Concavity of certain maps on positive definite matrices and applications to Hadamard products

If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ₁ and Φ₂ are concave maps among positive definite matrices, then the following map involving tensor products:

$\text{(A,B)?f[Φ}{}_1\text{(A)}{}^{\mathrm{?1}}\text{?Φ}{}_2\text{(B)]·(Φ}{}_1\text{(A)?I)}$

is proved to be concave. If Φ₁ is affine, it is proved without use of positivity that the map

$\text{(A,B)?f[Φ}{}_1\text{(A)?Φ}{}_2\text{(B)}{}^{\mathrm{?1}}\text{]·(Φ}{}_1\text{(A)?I)}$

is convex. These yield the concavity of the map

$\text{(A,B)?A}{}^{\mathrm{1?p}}\text{?B}{}^{\mathrm{p}}$

(0<p?1) (Lieb's theorem) and the convexity of the map

$\text{(A,B)?A}{}^{\mathrm{1+p}}\text{?B}{}^{\mathrm{?p}}$

(0<p?1), as well as the convexity of the map

$\text{(A,B)?(A·log[A])?I?A?log[B]}$

.These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices.     

Kolmogorov-Landau inequalities for monotone functions

Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetN_n (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g⁽ⁿ⁾(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofN_nis

$\text{g}{}_1\text{(x) = 0}\text{if}\text{?∞ < x < ?1, g}{}_1\text{(x) = (1 + x)}{}^{\mathrm{n}}\text{if}\text{?1 ? x ? 0}$

. Ifg(x) ∈ N_nis such that

$\text{g(0) ? g}{}_1\text{(0) = 1, g}{}^{\mathrm{(n)}}\text{(0) ? g}{}_1{}^{\mathrm{(n)}}\text{(0) = n!}$

, then

$\text{g}{}^{\mathrm{(v)}}\text{(0) ? g}{}_1{}^{\mathrm{(v)}}\text{(0)}$

for

1$\text{v = 1,…, n ? 1}$

. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if $\text{g(x) = g}{}_1\text{(x)}$ in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4).     

On an estimate for Bloch and Doob norm and covering problem in Doob's class

For any fixed 0 < π ? 2π, let D(π) be the family of all holomorphic functions in the unit disk Δ which satisfy (i)f(0) = 0 and (ii) $\text{lim inf}{}_{\mathrm{z\; \to \; \pi ¦f(z)¦\; ?\; 1}}$, for all π lying on some arc A_f ? ?Δ with arclength $\text{¦A}{}_{\mathrm{f}}\text{¦ ? π}$. We show that for each 0 < ε < 1, there is a π₀ > 0 such that for any f?D(π) with π < π₀, the Bloch and Doob norm respectively satisfy

$\text{6f6}{}_{\mathrm{B}}\text{=}\text{sup}\text{z?Δ}\text{|f′(z)| (1?|z|}{}^2\text{) > 2(1 ? ε)}\text{log}\text{1+}\text{cos}\text{(}\text{p}\text{2}\text{1?}\text{cos}\text{(}\text{p}\text{2}{}^{\mathrm{?1}}$

$\text{6f6}{}_{\mathrm{D}}\text{=}\text{sup}\text{z?Δ}\text{|f′(z)| (1?|z|) > (1 ? ε)}\text{log}\text{1}\text{1?}\text{cos}\text{(}\text{p}\text{2}{}^{\mathrm{?1}}$

These two estimates do not hold with ε = 0.     

Pairs of additive equations II. Large odd degree

Let k be an odd positive integer. Davenport and Lewis have shown that the equations

$\text{a}{}_1\text{x}{}_1{}^{\mathrm{k}}\text{+…+a}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}{}^{\mathrm{k}}\text{=0}$

with integer coefficients, have a nontrivial solution in integers x₁,…, x_N provided that

$\text{N?[36klog6k]}$

Here it is shown that for any ? > 0 and k > k₀(?) the equations have a nontrivial solution provided that

$\text{N?}\text{8}\text{log 2}\text{+?}\text{k log k.}$

     

On the number of zeros of diagonal cubic forms

Let M_s, be the number of solutions of the equation

$\text{X}{}_1{}^3\text{+ X}{}_2{}^3\text{+ … + X}{}_{\mathrm{s}}{}^3\text{=0}$

in the finite field GF(p). For a prime p ≡ 1(mod 3),

$\text{∑}\text{s=1}\text{∞}\text{M}{}_{\mathrm{s}}\text{X}{}^{\mathrm{s}}\text{=}\text{x}\text{1 ? px}\text{+}\text{x}{}^2\text{(p ? 1)(2 + dx)}\text{1 ? 3px}{}^2\text{? pdx}{}^3$

,

$\text{M}{}_3\text{= p}{}^2\text{+ d(p ? 1)}$

, and

$\text{M}{}_4\text{= p}{}^2\text{+ 6(p}{}^2\text{? p)}$

. Here d is uniquely determined by

$\text{4}{}_{\mathrm{p}}\text{= d}{}^2\text{+ 27b}{}^2\text{and}\text{d ≡ 1(}\text{mod}\text{3)}$

.     

Strong approximation theorems for density dependent Markov chains

A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form

$\text{X}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lY}{}_1\text{N ∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X}{}_{\mathrm{N}}\text{(s))ds}$

where $\text{l∈}\text{Z}{}^{\mathrm{t}}$, the Y₁ are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution).The corresponding deterministic model, satisfies

$\text{X(t)=x}{}_0\text{+ ∫}{}^{\mathrm{t}}{}_0\text{∑ lf}{}_1\text{(X(s))ds}$

Under very general conditions lim_N→∞X_N(t)=X(t) a.s. The process X_N(t) is compared to the diffusion processes given by

$\text{Z}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lB}{}_1\text{N∫}{}^{\mathrm{t}}{}_0\text{ft(Z}{}_{\mathrm{N}}\text{(s))ds}$

and

$\text{V(t)=∑ l∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X(s))}\text{d}\text{W}\text{?}{}_1\text{+∫}{}^{\mathrm{t}}{}_0\text{?F(X(s))·V(s)}\text{ds.}$

Under conditions satisfied by most of the applied probability models, it is shown that X_N,Z_N and V can be constructed on the same sample space in such a way that

$\text{X}{}_{\mathrm{N}}\text{(t)=Z}{}_{\mathrm{N}}\text{(t)+O}\text{logN}\text{N}$

and

$\text{N}\text{(X}{}_{\mathrm{N}}\text{(t)?X(t))=V(t)+O}\text{log N}\text{N}$

     

The nonorientable genus of the symmetric quadripartite graph

The nonorientable genus of K_4(n) is shown to satisfy:

$\text{γ}\text{(K}{}_{\mathrm{4(n)}}\text{)=2(n?1)}{}^2\text{for n ? 3}$

,

$\text{γ}\text{(K}{}_{\mathrm{4(2)}}\text{)=3,}\text{γ}\text{(K}{}_4\text{(1))=1}$

.     

On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

Minimax estimation of location parameters for certain spherically symmetric distributions

Families of minimax estimators are found for the location parameters of a p-variate distribution of the form

, where G(·) is a known c.d.f. on (0, ∞), p ≥ 3 and the loss is sum of squared errors. The estimators are of the form $\text{(1 ? ar(X′X)}\text{E}{}_0\text{(}\text{1}\text{X′X}\text{)X′X)X}$ where 0 ≤ a ≤ 2, r(X′X) is nondecreasing, and $\text{r(X′X)}\text{X′X}$ is nonincreasing. Generalized Bayes minimax estimators are found for certain G(·)'s.     

A complete set of invariants for finite groups and other results

For a finite group G and a set I ? {1, 2,…, n} let

${}_{\mathrm{G}}\text{(n,I) = ∑}{}_{\mathrm{g\; \in \; G}}\text{ε}{}_1\text{(g)?ε}{}_2\text{(g)???ε}{}_{\mathrm{n}}\text{(g)}$

,where

$\text{ε}{}_{\mathrm{i}}\text{(g)=g}\text{if}\text{i=∈ I,}$

$\text{ε}{}_{\mathrm{l}}\text{(g)=l}\text{if}\text{i=∈ I.}$

We prove, among other results, that the positive integers

$\text{tr}\text{(e}{}_{\mathrm{G}}\text{(n,I}{}_1\text{)+?+e}{}_{\mathrm{G}}\text{(n,I}{}_{\mathrm{r}}\text{))}{}^{\mathrm{k}}\text{:n,r,k,?1, I}{}_{\mathrm{j}}\text{?{1,…,n}, 1?|i}{}_{\mathrm{j}}\text{|?3}$

for 1 ? j ? r, I_j1 ∩ I_j2 ∩ I_j3 ∩ I_j4 = Ø for any 1 ? j₁ <j₂ <j₃ <j₄ ? r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups.     

On the qth convergence exponent of entire functions

In this paper we obtain a growth relation for entire functions of qth order with respect to the distribution of its zeros. We also derive certain relations between the qth convergence exponents of two or more entire functions. The most striking result of the paper is: If f(z) has at least one zero, then

$\text{lim sup}\text{r→∞}\text{log n}\text{(r)}\text{log}{}^{\mathrm{[q+1]}}{}_{\mathrm{r}}\text{=?(q)}$

, where n(r) is the number of zeros of f(z) in $\text{¦}\text{z}\text{¦ ? r}$ and

$\text{?(q)=g.l.b.}\text{α:α>0 and}\text{∑}\text{∞}\text{n=1}\text{(log}{}^{\mathrm{[q]}}\text{r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{-\alpha }}\text{<∞}$

.     

On the van der Waerden conjecture and zeros of polynomials

Let

$\text{F(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{A}{}_{\mathrm{n}}\text{≠ 0,}$

and

$\text{G(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{B}{}_{\mathrm{n}}\text{≠ 0,}$

be polynomials with real zeros satisfying A_n?1 = B_n?1 = 0, and let

$\text{H(x) =}\text{∑}\text{k=o}\text{n-2}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}\text{.}$

Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials.     

Generalized Schur complements

Let A be an n×n complex matrix. For a suitable subspace $\text{M}$ of Cⁿ the Schur compression A _$\text{M}$ and the (generalized) Schur complement A_/$\text{M}$ are defined. If A is written in the form

$\text{A=}\text{B}\text{C}\text{S}\text{T}$

according to the decomposition $\text{C}{}^{\mathrm{n}}\text{=}\text{M}\text{⊕}\text{M}{}^{\mathrm{\perp }}$ and if B is invertible, then

$\text{A}{}_{\mathrm{<mtext>M</mtext>}}\text{=}\text{B}\text{C}\text{S}\text{SB}{}^{\mathrm{?1}}\text{C}$

and

$\text{A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{=}\text{0}\text{0}\text{0}\text{T?SB}{}^{\mathrm{?1}}\text{C}\text{·}$

The commutativity rule for Schur complements is proved:

$\text{(A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{)}{}_{\mathrm{<mtext>/</mtext><mtext>N</mtext>}}\text{=(A)}{}_{\mathrm{<mtext>/</mtext><mtext>N</mtext>}}\text{)}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{·}$

This unifies Crabtree and Haynsworth's quotient formula for (classical) Schur complements and Anderson's commutativity rule for shorted operators. Further, the absorption rule for Schur compressions is proved:

$\text{(A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{)}{}_{\mathrm{<mtext>N</mtext>}}\text{=(A}{}_{\mathrm{<mtext>N</mtext>}}\text{)}{}_{\mathrm{<mtext>M</mtext>}}\text{=A}{}_{\mathrm{<mtext>M</mtext>}}\text{whenever}\text{M}\text{?}\text{N}$

.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

Periodic oscillations of coefficients of power series that satisfy functional equations

It is shown that the coefficients a_n of the power series f(z) = ∑^∞_n=1a_nzⁿ which satisfies the functional equation

$\text{f(z)=z+f(z}{}^2\text{+z}{}^3\text{)}$

display periodic oscillations; $\text{a}{}_{\mathrm{n}}\text{～ (}\text{ø}{}^{\mathrm{n}}\text{n}\text{) u(}\text{log}\text{n}$ as n → ∞, where $\text{ø =}\text{(1 + 5}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}\text{2}$ and u(x) is a positive, nonconstant, continuous function which is periodic with period log(4 ? ø). Similar results are obtained for a wide class of power series that satisfy similar functional equations. Power series of these types are of interest in combinatorics and computer science since they often represent generating functions. For example, the nth coefficient of the power series satisfying (1) enumerates 2, 3-trees with n leaves.     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

Critical exponents for the Pucci's extremal operatorsLes exposants critiques pour l'opérateur extrémal de Pucci

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

(1)$\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}\text{(D}{}^2\text{u)+u}{}^{\mathrm{p}}\text{=0,u?0}\text{in}\text{R}{}^{\mathrm{N}}\text{.}$

Here N?3, p>1 and $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}$ denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents $\text{1<p}{}^{\mathrm{s}}{}_{\mathrm{+}}\text{<p}{}^1{}_{\mathrm{+}}\text{<p}{}^{\mathrm{p}}{}_{\mathrm{+}}$, that satisfy: (i) If $\text{1<p<p}{}^1{}_{\mathrm{+}}$ then there is no nontrivial solution of ($\text{1}$). (ii) If $\text{p=p}{}^1{}_{\mathrm{+}}$ then there is a unique fast decaying solution of ($\text{1}$). (iii) If $\text{p}{}^1\text{<p?p}{}^{\mathrm{p}}{}_{\mathrm{+}}$ then there is a unique pseudo-slow decaying solution to ($\text{1}$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($\text{1}$). Similar results are obtained for the operator $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{?}}$. To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.     

Sur les équations de Lane–Emden avec opérateurs non linéaires

In this Note we consider nonnegative solutions for the nonlinear equation

$\text{M}{}^{\mathrm{+}}{}_{\mathrm{\lambda ,\Lambda }}\text{D}{}^2\text{u}\text{+|x|}{}^{\mathrm{\alpha }}\text{u}{}^{\mathrm{p}}\text{=0}$

in $\text{R}{}^{\mathrm{N}}$, where $\text{M}{}^{\mathrm{+}}{}_{\mathrm{\lambda ,\Lambda }}\text{(D}{}^2\text{u)}$ is the so called Pucci operator

$\text{M}{}^{\mathrm{+}}{}_{\mathrm{\lambda ,\Lambda }}\text{(M)=λ}\text{∑}\text{e}{}_{\mathrm{i}}\text{<0}\text{e}{}_{\mathrm{i}}\text{+Λ}\text{∑}\text{e}{}_{\mathrm{i}}\text{>0}\text{e}{}_{\mathrm{i}}\text{,}$

and the e_i are the eigenvalues of M et Λ?λ>0. We prove that if u satisfies the decreasing estimate

$\text{lim}\text{|x|→+∞}\text{|x|}{}^{\mathrm{\beta ?1}}\text{u(x)=0}$

for some β satisfying (β?1)(p?1)>2+α then u is radial. In a second time we prove that if $\text{p<}\text{N+2α+2}\text{N?2}$ and u is a nonnegative radial solution of (1), u(x)=g(r), such that g″ changes sign at most once, then u is zero. To cite this article: I. Birindelli, F. Demengel, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Some properties of Q-matrices

This paper deals with the class of Q-matrices, that is, the real n × n matrices M such that for every q ∈ $\text{R}$^n×1, the linear complementarity problem

$\text{Iw ? Mz = q}$

,

$\text{w ? 0, z ? 0, and w}{}^{\mathrm{T}}\text{z = 0}$

, has a solution. In general, the results are of two types. First, sufficient conditions are given on a matrix M so that M ∈ Q. Second, conditions are given so that M ? Q.