eigenvalues for infinite matrices, their computations and ... · matrices and fredholm integral...
TRANSCRIPT
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Eigenvalues for Infinite Matrices, TheirComputations and Applications
P.N.Shivakumar
Email: [email protected]
(Collaborator: Yang Zhang)
Department of Mathematics
University of Manitoba,Winnipeg, Canada R3T 2N2
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Abstract
In this talk, we discuss
• properties of eigenvalues which occur in infinite linear algebraic
systems of the form Ax = λx , where A is an infinite diagonally
dominant matrix and b is a bounded linear operator,
• the classical question “Can you hear the shape of a drum?”:
“YES” for certain convex planar regions with analytic
boundaries; “NO” for some polygons with reentrant corners,
• eigenvalues and van der Waerden infinite matrix.
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History
Earliest concept of infinity comes from Anaximander, a Greek and
consequently mathematical infinity is attributed to ZENO (400
B.C.). Also. The Indian text Surya Prajnapti (300-400 B.C.)
classified all numbers into three sets, namely, enumerable,
innumerable, and infinite. In 1655, Waller introduced the symbol
∞.
“Two things are infinite: the universe and human stupidity; I am
not sure about the universe.” ——– A. Einstein
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Discussion of infinite systems generally start with truncated finite
systems.
References:
(a) “A History of Infinite Matrices” by Michael Bernkopf
(b) “Infinite Matrices and Sequence Space” by R. G. Cooke
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Some earlier authors are Poincare (1984, Hill’s equation), Hilbert
(1906, Hilbert infinite quadratic forms, equivalent to infinite
matrices and Fredholm integral equations), John von Neumann
(1929, Infinite matrices as operators) and Richard Bellman (1947,
The boundedness of solutions of infinite systems of linear
equations).
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We give three different criteria for An×n = (aij) to be nonsingular.
(a) Diagonal dominance
σi |aii | =n∑
j=1, j 6=i
|aij |, 0 ≤ σi < 1, i = 1, 2, ..., n.
(b) A chain condition
For A = (aij)n×n, there exists, for each j /∈ J, a sequence of
nonzero elements of the form ai ,i1 , ai ,i2 , ..., ai ,ij with j ∈ J.
(c) Some sign patternsaij > 0 for i ≤ j ,
(−1)i+jaij > 0 for i > j ,(i , j = 1, 2, ..., n)
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Linear Algebraic Systems
Consider
∞∑j=1
aijxj = bi , i = 1, 2, ...,∞,
where
• the infinite matrix A = (aij),
• σi |aii | =∑∞
j=1 |aij |; 0 ≤ σi < 1, i = 1, 2, ...,∞,
• the sequence bi is bounded.
We establish sufficient conditions which guarantee the existence
and uniqueness of a bounded solution to the above system as well
as bounds for truncated systems.
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We have
|x (n+1)j − x
(n)j | ≤ Pσn+1 + Q|an+1,n+1|−1
for some positive constraints P and Q.
An estimate for the solution is given by
|x (n)| ≤n∏
k=1
1 + σk1− σk
n∑k=1
|bk ||akk |(1 + σk)
.
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Linear Eigenvalue Problem
To determine the location of eigenvalues for diagonally dominant
infinite matrices (which occur in (a) solutions of elliptic partial
differential equations and (b) solutions of second order linear
differential equations) and establishes upper and lower bounds of
eigenvalues, we define an eigenvalue of A to be any scalar λ for
which Ax = λx for some 0 6= x ∈ D(A).
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Assume A is an operator on l1.
E (1): aii 6= 0, i ∈ N and |aii | → ∞ as i →∞.
E (2): ∃ρ ∈ [0, 1), s.t. for each j ∈ N
Qj =∞∑
i=1,i 6=j
|aij | = ρj |ajj |, 0 ≤ ρj ≤ ρ.
E (3): For all i , j ∈ N, i 6= j , |aii − ajj | ≥ Qi + Qj .
E (4): For all i ∈ N, sup|aij | : j ∈ N <∞.
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Linear Differential System
We consider the infinite system
d
dtxi (t) =
∞∑j=1
aijxj(t) + fi (t), t ≥ 0, xi (0) = yi , i = 1, 2, ...
In particular, if A is a bounded operator on l1, then the
convergence of a truncated system has been established.
(Arley and Brochsenius (1944), Bellman (1947) and Shaw (1973)
have studied this problem. However, none of these papers yields
explicit error bounds for such a truncation. )
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Applications
1. Digital circuit dynamics
2. Conformal mapping of doubly connected regions
3. Fluid flow in pipes
4. Mathieu equation d2ydx2 + (λ− 2q cos 2x)y = 0 for a given q
with the boundary conditions: y(0) = y(π2 ) = 0
(Our interest: two consecutive eigenvalues merge and become
equal for some values of the parameter q. )
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5. Bessel Function
6. Eigenvalues of the Laplacian on an elliptic domain
∂2u
∂x2+∂2u
∂y2+ λ2u = 0 in D, u = 0 on ∂D,
where D is a plane region bounded by smooth curve ∂D.
Consider the domain bounded by the ellipse represented byx2
α2 + y2
β2 = 1, which can be expressed correspondingly in the
complex plane by
(z + z)2 = a + bzz ,
where a = 4α2β2
β2−α2 , b = 4α2
α2−β2 .
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After considerable manipulation, we get the value of u on the
ellipse as
u = 2a0 +∞∑n=1
A2n,0B0,nan +∞∑k=1
(−λ
2
4
)k2a0
k!k!(zz)k
+∞∑k=1
[k∑
n=1
(A2n,kb0,n +
n∑l=1
A2n,k−lbl ,n
)an
](zz)k
+∞∑k=1
[ ∞∑n=k+1
(A2n,kb0,n +
k∑l=1
A2n,k−lbl ,n
)an
](zz)k .
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For u = 0 on the elliptic boundary, we equate the powers of zz to
zero, where we arrive at an infinite system of linear equations of
the form
∞∑k=0
dknan = 0, n = 0, 1, 2, ...,∞,
where dkn’s are known polynomials of λ2. The infinite system is
truncated to n × n system and numerical values were calculated
and compared to existing results in literature. The method
derived here provides a procedure to numerically calculate the
eigenvalues.
8. Mathieu Equation
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Shape of A Drum
For the classical question, “Can you hear the shape of a drum?”,
the answer is known to be “yes” for certain convex planar regions
with analytic boundaries. The answer is also known to be “no” for
some polygons with reentrant corners. A large number of
mathematicians over four decades have contributed to the topic
from various approaches, theoretical and numerical.
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We develop a constructive analytic approach to indicate how a
preknowledge of the eigenvalues leads to the determination of the
parameters of the boundary. This approach is applied to a general
boundary and in particular to a circle, an ellipse, an annulus and a
square. In the case of a square, we obtain an insight into why the
analytical procedure does not, as expected, yield an answer.
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The pursuit of a “complete” solution to the question “Can one
hear the shape of a drum”, originally posed by Lipman Bers and
used as a title in 1966 by Mark Kac, has been a fascinating journey
and work can only be described as “work in progress”. The
research has involved many mathematicians and many tools
including Asymptotics, Probability Theory, Operator Theory,
infinite algebraic systems and inevitably intense computational
work involving approximate methods.
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Mathematically, the problem is, whether a preknowledge of the
eigenvalues of the Laplacian in a region Ω leads to the definition of
Γ, the closed boundary of Ω. Specifically, we have
uxx + uyy + λ2u = 0 in Ω, (1)
u = 0 on Γ. (2)
According to the maximum principle for linear elliptic partial
differential equations, the infinite eigenvalues λ2n, n = 1, 2, 3, ...,
are positive, real, ordered and satisfy
0 < λ21 < λ2
2 < · · · < λ2n < · · · <∞. (3)
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In the famous paper “Can you hear the shape of a drum?”, Mark
Kac makes an analysis using only asymptotic properties of large
eigenvalues and uses probability theory as the tool to establish that
one can hear the area of a polygonal drum and conjectures for
multiply connected regions.
A major contribution in 2000 is given by Zelditch, where a positive
answer “yes” is given for certain regions with analytic boundaries.
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Preliminaries
We express (1) in terms of complex variables z = x + iy ,
z = x − iy , and the system (1) and (2) becomes
uzz +λ2
4u = 0 in Ω, (4)
u = 0 on Γ. (5)
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We start the analysis using the completely integrated form of thesolution to (1) as given in Vekua by
u =
f0(z)−
∫ z
0f0(t)
∂
∂tJ0
(λ√
z(z − t))dt
+ conjugate, (6)
where f0(z) is an arbitrary analytic function which can be formallyexpressed as
f0(z) =∞∑n=0
anzn (7)
and J0 represents the Bessel function of first kind and order 0given by
J0
(λ√
z(z − t))
=∞∑q=0
(−λ
2
4
)zq(z − t)q
q!q!.
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From an identity given in Abramowitz and Stegun, we have, when
n is even,
zn + zn =
n2∑
m=0
cmn(z + z)n−2m(zz)m, n = 2, 4, 6, ...,
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Boundary Γ
(a) A general boundaryWe consider the parametrized analytical boundary Γ with biaxialsymmetry to be given by
(z + z)2n =∞∑n=0
dn1(zz)n
which yields, on using Cauchy products for infinite series,
(z + z)2np =∞∑n=0
dnp(zz)n,
where
dnp =n∑
l=0
dl p−1dn−l 1, p = 1, 2, 3, ....
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(b) Circular boundary
We consider the circular boundary given by
x2 + y2 = a2 or zz = a2.
(c) An elliptic boundary
We consider the elliptic boundary Γ given by
x2
α2+
y2
β2= 1 or (z + z)2 = a + bzz , α > 0.
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(d) A square boundary
We give this example to demonstrate why our analytical approach
does not yield information of the boundary with sharp corners from
a preknowledge of eigenvalues. Consider a square boundary given
by
x = ±a, y = ±a or z4 + z4 = 2(zz)2 − 16a2(zz) + 16a4
or z2 + z2 = 4(zz − 2a)2.
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We assume
f0(z) =∞∑n=0
a2nz2n,
u = 2a0J0
(λ√zz)
+∞∑n=1
a2n
∞∑k=0
(−λ
2
4
)A2n k(z2n + z2n)(zz)k ,
which on substitution gives
u = 2a0J0(λ√zz)+
∞∑n=1
a2n
∞∑k=0
(−λ
2
4
)A2n k
n∑m=0
(−1)m2n(2n −m − 1)!
m!(2n − 2m)!(z + z)2(n−m)(zz)m.
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(a) General boundary Γ
After rearrangement of summations, we get
u = 2a0
+∞∑n=1
a2nDn 0 0 0 +
[2a0
(−λ
2
4
)+∞∑n=1
a2n
1∑
i=0
1−i∑p=0
Dn p 1−i−p i
]zz
+∞∑q=2
2a0
(−λ
2
4
)q1
q!q!+
q−1∑n=1
a2n
[n∑
i=0
q−1∑p=0
Dn p q−i−p i
]
+∞∑
a2n
[q∑ q−1∑
Dn p q−i−p i
](zz)q.
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Now we equate the coefficients of (zz)q, q = 0, 1, 2, ... to zero to
obtain the infinite linear algebraic system. Writing the resulting
system as
∞∑n=0
fqna2n, q = 0, 1, 2, ...,
we note that fqn is a polynomial of degree q in µ and contains only
d01, d11, d21,.... The values of µ are determined by formally
setting D = det(fqn) = 0, which can be expressed as an infinite
series in µ given by
D =∞∑i=0
Diµi
and µi ’s are given by D = 0.
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We note that we can write
D = D0
∞∏i=1
(1− µ
µi
)
formally suggesting that DiD0
is the sum of the inverse products of
µ1, µ2,... taken i at a time. In particular,
D1
D0= −
∞∑i=1
1
µi,
D2
D0=∞∑i=1
∞∑j=i+1
1
µiµj
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(b) A circle
It is well known that the solutions to (1) and (2) is given by
u = a0J0
(λ√zz)
and the eigenvalues are given by the zeros of J0(λa) = 0, namely,
λi =j0ia, i = 1, 2, ....
We can write
∞∑q=0
(−λ
2
4
)q1
q!q!a2q =
∞∏j=1
(1− λ2
λ2j
).
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We obtain
a2 = 4∞∑j=1
1
λ2j
,
thus establishing uniquely the value of a if all the eigenvalues λ1,
λ2,... are known. In fact, substitute for λi from λi = j0ia we obtain
a2
4=∞∑j=1
a2
j20 j
which checks with the well-known fact that
∞∑j=1
1
j20 j
= 4.
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(c) An ellipse
Here we develop the solution in the elliptic domain. We get
∞∑n=0
fnqa2n = 0, q = 0, 1, 2, ....
We note that fnq is a polynomial of degree q in µ. The
determinant D1 is the sum of an infinite determinants derived from
differentiating D with respect to µ and setting µ = 0.
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(d) A square boundary
We give this example to demonstrate how our approach does notyield information of the boundary from a preknowledge of theeigenvalues in the case of a boundary containing sharp corners.Consider the square boundary
z2 + z2 = 4(zz − 2a)2.
We use f0(z) =∑∞
n=0 a4nz4n to obtain
u = 2a0J0
(λ√zz)
+∞∑n=1
∞∑k=0
(−λ
2
4
)k
A4n k(z4n + z4n)a4n(zz)k ,
z4n + z4n =n∑
m=0
bmn(z2 + z2)2n−2m(zz)2m,
which gives z4n + z4n =∑n
m=0 bmn4n−m(zz − 2a)n−m.
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It is well-known that for a square, the eigenvalues λ2 are given bym2π2
a2 + n2π2
a2 and using µ = −λ2
4 yields
D1
D0= −
∞∑i=1
1
µi=
4a2
π2
∞∑m=1
∞∑n=m+1
1
m2 + n2.
The double series is clearly divergent and hence the failure of the
analytic approach, although det[gqn] = 0 yields the eigenvalues.
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Conclusions
In this article, we have demonstrated a constructive approach when
the answer is “yes”. We give an interpretation to “a preknowledge
of eigenvalues” as knowing the sums of the inverse products of the
eigenvalues µi , i = 1, 2, ..., taken i at a time. In the case of an
ellipse, we express the parameters of the ellipse in terms of the
eigenvalues. Our conjecture is that for the answer to be “yes”, it is
necessary for the sums of the inverse products of the eigenvalues
µi , i = 1, 2, ..., taken i at a time to be convergent series. Another
conjecture is that all analytic curves (curvilinear polygons for
example) yield the answer “yes”.
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Hearing the Shape of a Drum with a Hole
We are interested in the Helmholtz equation
d2u
dx2+
d2u
dy2+ λu = 0 in Ω
where Ω is the doubly connected region with the boundary
conditions
u = 0 on π1 : x2 + y2 = a2
u = 0 on π2 : x2 + y2 = b2 a > b
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We need to show that if the eigenvalues of λ are known, the ratio
of the annulus is uniquely determined.
Equation d2udx2 + d2u
dy2 + λu = 0 in polar coordinates (r , θ) and noting
that u = u(r), we get the Bessel equation
d2u
du2+
du
dr+ λu = 0
whose solution is given in Ω [Garabedian].
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U = c1J0(√λr) + c2
J0(√λr) ln(
√λr)−
∞∑n=1
anHnλnr2n
where an = (−1)n
2nn! and Hn = 1 + 12 + 1
3 + · · ·+ 1n .
Using the previous equations, we get
c1J0(√λa) + c2
J0(√λa) ln(
√λa)−
∞∑n=1
anHnλna2n
= 0
c1J0(√λb) + c2
J0(√λb) ln(
√λb)−
∞∑n=1
anHnλnb2n
= 0
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For non-zero c1 and c2, we need
J0(√λa)J0(
√λb) ln
(b
a
)+
J0(√λb)
∞∑n=1
anHnλna2n − J0(
√λa)
∞∑n=1
anHnλnb2n
= 0
Equation above becomes an infinite series in λ. In fact, we can
express the infinite series in the above equation by
∞∏j=1
(1− λ
λj
)ln
(b
a
)
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Equating the constants, we get ln(ba
)= ln
(ba
). And equating the
coefficient of λ, we get
a1(a2 + b2) ln
(b
a
)+ a1H1(a2 − b2) = −
∞∑j=1
1
λjln
(b
a
).
Using H1 = 1, a = 1 and a1 = − 124 , we get
1
24(1 + b2) +
1− b2
ln(b)=∑ 1
λj= P.
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Knowing P, we can numerically solve for b. We also note that
conversely, if b is known, we get the value of
∞∑j=1
1
λj.
Hence, we can hear the shape of the drum which is an annulus if
the eigenvalues are known.
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Zeros of Infinite Series and Ax = b
For the infinite series in λ, we will assume that the zeros of the
series λn, n = 1, 2, 3, ...; we can write, with λ2 replaced by x as
y(x) =∞∑n=0
cnx2n =
∞∏n=1
(1 + anx)
where c0 = 1, c1 =∞∑k=1
aky′(0) and λn = − 1
an.
We also have 0 < λ21 < λ2
2 < λ23 < · · · . We seek to connect a’s
with cn’s.
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The solution to
y ′′ + f (x)y = 0, (8)
when f (x) is an arbitrary function with continuous derivatives, is
expressed by
∞∑k=0
cnxn, (9)
where cn’s are explicitly dependent on derivatives of f (x) and the
initial conditions on y(x), namely, y(0) and y ′(0).
In fact, y(0) = c0 and y ′(0) = c1.
Reference: P. N. Shivakumar and Y. Zhang, On series solutions of y ′′–f (x)y = 0 and
applications, Advances in difference equations, 2013, 201: 47.
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It can be easily shown that, with
f (x) =
( ∞∑n=1
an1 + anx
)2
−∞∑n=1
a2n
(1 + anx)2, (10)
solution to (8) is given by (9) in terms of Qp.
We will use the notation Qp =∞∑k=1
apk , p = 1, 2, 3, ...
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Finally, we get
∞∑k=1
apk = ep, p = 1, 2, ..., (11)
where ep’s are explicitly obtained as combinations of cn’s.
Rewriting the above equations as
∞∑k=1
apkxk = ep, p = 1, 2, ...,
we note that the system is of the form Ax = e, where x is a unite
vector and A is the well-known Vandermonde infinite matrix. We
now get x = A−1e. In all the discussion above, infinity can be
replaced by a positive integer.
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As a simple example, we consider a polynomial of degree 3 with
increasing positive roots. By the above analysis, we arrive at the
system
x1 + x2 + x3 = c1,
x21 + x2
2 + x23 = c2,
x31 + x3
2 + x33 = c3,
where c1, c2 and c3 are determined by the coefficients of the
polynomial. We arrive at the bound
0 < x1 <c2
c1< x2 <
c3
c2< x3 <
c1
3.
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References
• P. N. Shivakumar and Y. Wu, Eigenvalues of the Laplacian on
an elliptic domain, Comput. Math. Appl. 55(2008)1129-1136.
• P. N. Shivakumar, Diagonally dominant infinite matrices in
linear equations, Utilitas Mathematica, Vol. 1, 1972, 235-248.
• P. N. Shivakumar and R. Wong, Linear equations in infinite
matrices, Journal of Linear Algebra and its Applications, Vol. 7,
1973, No. l, 553-562.
• P. N. Shivakumar and K. H. Chew, A sufficient condition for the
vanishing of determinants, Proceedings of the American
Mathematical Society, Vol. 43, No. 1, 1974, 63-66.
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• P. N. Shivakumar, K. H. Chew and J. J. Williams, Error bounds
for the truncation of infinite linear differential systems, Journal
of the Institute of Mathematics and its Applications, Vol. 25,
1980, 37-51.
• P. N. Shivakumar, N. Rudraiah and J. J. Williams, Eigenvalues
for infinite matrices, Journal of Linear Algebra and its
Applications,1987 Vol. 96, 35-63.
• P. N. Shivakumar and J. J. Williams, An iterative method with
truncation for infinite linear systems, Journal of Computational
and Applied Mathematics, 24, 1988, 109-207.
• P. N. Shivakumar and C. Ji, On Poisson’s equation for doubly
connected regions , Canadian Applied Mathematics Quarterly,
Vol.1, No. 4, Fall 1993
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• P. N. Shivakumar and C. Ji, Upper and lower bounds for inverse
elements of finite and infinite tri-diagonal matrices, Linear
Algebra and its Applications 247:297-316, 1996
• P. N. Shivakumar and Q. Ye, Bounds for the width of instability
intervals in the Mathieu equation, accepted for publication in
the Proceedings of the International Conference on Applications
of Operator Theory, Birkhauser series “Operator Theory:
Advances and Applications”, Vol.87, 348- 357, 1996.
• P. N. Shivakumar, J. J. Williams, Q. Ye and C. Marinov, On
two-sided bounds related to weakly diagonally dominant
M-matrices with application to digital circuit dynamics, accepted
for publication in Siam J. Matrix Analysis and Applications,
Vol.17, No.2, 298-312, April 1996.
BIRS Workshop March 22-27, 2015 P.N.Shivakumar 50
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• P. N. Shivakumar and C. Ji, On the nonsingularity of matrices
with certain sign patterns, Linear Algebra and its Applications
273:29-43, 1998.
• P. N. Shivakumar and A. G. Ramm, Inequalities for the minimal
eigenvalue of the Laplacian in an annulus, Mathematical
Inequalities & Applications, Vol.1, No. 4, 559-563. 1998.
• P. N. Shivakumar and J. Xue, On the double points of a
Mathieu equation, Journal of Computational and Applied
Mathematics, 109, 111-125, 1999.
BIRS Workshop March 22-27, 2015 P.N.Shivakumar 51
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• P. N. Shivakumar, Y. Wu and Y. Zhang, Shape of a drum, a
constructive approach, Transactions on Mathematics, WSEA,
2011.
• P. N. Shivakumar and Y. Zhang, On series solutions of
y ′′–f (x)y = 0 and applications, Advances in difference
equations, 2013, 201: 47.
• P. N. Shivakumar and K. C. Sivakumar, A review of infinite
matrices and their applications, Linear Algebra and its
Applications 430 (2009) 976-998.
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Thank you!
BIRS Workshop March 22-27, 2015 P.N.Shivakumar 53