Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

We compute the bi-free max-convolution which is the operation on bivariate distribution functions corresponding to the max-operation with respect to the spectral order on bi-free bipartite two-faced pairs of Hermitian non-commutative random variables. With the corresponding definitions of bi-free max-stable and max-infinitely divisible laws, their determination becomes in this way a classical analysis question.     

Random matrix models of stochastic integral type for free infinitely divisible distributions

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ?? ₀ ^?? e ^?1 d?? _t ^d , d ?? 1, where ?? _t ^d is a d × d matrix-valued Lévy process satisfying an I _log condition.     

The integral part of a nonlinear form with five cubes of primes*

This paper shows that if μ ₁ , . . . , μ ₅ are nonzero real numbers, not all negative, at least one of μ _j $ \left( {1\leqslant j\leqslant 5} \right) $ is irrational, and k is a positive integer, then there exist infinitely many primes p ₁ , . . . , p ₅ , p such that $ \left[ {{\mu_1}p_1^3+\cdots +{\mu_5}p_5^3} \right]=kp $ . In particular, $ \left[ {{\mu_1}p_1^3+\cdots +{\mu_5}p_5^3} \right] $ represents infinitely many primes.     

A free analogue of Hincin's characterization of infinite divisibility

Hincin characterized the class of infinitely divisible distributions on the line as the class of all distributional limits of sums of infinitesimal independent random variables. We show that an analogue of this characterization is true in the addition theory of free random variables introduced by Voiculescu.

     

Arithmetical progressions formed from three different Euler pseudoprimes for the odd basea

An odd compositen is an Euler pseudoprime for the basea if (a, n)=1 and \(a^{\left( {n - 1} \right)/2} \equiv \left( {\frac{a}{n}} \right)\) (modn), where \(\left( {\frac{a}{n}} \right)\) is the Jacobi symbol. It is shown that for every odda≥3 there exist infinitely many arithmetical progressions consisting of three different Euler pseudoprimes for the basea.     

Remarks on characterization of normal and stable distributions

LetX, Y, Z be independent identically distributed (i.i.d.) random variables. Suppose $$E\left| {tX + uY + vZ} \right|^p = A(\left| t \right|^q + \left| u \right|^q + \left| v \right|^q )^{{p \mathord{\left/ {\vphantom {p q}} \right. \kern-\nulldelimiterspace} q}} $$ for all realt, u, v, whereq=2 andp≠2m (m=1, 2,...) or 0<p<q<2. It was proved by the author this impliesX, Y, Z have the symmetricq-stable distribution. For two random variables such result is not true. One may suppose that the condition $$E\left| {tX + uY} \right|^p = A(\left| t \right|^q + \left| u \right|^q )^{{p \mathord{\left/ {\vphantom {p q}} \right. \kern-\nulldelimiterspace} q}} $$ and additional assumption on the behavior ofP{|X|≥x} (x→∞) implyX, Y are stable. In this paper we show it is not valid. The second result is: if the last relation holds for two different exponents andq=2, thenX andY are normal.     

Multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

If all prime closed geodesics on (Sⁿ, F) with an irreversible Finsler metric F are irrationally elliptic, there exist either exactly 2 \(\left[ {\frac{{n + 1}}{2}} \right]\) or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler (S³, F) if any prime closed geodesic has non-zero Morse index.     

Classification of arrangements by the number of their cells

The possible pairs (n, f) of integers for which there are arrangements withn lines andf cells are determined. The pair (n, f) corresponds to an arrangement if and only if there is an integerk with 0≤k≤n?2 such that $$(n - k)(k + 1) + \left( {_2^k } \right) - \min \left\{ {n - k,\left( {_2^k } \right)} \right\} \leqslant f \leqslant (n - k)(k + 1) + \left( {_2^k } \right).$$      

Some new asymptotic properties for the zeros of Jacobi,Laguerre, and Hermite polynomials

For the generalized Jacobi, Laguerre, and Hermite polynomials $P_n^{\left( {\alpha _n ,\beta _n } \right)} \left( x \right),L_n^{\left( {\alpha _n } \right)} \left( x \right),H_n^{\left( {\gamma _n } \right)} \left( x \right)$ , the limit distributions of the zeros are found, when the sequences α _n or β _n tend to infinity with a larger order thann. The derivation uses special properties of the sequences in the corresponding recurrence formulas. The results are used to give second-order approximations for the largest and smallest zero which improve (and generalize) the limit statements in a paper by Moak, Saff, and Varga [11].     

Randomly weighted sums of dependent subexponential random variables*

We consider the randomly weighted sums $ \sum\nolimits_{k = 1}^n {{\theta_k}{X_k},n \geqslant 1} $ , where $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ are n real-valued random variables with subexponential distributions, and $ \left\{ {{\theta_k},1 \leqslant k \leqslant n} \right\} $ are other n random variables independent of $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ and satisfying $ a \leqslant \theta \leqslant b $ for some $ 0 < a \leqslant b < \infty $ and all $ 1 \leqslant k \leqslant n $ . For $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ satisfying some dependent structures, we prove that $$ {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant m \leqslant n} \sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant k \leqslant n} {\theta_k}{X_k} > x} \right)\sim \sum\limits_{k = 1}^m {{\text{P}}\left( {{\theta_k}{X_k} > x} \right)} $$ as x??????.     

Multivariate polynomials: A spanning question

The main result of this paper is the following. Ifg is any given polynomial of two variables, then $$span\left\{ {\left( {g\left( {. - a,. - b} \right)} \right)^k :\left( {a,b} \right) \in R^2 ,k \in {\rm Z}_ + } \right\}$$ contains all polynomials if and only if $$span\left\{ {g\left( {. - a,. - b} \right):\left( {a,b} \right) \in R^2 } \right\}$$ separates points. This result is not valid inR ^d ford≥4.     

A Certain Functional Equation

In this paper, we use a simpler argument and solve the following more general function equation: $$\left| {f\left( {z + w} \right)} \right| + \left| {g\left( {z - w} \right)} \right| = \left| {h\left( {z + \bar w} \right)} \right| + \left| {k\left( {z - \bar w} \right)} \right|$$ where f, g, h, k are unknown entire functions and z, w are complex variables.     

On the Wick theorem for mixtures of centered Gaussian distributions

We consider centered conditionally Gaussian d-dimensional vectors X with random covariance matrix Ξ having an arbitrary probability distribution law on the set of nonnegative definite symmetric d × d matrices M _d ⁺. The paper deals with the evaluation problem of mean values \( E\left[ {\prod\nolimits_{i = 1}^{2n} {\left( {{c_i},X} \right)} } \right] \) for c _i ∈ ? ^d , i = 1, …, 2n, extending the Wick theorem for a wide class of non-Gaussian distributions. We discuss in more detail the cases where the probability law ?(Ξ) is infinitely divisible, the Wishart distribution, or the inverse Wishart distribution. An example with Ξ \( = \sum\nolimits_{j = 1}^m {{Z_j}{\sum_j}} \), where random variables Z _j , j = 1, …, m, are nonnegative, and Σ _j ∈ M _d ⁺, j = 1, …, m, are fixed, includes recent results from Vignat and Bhatnagar, 2008.     

The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions

It is shown that the limits of the nested subclasses of five classes of infinitely divisible distributions on ${\mathbb{R}^{d}}$ , which are the Jurek class, the Goldie– Steutel–Bondesson class, the class of selfdecomposable distributions, the Thorin class and the class of generalized type G distributions, are identical with the closure of the class of stable distributions. More general results are also given.     

On a novel eccentricity-based invariant of a graph

In this paper,for the purpose of measuring the non-self-centrality extent of non-selfcentered graphs,a novel eccentricity-based invariant,named as non-self-centrality number(NSC number for short),of a graph G is defined as follows:N(G)=∑v_i,v_j∈V(G)|e_i-e_j| where the summation goes over all the unordered pairs of vertices in G and e_i is the eccentricity of vertex v_i in G,whereas the invariant will be called third Zagreb eccentricity index if the summation only goes over the adjacent vertex pairs of graph G.In this paper,we determine the lower and upper bounds on N(G) and characterize the corresponding graphs at which the lower and upper bounds are attained.Finally we propose some attractive research topics for this new invariant of graphs.     

Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus

We investigate properties of a fundamental solution and establish the correct solvability of the Cauchy problem for one class of degenerate Kolmogorov-type equations with { p^?,h^? } \left\{ {\overrightarrow p, \overrightarrow h } \right\} -parabolic part with respect to the main group of variables and with positive vector genus in the case where solutions are infinitely differentiable functions and their initial values may be generalized functions of Gevrey ultradistribution type.     

Lorentzian Sasaki-Einstein metrics on connected sums of <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

We settle completely an open problem formulated by Boyer and Galicki in [5] which asks whether or not #kS ² × S ³ are negative Sasakian manifolds for all k. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki η-Einstein and Lorentzian Sasaki-Einstein metrics on these five-manifolds for all k and moreover all of these can be realized as links of isolated hypersurface singularities defined by weighted homogenous polynomials. The key step is to construct infinitely many hypersurfaces in weighted projective space that contain branch divisors \({\Delta=\sum_{i}\left(1-\frac{1}{m_{i}}\right)D_i}\) such that the D _i are rational curves.     

Cauchy problem for a class of degenerate kolmogorov-type parabolic equations with nonpositive genus

We study the properties of the fundamental solution and establish the correct solvability of the Cauchy problem for a class of degenerate Kolmogorov-type equations with { p^?,h^? } \left\{ {\overrightarrow p, \overrightarrow h } \right\} -parabolic part with respect to the main group of variables and nonpositive vector genus in the case where the solutions are infinitely differentiable functions and their initial values are generalized functions in the form of Gevrey ultradistributions.     

Shrinking target problems for beta-dynamical system

For any β>1,let([0,1],Tβ) be the beta dynamical system.For a positive function ψ:N→R+ and a real number x0 ∈[0,1],we define D(Tβ,ψ,x0) the set of ψ-well approximable points by x0as {x∈[0,1]:|Tβnx-x0|<ψ(n) for infinitely many n∈N}.In this note,by proving a structure lemma that any ball B(x,r) contains a regular cylinder of comparable length with r,we determine the Hausdorff dimension of the set D(Tβ,ψ,x0) completely for any β>1 and any positive function ψ.     

Realization of Boolean functions by formulas in continuous bases containing a continuum of constants

For an arbitrary Boolean function of n variables, we show how to construct formulas of complexity O(2 ^n/2) in the bases $$\left\{ {x - y,xy,\left| x \right|} \right\}\bigcup {\left[ {0,1} \right], } \left\{ {x - y,x*y,2x,\left| x \right|} \right\}\bigcup {\left[ {0,1} \right],}$$ , where x * y = max(?1, min(1, x))max(?1, min(1, y)). The obtained estimates are, in general, order-sharp.