for X and is denoted by nw(X). Denote by A theclass of topological spaces X that are completely regular and nw(F ) = w(F )for every closed subset F of X.

Theorem 1. Let f : X → Y be a closed continuous mapping of a regularspace X of a countable pseudocharacter onto a q-space space Y . Then thereexists a closed subspace Z of X such that Z is first countable, f(Z) = Y andf−1(y) ∩ Y is a bounded subset of Z and X for every point y ∈ Y . Moreoverthe space Y is Frechet-Urysohn.

Theorem 2. Let X ∈ A and Y be a dense metrizable subspace of the spaceX. Then X \ Y ∈ A and nw(X \ Y ) = w(X \ Y ).

In particular, we obtain the following assertion which contains a positiveanswer to Question 2.15 from [1].

Corollary 2. If X is a compactification of a metrizable space Y , thenX \ Y ∈ A and nw(X \ Y ) = w(X \ Y ).

References

[1] A.V. Arhangel’skii and A. Bella, Cardinal invariants in remainders andvariations of tightness, Proc. Amer. Math. Soc. 119, No 2 (1993), 947-954.

[2] E. Michael, A note on closed maps and compact sets, Israel J. Math. 2(1964), 173-176.

1Partially supported by Contract N 149/ 2013 with CNIP - UACEG2Partially supported by Contract N 149/ 2013 with CNIP - UACEG



EXISTENCE OF HOLOMORPHIC FUNCTIONS

ON TWISTOR SPACES

Johann Davidov 1 and Oleg Mushkarov 2


“Acad. G. Bontchev” Str. Block 8, Sofia - 1113, BULGARIAe-mails: 1 [email protected] , 2 [email protected]

The twistor space Z of an oriented Rienannlan 4-manifold M is a 2-spherebundle over M parametrizing the complex structures on the tangent spaces of Mcompatible with its metric and the opposite orientation. The 6-manifold Z ad-mits two natural almost complex structures J1 and J2 introduced, respectively,by Atiyah-Hitchin-Singer and Eells-Salamon. It is a result of Atiyah-Hitchin-Singer that J1 is integrable (i.e. comes from a complex structure) if and onlyif the base manifold M is self-dual. On the other hand, the almost complexstructure J2 is never integrable by a result of Eells-Salamon.

The main purpose of this talk is to discuss the problem of local existenceof holomorphic functions on the almost complex manifolds (Z, J1) and (Z, J2).Recall that a smooth complex-valued function on an almost complex manifold(X,J) is said to be holomorphic (or pseudo-holomorphic) if its differential iscomplex linear with respect to J . In contrast to the case of a complex mani-fold, there are almost complex manifolds that do not admit any non-constantholomorphic function even locally. Examples of such manifolds are the 6-sphereS6 or, more generally, every isotropy irreducible homogeneous almost complexspace with non-integrable structure, every compact hypersurface in R7 withthe Calabi almost complex structure, etc. Given an almost complex manifold(X,J), denote by Fn(J) the (possibly empty) set of points p ∈ X such thatn is the maximal number of local holomorphic functions with C-linearly inde-pendent differentials at p. Then X is the disjoint union of the sets Fn(J) forn = 0, 1, 2, ..., 1

2dim X. In these terms the Newlander-Nirenberg theorem saysthat J is integrable if and only if X = Fn(J) for n = 1

2dimX, while X = F0(J)means that there are no non-constant local holomorphic functions.

Our main results give descriptions of the subsets Fn(J1) and Fn(J2) of thetwistor space Z in terms of the curvature of the base manifold M . They canbe considered as refinements of the Atiyah-Hitchin-Singer and Eells-Salamontheorems mentioned above.

While the fibres of the classical twistor spaces are 2-spheres, the fibres ofthe so-called hyperbolic twistor spaces over oriented 4-manifolds with metricsof signature (2, 2) are hyperbolic 2-planes. The problem of existence of holo-morphic functions with respect to the analogs of the Atiyah-Hitchin-Singer andEells-Salamon almost complex structures on hyperbolic twistor spaces will alsobe discussed. We show that, in contrast to the classical case, there can be anabundance of global holomorphic functions on these twistor spaces.

MSC 2010 : 53C28, 32L25, 53C15Key Words and Phrases: almost complex manifolds, holomorphic functions,

twistor spaces



FROM STARLIKE TOWARDS CLOSE-TO-CONVEX

FUNCTIONS

Georgi Dimkov

Department “Analysis, Geometry and Topology”Institute of Mathematics and Informatics

Bulgarian Academy of Sciences8, Acad. G. Bonchev str., 1113 – Sofia, BULGARIA


We consider subclasses of the general class S of all normalized functionsf(z) = z + a2z

2 + . . . that are analytic and univalent in the unit disc ∆ = z :|z| < 1.

Let α ∈ [0, 1). If a function f ∈ S satisfies the condition

Rez f

′(z)

f(z)> α, |z| < 1,

it is called starlike of order α - subclass, S∗(α), S∗(0) ≡ S being the generalclass of starlike functions.

If a function f ∈ S satisfies the condition

θ2∫

θ1

Re

1 +

z f′′(z)

f ′(z)

dθ > −π, z = reiθ, θ1 < θ2, r < 1,

it is called close-to-convex - subclass CC.In our talk we construct close-to-convex functions using special sets of func-

tions starlike of order α.

MSC 2010 : 30C45Key Words and Phrases: univalent functions, starlike and close-to-convex

functions



NEW ASPECTS OF MULTIREAL AND

MULTICOMPLEX VARIABLES

Stancho Dimiev

Department of Analysis, Geometry and TopologyInstitute of Mathematics and Informatics

Bulgarian Academy of Sciences”Acad. G. Bontchev” Str., Block 8

Sofia – 1113, BULGARIA


The notion of anticirculant matrix unifies these ones of vector a = (a0, a1, a2,. . . , am) and matrix A = j0a0 + ja1 + j2a2 + . . . + jmam, j being the basic an-ticirculant matrix, j0 = 1 - the unit matrix. So all linear groups, differentialforms etc. of finite vector space theory are adapted by the commutative anti-circulant algebra’s theory. This presents a base for a generalized sheaf theory:GScont ⊃ GSdiff ⊃ GShol ⊃ GSalg. It is to remark that in the case of anticircu-lant algebras (higher dimensions) appear some new singularities in comparisonwith the classical cases of R(1, j) = C and Cn. The corresponding functiontheory holds also for tangent vector fields, especially in the smooth case C∞

for the so called almost complex structures, in the integrable case of bicomplextype.

Closely related is the bicomplex algebraic geometry and its higher gener-alizations. Based on a generalization of the fundamental theorem of the alge-bra, one develop bicomplex Weierstrass polynomials and bicomplex Weierstrasspreparation theorem with some application of bicomplex geometric character.

I am very grateful to my friends Lilia Apostolova, Marin S. Marinov, Nedel-cho Milev, Peter Stoev, Stanislava Stoilova (some of them my PhD students)for the help in the research work during the last years. But especially to Dr.Lilia Apostolova for her help in the preparation of this manuscript.

MSC 2010 : 32C05, 32B05, 32C35Key Words and Phrases: anticirculant matrix; partition analogous; bicom-

plex algebraic geometry; bicomplex preparation theorem



COMMUTANT OF A CHEREDNIK TYPE OPERATOR

ON THE REAL LINE

Ivan H. Dimovski 1, Valentin Z. Hristov 2


“Acad. G. Bontchev” Str. Block 8, Sofia - 1113, BULGARIAe-mails: 1 [email protected], 2 [email protected]

We characterize the commutant of the following Cherednik type singlardifferential-difference operator on the real line due to Mourou

Λy(x) =df

dx+

A′(x)A(x)

(y(x)− y(−x)

2

)− ρy(−x),

where A(x) = |x|2α+1B(x), α > −12, B being a positive C∞ function on R, and

ρ ≥ 0. This operator generalizes both Dunkl and Cherednik operators.

MSC 2010 : 47B38; 47B39; 47B99Key Words and Phrases: commutant of linear operator



EXTENSION OF DUHAMEL PRINCIPLE FOR LINEAR

NONLOCAL INITIAL-BOUNDARY VALUE PROBLEMS

Ivan Dimovski 1, Margarita Spiridonova 2

1 Institute of Mathematics and Informatics – Bulgarian Academy of SciencesAcad. G. Bonchev Str. Bl. 8, Sofia 1113, BULGARIA

e-mail: [email protected] Institute of Mathematics and Informatics – Bulgarian Academy of Sciences

Acad. G. Bonchev Str. Bl. 8, Sofia 1113, BULGARIAe-mail:mspirid@ math.bas.bg

Local and non-local boundary value problems (BVPs) for the classical equa-tions of Mathematical Physics in rectangular domains often are solved by Fouriermethod or some of it extensions intended for the non-local case.

We aim to make effective the Fourier method for general classes of non-localBVPs for the heat and wave equations in a strip. To this end the method iscombined with an extension of the Duhamel principle to the space variable.A non-classical operational calculi, custom-tailored for the specific problem isused. Thus explicit solutions of the considered problems are obtained.

MSC 2010 : 44A35, 44A40, 35K05, 35K20, 35L05, 35L20Key Words and Phrases: convolution, calculus of Mikusinski and other op-

erational calculi, Fourier method, Duhamel principle, boundary value problem,wave equation, heat equation



RESONANCE CASES FOR NONLOCAL CAUCHY PROBLEMS

Ivan Dimovski 1, Yulian Tsankov 2

1 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences”Acad. G. Bonchev” Str., Block 8, Sofia - 1113, BULGARIA

e-mail: [email protected] Faculty of Mathematics and InformaticsSofia University ”St. Kliment Ohridsky”

J. Boucher Str., No 5, Sofia - 1164, BULGARIAe-mail: [email protected]

Consider the nonlocal Cauchy problem for ODEs of the form

P

(d

dt

)y(t) = f(t), where P is a polynomial with constant coefficients,

with homogeneous boundary value conditions of the form χy(k)(t) = 0, k =0, 1, ..., deg P−1, where χ is an arbitrary linear functional.

By means of a Mikusinski type operational calculus the problem reduces toan algebraic equation of the form P (s) y = f . In Dimovski and Spiridonova (I.H.Dimovski, M. Spiridonova, Operational calculus approach to nonlocal Cauchyproblems, Math. Comput. Sci., Vol. 4, 2010, p. 243-258) the case when P (s)is a non-divisor of zero is treated. Here we solve completely the resonance casewhen P (s) is a divisor of zero (the resonance case).

MSC 2010 : 44A40, 44A35, 34A12, 42A75Key Words and Phrases: nonlocal Cauchy problem, non-classical convolu-

tion, convolution fraction, Duhamel principle



CANONICAL REPRESENTATIONS OF MINIMAL SURFACES

IN EUCLIDEAN OR MINKOWSKI SPACES

Georgi GanchevInstitute of Mathematics and Informatics – Bulgarian Academy of Sciences

8, Acad. G. Bonchev str., 1113 – Sofia, BULGARIAe-mail: [email protected]

Using the fact that any minimal surface in the three dimensional Euclideanspace carries locally canonical principal parameters, we obtain a canonical prin-cipal representation of these surfaces. Similarly, using the fact that any minimalsurface locally admits canonical asymptotic parameters, we obtain a canonicalasymptotic representation of these surfaces. These results allow us to obtainexplicitly the solutions of the natural partial differential equation of minimalsurfaces. As an application, we find all minimal surfaces, whose asymptoticlines are generalized helices.

Using the fact that any space-like surface with H = 0 in the three-dimensionalMinkowski space carries locally canonical principal parameters, we obtain acanonical principal representation of these surfaces. Similarly, using the factthat any space-like surface with H = 0 carries locally canonical asymptotic pa-rameters, we obtain a canonical asymptotic representation of the surfaces withH = 0. These results allow us to obtain explicitly the solutions of the naturalpartial differential equation of space-like surfaces with H = 0. As an applica-tion, we find: 1) all space-like surfaces with H = 0 whose principal lines areplane curves; 2) all space-like surfaces with H = 0 whose asymptotic lines aregeneralized helices.

Minimal time-like surfaces in the three dimensional Minkowski space aredivided into two basic classes: with negative or positive Gauss curvature. Weprove that any minimal time-like surface with negative Gauss curvature carrieslocally canonical principal parameters and using this result we obtain a canoni-cal principal representation of these surfaces. For the minimal time-like surfaceswith positive Gauss curvature, we prove that they carry locally canonical asymp-totic parameters, and obtain a canonical asymptotic representation of minimaltime-like surfaces. These results allow us to obtain explicitly the solutions ofthe natural partial differential equation of minimal time-like surfaces.

MSC 2010 : 53A10, 53A05; Key Phrases: canonical principal parameters,canonical asymptotic parameters, canonical representation of minimal surfaces



A HISTORICAL SURVEY ON THE PLACE AND ROLE

OF TASKS IN MATHEMATICS TEACHING

Valentina Gogovska

Institute of Mathematics, Faculty of Natural Sciences and MathematicsUniversity “Ss. Cyril and Methodius”

1000 Skopje, MACEDONIAe-mail: [email protected]

Mathematical tasks are a main tool to achieve educational, practical andinstructional aims of mathematics teaching. In order to achieve long-term, aswell as comprehensive adoption of the prescribed material, it is necessary tosolve a significant number of tasks. Trying to emphasize the significance ofmathematical tasks, it is sufficient to ask ourselves the following question: “Ismathematics teaching possible without mathematical tasks?” In the beginningwe will try to point out the role of mathematical tasks through history.

Mathematical tasks were the basic tool for strengthening mathematical knowl-edge in pre-Greek period. However, over time, tasks were replaced first by the-orems and than by concepts. Therefore, historically there is a certain dynamicsbetween the set of theorems and set of tasks. Does this mean that with thechange of the position of tasks through history their significance has been losttoo?

References

[1] J. Dewey, How We Think: A Restatement of the Relation of ReflectiveThinking to the Educative Process, Boston Heat, 1933.

[2] I. Ganchev, Basic School Activities During Mathematical Lesson, Modul,Sofia, 1996.

[3] S. Grozdev, For High Achievements in Mathematics. The Bulgarian Expe-rience (Theory and Practice), Sofia, 2007.

[4] V.P. Panov, Mathematics, Ancient and Now, Moscow, 2006.[5] History of Mathematics.

MSC 2010 : 97A30, 97D50Key Words and Phrases: mathematical tasks, concepts and theorems,

axioms, definitions, didactic tools



COMPACTNESS OF QUASICONFORMAL MAPPINGS

IN HIGHER DIMENSIONAL SPACES

Jianhua Gong

Department of Mathematical Sciences – United Arab Emirates UniversityPO Box 15551, Al Ain, UNITED ARAB EMIRATES


The quasiconformal mappings in higher dimensional spaces are transforma-tions of subdomains Ω of the extended Euclidean space Rn = Rn ∪ ∞, n ≥ 2,which have uniformly bounded distortion. They provide a class of mappingsthat lie between homeomorphisms and conformal mappings. The compactnessproperties of quasiconformal mappings play a crucial role there.

We present some compactness properties of quasiconformal mappings in thistalk. For example, each K-quasiconformal group stabilizing a compact subsetof Ω is a Lie group acting on Ω.

MSC 2010 : 30C60Key Words and Phrases: quasiconformal mappings, quasiconformal group,

locally compact, compact, Lie group



ON THE MATRIX APPROACH IN THE COMPLEX ANALYSISAND ITS GEOMETRIC APPLICATIONS

Milen HristovDepartment of Algebra and Geometry – University of Veliko Tarnovo

2 T.Tarnovski Str., Veliko Tarnovo - 5003, BULGARIAe-mail: [email protected]

The main reason for the analysis of the functions of complex variable tobe translated into matrix form is its computer implementation. We follow thefollowing steps.

Step 1. We consider the well known isomorphism between the field C of

complex numbers and the unitary groupU=

Z =(

a b−b a

)=aE+bJ, a, b ∈ R

.

In this way all the algebraic operations in C have their U-matrix equivalents.Step 2. The U-matrix equivalent of the Euclidean metric in C is of the form

(ε)g (Z1, Z2) =

√det(Z1 − Z2) =

(ε)g (JZ1, JZ2), Z1, Z2 ∈ U.

Analogously to the complex extended plane C∗ = C ∪ ∞, we consider the ex-tension U∗ of U by adding the matrix at infinity – the analogue of the complexpoint at infinity (∞). By means of the stereographic projection of the C-planeover the Riemann sphere, U∗ is endowed with the spherical metric

(σ)g (Z1, Z2) = [det(Z1 − Z2)]

12 [(1 + detZ1)(1 + detZ2)]−

12 =

(σ)g (JZ1, JZ2).

Thus the topology of C (C∗) induces a topology of U (U∗).Step 3. We consider U-curves in U as analogues of the continues C-

valued functions of real argument. For such smooth matrix curves we get theU-matrix description of the Frennet formulae for the matrix tangent and thematrix normal and express its signed curvature.

Step 4. As an analogue of the C-valued functions of C-argument weconsider U-valued functions of U-argument. We give the U-matrix analogues ofthe complex derivative and of the Cauchy-Riemann equations. As an examplewe give the U-matrix analogues of the fractional-linear and of the Zhukovski-typefunctions and the corresponding conformal transformations to rational Beziercurves are applied.

MSC 2010 : 30C20, 30E05, 51B10, 53A04, 65D17Key Words and Phrases: functions of one complex variable; unitary group;

plane unitary matrix curves; conformal transformation; rational Bezier curves



GENERALIZED PADE APPROXIMANTS

FOR PLANE CONDENSER

Nikolay Ikonomov

Institute of Mathematics and InformaticsBulgarian Academy of SciencesAcad. G. Bonchev Str., Bl.81113 – Sofia, BULGARIA


Given a regular plane condenser (E, F ), ∞ ∈ F , let αn,k and βn,k,n = 1, 2, . . ., k = 1, . . . , n, be two triangular tables of points, with accumulationpoints belonging to E and F , respectively.

Given a function f(z), holomorphic on E, let π(α,β)n,m be the generalized Pade

approximant of order (n,m) with respect to (α, β). Let Γ be a canonical curvein C \ (E ∪ F ), let Q(z) be a polynomial of degree n with zeros inside Γ.

In the present paper we show the following: if the generalized Pade approx-imant π

(α,β)n,m of the function ft(z) := 1

(t−z)Q(z) behaves regularly for every t ∈ Γ,then the points 〈αn,k, βn,k〉 have an extremal distribution with respect to thecondenser.

MSC 2010 : 30E10, 31A05Key Words and Phrases: Pade approximants, maximal convergence, plane

condenser



THE RAMANUJAN ENTIRE FUNCTION

Mourad E.H. Ismail

University of Central Florida, Orlando, FL 32816 – USAand King Saud University, Riyadh, SAUDI ARABIA


Ramanujan was a self-educated college drop out who did some of the bestmathematics of the twentieth century. He extensively worked on the

F (z) = 1 +∞∑

n=1

(−z)nqn2

(1− q)(1− q2) . . . (1− qn),

which we refer to as the Ramanujan entire function. We demonstrate the sig-nificance of this function in number theory and analysis and give a new inter-pretation of the statement

1 +∞∑

n=1

znqn2

(1− q)(1− q2) . . . (1− qn)=

∞∏

n=1

(1 +

zq2n−1

1− c1qn − c2q2n − . . .

)

in Ramanujan’s lost note book.

The coefficients c1, c2, . . . turned out to have very interesting patterns.

MSC 2010 : 30D10, 30D15, 30D20; 30C15; 11-XXKey Words and Phrases: Ramanujan entire function, number theory, anal-

ysis



THE OBATA SPHERE THEOREMS ON AQUATERNIONIC CONTACT MANIFOLD

OF DIMENSION BIGGER THAN SEVEN

Stefan Ivanov 1,§, Alexander Petkov 1, Dimiter Vassilev 2

1 Sofia University ”St. Kl. Ohridski” Fac. Math. & Inf. Sofia, BULGARIAe-mail: [email protected]

2 University of New Mexico, Dept. Math. Stat., Albuquerque, NEW MEXICOe-mail: [email protected]

Quaternionic contact versions of the Lichnerowicz [5] and Obata [6] spheretheorems are given:

Theorem 1 ([3]) Let (M,η, g,Q) be a compact quaternionic contact (qc) man-ifold of dimension bigger than seven whose Ricci tensor satisfies the Lichnerow-icz type lower bound estimate ([2]). Then the first positive eigenvalue of thesub-Laplacian takes the smallest possible value [2] if and only if the qc manifoldis qc equivalent to the standard 3-Sasakian sphere.

Theorem 1 follows from the following

Theorem 2 ([3]) Let (M, η, g,Q) be a quaternionic contact manifold of dimen-sion 4n + 3 > 7 which is complete with respect to the associated Riemannianmetric h = g+(η1)2+(η2)2+(η3)2. If there exists a non-constant smooth functionf whose horizontal Hessian with respect to the Biquard connection satisfies

∇df(X, Y ) = −fg(X, Y )−3∑

s=1

df(ξs)ωs(X, Y ),

then the qc manifold (M, η, g,Q) is qc equivalent to the unit 3-Sasakian sphere.

The work relies on considerations in [1]. We achieve the proof of Theorem 2by showing first that M is isometric to the unit sphere S4n+3 and then that Mis qc-equivalent to the standard 3-Sasakian structure on S4n+3. To this effectwe show that the torsion of the Biquard connection vanishes and in this case the

Riemannian Hessian satisfies(∇h)2f = −fh after which we invoke the classicalObata theorem showing that M is isometric to the unit sphere. In order to provethe qc-equivalence part we show that the qc-conformal curvature vanishes, whichgives the local qc conformal equivalence with the 3-Sasakian sphere due to [4,Theorem 1.3], and then prove a Liouville-type result, which implies the existenceof a global qc-conformal map between M and the 3-Sasakian sphere.

MSC 2010 : 53C26, 53C25, 58J60Key Words and Phrases: quaternionic contact structures, qc conformal flat-

ness, qc conformal curvature, Einstein metrics, sub-Laplacian, Obata spheretheorem

References

[1] S. Ivanov, I. Minchev, D. Vassilev, Quaternionic contact Einstein structuresand the quaternionic contact Yamabe problem, To appear in: MemoirsAmer. Math. Soc.

[2] S. Ivanov, A. Petkov, D. Vassilev, The sharp lower bound of the first eigen-value of the sub-Laplacian on a quaternionic contact manifold, J. Geom.Anal., DOI 10.1007/s12220-012-9354-9; arXiv :1112.0779.

[3] S. Ivanov, A. Petkov, D. Vassilev, The Obata sphere theorems on a quater-nionic contact manifold of dimension bigger than seven, arXiv :1303.0409.

[4] S. Ivanov, D. Vassilev, Conformal quaternionic contact curvature and thelocal sphere theorem, J. Math. Pures Appl. 93 (2010), 277–307.

[5] A. Lichnerowicz, Geometrie des groupes de transformations. Travaux etRecherches Mathematiques, III. Dunod, Paris 1958.

[6] M. Obata, Certain conditions for a Riemannian manifold to be iosometricwith a sphere, J. Math. Soc. Japan 14, No 3 (1962), 333–340.



MEASURES OF MATHEMATICAL KNOWLEDGE

FOR TEACHING AND UNIVERSITY

MATHEMATICS COURSES DESIGN

Slagjana Jakimovik

Faculty of Pedagogy “St. Kliment Ohridski”University “Ss. Cyril and Methodius” - Skopje

Bul. Partizanski Odredi bb, Skopje - 1000, Republic of MACEDONIAe-mail: [email protected]

During the past few decades a significant body of research has been accumu-lated worldwide in the area of mathematical knowledge for teaching in primaryschools. Numerous theoretical and empirical research studies have paved theway towards defining what it is that teachers need to know and be able todo to produce positive effects on the development of primary school students’mathematics competencies.

Taking into consideration results from a previous study on students’ mathe-matics competencies when entering university teacher education studies, impli-cations on the kind of mathematics courses that need to be developed withinuniversity studies for primary school teachers have been hypothesized in this pa-per. A pilot study has been conducted designed to illuminate the way towardsa larger study on the correlation between mathematical knowledge for teachingbuilt in university mathematics courses and teacher performance. Initial find-ings from the pilot study are discussed in the paper and recommendations forfurther explorations are formulated.

MSC 2010 : 97B10, 97B40, 97B50, 97B70, 97C70Key Words and Phrases: mathematical knowledge for teaching, mathemat-

ics course design, university education, primary school teachers



ON THE GROWTH OF THE CONVOLUTION IN VARIOUS

SPACES OF GENERALIZED FUNCTIONS

Andrzej Kaminski 1,§, Svetlana Mincheva-Kaminska 2

1 Institute of Mathematics – University of RzeszowRejtana 16 A, 35-959 Rzeszow, POLAND

e-mail: [email protected] Institute of Mathematics – University of Rzeszow

Rejtana 16 A, 35-959 Rzeszow, POLANDe-mail: [email protected], [email protected]

Certain theorems concerning the existence of the convolution as well as itsgrowth in various spaces of functions and generalized functions are essentiallysharpened and extended. The results are applied in the theory of integral trans-formations.

MSC 2010 : 46F10, 46F05, 46F12

Key Words and Phrases: convolution of functions, convolution of distribu-tions and ultradistributions



AN OPEN PROBLEM OF LJUBOMIR ILIEV

RELATED TO THE MITTAG-LEFFLER FUNCTION

AND FRACTIONAL CALCULUS OPERATORS

Virginia Kiryakova

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences”Acad. G. Bontchev” Str., Block 8, Sofia – 1113, BULGARIA


Dedicated to the 100th Anniversary of Acad. Ljubomir Iliev

This is a short survey on an open problem posed by Academician LjubomirIliev [4], [5] in his studies on constructive theory of Laguerre functions. It isstated in notions as the Jensen polynomials and zeros of polynomials and en-tire functions. However, the linear differential operator Dα introduced by Ilievand involved in this problem is related to the most popular special function ofthe fractional calculus (FC) – the Mittag-Leffler (M-L) function, and the op-erator itself can be considered as a special case of the operators of FC and ofthe Gelfond-Leontiev (G-L) operators of generalized differentiation. From thispoint of view, Iliev’s open problem can be formulated also in a more generalsetting, related to the multi-index M-L functions ([7], [8]) and to the operatorsof generalized FC ([7]). Together with the problem as stated by Iliev in [4], [5],and its alternative interpretations, we provide the readers with some additionalliterature closely related to techniques and results possibly helpful in under-standing and solving the open problem, the basic among them – by Craven andCsordas [1], [2], Dzrbashjan [3], Ostrovskii and Peresyolkova [9], Popov [10].

Shortly, Iliev introduced a linear differential operator Dα, α > 0, acting as

f(z) = a0 +a1z+a2z2 + ..., |z| < R −→ Dαf(z) =

∞∑

k=0

akΓ(α(n+1))

Γ(α(n−k+1))zn−k.

The following notations are also used: Eα(z) := Eα,α(z) =∞∑

k=0

zk/Γ(α(k + 1))

for the M-L function;(

n

k

)

α

:= Γ(α(n + 1))/Γ(α(k + 1)) Γ(α(n− k + 1)) for the

“fractional” order binomial coefficients; then the Jensen polynomial of Eα,α(z)with respect to the operator Dα can be represented as:

Jαn (Eα,α, z)=

(n

0

)

α

zn+(

n

1

)

α

zn−1+(

n

2

)

α

zn−2+ . . .+(

n

n

)

α

=n∑

k=0

(n

k

)

α

zn−k.

As Iliev mentioned, if α is a positive integer, then Γ(k+1)/Γ(α(k+1))is an α-sequence (here α has different meaning), or in terms of [1], [2] – amultiplier sequence, and in this case the zeros of Jα

n (Eα,α, z) are real. Thequestion (Problem 2.5) which is the domain Aα, where all the zeros of the Jensenpolynomials Jα

n (Eα,α, z), n = 1, 2, . . ., lie, for non-integer α > 0, remains open.After giving some background in Sections 1 and 3, we emphasize that Iliev’s

operator Dα can be considered also a Gelfond-Leontiev (G-L) operator for gener-alized differentiation generated by the M-L function Eα,α, as well as an Erdelyi-Kober (E-K) operator for differentiation of fractional order α. We suggest thatthe tools of the M-L functions, of the G-L and E-K operators and of the frac-tional calculus could be explored to resolve Problem 2.5, and even to state it ina more general setting. In Section 4 we provide some additional readings, in-cluding also more recent results (on the multiplies sequences and their relationto the special functions, their zeros distribution, etc.) that could be useful.

MSC 2010 : 30C15, 30D20, 33E12Key Words and Phrases: zeros of polynomials and entire functions; Jensen

polynomials; Mittag-Leffler function; multiplier sequences, Gelfond-Leontiev op-erators; operators of fractional calculus

References: [1] Th. Craven, G. Csordas, Problems and theorems in thetheory of multiplier sequences, Serdica Math. J. 22 (1996), 515–524; [2] -//-,The Fox-Wright functions and Laguerre multiplier sequences, J. Math. Anal.Appl. 314 (2006), 109-125; [3] M.M. Dzrbashjan, Integral Transforms and Rep-resentations of Functions in the Complex Domain (in Russian), Nauka, Moscow(1966); [4] L. Iliev, Zeros of Entire Functions, Publ. House of Bulg. Acad.Sciences, Sofia (1979), In Bulgarian; [5] -//-, Laguerre Entire Functions, Publ.House of Bulg. Acad. Sciences, Sofia (1987); [6] V. Kiryakova, GeneralizedFractional Calculus and Applications, Longman & J. Wiley, Harlow - N. York(1994); [7] -//-, Multiple (multiindex) Mittag-Leffler functions and relations togeneralized fractional calculus, J. Comput. Appl. Math. 118 (2000), 241–259;[8] -//-, The multi-index Mittag-Leffler functions as important class of specialfunctions of fractional calculus, Computers and Math. with Appl. 59, No 5(2010), 1885–1895; [9] I.V. Ostrovskii, I.N. Peresyolkova, Nonasymptotic re-sults on distribution of zeros of the function Eρ(z, µ), Analysis Mathematica 23(1997), 283–296; [10] A.Yu. Popov, On the zeros of a family of Mittag-Lefflerfunctions, In: Contemporary Mathematics and its Applications 35: Proc. SpringMath. School - Voronezh, 2003, Part 2 (2005), 28–30, in Russian.



ISOMETRIES BETWEEN FUNCTION SPACES

H. Koshimizu 1, T. Miura 2,§, H. Takagi 3, S.-E. Takahasi 4

1 Yonago National College of Technology4448 Hikona, Yonago, Tottori 683-8502, JAPAN

e-mail: [email protected] Department of Mathematics, Faculty of Science, Niigata University

Niigata 950-2181, JAPANe-mail: [email protected]

3 Department of Mathematical Sciences, Faculty of ScienceShinshu University, Matsumoto 390-8621, JAPAN

e-mail: [email protected] Toho University, Yamagata University (Professor Emeritus)

Chiba 273-0866, JAPANe-mail: sin [email protected]

Let A be a complex linear subspace of C0(X), the Banach space of all com-plex valued continuous functions on a locally compact Hausdorff space X, whichvanish at infinity. We say that A is strongly separating if for each distinct pointsx, y ∈ X there exists f ∈ A such that |f(x)| 6= |f(y)|. A is strongly triple sepa-rating provided that for each distinct points x, y, z ∈ X there exists f ∈ A suchthat |f(x)| 6= |f(y)| and f(z) = 0. We characterize surjective (not necessarilylinear) isometries between such function spaces.

Theorem. Let A be a strongly triple separating, complex linear subspaceof C0(X), and B a strongly separating, complex linear subspace of C0(Y ). IfS : A → B is a surjective isometry with S(0) = 0, then there exist a (possiblyempty) closed and open subset K of Ch(B), a homeomorphism ϕ : Ch(B) →Ch(A) and a continuous function α : Ch(B) → z ∈ C : |z| = 1 such that

S(f)(y) =

α(y)f(ϕ(y)) y ∈ K

α(y)f(ϕ(y)) y ∈ Ch(B) \K

for all f ∈ A, where Ch(·) denotes Choquet boundary.

MSC 2010 : 46J10Key Words and Phrases: function spaces, isometries



MULTIPOINT PADE APPROXIMANTS AND UNIFORM

DISTRIBUTION OF INTERPOLATION NODES

Ralitza Kovacheva

Institute of Mathematics and Informatics, Bulgarian Academy of SciencesAcad. G. Bonchev Str. 8, 1113 Sofia, BULGARIA


Given a function f , analytic on a regular compact set E in the complex planeand a triangle table of points β := βn,kk=1,...,n; n=1,2,... with no accumulationpoints outside of E, let πβ

n,m be the multipoint Pade appproximant of f of order(n,m) with respect to β. Assume that the rational functions πβ

n,m are maximallyconvergent to f as n →∞ and m is fixed. We pose the question about the natureof the the points βn,k. In the present talk, we will prove that βn,k distribute, asn →∞ regularly with respect to the equilibrium measure of E.

MSC 2010 : 30E10, 31A05Key Words and Phrases: Pade approximants, interpolation nodes



HARMONICITY, HOLOMORPHY, QUATERNIONS,

AND CRYSTAL STATISTICS

Julian Lawrynowicz

Faculty of Physics and Applied Informatics, University of LodzPomorska 149/153, PL-90-236 Lodz, POLAND

& Institute of Mathematics, Polish Academy of SciencesSniadeckich 8, P.O.B. 21, PL-00-956 Warszawa, POLAND


Various aspects of harmonicity, holomorphy, and quaternionic structures areshown to be important in investigating crystal statistics. This concerns [1, 2]:

1) one-to-one correspondence between spinor solutions of some structurespinor equations and harmonic forms with respect to, e.g., the (1,1)-metric and(0,4)-metric,

2) one-to-one correspondence between the space of holomorphic solutions ofthe above mentioned structure spinor equations, and a suitable one-dimensionalDolbeault cohomology group,

3) existence of holomorphic embeddings of C2 to the Grassmannian G(2,4)and C4 to G(8,16) being real parts of holomorphic mappings in the classicalsense.

The (0,4)-embeddings resp. (0.8)-embeddings depend explicitly on the quater-nionic resp. para-quaternionic structure. These results are important in crys-tal statistics, e.g. when studying relaxation problems leading to Oguchi-typeparabolic differential equations of the first type and and of the second type.

On the other hand, the physical demands inspire research on harmonicity,holomorphy, and quaternionic structures.

References

[1] J. Lawrynowicz and M. Vaccaro, Structure fractals and para-quaternionicgeometry, Ann. Univ. Mariae Curie-Sklodowska Sect. A Math. 65, No 2(2012), 63–73.

[2] J. Lawrynowicz, S. Marchiafava, F.L. Castillo Alvarado, and A. Niem-czynowic, (Para)quarter-nionic geometry, harmonic forms, and stochasticalrelaxation, Publ. Math. Debrecen, 17 pp., to appear.

MSC 2010 : 32L25, 15A66, 52A50, 81R25; Key Phrases: (para)quaternionicstructure, parabolic equations, relaxation, holomorphy, harmonicity



EVOLUTION EQUATIONS FOR THE STEFAN PROBLEM

Martin Lukarevski

Department of Mathematics and Statistics – University ”Goce Delcev”,Faculty of Informatics

Krste Misirkov 10-A, Stip 2000, MACEDONIAe-mail: [email protected]

The Stefan problem is a particular kind of a free boundary value problemwhich models phase transition phenomena, for example melting of ice and freez-ing of water. We study a quasi-steady variant and propose in our model aboundary condition with surface tension and kinetic undercooling that reflectsthe relaxation dynamics. Our approach to the problem is by using the theoryof abstract parabolic evolution equations. We obtain results in a special kind ofSobolev spaces.

MSC 2010 : 35K90, 65J08, 37L05Key Words and Phrases: Stefan problem, boundary value problem, abstract

parabolic evolution equations



ON INTEGRABILITY AND CONVOLVABILITY

OF GENERALIZED FUNCTIONS

Svetlana Mincheva-Kaminska

Institute of Mathematics – University of RzeszowRejtana 16 A, 35-959 Rzeszow, POLAND

e-mail: [email protected], [email protected]

General sequential conditions of integrability and convolvability in variousspaces of generalized functions, expressed in terms of different classes of approx-imate units, are discussed. The equivalence of the respective convolutions ofgeneralized functions is proved.

MSC 2010 : 46F05 (primary); 46F10, 46F12 (secondary)

Key Words and Phrases: convolution of distributions, special approximateunit, special upper approximate unit, unit support of a function, b-bounded set



MIKUSINSKI’S OPERATIONAL CALCULUS APPROACH TOTHE DISTRIBUTIONAL STIELTJES TRANSFORM

Dennis Nemzer

Department of Mathematics – California State University, StanislausOne University Circle, Turlock, CA – 95382, USA


The ring of continuous complex-valued functions on the real line which van-ish on (−∞, 0), denoted by C+(R), with addition and convolution has no zerodivisors by Titchmarch’s theorem. The quotient field of C+(R) is called the fieldof Mikusinski operators.

Yosida constructed a space M in order to provide a simplified version forMikusinski’s operational calculus without using Titchmarch’s convolution the-orem. Even though the space M does not give the full space of Mikusinskioperators, it contains many of the important operators needed for applications.

In this note, we use the space M(r) ⊂ M to extend the classical Stieltjestransform. It turns out that M(r) is isomorphic to the space of distributionsJ ′(r).

The space J ′(r) , and variations of J ′(r), have been investigated by severalauthors (Carmichael, Lavoine, Misra, Pilipovic, Stankovic, and Takaci to namea few) in regards to extending the Stieltjes transform.

While the construction of J ′(r) requires a space of testing functions, theconcept of a dual space, and functional analysis, the construction of M(r) isalgebraic, elementary, and only requires elementary calculus.

To illustrate the simplicity of this construction, we present some Abeliantype theorems for the Stieltjes transform.

MSC 2010 : 44A15, 44A40, 46F10, 46F12.Key Words and Phrases: Abelian theorems; generalized function; Mikusinski

operational calculus; Stieltjes transform



A CONVERSE OF THE GAUSS-LUCAS THEOREM

Nikolai Nikolov 1,§, Blagovest Sendov 2

1 Institute of Mathematics and InformaticsBulgarian Academy of Sciences

“Acad. G. Bontchev” Str. Block 8, Sofia - 1113, BULGARIAe-mail: [email protected]

2 Institute of Information and Communication TechnologiesBulgarian Academy of Sciences

“Acad. G. Bonchev” Str. Block 25A, Sofia - 1113, BULGARIAe-mail: [email protected]

All linear operators L : C[z] → C[z] which decrease the diameter of the zeroset of any P ∈ C[z] are found.

This talk is based on our joint paper in Amer. Math. Monthly (to appearin 2014).

MSC 2010 : 30C15Key Words and Phrases: Gauss-Lucas theorem, zeros of polynomials



ABOUT THE SOLUTION OF THE FRACTIONAL

COULOMB EQUATION

Yanka Nikolova

Faculty of Applied Mathematics and InformaticsTechnical University of Sofia

8 ”Kliment Ohridski” Bul. , Sofia – 1000, BULGARIA


In this paper the Riemann-Liouville operator for fractional differentiationis applied to treat a generalization of the Coulomb wave equation. By meansof the Frobenious method, the fractional Coulomb equation is solved and thesolution is obtained in power series form. It is shown that this solution containsthe regular solution of the classical Coulomb equation as a particular case.

MSC 2010 : 26A33, 44A99, 35R11, 44A10, 44A15Key Words and Phrases: Riemann-Liouville fractional derivative; Kummer

functions; Kummer equation; differential equation of fractional order



COMPARISON BETWEEN THE CONVERGENCEOF POWER AND GENERALIZED MITTAG-LEFFLER SERIES

Jordanka Paneva-Konovska1 Fac. of Applied Mathematics and Informatics - Technical University of Sofia,

8 ”Kliment Ohridski” bul., Sofia – 1000, BULGARIAe-mail: [email protected]

2 Associate at:Institute of Mathematics and Informatics - Bulgarian Academy of Sciences

”Acad. G. Bontchev” Str., Block 8, Sofia – 1113, BULGARIA

The paper deals with the family of three-index generalizations of the clas-sical Mittag-Leffler functions, introduced by Prabhakar. We consider series inthese special functions in the complex plane and study their convergence. Moreprecisely, we determine where the series converges and where it does not, wherethe convergence is uniform, which is the domain of convergence, what is thebehaviour of the series ”near” the boundary of this domain of convergence, andon itself. We provide analogues of the classical Cauchy-Hadamard, Abel andFatou theorems for the power series. Finally, we compare the obtained resultswith the classical ones for the conventional case of power series.

Same type of results have been previously obtained for series in other specialfunctions, for example: for series in Laguerre and Hermite polynomials, by P.Rusev, and resp. by the author – for series in Bessel functions, their Wright’s2-, 3-, and 4-indices generalizations, and the more general multi-index Mittag-Leffler functions (in the sense of Yu. Luchko - V. Kiryakova). See for examplethe papers by J. Paneva-Konovska as listed below.

MSC 2010 : 40A30, 33E12, 31A20, 30D15, 30B30, 30B10Key Words and Phrases: Mittag-Leffler functions and generalizations, series

in generalized Mittag-Leffler functions, convergence and divergence, Cauchy-Hadamard, Abel and Fatou type theorems

References: – Cauchy-Hadamard and Abel type theorems for Bessel func-tions series, In: Proc. 19-th Summer School ”Appls. of Math. in Engineering,Varna, 24.08.-02.09, 1993”, Sofia (1994), 165-170; – Series in Mittag-Lefflerfunctions: Geometry of convergence, Adv. Math. Sci. Journal, 1, No 2 (2012),73-79; – Convergence of series in three parametric Mittag-Leffler functions, Toappear in: Math. Slovaca (Accept. 2012); – The convergence of series in multi-index Mittag-Leffler functions, Integr. Transf. Spec. Funct. 23 (2012), 207-221.



CERTAIN CLASSES OF FUNCTIONS

WITH NEGATIVE COEFFICIENTS

Donka PashkoulevaInstitute of Mathematics and Informatics – Bulgarian Academy of Sciences

8, Acad. G. Bonchev str., 1113 – Sofia, BULGARIAe-mail: donka zh [email protected]

Let S denote the class of functions of the form:

f(z) = z +∞∑

k=2

akzk (1)

which are analytic and univalent in the open unit disk U = z : |z| < 1.For a function f ∈ S, we define:

D0f(z) = f(z); D1f(z) =f(z) + zf ′(z)

2= Df(z);

Dnf(z) = D(Dn−1f(z)

); n ∈ N = 1, 2, . . ..

For β ≥ 0,−1 ≤ α ≤ 1 and n ∈ N0, let S(n, α, β) denote the subclass of Sconsisting of functions of the form (1) and satisfying the analytic condition

<

z(Dnf(z))′

Dnf(z)− α

> β

∣∣∣∣z(Dnf(z))′

Dnf(z)− 1

∣∣∣∣ .

We denote by T the subclass of S consisting of functions of the form

f(z) = z −∞∑

k=2

akzk, ak ≥ 0. (2)

Further, we define the class ST (n, α, β) by ST (n, α, β) = S(n, α, β) ∩ T .Theorem 1. A necessary and sufficient condition for the function f(z) of

the form (2) to be in the class ST (n, α, β) is that∞∑

k=1

[k(1 + β)− (α + β)](

k + 12

)n

ak ≤ 1− α,

where −1 ≤ α < 1, β ≥ 0 and n ∈ N0.Theorem 2. Let the function f(z) defined by (2) be in the class ST (n, α, β).

Then,

|Dif(z)| ≥ |z| − 1−α

2−α+β

(23

)n−i

|z|2, |Dif(z)| ≤ |z|+ 1−α

2−α+β

(23

)n−i

|z|2

for z ∈ U , where 0 ≤ i ≤ n.MSC 2010 : 30C45; Key Phrases: univalent, convex, starlike functions



THE CONTRIBUTIONS OF ACADEMICIAN LJUBOMIR ILIEV

TO THE EDUCATION IN MATHEMATICS IN BULGARIA

Donka Pashkouleva


“Acad. G. Bonchev” Str., Bl. 8, BG-1113 Sofia, BULGARIAe-mail: donka zh [email protected]

In 1936 Academician Ljubomir Iliev graduated in Mathematics from theSofia University ”St. Kliment Ohridski”. After specialization for 2 years withAcademician Nikola Obreshkov and defending a Doctorate dissertation in 1938,he became a teacher in Third Sofia Male Gymnasium. During the period 1941– 1947 he was an Assistant Professor in the Sofia University, and during 1947– 1952, an Associate Professor and a Deputy Dean of the Faculty of NaturalSciences and Mathematics there. He wrote 233 scientific publications and 9publications for the education in Mathematics. Out of them, 4 are textbooksfor the secondary school, and 5 are for university students. Also he published3 books with mathematical problems. He also translated 2 textbooks – oneon Methodology of Geometry and another on Geometry. He set and solvednew mathematical problems as well. Acad. Iliev also published methodologicalarticles in the Bulgarian journal of the Society in Mathematics and Physics.

To the present report I enclose a full list of all the scientific publications ofAcademician Ljubomir Iliev, as well as a list of his publications on education inMathematics.

MSC 2010 : 01A70, 97-30, 97A30Key Words and Phrases: education in mathematics, textbooks, elementary

mathematics, higher education



THE LICHNEROWICZ TYPE RESULTS

ON THE QUATERNIONIC CONTACT MANIFOLDS

Alexander Petkov,

joint work with Stefan Ivanov and Dimiter Vassilev

Faculty of Mathematics and InformaticsUniversity Of Sofia “St. Kliment Ohridsky”5 “J. Baucher” Bul., Sofia 1126, BULGARIA

e-mail: a petkov [email protected]

In this talk we give analogues of the classical theorem of Lichnerowicz in thecase of quaternionic contact manifolds. The Lichnerowicz-type theorem says,that under some condition imposed on the Ricci tensor and the torsion tensorof the Biquard connection of a compact quaternionic contact manifold, it isobtained a sharp lower bound of the first eigenvalue of the sub-Laplacian. Theestimate is sharp in the sense, that the equality in the corresponding inequalityis obtained on the 3-Sasakian sphere. To obtain the estimate on the seven-dimensional quaternionic contact manifolds, we need by an exstra assumption,concerning the non-negativity of the P -function of any eigenfunction.

MSC 2010 : 35P15, 53C17, 53C21, 53C25, 53C26, 53D10, 58J60Key Words and Phrases: quaternionic contact structure, sub-Riemann ge-

ometry, sub-Laplacian, first eigenvalue, Lichnerowicz inequality, Paneitz opera-tor, P-function, 3-Sasacian structure

References

[1] S. Ivanov, A. Petkov, D. Vassilev, The sharp lower bound of the first eigen-value of the sub-Laplacian on a quaternionic contact manifold, Journal ofGeometric Analysis (2012), 1–23, doi: 10.1007/s12220-012-9354-9.

[2] S. Ivanov, A. Petkov, D. Vassilev, The sharp lower bound of the first eigen-value of the sub-Laplacian on a quaternionic contact manifold in dimensionseven, arXiv:1210.6932 [math.DG].



WAVE FRONT SETS FOR GEVREY CLASSES

Stevan Pilipovic

Department of Mathematics and Informatics, Faculty of SciencesUniversity of Novi Sad

Trg Dositeja Obradovica 4, 21000 Novi Sad, SERBIAe-mail: [email protected]

In the first part of the talk, we recall our results concerning wave-frontsets of Fourier-Lebesgue and modulation space types. Then we present variousdefinitions of wave-front sets, including the discrete version, and compare themwith the usual wave-front sets of ultradistributions. Especially, we describethe quasi-analytic wave front sets which correspond to the Gevrey sequenceMp = p!s, 1/2 < s < 1. It is quite different from the case which corresponds tothe Gevrey sequence Mp = p!.

The talk is based on joint work with J. Toft (Linnæus University, Vaxjo,Sweden).

MSC 2010 : 46F05, 35A18Key Words and Phrases: topological linear spaces of test functions, distri-

butions and ultradistributions; wave front sets; modulation space types; Gevreyclasses

Acknowledgements. This paper is on the working program of bilateral projectbetween Bulgarian and Serbian Academies of Sciences, ”Mathematical modelingby means of integral transform methods, partial differential equations, specialand generalized functions”.

References

[1] K. Johansson, S. Pilipovic, N. Teofanov, J. Toft, Gabor pairs, and a dis-crete approach to wave-front sets, Monatsh. Math., 166, No 2 (2012), 181-199; DOI 10.1007/s00605-011-0288-2.

[2] K. Johansson, S. Pilipovic, N. Teofanov, J. Toft, Analytic wave-front setsin Fourier Lebesgue and modulation spaces, In preparation.

[3] S. Pilipovic, Microlocal analysis of ultradistributions, Proc. Amer. Math.Soc., 126 (1998), 105–113.

[4] S. Pilipovic, N. Teofanov, J. Toft, Micro-local analysis in Fourier Lebesgueand modulation spaces. Part I, J. Fourier Anal. Appl., 17, No 3 (2011),374–407.

[5] S. Pilipovic, N. Teofanov, J. Toft, Micro-local analysis in Fourier Lebesgueand modulation spaces. Part II, J. Pseudo-Differ. Oper. Appl., 1, No 3(2010), 341–376.



PSEUDO-DIFFERENTIAL OPERATORS OF PRINCIPAL

TYPE, SUBELLIPTIC ESTIMATES FOR SCALAR

OPERATORS AND FOR THE ∂-NEUMANN PROBLEM

AND SOME APPLICATIONS

Peter R. Popivanov


”Acad. G. Bontchev” Str., Block 8Sofia – 1113, BULGARIA

Dedicated to the 100th anniversaryof my teacher Professor Ljubomir Iliev

This survey talk deals with pseudodifferential operators of principal typeincluding their local (non) solvability and subelliptic estimates. The main resultsare due to L. Hormander, Y.V. Egorov, L. Nirenberg, F. Treves, N. Lerner, N.Dencker, see [1]-[13]. In the second part of the talk we discuss Catlin’s resultson subellipticity of the ∂-Neumann problem for (0, q), 1 ≤ q ≤ N−1 forms inCN .

A shortened variant of this survey can be found in the Conference Proceed-ings.

MSC 2010 : 35S05, 35H20, 35A07, 32T27, 32W05, 32W25

Key Words and Phrases: pseudodifferential operator, operator of principaltype, local solvability, regularity in Sobolev spaces, subelliptic estimates, ∂-Neumann problem, loss and sharp loss of regularity

Acknowledgements. This paper is performed in the frames of the Bilat-eral Research Project ”Mathematical modelling by means of integral transformmethods, partial differential equations, special and generalized functions” be-tween Bulgarian Academy of Sciences and Serbian Academy of Sciences andArts (2012-2014).

References

[1] R. Beals, Ch. Fefferman, On local solvabiliy of linear partial differentialoperators. Ann. of Math. 97 (1973), 482–498.

[2] D. Catlin, Necessary conditions for subellipticity of the ∂-Neumann prob-lem. Ann. of Math. 117 (183), 147–171.

[3] D. Catlin, Boundary invariants of pseudoconvex domains. Ann. of Math.120 (1984), 529–586.

[4] D. Catlin, Subelliptic estimates for the ∂ Neumann problem on pseudo-convex domains. Ann. of Math. 126 (1987), 131–191.

[5] D. Catlin, J. D’Angelo, Subelliptic estimates. arXiv: 0906.0009.v1[math.CV].

[6] J. D’Angelo, Subelliptic estimates and failure of semicontinuity of ordersof contact. Duke Math. J. 47 (1980), 955–957.

[7] J. D’Angelo, Real hypersurfaces, orders of contact and applications. Ann.of Math. 115 (1982), 615–637.

[8] N. Dencker, The resolution of Nirenberg-Treves conjecture. Ann. of Math.163 (2006), 405–444.

[9] Y.V. Egorov, Linear Partial Differential Operators of Principal Type.Nauka, Moscow, 1984 (in Russian).

[10] L. Hormander, An Introduction to Complex Analysis in Several Variables.D. van Nostrand Publ. Co., Princeton, 1966.

[11] L. Hormander, The Analysis of Linear Partial Differential Operators, IV.Springer, 1985.

[12] J.J. Kohn, Boundary behaviour of ∂ on weakly pseudoconvex manifolds ofdimension 2. J. of Diff. Geom. 6 (1972), 523–542.

[13] N. Lerner, Non-solvability in L2 for a first order operator satisfying condi-tion (ψ). Ann. of Math. 139 (1994), 363–393.



THE MEAN-VALUE THEOREM FOR

HOLOMORPHIC FUNCTIONS

Elena Radzievskaya

National University of Food Technology UkraineVolodymyrska Str. 68, Kyiv, UKRAINE


There are many articles devoted to local mean value theorems for vector-valued functions, in particular holomorphic functions. This report relates di-rectly to this area and will be studied the problem of representing the remainderof the Taylor expansion for a holomorphic function in Lagrange form. We findout when and how the expansion well-known in the case of real-valued functionson an interval of the real axis can be transferred to holomorphic functions ina complex domain. Our theorems not only cover the some known results butalso imply the following intuitively clear fact: If f is a holomorphic functionin a neighborhood of the real axis and f takes real values at real values of theargument then the mean value in the remainder of the Taylor expansion, writtendown in Lagrange form, can be localized more precisely than without using theholomorphy of f . Henceforth f is a holomorphic function in a domain D ofthe complex plane C, is the boundary of D, and is the closure of D. We denoteby U(α; r) := z ∈ C : |z − α| < r the open disk of radius r > 0 centered atα and let arg z — stand for the argument of a nonzero complex number z and−π < arg z ≤ π.We suppose that the points z0 and z1 belong to D. In thisreport we study the following question: When is the remainder Qn(z0; z1; f) ofthe Taylor expansion

f(z1) =n−1∑

k=0

(z1 − z0)k

k!f (k)(z0) + Qn(z0; z1; f),

representable in Lagrange form Qn(z0; z1; f) = (z1−z0)n

n! f (n)(ξ) and where doesξ lie?

MSC 2010 : 30K05Key Words and Phrases: holomorphic functions, Taylor expansion, mean

value theorem, remainder



ON ORTHOGONAL POLYNOMIALS WITH RESPECT TOTHE WEIGHT DEPENDING ON POLYNOMIAL’S DEGREE

Predrag Rajkovic1,§, Sladjana Marinkovic2, Miomir Stankovic3

1 Depart. of Mathematics, Fac. of Mechanical Engineering – University of NisBeogradska 14, Nis - 18 000, SERBIA

e-mail: [email protected] Depart. of Mathematics, Fac. of Electronic Engineering – University of Nis

Beogradska 12, Nis - 18 000, SERBIAe-mail: [email protected]

3 Depart. of Mathematics, Fac. of Occupational safety – University of NisCarnojeviceva 10a, Nis - 18 000, SERBIAe-mail: [email protected]

The standard theory of orthogonal polynomials understand the weight func-tion like a positive function on a support depending only on a variable and maybea few parameters. The assumption that degree of the orthogonal polynomial canbe included into the weight function brought to a controversy between a fewspecialists in this theory. A few such sequences, named the relativistic Hermitepolynomials, were recognized like classical sequences and the idea was rejectedlike a contribution. But, a recent research in the theoretical physics, remindus to revise this question. Namely, some differential equations occurred whoseeigenfunctions are polynomials orthogonal with respect to the weight dependingon their degrees. It is proven that they exist and how can be constructed, but allother nice properties of the orthogonal polynomial sequences seemed lost. Wewill examine this problem in a few directions considering recurrence relations,zeros and differential properties.

MSC 2010 : 33C45, 42C05, 05E35.Key Words and Phrases: weight functions; orthogonal polynomials; recur-

rence relations; zeros of polynomials; differential properties



INVARIANTS OF FEW DISCRETE TRANSFORMS

Predrag Rajkovic 1,§, Paul Barry 2, Natasa Savic 3

1 Depart. of Mathematics, Fac. of Mechanical Engineering – University of NisBeogradska 14, Nis - 18 000, SERBIA

e-mail: [email protected] School of Science, Waterford Institute of Technology

Cork Road, Waterford, IRELANDe-mail: [email protected]

3 High Technical School, A. Medvedeva 20, Nis, SERBIAe-mail: [email protected]

In this paper, we will expose the algorithms for constructing the invariantsequences for few discrete transforms. Such sequences are their fixed pointsand therefore they are of the special concerning. Especially, we consider thebinomial, invert, Laguerre, Catalan and Hankel transform of number sequences.The indicated considerations are illustrated by examples. In addition, we discussthe operations which are invariants for the Riordan array.

MSC 2010 : 11B83, 44A55, 47B35Key Words and Phrases: sequences, invariants, binomial transform, Hankel

determinants, Riordan arrays



ON A GENERALIZATION OF CONTIGUOUS WATSON’S

THEOREM FOR THE SERIES 3F2(1)

Medhat A. Rakha 1, Arjun K. Rathie 2, Ujjwal Pandey 3

1 Department of Mathematics and Statistics, College of ScienceSultan Qaboos University, P.O. Box 36 (123) - Alkhoud - Muscat

Sultanate of OMANe-mail: [email protected]

2 Department of Mathematics, School of Mathematical and Physical Sciences,Central University of Kerala, Riverside Transit Campus

Kasaragod - 671 328, Kerala - INDIAe-mail: [email protected]

3 Mathematics Department, Marudhar Engineering CollegeRajasthan Technical University

NH-11, Jaipur Road, Bikaner - 334803, Rajasthan State, INDIAe-mail: [email protected]

The aim of this research paper is to establish explicit expressions of

3F2

a, b, c,; 1

12(a + b + i + 1), 2c + j

for i = 0,±1,±2, . . . ,±5; j = 0,±1,±2.For i = j = 0, we get the well known Watson’s theorem for the series 3F2(1).

Several new and known results are obtained as special cases of our main findings.

MSC 2010 : 33C05, 33C20, 33C70Key Words and Phrases: generalized hypergeometric functions; Watson’s

transformation theorem



HOLOMORPHIC CLIFFORDIAN FUNCTIONSAS A NATURAL EXTENSION OF MONOGENIC AND

HYPERMONOGENIC FUNCTIONS

Ivan Pierre Ramadanoff

Laboratoire de Mathematiques Nicolas Oresme, Universite de CaenBasse – Normandie, FRANCE


This is an expository paper which aim is to defend the notion of holomor-phic Cliffordian functions which was introduced by G. Laville and the authorin 1998. The way to argue for is to exhibit non-trivial applications. Some ofthem were known earlier: our contributions on the construction of elliptic Clif-fordian functions. The most recent interesting development is the paper of D.Constales, D. Grob and R.S. Krausshar (2013) in which a new class of Cliffordvalued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen groupis obtained.

At the beginning, we will recall the fundamental definitions and propertiesof Clifford algebras, especially those of anti-euclidean type.

Then, we will make a brief overview of the different theories of ”hypercom-plex” variables, namely the classical theory of monogenic functions of F. Brackx,R. Delanghe and F. Sommen (1982), the holomorphic Cliffordian functions, andfinally the hypermonogenic functions studied by H. Leutwiler and S.-L. Eriksson-Bique (1992). A careful analysis of the connections between those three classesof functions argues for the holomorphic Cliffordian ones. This is a set which isendowed with many function theoretical tools that are also offered for complexholomorphic functions. Basically, they were introduced in order to contain thefunctions x 7→ xn(n ∈ N, x a paravector) and to be stable under any direc-tional derivation. Consequently, they form a class of functions containing thetwo others.

Finally, we will deal with the problem how to construct Cliffordian analoguesof the Weierstrass ζ and ℘ functions, as well as the Jacobi cn, dn and sn. Wewill end with the construction of a new class of hypercomplex analytic cuspforms in which clearly one see how holomorphic Cliffordian functions are ableto solve a problem which was unsatisfactory solved before.

MSC 2010 : 30G35, 30G30, 30A05, 33E05, 33



TURAN TYPE CONVERSE MARKOV INEQUALITIES

ON CONVEX SUBSETS OF C

Szilard Gy. Revesz

Department of Analysis – Alfred Renyi Institute of MathematicsHungarian Academy of Sciences

Realtanoda street 13, Budapest – 1053, HUNGARYe-mail: reveszrenyi.mta.hu

This lecture is a survey of results obtained in the last decade.For a convex domain K ⊂ C the well-known general Bernstein-Markov in-

equality holds: a polynomial p of degree n must have ‖p′‖ ≤ c(K)n2‖p‖. How-ever, for polynomials in general, ‖p′‖ can be arbitrarily small, compared to ‖p‖.

Turan investigated the situation under the condition that p have all its zeroesin the convex body K. With this assumption he proved ‖p′‖ ≥ (n/2)‖p‖ for theunit disk D and ‖p′‖ ≥ c

√n‖p‖ for the unit interval I := [−1, 1]. Levenberg and

Poletsky provided general lower estimates of order√

n, and there were certainclasses of domains with order n lower estimates.

We show that for all compact and convex domains K and polynomials pwith all their zeroes in K ‖p′‖ ≥ c(K)n‖p‖ holds true, while ‖p′‖ ≤ C(K)n‖p‖occurs for arbitrary compact connected sets K ⊂ C. Moreover, the dependenceon width and diameter of the set K is found up to a constant factor. Note thatif K is not a domain (intK = ∅), then the order is only

√n.

Erod observed that in case the boundary of the domain is smooth and thecurvature exceeds a constant κ > 0, then we can get an order n lower estimationwith the curvature occurring in the implied constant. Elaborating on this ideaseveral extensions of the result are given. Again, geometry is in focus, includinga new, strong “discrete” version of the classical Blaschke Rolling Ball Theorem.

MSC 2010 : Primary 41A17; Secondary 30E10, 52A10Key Words and Phrases: Bernstein-Markov Inequality, Turan’s lower es-

timate of derivative norm, logarithmic derivative, Chebyshev constant, convexsets and domains, width of a set, circular domains, convex curves, smooth convexbodies, curvature, osculating circle, Blaschke’s rolling ball theorem, subdiffer-ential or Lipschitz-type lower estimate of increase



QUANTIZATION OF UNIVERSAL TEICHMULLER SPACE

Armen Sergeev

Steklov Mathematical InstituteGubkina 8, Moscow - 119991, RUSSIA


Universal Teichmuller space T is the quotient of the group QS(S1) of qua-sisymmetric homeomorphisms of the unit circle S1 (i.e. homeomorphisms of S1

extending to quasiconformal homeomorphisms of the unit disc) modulo Mobiustransformations. It contains the quotient S of the group Diff+(S1) of diffeomor-phisms of S1 modulo Mobius transformations. Both groups act naturally on theSobolev space H := H

1/20 (S1,R) of half-differentiable functions on S1.

Quantization problem for T and S arises in string theory where these spacesare considered as phase manifolds. To solve the problem for a given phase spacemeans to fix a Lie algebra of functions (observables) on it and construct itsirreducible representation in a Hilbert (quantization) space.

For the space S of diffeomorphisms of S1 the algebra of observables coincideswith the Lie algebra Vect(S1) of Diff+(S1). Its quantization space is identifiedwith the Fock space F (H), associated with the Sobolev space H. Infinitesimalversion of the Diff+(S1)-action on H generates an irreducible representation ofVect(S1) in F (H), yielding a quantization of S.

For the universal Teichmuller space T the situation is more subtle sinceQS(S1)-action on T is not smooth. Respectively, there is no classical Lie al-gebra, associated to QS(S1). However, we can define a quantum Lie algebraof observables Derq(QS), generated by quantum differentials, acting on F (H).These differentials arise from integral operators dqh on H with kernels, givenessentially by finite-difference derivatives of h ∈ QS(S1).

MSC 2010 : 32Q15, 53Z05, 81T30Key Words and Phrases: universal Teichmuller space, quasiconformal maps,

geometric quantization



FROM HILBERT FRAMES TO GENERAL FRECHET FRAMES

Diana T. Stoeva

Acoustics Research Institute, Wohllebengasse 12-14, Vienna 1040, AUSTRIAand

Department of Mathematics – University of Architecture, Civil Engineering,and Geodesy, Blvd ”Hristo Smirnenski” 1, Sofia 1046, BULGARIA


Frame theory is one of the current scientific topics, which are very importantboth for theory and applications. Frames are used in many real-life applications,for example in signal and image processing.

In the present talk we consider frames in Hilbert, Banach, and Frechetspaces. The main aim is to present results concerning series expansions viasuch sequences.

The concept of (Hilbert) frame extends the concept of orthonormal basis.While orthonormal bases allow representation of the space-elements in a uniqueway, overcomplete frames allow many representations of the space-elements(which are very useful in applications, giving the possibility to choose an ”appro-priate” representation among the many existing ones). A natural extension offrames to Banach spaces are the so-called Xd-frames. In contrary to the Hilbertframe case, Xd-frame does not necessarily lead to series expansions. Further,we consider projective and inductive limits of Banach spaces (for example, theSchwartz space S(Rn) of rapidly decreasing functions and its dual, the space oftempered distributions S ′(Rn)) and Frechet frames for such spaces. We presentsome recent results on General Frechet frames and sufficient conditions for seriesexpansions in the projective and inductive limits, as well as in the generatingBanach spaces.

The results concerning General Frechet frames are joint work with StevanPilipovic.

MSC 2010 : 42C15, 46A13, 46B15Key Words and Phrases: frames; Xd-frames; Banach frames; General Frechet

frames; series expansions



ON THE APPROXIMATE SOLUTIONS OF A FRACTIONAL

DIFFERENTIAL EQUATION

Arpad Takaci 1,§, Djurdjica Takaci 2, Mirjana Stojanovic 3

Department of Mathematics and Informatics, Faculty of Sciences– University of Novi Sad

Trg Dositeja Obradovica 4, 21000 Novi Sad, SERBIAe-mails: 1 [email protected],

2 [email protected], 3 [email protected]

We consider the fractional differential equation

∂βu(x, t)∂tβ

=∂αu(x, t)

∂xα+

2(x2 + 1

)t2−β

Γ(3− β)− 2

(t2 + 1

)x2−α

Γ(3− α),

in the cases 0 < β < 1, 0 < α < 1, and 1 < β < 2, 1 < α < 2, for0 < t < T, 0 < x < X, and with the appropriate initial conditions. We treatthe problems in the frame of two-dimensional operational calculus to determinethe exact and the approximate operational solutions for each of the two cases.

MSC 2010 : 26A33, 44A45, 44A40, 65J10Key Words and Phrases: fractional calculus, operational calculus, Mikusinski

operators



ON THE SOLUTIONS OF FUZZY FRACTIONAL

DIFFERENTIAL EQUATION WITH FUZZY COEFFICIENTS

Djurdjica Takaci 1,§, Aleksandar Takaci 2

1 Department of Mathematics and Informatics, Faculty of Sciences –University of Novi Sad

Trg Dositeja Obradovica 4, 21000 Novi Sad, SERBIAe-mail: [email protected]

2 Faculty of Technology – University of Novi SadTrg Dositeja Obradovica 4, 21000 Novi Sad, SERBIA


A fuzzy fractional differential equation is studied in the frames of the Mikusin-ski calculus. The exact and the approximate solutions of the considered problemare constructed and their character is analyzed.

MSC 2010 : 34A07, 34A12,34A25, 44A40Key Words and Phrases: fuzzy calculus, operational calculus, fuzzy differ-

ential equations, fractional calculus, Mikusinski operators



ON HOMOCLINIC SOLUTIONS OF SOME CLASSES OFP-LAPLACIAN DIFFERENCE EQUATIONS

Stepan Tersian

Department of Mathematics – University of Ruse8 Studentska, Ruse 7017, BULGARIA


We study the existence of homoclinic solutions for the p−Laplacian differ-ence equation

∆2pu (k − 1)− V (k) u (k) |u (k)|q−2 + λf (k, u (k)) = 0,

u(k) → 0, |k| → ∞,

where 1 < p ≤ q, (u(k)), k ∈ Z is a sequence or real numbers, ∆ is the differenceoperator ∆u (k) = u (k + 1)− u (k),

∆2pu (k − 1) = ∆u (k) |∆u (k)|p−2 −∆u (k − 1) |∆u (k − 1)|p−2

is referred as the p-Laplacian difference operator and the functions V (k) andf(k, x) satisfy suitable conditions.

The presentation is based on results published in papers [1] and [2], wherethe functions V (k) and f(k, x) are T-periodic in k, and [3], where the functionV (k) is coercive, i.e. V (k) →∞ as k →∞. See References below:

[1] A. Cabada, C. Li, S. Tersian, On homoclinic solutions of a semilinear p-Laplacian difference equation with periodic coefficients, Adv. DifferenceEqu. 2010 (2010), 17 pp.

[2] M. Mihilescu, V. Radulescu, S. Tersian, Homoclinic solutions of differenceequations with variable exponents, Topological Methods in Nonlinear Anal-ysis, No 38 (2011), 277-289.

[3] A. Iannizzotto, S. Tersian, Multiple homoclinic solutions for the discretep-Laplacian via critical point theory, J. of Mathematical Analysis and Ap-plications 403, Issue 1 (1 July 2013), 173-182.

MSC 2010 : 34C37, 34K28, 49J40Key Words and Phrases: p-Laplacian difference equation, homoclinic orbit,

variational method, Brezis-Nirenberg’s theorem, Pucci-Serrin theorem



PROPER HOLOMORPHIC MAPPINGS, BELL’S FORMULA

AND THE LU QI-KENG PROBLEM ON THE TETRABLOCK

Maria TrybuÃla

Institute of Mathematics, Faculty of Mathematics and Computer ScienceJagiellonian University

ÃLojasiewicza 6, 30-348 Krakow, POLANDe-mail: [email protected]

We prove some generalization of Rudin’s Theorem. Namely: Let D ⊂ Cn

be any domain. Let π : D → Cn be a holomorphic map. Assume thereexists a finite group of homeomorphic transformations U of D such that Dis precisely U-invariant, that is for z, w ∈ D we have that π(z) = π(w) if andonly if Uz = w for some U ∈ U . Then π(D) is a domain and π : D → π(D)is a proper mapping.

MSC 2010 : 32A25, 32A70Key Words and Phrases: functions of several complex variables, proper

holomorphic mappings, Bell’s formula, Lu Qi-Keng problem



BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS

— REAL AND COMPLEX METHODS IN COMPARISON

Wolfgang Tutschke

Department of Computational Mathematics – Graz University of TechnologySteyrergasse 30, 8010 Graz, AUSTRIA


1) Let Ω be a (bounded) domain in Rn and L an elliptic differential operatorin Ω. Then the boundary value problems

Lu = F (x, u) in Ωlu = g on ∂Ω

can be reduced to fixed-point problems. Provided E(x, ξ) is a fundamentalsolution of Lu = 0, solution of the boundary value problem are fixed points ofthe operator

U(x) = u0(x) + u(x) +∫

Ω

E(x, ξ)F (ξ, u(ξ))dξ, (1)

and vice versa (u0 is a solution of the given boundary value problem for thesimplified equation Lu = 0, and u compensates the boundary values of theintegral term to zero).

2) The first order systems

Hj(x, y, u, v, ∂xu, ∂yu, ∂xv, ∂yv) = 0, j = 1, 2,

in the complex plane can (under a solvability condition) be reduced to thecomplex normal form

∂zw = F (z, w, ∂zw), (2)

where 2∂z = ∂x − i∂y and 2∂z = ∂x + i∂y. Since the Cauchy kernel is a funda-mental solution of the Cauchy-Riemann system, boundary value problems forsystems (2) can be reduced to operators of type (1):

W (z) = w0(z) + w(z)− 1π

∫∫

Ω

F (ζ, w(ζ), ∂z(ζ))dξdη,

w0 is a holomorphic solution of the boundary value problem, and the holomor-phic function w compensates the boundary values of the integral term, ζ = ξ+iη.

3) Using the Cauchy kernel

E(x, ξ) =1

ωn+1· x− ξ

|x− ξ|n+1

of Clifford analysis in R1+n, similar constructions are also possible for first-ordersystems

Hj(x, u0, ..., u12...n, ∂x0u0, ..., ∂xnu12...n) = 0, j = 1, ..., 2n,

in higher dimensions.

4) The related fixed-point problems can be solved by the contraction-mappingprinciple if the right-hand sides of the differential equations satisfy a Lipschitzcondition (with respect to the desired solution) and, further, if the Lipschitzconstants are small enough. In case the right-hand sides are only continuous,then the second version of the Schauder fixed-point theorem is applicable.

5) In case the right-hand sides are only locally continuous (or Lipschitz-continuous), the fixed-point theorems cannot be applied in the whole functionspace but only in subsets such as balls. An optimal choice of the radius leads tothe solution of the boundary value problem under conditions which are as weakas possible.

6) Since the real-valued components of monogenic u functions in R1+n aresolutions of the Laplace equation, a monogenic function can be recovered from itsboundary values. A monogenic function in R1+n has 2n real-valued components.However, in order to recover u, one needs only to know the boundary valuesof 2n−1 components on the whole boundary. Then the other components cancompletely be determined from their values on some lower-dimensional parts ofthe boundary. These parts are the so-called distinguishing parts of the boundary.In cylindrical domains of R3, for instance, one has to know two componentson the whole boundary, one component on the lower covering surface of thecylindrical domain, and the fourth components is already uniquely determinedby its value at one point.

MSC 2010 : 35G30, 35F60, 31B05, 30G20, 30G35Key Words and Phrases: reduction of boundary value problems for ellip-

tic equations and systems to fixed-point methods using fundamental solutions;complex and Clifford-analytic normal forms of real systems; distinguishing partsof the boundary; optimization of fixed-point methods



UNIQUENESS OF POSITIVE RADIAL SOLUTIONS

TO CHOQUARD’S EQUATION

George Venkov

Department of Applied Mathematics and Informatics– Technical University of Sofia

Kliment Ohridski Blvd, Sofia - 1000, BULGARIAe-mail: [email protected]

We are interested in uniqueness of positive radial solutions of the followingChoquard type equation in Rn, n ≥ 3

−∆u + ω|u|p−2u =(

1|x|(n−2)

∗ |u|p)|u|p−2u, 2 ≤ p < 1 +

4n− 2

(1)

with fixed Lp-norm ∫

Rn

|u(x)|pdx = 1. (2)

The natural energy functional associated with this problem is given by

E(u) =12‖∇u‖2

L2 − 12p

D(|u|p, |u|p), (3)

where we denoteD(f, g) =

∫

Rn

∫

Rn

f(x)f(y)|x− y|(n−2)

dydx. (4)

The existence of nonnegative solutions follows from the fact that the infimum

infE(u);u ∈ H1 ∩ Lp, ‖u‖pLp = 1 (5)

is attained. Any minimizer, after scaling and multiplication by a constant,yields a nonnegative solution. Then, by rearrangement inequalities it is easy toshow that the Choquard equation has a positive solution which is sphericallysymmetric and strictly decreasing.

The most delicate result we would like to address here concerns uniquenessof positive radial solutions vanishing at infinity. Our proof is based on a carefulODE analysis of the corresponding Wronskian of solutions.

MSC 2010 : 35Q55, 35Q40, 47J10Key Words and Phrases: Choquard equation, energy minimizers, unique-

ness.



GENERAL STIELTJES MOMENT PROBLEMS FOR

RAPIDLY DECREASING SMOOTH FUNCTIONS

Jasson Vindas

Department of Mathematics – Ghent UniversityKrijgslaan 281 Building S22, B 9000 Gent, BELGIUM


The problem of moments, as its generalizations, is an important mathemat-ical problem which has attracted much attention for more than a century. Itwas first raised and solved by Stieltjes for positive measures. Boas and Polya,independently, showed later that given an arbitrary sequence an∞n=0 there isalways a function of bounded variation F such that

an =∫ ∞

0xndF (x) , n ∈ N . (1)

A major improvement to this result was achieved by Duran, who was able toshow the existence of regular solutions to (1). He proved (Proc. Amer. Math.Soc. 107 (1989), 731–741) that every Stieltjes moment problem

an =∫ ∞

0xnφ(x)dx , n ∈ N , (2)

admits a solution φ ∈ S(0,∞), that is, a solution in the Schwartz class of rapidlydecreasing smooth functions with suppφ ⊆ [0,∞).

In this talk we discuss a result which significantly improves Duran’s theoremquoted above. We shall replace the sequence of monomials in (2) by a rathergeneral sequence of distributions fn∞n=0 with supp fn ⊆ [0,∞) and presentconditions which ensure that every generalized Stieltjes moment problem

an = 〈fn, φ〉 , n ∈ N ,

has a solution φ ∈ S(0,∞).

MSC 2010 : 30E05, 44A60, 47A57Key Words and Phrases: Stieltjes moment problems; rapidly decreasing

smooth solutions

Complex Analysis and Applications '13 (Proc. of International Conference, Sofia, 31 Oct.-2 Nov. 2013)

Preliminary List of Participants

Praween Agarwal (India) Lilia Apostolova (Bulgaria) Stoyu Barov (Bulgaria) Emilia Bazhlekova (Bulgaria) Nikolai Bozhinov (Bulgaria) Melkana Brakalova-Trevithick (USA) Johann Davidov (Bulgaria) Georgi Dimkov (Bulgaria) Stancho Dimiev (Bulgaria) Ivan Dimovski (Bulgaria) Georgi Ganchev (Bulgaria) Valentina Gogovska (Macedonia) Jianhua Gong (UAE) Vanja Hadzijski (Bulgaria) Milen Hristov (Bulgaria) Valentin Hristov (Bulgaria) Nikolay Ikonomov (Bulgaria) Georgi Iliev (Bulgaria) Mourad EH Ismail (USA) Stefan Ivanov (Bulgaria) Slagjana Jakimovik (Macedonia) Andrzej Kaminski (Poland) Virginia Kiryakova (Bulgaria) Ralitza Kovacheva (Bulgaria) Julian Lawrynowicz (Poland) Rumyan Lazov (Bulgaria) Martin Lukarevski (Macedonia) Sladjana Marinkovic (Serbia) Ekaterina Mihaylova (Bulgaria) Velichka Milusheva (Bulgaria) Ivan Minchev (Bulgaria) Svetlana Мincheva-Kaminska (Poland)

Takeshi Miura (Japan) Oleg Muskarov (Bulgaria) Dennis Nemzer (USA) Nikolai Nikolov (Bulgaria) Yanka Nikolova (Bulgaria) Ludmila Nikolova (Bulgaria) Inna Nikolova (Bulgaria) Jordanka Paneva-Konovska (Bulgaria) Donka Pashkouleva (Bulgaria) Alexander Petkov (Bulgaria) Marko Petkovic (Serbia) Stevan Pilipovic (Serbia) Peter Popivanov (Bulgaria) Predrag Rajkovic (Serbia) Medhat Rakha (Oman) Szilárd Révész (Hungary) Peter Rusev (Bulgaria) Natasa Savic (Serbia) Blagovest Sendov (Bulgaria) Armen Sergeev (Russia) Margarita Spiridonova (Bulgaria) Miomir Stankovic (Serbia) Diana Stoeva (Bulgaria) Arpad Takaci (Serbia) Djurdjica Takaci (Serbia) Stepan Terzian (Bulgaria) Maria Trybula (Poland) Yulian Tsankov (Bulgaria) Wolfgang Tutschke (Austria) Georgi Venkov (Bulgaria) Jasson Vindas (Belgium) Simeon Zamkovoy (Bulgaria)

Contents of CAA ’13 Book of Abstracts

Preface (V. Kiryakova: L. Iliev & the Day of National Leaders) . . 1Bl. Sendov (Acad. L. Iliev - Leader of ... Math. Community) . . . . 6P. Rusev (Acad. L. Iliev and ... Complex Analysis) . . . . . . . . . . . . . . 10List of Publications of L. Iliev (by D. Pashkouleva) . . . . . . . . . . . . . . 17

P. Agarwal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33L. Apostolova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34S. Barov, J.J. Dijkstra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35E. Bazhlekova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36R. Bojicic, M. Petkovic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37N. Bozhinov, I. Dimovski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38M. Brakalova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39M. Choban, E. Mihaylova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40J. Davidov, O. Muskarov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41G. Dimkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43S. Dimiev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44I. Dimovski, V. Hristov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45I. Dimovski, M. Spiridonova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46I. Dimovski, Yu. Tsankov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47G. Ganchev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48V. Gogovska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49J. Gong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50M. Hristov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51N. Ikonomov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52M.E.H. Ismail (plenary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53S. Ivanov, A. Petkov, D. Vassilev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Sl. Jakimovik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56A. Kaminski, S. Mincheva-Kaminska . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57V. Kiryakova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58H. Koshimizu, T. Miura, H. Takagi, S.E. Takahasi . . . . . . . . . . . . . . . 60R. Kovacheva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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ISBN 978-954-8986-37-3 Contents, continued:

J. Lawrynowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62M. Lukarevski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63S. Mincheva-Kaminska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64D. Nemzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65N. Nikolov, Bl. Sendov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Y. Nikolova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67J. Paneva-Konovska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68D. Pashkouleva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69D. Pashkouleva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A. Petkov (with S. Ivanov, D. Vassilev) . . . . . . . . . . . . . . . . . . . . . . . . . 71S. Pilipovic (plenary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72P. Popivanov (plenary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74E. Radzievskaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76P. Rajkovic, Sl. Marinkovic, M. Stankovic . . . . . . . . . . . . . . . . . . . . . . . 77P. Rajkovic, P. Barry, N. Savic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78M. Rakha, A.K. Rathie, U. Pandey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79I. Ramadanoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Sz. Gy. Revesz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A. Sergeev (plenary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82D. Stoeva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A. Takaci, Dj. Takaci, M. Stojanovic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Dj. Takaci, Al. Takaci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85S. Terzian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86M. Trybula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87W. Tutschke (plenary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88G. Venkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90J. Vindas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Preliminary List of participants: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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