Certain decompositions of matrices over Abelian rings

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ?. We prove that M ⁿ (R) is nil clean if and only if R/J(R) is Boolean and M _n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ?₃, B or ?₃ ⊕ B where B is a Boolean ring, and that M _n (R) is weakly nil clean if and only if M _n (R) is nil clean for all n ≥ 2.     

From Grassmann Necklaces to Restricted Permutations and Back Again

We study the commutative algebras Z _{J K} appearing in Brown and Goodearl’s extension of the \(\mathcal {H}\)-stratification framework, and show that if A is the single parameter quantized coordinate ring of M _m,n , G L _n or S L _n , then the algebras Z _{J K} can always be constructed in terms of centres of localizations. The main purpose of the Z _{J K} is to study the structure of the topological space s p e c(A), which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real m × n matrices in terms of its restricted permutation.     

Rectangular <Emphasis Type="Italic">R</Emphasis>-Transform as the Limit of Rectangular Spherical Integrals

In this paper, we connect rectangular free probability theory and spherical integrals. We prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maïda proved for Hermitian matrices in (J. Funct. Anal. 222(2):435–490, 2005). More specifically, we study the limit, as n and m tend to infinity, of \(\frac{1}{n}\log\mathbb{E}\{\exp[\sqrt{nm}\theta X_{n}]\}\), where θ∈?, X _n is the real part of an entry of U _n M _n V _m and M _n   is a certain n×m deterministic matrix and U _n and V _m are independent Haar-distributed orthogonal or unitary matrices with respective sizes n×n and m×m. We prove that when the singular law of M _n converges to a probability measure μ, for θ small enough, this limit actually exists and can be expressed with the rectangular R-transform of μ. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.     

On a classical theorem on the diameter and minimum degree of a graph

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.We use these bounds in order to study hyperbolic graphs(in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ_0) be the set of graphs G with n vertices and minimum degree δ_0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ_0) = min{δ(G) | G ∈ H(n, δ_0)}, b(n, δ_0) = max{δ(G) |G ∈ H(n, δ_0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ_0) and we compute the precise value of a(n, δ_0), α(n, Δ) andβ(n, Δ) for all values of n, δ_0 and Δ, respectively.     

Classification of rings with toroidal Jacobson graph

Let R be a commutative ring with nonzero identity and J(R) the Jacobson radical of R. The Jacobson graph of R, denoted by J_R, is defined as the graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 ? xy is not a unit of R. The genus of a simple graph G is the smallest nonnegative integer n such that G can be embedded into an orientable surface S_n. In this paper, we investigate the genus number of the compact Riemann surface in which J_R can be embedded and explicitly determine all finite commutative rings R (up to isomorphism) such that J_R is toroidal.     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

Non-uniform hyperbolicity in complex dynamics

We say that a rational function F satisfies the summability condition with exponent α if for every critical point c which belongs to the Julia set J there exists a positive integer n _c so that \(\sum_{n=1}^{\infty} |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles. Let μ _max be the maximal multiplicity of the critical points.The objective is to study the Poincaré series for a large class of rational maps and establish ergodic and regularity properties of conformal measures. If F is summable with exponent \(\alpha<\frac{\delta_{\textit{Poin}}(J)}{\delta_{\textit{Poin}}(J)+\mu_{\textit{max}}}\) where δ _Poin (J) is the Poincaré exponent of the Julia set then there exists a unique, ergodic, and non-atomic conformal measure ν with exponent δ _Poin (J)=HDim(J). If F is polynomially summable with the exponent α, \(\sum_{n=1}^{\infty}n |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles, then F has an absolutely continuous invariant measure with respect to ν. This leads also to a new result about the existence of absolutely continuous invariant measures for multimodal maps of the interval.We prove that if F is summable with an exponent \(\alpha< \frac{2}{2+\mu_{\textit{max}}}\) then the Minkowski dimension of J is strictly less than 2 if \(J\neq\hat{\mathbb{C}}\) and F is unstable. If F is a polynomial or Blaschke product then J is conformally removable. If F is summable with \(\alpha<\frac{1}{1+\mu_{\textit{max}}}\) then connected components of the boundary of every invariant Fatou component are locally connected. To study continuity of Hausdorff dimension of Julia sets, we introduce the concept of the uniform summability.Finally, we derive a conformal analogue of Jakobson’s (Benedicks–Carleson’s) theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points c from the Mandelbrot set with respect to the harmonic measure.     

Continuous time random walks and the Cauchy problem for the heat equation

We deal with anomalous diffusions induced by continuous time random walks - CTRW in ?ⁿ. A particle moves in ?ⁿ in such a way that the probability density function u(·, t) of finding it in region Ω of ?ⁿ is given by ∫_Ωu(x, t)dx. The dynamics of the diffusion is provided by a space time probability density J(x, t) compactly supported in {t ≥ 0}. For t large enough, u satisfies the equation

$$u\left( {x,t} \right) = \left[ {\left( {J - \delta } \right)*u} \right]\left( {x,t} \right)$$

, where δ is the Dirac delta in space-time. We give a sense to a Cauchy type problem for a given initial density distribution f. We use Banach fixed point method to solve it and prove that under parabolic rescaling of J, the equation tends weakly to the heat equation and that for particular kernels J, the solutions tend to the corresponding temperatures when the scaling parameter approaches 0.

     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

On structural decompositions of finite frames

A frame in an n-dimensional Hilbert space H _n is a possibly redundant collection of vectors {f _i } _i∈I that span the space. A tight frame is a generalization of an orthonormal basis. A frame {f _i } _i∈I is said to be scalable if there exist nonnegative scalars {c _i } _i∈I such that {c _i f _i } _i∈I is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame {f _i } _i∈I to be a collection of subsets of I ordered by inclusion so that nonempty J?I is in the factor poset iff {f _j } _j∈J is a tight frame for H _n . We study various properties of factor posets and address the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset; in doing so we present a bridge between erasure resilience (as studied via prime tight frames) and scalability.     

Jacobi Maaß forms

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J _k,m ^nh ) of weight k∈? and index m∈? as eigenfunctions of a degree three differential operator \(\mathcal{C}^{k,m}\). We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J _k,m ^nh . We construct new examples of cuspidal Jacobi Maaß forms F _f of weight k∈2? and index 1 from weight k?1/2 Maaß forms f with respect to Γ₀(4) and show that the map f ? F _f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of J _k,m ^nh can be “essentially” obtained from scalar or vector valued half integer weight Maaß forms.     

On the metric type of measurable functions and convergence in distribution

In the present paper, sequences of real measurable functions defined on a measure space ([0, 1], µ), where µ is the Lebesgue measure, are studied. It is proved that for every sequence f_n that converges to f in distribution, there exists a sequence of automorphisms S_n of ([0, 1], µ) such that f_n(S_n(t)) converges to f(t) in measure. Connection with some known results is also discussed.     

Classification of finite commutative rings with planar,toroidal, and projective line graphs associated with Jacobson graphs

Let R be a commutative ring with nonzero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by J _R , is a graph with vertex-set R J(R), such that two distinct vertices a and b in R J(R) are adjacent if and only if 1 ? ab is not a unit of R. Also, the line graph of the Jacobson graph is denoted by L(J _R ). In this paper, we characterize all finite commutative rings R such that the graphs L(J _R ) are planar, toroidal or projective.     

Kazhdan-Lusztig cells in some weighted Coxeter groups

Let(W,S) be a Coxeter group with S = I■J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W_I of W with w_0 the longest element of W_I. We describe all the left cells and two-sided cells of the weighted Coxeter group(W,S,L) that have non-empty intersection with W_J,where the weight function L of(W, S) is in one of the following cases:(i) max{L(s) | s ∈J} min{L(t)|t∈I};(ii) min{L(s)|s ∈J} ≥L(w_0);(iii) there exists some t ∈ I satisfying L(t) L(s) for any s ∈I-{t} and L takes a constant value L_J on J with L_J in some subintervals of [1, L(w_0)-1]. The results in the case(iii) are obtained under a certain assumption on(W, W_I).     

ABELIAN BALANCED HERMITIAN STRUCTURES ON UNIMODULAR LIE ALGEBRAS

Let g be a 2n-dimensional unimodular Lie algebra equipped with a Hermitian structure (J; F) such that the complex structure J is abelian and the fundamental form F is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of SU(n – k), being 2k the dimension of the center of g. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent.     

G-matrices,J-orthogonal matrices,and their sign patterns

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D₁ and D₂ such that A^?T = D₁AD₂, where A^?T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or ?1. A nonsingular real matrix Q is called J-orthogonal if Q^TJQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.     

Asymptotics and Arithmetical Properties of Complexity for Circulant Graphs

Abstract—We study analytical and arithmetical properties of the complexity function for infinite families of circulant C _n (s₁, s_2,…, s _k ) C_2n(s₁, s_2,…, s _k , n). Exact analytical formulas for the complexity functions of these families are derived, and their asymptotics are found. As a consequence, we show that the thermodynamic limit of these families of graphs coincides with the small Mahler measure of the accompanying Laurent polynomials.     

The rate of the smallest value of the weighted measure of the nonnegativity set for polynomials with zero mean value on a closed interval

Let P _n (α) be the set of algebraic polynomials p _n of order n with real coefficients and zero weighted mean value with ultraspherical weight \(\phi ^{(\alpha )} (t) = (1 - t^2 )^\alpha \) on the interval \([ - 1,1]:\int_{ - 1}^1 {\phi ^{(\alpha )} (t)p_n (t)dx = 0} \). We study the problem on the smallest value µ _n = inf{m(p _n ): p _n ∈ P _n (α)} of the weighted measure \(m(p_n ) = \int_{\chi (p_n )} {\phi ^{(\alpha )} (t)dt} \) of the set where p _n is nonnegative. The order of µ _n with respect to n is found: it is proved that \(\mu _n (\alpha ) \asymp n^{ - 2(\alpha + 1)} \) as n→∞.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”