Almost commuting matrices

It is shown that if A and B are n × n complex matrices with $\text{A = A}{}^1\text{and}\text{∥AB ? BA∥</}\text{2?}{}^2\text{(n ? 1)}$, then there exist n × n matrices A′ and B′ with $\text{A′ = A′}{}^1\text{such that}\text{A′B′ = B′A′}\text{and}\text{∥A ? A′∥? ?, ∥B ? B′∥? ?}$.     

Maximum zero strings of Bell numbers modulo primes

For any prime p, the sequence of Bell exponential numbers B_n is shown to have p ? 1 consecutive values congruent to zero (mod p), beginning with B_m, where $\text{m ≡ 1 ?}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}{}^2$ ($\text{mod}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}$). This is an improvement over previous results on the maximal strings of zero residues of the Bell numbers. Similar results are obtained for the sequence of generalized Bell numbers A_n generated by .     

On the convergence of the Neumann series in interval analysis

Let $\text{A}$ be a real or complex n × n interval matrix. Then it is shown that the Neumann series $\text{Σ}{}^{\mathrm{\infty }}{}_{\mathrm{k=0}}\text{A}{}^{\mathrm{k}}$ is convergent iff the sequence {$\text{A}$^k} converges to the null matrix $\text{O}$, i.e., iff the spectral radius of the real comparison matrix $\text{B}$ constructed in [2] is less than one.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Transcendental continued fractions

In a previous paper it was proven that given the continued fractions

$\text{A = a}{}_1\text{+}\text{1}\text{a}{}_2\text{+}\text{1}\text{a}{}_3\text{+… and B = b}{}_1\text{+}\text{1}\text{b}{}_2\text{+}\text{1}\text{b}{}_3\text{+…}$

where the a's and b's are positive integers, then A, B, A ± B, $\text{A}\text{B}$ and AB are irrational numbers if $\text{a}{}_{\mathrm{n}}\text{2}\text{> b}{}_{\mathrm{n}}\text{> a}{}_{\mathrm{n?1}}{}^{\mathrm{5n}}$ for all n sufficiently large, and transcendental numbers if for all n sufficiently large. Using a more direct approach it is proven in this paper that A, B, A ± B, $\text{A}\text{B}$ and AB are transcendental numbers if a_n > b_n > a_n?1^(n?1)2 for all n sufficiently large.     

Bounds for iterated normal matrices

Given a normal matrix A, asymptotic bounds are obtained for $\text{|}\text{A}{}^{\mathrm{m}}\text{|}{}_{\mathrm{\infty }}$ in terms of the spectral radius of A, the number of eigenvalues of A with modulus equal to the spectral radius of A, and the order of A. These results are extended to provide bounds for $\text{|}\text{A}{}^{\mathrm{m}}\text{|}{}_{\mathrm{\infty }}$ for all m ? 1.     

Lower bounds for the norm of the symmetric product

Suppose each of m, n, and k is a positive integer, k ? n, A is a (real-valued) symmetric n-linear function on E_m, and B is a k-linear symmetric function on E_m. The tensor and symmetric products of A and B are denoted, respectively, by A ?B and A?B. The identity

$\text{6A · B6}{}^2\text{=}\text{∑}\text{q=0}\text{n}\text{(}\text{n}\text{k}\text{)}\text{(}\text{n+k}\text{k}\text{)}\text{6A?}{}_{\mathrm{q}}\text{B6}{}^2$

is proven by Neuberger in [1]. An immediate consequence of this identity is the inequality

$\text{6A · B 6}{}^2\text{?}\text{n+k}\text{n}{}^{\mathrm{?1}}\text{6A · B 6}{}^2$

In this paper a necessary and sufficient condition for

$\text{6A · B 6}{}^2\text{=}\text{n+k}\text{n}{}^{\mathrm{?}}\text{6A · B 6}{}^2$

is given. It is also shown that under certain conditions the inequality can be considerably improved. This improvement results from an analysis of the terms 6A?_qB6, 1?q?n, appearing in the identity.     

Continuity of positive functionals on topological 1-algebras

It is shown that the set $\text{C}$_{m × n} of complex m × n matrices forms a lower semilattice under the partial ordering A ? B defined by $\text{A}{}^1\text{A = A}{}^1\text{B,}{}^1\text{AA}{}^1\text{= BA}{}^1\text{,}\text{where}\text{A}{}^1$ denotes the conjugate transpose of A. As a special case of a result for division rings, it is further shown that, over any field F, form = n = 2 and any proper involution 1 of F_{2 × 2}, the corresponding intersections A ∩ B all exist.     

Interpolation and inner maps that preserve measure

It is shown, for n ? m ? 1, that there exist inner maps Φ: Bⁿ → B^m with boundary values $\text{Φ}{}_1\text{: B}{}^{\mathrm{n}}\text{→ B}{}^{\mathrm{m}}$ such that $\text{σ}{}_{\mathrm{m}}\text{(A) = σ}{}_{\mathrm{n}}\text{(Φ}{}_1{}^{\mathrm{?1}}\text{(A))}$. where σ_n and σ_m are the Haar measures on ?Bⁿ and ?B^m, respectively, and A ? Bⁿ is an arbitrary Borel set.     

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

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.     

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.     

Kato's inequality and the comparison of semigroups

Let A be the generator of a positivity preserving semigroup and let B be another semibounded self-adjoint operator. We give necessary and sufficient conditions in terms of the generators for the inequality $\text{¦ e}{}^{\mathrm{?tB}}\text{u ¦ ? e}{}^{\mathrm{?tA}}\text{¦ u ¦}$ to hold pointwise.     

A regularity result for singular nonlinear elliptic systems in inverse-power weighted Sobolev spaces

The compactness method to weighted spaces is extended to prove the following theorem:Let H_2,s¹(B₁) be the weighted Sobolev space on the unit ball in Rⁿ with norm

$\text{6ν6}{}^1{}_{\mathrm{2,s}}\text{=}\text{∫}{}_{\mathrm{B1}}\text{(}\text{1}\text{r}{}^{\mathrm{s}}\text{)|ν|}{}^2\text{dx + ∫}{}_{\mathrm{B1}}\text{(}\text{1}\text{r}{}^{\mathrm{s}}\text{)|Dν|}{}^2\text{dx.}$

Let n ? 2 ? s < n. Let u? [H_2,s¹(B₁) ∩ L^∞(B₁)]^N be a solution of the nonlinear elliptic system

$\text{∫}{}_{\mathrm{B1}}\text{1}\text{r}{}^{\mathrm{s}}\text{,}\text{∑}\text{i,j=1}\text{n}\text{,}\text{∑}\text{h,K=1}\text{N}\text{A}{}^{\mathrm{hK}}{}_{\mathrm{ij}}\text{(x,u) D}{}^{\mathrm{i}}\text{u}{}_{\mathrm{h}}\text{D}{}^{\mathrm{j\psi }}{}_{\mathrm{K}}\text{dx=0}$

, $\text{∨}\text{ψ}\text{?}\text{¦C}{}_0{}^1\text{(B}{}_1\text{)¦}{}^{\mathrm{N}}\text{,}\text{where}\text{¦A}{}_{\mathrm{ij}}{}^{\mathrm{hk}}\text{¦ ? L, A}{}_{\mathrm{ij}}{}^{\mathrm{hk}}$ are uniformly continuous functions of their arguments and satisfy:

$\text{|η|}{}^2\text{=}\text{∑}\text{i=1}\text{n}\text{,}\text{∑}\text{j=1}\text{N}\text{|η}{}^{\mathrm{i}}{}_{\mathrm{j}}\text{|}{}^2\text{?}\text{∑}\text{i,j=1}\text{n}\text{,}\text{1}\text{r}{}^{\mathrm{s}}\text{,}\text{∑}\text{h,K=1}\text{N}\text{A}{}^{\mathrm{hK}}{}_{\mathrm{ij}}\text{η}{}^{\mathrm{i}}{}_{\mathrm{h}}\text{η}{}^{\mathrm{i}}{}_{\mathrm{k}}\text{,}\text{?η?R}{}^{\mathrm{Nn}}$

. Then there exists an R₁, 0 < R₁ < 1, and an α, 0 < α < 1, along with a set $\text{Ω ? B}{}_1$ such that (1) $\text{H}{}_{\mathrm{n\; ?\; 2}}\text{(Ω) = 0}$, (2) Ω does not contain the origin; Ω does not contain B_R1, (3) $\text{B}{}_1\text{? Ω}$ is open, (4) u is $\text{Lip}{}_{\mathrm{\alpha }}\text{(B}{}_1\text{? Ω)}$; u is Lip_αB_R1.     

Structure of quadratic inequalities

When A and B are n × n positive semi-definite matrices, and C is an n × n Hermitian matrix, the validity of a quadratic inequality $\text{(x}{}^1\text{Ax)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(x}{}^1\text{Bx)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? ¦x}{}^1\text{Cx¦}$ is shown to be equivalent to the existence of an n × n unitary matrix W such that $\text{A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{WB}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ B}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{W}{}^1\text{A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 2C.}$ Some related inequalities are also discussed.     

Some monotonicity properties of Schur powers of matrices and related inequalities

A series of inequalities are developed relating the spectral radius ?(A ° B) of the Schur product A ° B of two nonnegative matrices A and B with those of ?(A ° A) and ?(B ° B) yielding $\text{?(A ° B) ? [?(A ° A)?(B ° B)]}\text{1}\text{2}$. As a corollary it is proved that the spectral radius of the Schur powers ?_r = ?(A^[r]), A^[r] = A ° A °?°A (r factors) satisfies $\text{(}\text{1}\text{r}\text{)}\text{log}\text{?}{}_{\mathrm{r}}$ is decreasing while $\text{(}\text{1}\text{r?1}\text{)}\text{log}\text{?}{}_{\mathrm{r}}$ is increasing, the latter provided A is a stochastic matrix. The entropy of a finite stationary Markov chain is identified with $\text{d?}{}_{\mathrm{r}}\text{dr|}{}_{\mathrm{r=1}}$. A number of majorization comparisons for the spectral radius of Schur powers is given.     

An inequality involving permanents of certain direct products

Let A denote a decomposable symmetric complex valued n-linear function on C^m. We prove

$\text{6A·A6}{}^2\text{?2}{}^{\mathrm{n}}\text{2n}\text{n}{}^{\mathrm{?1}}\text{6A?A6}{}^2$

, where · denotes the symmetric product and ? the tensor product. As a consequence we have per

$\text{M}\text{M}\text{M}\text{M}\text{?2}{}^{\mathrm{n}}\text{[per(M)]}{}^2$

, where M is a positive semidefinite Hermitian matrix and per denotes the permanent function. A sufficient condition for equality in the matrix inequality is that M is a nonnegative diagonal matrix.     

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

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