the irresistible inequality of milne - elteabesenyei.web.elte.hu/publications/milne.pdf•james...

The irresistible inequality of Milne

Adam [email protected]

Department of Applied Analysis and Computational Mathematics,Eotvos Lorand University, Budapest

CIA2016Hajduszoboszlo, September 1, 2016

mailto:[email protected]

An inspirational quotation

Vladimir Arnold(1937–2010)

“Mathematics is a part of physics. Physics is an experimentalscience, a part of natural science. Mathematics is the part ofphysics where experiments are cheap.”

On teaching mathematics (Paris, March 7, 1997)

Adam BESENYEI (ELTE) Milne’s inequality CIA2016, August, 2016 1 / 17

A brief lesson on circuits

Resistors connected in series:

I

U1

I

U2

I I

Un

IR1 R2 Rn

+ −

U

• Kirchhoff’s voltage law (Konigsberg, 1845):

U = U1 + U2 + . . . + Un,

• Ohm’s law (1827): current = voltage / resistance

=⇒ RtotalI = R1I + R2I + . . . + RnI

=⇒ Rtotal = R1 + R2 + . . . + Rn



Resistors connected in parallel:

−

+U

I

I I1

R1

I1

U1

I2

R2

I2

U2

In

Rn

In

Un

• Kirchhoff’s current law (Konigsberg, 1845):

I = I1 + I2 + · · ·+ In.

• Ohm’s law (1827): current = voltage / resistance

=⇒ U

Rtotal= U

R1+ U

R2+ · · ·+ U

Rn

=⇒ 1Rtotal

= 1R1

+ 1R2

+ · · ·+ 1Rn



Rayleigh’s monotonicity principle:

resistance of any part increases =⇒ total resistance does not decrease

• John William Strutt, 3rd Baron Rayleigh (1842–1919):On the Theory of Resonance (1871), On the Theory of Sound (1877)

• James Clerk Maxwell (1831–1879):A Treatise on Electricity and Magnetism (1873)


Circuits and means

Monotonicity Principle =⇒ Arithmetic Mean ≥ Harmonic Mean

R1

R2

R2

R1

Ropentotal = 1

1R1+R2

+ 1R1+R2

= R1 + R22

R1

R2

R2

R1⇐⇒

R1

R2 R1

R2

Rclosedtotal = 1

1R1

+ 1R2

+ 11

R1+ 1

R2

= 21

R1+ 1

R2


Circuits and means


R1

R2

R2

R1

Ropentotal = 1

1R1+R2

+ 1R1+R2

= R1 + R22

R1

R2

R2

R1

⇐⇒

R1

R2 R1

R2

Rclosedtotal = 1

1R1

+ 1R2

+ 11

R1+ 1

R2

= 21

R1+ 1

R2


Circuits and means


R1

R2

R2

R1

Ropentotal = 1

1R1+R2

+ 1R1+R2

= R1 + R22

R1

R2

R2

R1⇐⇒

R1

R2 R1

R2

Rclosedtotal = 1

1R1

+ 1R2

+ 11

R1+ 1

R2

= 21

R1+ 1

R2


Circuits and means


R1

R2

R2

R1

Ropentotal = 1

1R1+R2

+ 1R1+R2

= R1 + R22

R1

R2

R2

R1⇐⇒

R1

R2 R1

R2

Rclosedtotal = 1

1R1

+ 1R2

+ 11

R1+ 1

R2

= 21

R1+ 1

R2

>


Circuits and means

Monotonicity Principle =⇒ Arithmetic ≥ Geometric ≥ Harmonic

R3

R1

R2

R2

R1

R3 =∞ =⇒ Ropentotal = R1 + R2

2

R3 =√

R1R2 =⇒ Rbetweentotal =

√R1R2

R3 = 0 =⇒ Rclosedtotal = 2

1R1

+ 1R2


Circuits and means

Monotonicity Principle =⇒ Arithmetic ≥ Geometric ≥ Harmonic

R3

R1

R2

R2

R1

R3 =∞ =⇒ Ropentotal = R1 + R2

2

R3 =√

R1R2 =⇒ Rbetweentotal =

√R1R2

>

R3 = 0 =⇒ Rclosedtotal = 2

1R1

+ 1R2

>


Circuits and inequalities

A generalization:a1

b1

a2

b2

a3

b3

an

bn

Ropentotal = (a1 + . . . + an)(b1 + . . . + bn)

a1 + . . . + an + b1 + . . . + bn

a1

b1

a2

b2

a3

b3

an

bn

Rclosedtotal = a1b1

a1 + b1+ . . . + anbn

an + bn



A generalization:a1

b1

a2

b2

a3

b3

an

bn

Ropentotal = (a1 + . . . + an)(b1 + . . . + bn)

a1 + . . . + an + b1 + . . . + bn

a1

b1

a2

b2

a3

b3

an

bn

Rclosedtotal = a1b1

a1 + b1+ . . . + anbn

an + bn

>


Let us switch to Mathematics

Theorem (Milne’s inequality)

ai, bi > 0 (i = 1, . . . , n) =⇒n∑

i=1

aibi

ai + bi≤

n∑i=1

ai ·n∑

i=1bi

n∑i=1

(ai + bi)

Proof:n∑

i=1ai ·

n∑i=1

bi −n∑

i=1(ai + bi) ·

n∑i=1

aibi

ai + bi=

∑1≤i<j≤n

(aibj − ajbi)2

(ai + bi)(aj + bj)

Edward Arthur Milne (1896–1950):

• British astrophysicist and mathematician

• in 1925 he proved the integral version of the inequality



Brief history:

• Alfred Lehman (1931–2006): SIAM Review Problem Section, 1960.nonlinear Ohm’s law U = R · I−1/p =⇒ Minkowski’s inequality

• William Niles Anderson, Richard James Duffin (1909–1996)Series and parallel addition of matrices, 1968.

• appeared in many books, popular papers:

– P. G. Doyle and J. L. Snell’s book: Random Walks and ElectricNetworks, 1984.

– Mark Levi’s book: The Mathematical Mechanic: Using PhysicalReasoning to Solve Problems, 2009.

– Alfred Witkowski’s paper in the Mathematics Magazine, 2014.– Adam Besenyei’s paper in the Hungarian Mathematical Journal

for Secondary Schools (KoMaL), 2016.

• Zoltan Bertalan: A ≥ G ≥ H, KoMaL, Problem P. 4813., 2016.


Entropy

Physics

• Rudolph Clausius (1822–1888) in 1865 in classical thermodynamics

• Josiah Willard Gibbs (1839–1903), Ludwig Boltzmann (1844–1906)

Information Theory

• Claude Shannon (1916–2001) in 1948 in his paper at Bell Telephone:A Mathematical Theory of Communication

Definition (Shannon entropy)For a discrete random variable X with possible values of outcome{x1, . . . , xn} and probability distribution p(x) = P (X = x):

H(X) = −n∑

i=1p(xi) log p(xi)

with the convention 0 log 0 = 0. (The letter H is the Greek capital eta.)


Entropy

What is entropy?

• Greek: “a turning towards”

• measures the “uncertainty” of the random variable

• John von Neumann to Shannon in 1949:

“You should call it entropy, for two reasons. In the first placeyour uncertainty function has been used in statistical mechanicsunder that name, so it already has a name. In the second place,and more important, nobody knows what entropy really is, so ina debate you will always have the advantage.”


Subadditivity of Shannon entropy

Joint entropy: for a pair (X, Y ) of discrete random variables

H(X, Y ) = −n∑

i=1

m∑j=1

p(xi, yj) log p(xi, yj).

Theorem1. Subadditivity of entropy:

H(X, Y ) ≤ H(X) + H(Y ).(Joint entropy not greater than the sum of the individual entropies.)

2. Strong subadditivity of entropy:H(X, Y, Z) + H(Y ) ≤ H(X, Y ) + H(Y, Z).

(Conditional entropies, X = “past”, Y = “present”, Z = “future”.)


q-entropy

DefinitionFor a discrete random variable X and q 6= 1 the q-entropy is defined as

Sq(X) = −n∑

i=1p(xi) logq p(xi) = 1−

∑ni=1 p(xi)q

q − 1 ,

where the q-logarithm function is

logq x = xq−1 − 1q − 1 (q 6= 1).

History:

• Zoltan Daroczy (1938–):Generalized information functions, 1970.

• Constantino Tsallis (1943–):A possible generalization of Boltzmann–Gibbs statistics, 1988.


Strong subadditivity of Tsallis entropy

Theorem (Daroczy, 1970)If q > 1, then the q-entropy is subadditive:

Sq(X, Y ) ≤ Sq(X) + Sq(Y ).Equality holds if and only if q = 1 and X, Y are independent.

A special case:

(x+y)q +(z+v)q +(x+z)q +(y+v)q ≤ xq +yq +zq +vq +(x+y+z+v)q.

A similar inequality: proposed in Amer. Math. Monthly, 2014.


Strong subadditivity of Tsallis entropy

Theorem (Shigeru Furuichi, 2004)If q > 1, then the q-entropy is strongly subadditive:

Sq(X, Y, Z) + Sq(Y ) ≤ Sq(X, Y ) + Sq(Y, Z).

Reformulation: notation pijk := p(xi, yj , zk), p−j− = p(yj) etc.r∑

k=1

m∑j=1

n∑i=1

pijk

(logq pijk + logq p−j− − logq pij− − logq p−jk

)≥ 0.

Theorem (B.–Petz, 2013)For q > 1 and fixed 1 ≤ j ≤ n, 1 ≤ k ≤ r the following inequality holds:

n∑i=1

pijk

(logq pijk + logq p−j− − logq pij− − logq p−jk

)≥ 0.


Partial strong subadditivity of Tsallis entropy

Proof: The inequality is equivalent ton∑

i=1pijk

(pq−1

ij− − pq−1ijk

)≤ p−jk

(pq−1

−j− − pq−1−jk

).

With the notation ai = pijk, bi = pij− − pijk (i = 1, . . . , n) andA =

∑ni=1 ai = p−jk, B =

∑ni=1 bi = p−j− it reduces to

n∑i=1

ai

((ai + bi)q−1 − aq−1

i

)≤ A

((A + B)q−1 −Aq−1

),

or equivalently

(q − 1)∫ 1

0

n∑i=1

aibi(ai + tbi)q−2 dt ≤ (q − 1)∫ 1

0AB(A + tB)q−2 dt,

where by Milne’s inequalityn∑

i=1aitbi(ai+tbi)q−1 ≤

n∑i=1

ai · tbi

ai + tbi(A+tB)(A+tB)q−2 ≤ tAB(A+tB)q−2.


References

Adam Besenyei, Denes Petz, Partial subadditivity of entropies, LinearAlgebra Appl., 439 (2013), 3297–3305.

Z. Daroczi, General information functions, Information and Control,16 (1970), 36–51.

S. Furuichi, Information theoretical properties of Tsallis entropies,J.Math.Phys., 47 (2006), 023302.

E. A. Milne, Note on Rosseland’s integral for the stellar absorptioncoefficient, Mon. Not. R. Astron. Soc., 85 (1925) 979–984. http://mnras.oxfordjournals.org/content/85/9/979.full.pdf

A. Witkowski, Proof Without Words: An Electrical Proof of theAM-HM Inequality, Math. Mag., 87 (2014), 275.http://www.jstor.org/stable/10.4169/math.mag.87.4.275


http://mnras.oxfordjournals.org/content/85/9/979.full.pdf

http://mnras.oxfordjournals.org/content/85/9/979.full.pdf

http://www.jstor.org/stable/10.4169/math.mag.87.4.275

Thank you for your attention!

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