On the nonexistence of solutions of some anisotropic elliptic and backward parabolic inequalities

We study the nonexistence of weak solutions of higher-order elliptic and parabolic inequalities of the following types: \(\sum {_{i = 1}^N\sum\nolimits_{{e_i} \leqslant {\alpha _i} \leqslant {m_i}} {D_{{x_i}}^{{\alpha _i}}\left( {{A_{{\alpha _i}}}\left( {x,u} \right)} \right)} \geqslant f\left( {x,u} \right),} x \in {\mathbb{R}^N}\), and \({u_t} + \sum {_{i = 1}^N\sum\nolimits_{{k_i} \leqslant {\beta _i} \leqslant {n_i}} {D_{{x_i}}^{{\beta _i}}\left( {{B_{{\beta _i}}}\left( {x,t,u} \right)} \right)} > g\left( {x,t,u} \right),\left( {x,t} \right)} \in {\mathbb{R}^N} \times {\mathbb{R}_ + }\), where l _i , m _i , k _i , n _i ∈ N satisfy the condition l _i , k _i > 1 for all i = 1,..., N, and A _αi (x, u), B _βi (x, t, u), f(x, u), and g(x, t, u) are some given Carathéodory functions. Under appropriate conditions on the functions A _αi , B _βi , f, and g, we prove theorems on the nonexistence of solutions of these inequalities.     

Stabilization by output of continuous and pulse-modulated uncertain systems

The system ? _i = ? _i (?) + x _i+2, \(i \in \overline {1,n - 2} \), ? _n?1 = ? _n?1(?) + u ₁, ? _n = ? _n (?) + u ₂,where ? _i (·) are nonanticipating functionals of an arbitrary nature with the following properties—\(\left| {{\varphi _i}\left( \cdot \right)} \right| \leqslant c\sum\nolimits_{k = 1}^i {\left| {{x_k}\left( t \right)} \right|} \), \(i \in \overline {1,n} \), c = const—and u ₁ and u ₂ are the controls is considered. It is assumed that only the outputs x ₁ and x ₂ are measurable. The problem of synthesis of both continuous and impulsive controls u1 and u2, which make the system globally asymptotically stable, is solved. The solution of the problem is based on the construction of the observer-based equations, the quadratic Lyapunov function, and the averaging method.     

Matrix polynomial generalizations of the sample variance-covariance matrix when <Emphasis Type="Italic">pn</Emphasis><Superscript>−1</Superscript> → <Emphasis Type="Italic">y</Emphasis> ∈ (0, ∞)

Let {Z _u = ((ε_{u, i, j}))_p×n} be random matrices where {ε_{u, i, j}} are independently distributed. Suppose {A _i }, {B _i } are non-random matrices of order p × p and n × n respectively. Consider all p × p random matrix polynomials \(P = \prod\nolimits_{i = 1}^{k_l } {\left( {n^{ - 1} A_{t_i } Z_{j_i } B_{s_i } Z_{j_i }^* } \right)A_{t_{k_l + 1} } }\). We show that under appropriate conditions on the above matrices, the elements of the non-commutative *-probability space Span {P} with state p^?1ETr converge. As a by-product, we also show that the limiting spectral distribution of any self-adjoint polynomial in Span{P} exists almost surely.     

Stability of uncertain discrete systems

The uncertain system

$x_{n + 1} = A_n x_n , n = 0,1,2, \ldots ,$

is considered, where the coefficients a _ij (n) of the m×m matrix A _n are functionals of any nature subject to the constraints

$\begin{array}{*{20}c} {\left| {a_{i,i} (n)} \right| \leqslant \alpha _ * < 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \alpha _0 for j \geqslant i + 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \delta for j < i.} \\ \end{array} $

Such systems include, in particular, switched-type systems, whose matrix A can take values in a given finite set.By using a special Lyapunov function, a bound δ ≤ δ(α₀,α_*) ensuring the global asymptotic stability of the system is found. In particular, the system is stable if the last inequality is replaced by a _i,j (n) = 0 for j < i.It is shown that pulse-width modulated systems reduce to the uncertain systems under consideration; moreover, in the case of a pulse-width modulation of the first kind, the coefficients of the matrix A are functions of x(n), and in the case of a modulation of the second kind, they are functionals.     

K?hler Manifolds with Almost Non-negative Ricci Curvature

Compact Kähler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell’s work, if M is a compact Kähler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{M} \cong X_{1} \times \cdots \times X_{m} \), where X _j is a Calabi-Yau manifold, or a hyperKähler manifold, or X _j satisfies H ⁰(X _j , Ω ^p ) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kähler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ∈ > 0, there exists a Kähler structure (J _∈, g _∈) on M such that the volume \({\text{Vol}}_{{g_{ \in } }} {\left( M \right)} < V\), the sectional curvature |K(g _∈)| < Λ², and the Ricci-tensor Ric(g _∈)> ?∈g _∈, where V and Λ are two constants independent of ∈. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{X} \cong X_{1} \times \cdots \times X_{s} \), where X _i is a Calabi-Yau manifold, or a hyperKähler manifold, or X _i satisfies H ⁰(X _i , Ω ^p ) = {0}, p > 0.     

Boundedness of Hausdorff operators on the power weighted Hardy spaces

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H_(|x|α)~1(R~2) ( -1 ≤α≤0), defined by H_(Φ,A)f(x)=∫R~2Φ(u)f(A(u)x)du,where Φ∈L_loc~1(R~2),A(u) = (α_(ij)(u))_(i,j=1)~2 is a 2×2 matrix, and each α_(i,j) is a measurablefunction.We obtain that HΦ,A is bounded from H_(|x|~α)~1(R~2) ( -1≤α≤0) to itself, if∫R2|Φ(u)‖det A~(-1)(u)|‖A(u)‖~(-α)ln(1+‖A~(-1)(u)‖~2/|det A~(-1)(u)|)du∞.This result improves some known theorems, and in some sense it is sharp.     

The smallest singular value of random rectangular matrices with no moment assumptions on entries

Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N₀ depending only on β and δ with the following property: for any N,n such that N ≥ max(N₀, δ_n), any N × n random matrix A = (a_ij) with i.i.d. entries satisfying \({\sup _{\lambda \in \mathbb{R}}}P\left\{ {\left| {{a_{11}} - \lambda } \right| \leqslant 1} \right\} \leqslant 1 - \beta \) and any non-random N × n matrix B, the smallest singular value s_n of A + B satisfies \(P\left\{ {{s_n}\left( {A + B} \right) \leqslant u\sqrt N } \right\} \leqslant \exp \left( { - vN} \right)\). The result holds without any moment assumptions on the distribution of the entries of A.     

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.     

On criteria of estimation of the errors of cubature formulas

The paper considers cubature formulas for calculating integrals of functions f(X), X = (x ₁, …, x _n ) which are defined on the n-dimensional unit hypercube K ⁿ = [0, 1] ⁿ and have integrable mixed derivatives of the kind \(\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)\), 0 ≤ α _j ≤ 2. We estimate the errors R[f] = \(\smallint _{K^n } \) f(X)dX ? Σ _{k = 1} ^N c _k f(X(k)) of cubature formulas (c _k > 0) as functions of the weights c _k of nodes X(k) and properties of integrable functions. The error is estimated in terms of the integrals of the derivatives of f over r-dimensional faces (r≤n) of the hypercube K ⁿ : |R(f)| ≤ \(\sum _{\alpha _j } \) G(α _j )\(\int_{K^r } {\left| {\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)} \right|} \) dX _r , where coefficients G(α _j ) are criteria which depend only on parameters c _k and X(k). We present an algorithm to calculate these criteria in the two- and n-dimensional cases. Examples are given. A particular case of the criteria is the discrepancy, and the algorithm proposed is a generalization of those used to compute the discrepancy. The results obtained can be used for optimization of cubature formulas as functions of c _k and X(k).     

Meromorphic Jost functions and asymptotic expansions for Jacobi parameters

We show that the parameters a _n , b _n of a Jacobi matrix have a complete asymptotic expansion

$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $

, where 1 < |µ_j| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ^?1) is an entire meromorphic function. We relate the poles of u to the µ_j’s.

     

Circular Law for Noncentral Random Matrices

Let (X _jk )_j,k≥1 be an infinite array of i.i.d. complex random variables with mean 0 and variance 1. Let λ _n,1,…,λ _n,n be the eigenvalues of \((\frac{1}{\sqrt{n}}X_{jk})_{1\leqslant j,k\leqslant n}\). The strong circular law theorem states that, with probability one, the empirical spectral distribution \(\frac{1}{n}(\delta _{\lambda _{n,1}}+\cdots+\delta _{\lambda _{n,n}})\) converges weakly as n→∞ to the uniform law over the unit disc {z∈?,|z|≤1}. In this short paper, we provide an elementary argument that allows us to add a deterministic matrix M to (X _jk )_{1≤ j,k ≤ n} provided that Tr(MM ^*)=O(n ²) and rank(M)=O(n ^α ) with α<1. Conveniently, the argument is similar to the one used for the noncentral version of the Wigner and Marchenko–Pastur theorems.     

Sums of Fourier coefficients of cusp forms of level <Emphasis Type="Italic">D</Emphasis> twisted by exponential functions

Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let a _g (n) be its n-th Fourier coefficient. We consider the sum \({S_1} = \sum {_{X < n \leqslant 2X}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)}\) and prove that S ₁ has an asymptotic formula when β = 1/2 and α is close to \(\pm 2\sqrt {q/D}\) for positive integer q ≤ X/4 and X sufficiently large. And when 0 < β < 1 and α, β fail to meet the above condition, we obtain upper bounds of S ₁. We also consider the sum \({S_2} = \sum {_{n > 0}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)\phi \left( {n/X} \right)}\) with ø(x) ∈ C _c ^∞ (0,+∞) and prove that S ₂ has better upper bounds than S ₁ at some special α and β.     

A characterization of the normal distribution using stationary max-stable processes

Consider a max-stable process of the form \(\eta (t) = \max _{i\in \mathbb {N}} U_{i} \mathrm {e}^{\langle X_{i}, t\rangle - \kappa (t)}\), \(t\in \mathbb {R}^{d}\), where \(\{U_{i}, i\in \mathbb {N}\}\) are points of the Poisson process with intensity u ^?2du on (0,∞), X _i , \(i\in \mathbb {N}\), are independent copies of a random d-variate vector X (that are independent of the Poisson process), and \(\kappa :\mathbb {R}^{d} \to \mathbb {R}\) is a function. We show that the process η is stationary if and only if X has multivariate normal distribution and κ(t)?κ(0) is the cumulant generating function of X. In this case, η is a max-stable process introduced by R. L. Smith.     

A radial symmetry and Liouville theorem for systems involving fractional Laplacian

We investigate the nonnegative solutions of the system involving the fractional Laplacian:

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$

where 0 < α < 1, n > 2, f _i (u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree p _i ≥ 0 and nondecreasing with respect to the independent variables u ₁, u ₂,..., u _m . By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x ₀ if p _i = (n + 2α)/(n ? 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < p _i < (n + 2α)/(n ? 2α) for all 1 ≤ i ≤ m.

     

Divided differences for functions of two variables for irregularly spaced arguments

Divided differences forf (x, y) for completely irregular spacing of points (x _i ,y _i ) are developed here by a natural generalization of Newton's scheme. Existing bivariate schemes either iterate the one-dimensional scheme, thus constraining (x _i ,y _i ) to be at corners of rectangles, or give polynomials Σa _jk x ^j y ^k having more coefficients than interpolation conditions. Here the generalizedn ^th divided difference is defined by (1)\(\left[ {01... n} \right] = \sum\limits_{i = 0}^n {A_i f\left( {x_i , y_i } \right)} \) where (2)\(\sum\limits_{i = 0}^n {A_i x_i^j , y_i^k = 0} \), and 1 for the last or (n+1)^th equation, for every (j, k) wherej+k=0, 1, 2,... in the usual ascending order. The gen. div. diff. [01...n] is symmetric in (x _i ,y _i ), unchanged under translation, 0 forf (x, y) an, ascending binary polynomial as far asn terms, degree-lowering with respect to (X, Y) whenf(x, y) is any polynomialP(X+x, Y+y), and satisfies the 3-term recurrence relation (3) [01...n]=λ{[1...n]?[0...n?1]}, where (4) λ= |1...n|·|01...n?1|/|01...n|·|1...n?1|, the |...i...| denoting determinants inx _i ^j y _i ^k . The generalization of Newton's div. diff. formula is (5)

$$\begin{gathered} f\left( {x, y} \right) = f\left( {x_0 , y_0 } \right) - \frac{{\left| {\alpha 0} \right|}}{{\left| 0 \right|}}\left[ {01} \right] + \frac{{\left| {\alpha 01} \right|}}{{\left| {01} \right|}}\left[ {012} \right] - \frac{{\left| {\alpha 012} \right|}}{{\left| {012} \right|}}\left[ {0123} \right] + \cdots + \hfill \\ + \left( { - 1} \right)^n \frac{{\left| {\alpha 01 \ldots n - 1} \right|}}{{\left| {01 \ldots n - 1} \right|}}\left[ {01 \ldots n} \right] + \left( { - 1} \right)^{n + 1} \frac{{\left| {\alpha 01 \ldots n} \right|}}{{\left| {01 \ldots n} \right|}}\left[ {01 \ldots n} \right], \hfill \\ \end{gathered} $$     

Hyperbolas over two-dimensional Fibonacci quasilattices

For the number n _s (α, β; X) of points (x ₁ , x ₂) in the two-dimensional Fibonacci quasilattices \( \mathcal{F}_m^2 \) of level m?=?0, 1, 2,… lying on the hyperbola x ₁ ² ? ??αx ₂ ² ?=?β and such that 0?≤?x ₁? ≤?X, x ₂? ≥?0, the asymptotic formula

$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $

is established, and the coefficient c _s (α, β) is calculated exactly. Using this, we obtain the following result. Let F _m be the Fibonacci numbers, A _i ∈ \( \mathbb{N} \), i?=?1, 2, and let \( \overleftarrow {{A_i}} \) be the shift of A _i in the Fibonacci numeral system. Then the number n _s (X) of all solutions (A ₁ , A ₂) of the Diophantine system

$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $

0?≤?A ₁? ≤?X, A ₂? ≥?0, satisfies the asymptotic formula

$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $

Here τ?=?(?1?+?√5)/2 is the golden ratio, and c _s ?=?1/2 or 1 for s?=?0 or s?≥?1, respectively.

     

Ramanujan-type congruences modulo powers of 5 and 7

Let b _? (n) denote the number of ?-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of ?. In particular, they showed that for α, n ≥ 0, b ₂₅ (3^2α+3 n+2 · 3^2α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where c _N (n) counts the number of bipartitions (λ₁,λ₂) of n such that each part of λ₂ is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, \({c_{25}}\left( {{5^{2j}}n + \frac{{11 \cdot {5^{2j}} + 13}}{{12}}} \right) \equiv 0\) (mod 5 ^j+1), \({c_{49}}\left( {{7^{2j}}n + \frac{{11 \cdot {7^{_{2j}}} + 25}}{{12}}} \right) \equiv 0\) (mod 7 ^j+1) and b ₂₅ (3^2α+3 · n+2 · 3^2α+2-1) ≡ 0 (mod 3 · 5^2j-1).     

Ideal counting function in cubic fields

For a cubic algebraic extension K of ?, the behavior of the ideal counting function is considered in this paper. More precisely, let a _K (n) be the number of integral ideals of the field K with norm n, we prove an asymptotic formula for the sum \(\sum\nolimits_{n_1^2 + n_2^2 \leqslant x} {a_K \left( {n_1^2 + n_2^2 } \right)} \).     

An upper bound of the heat kernel along the harmonic-Ricci flow

Let \({c : C \rightarrow X \times X}\) be a correspondence with C and X quasi-projective schemes over an algebraically closed field k. We show that if \({u_\ell : c_1^*\mathbb{Q}_\ell \rightarrow c_2^!\mathbb{Q}_\ell}\) is an action defined by the localized Chern classes of a c ₂-perfect complex of vector bundles on C, where ? is a prime invertible in k, then the local terms of u _? are given by the class of an algebraic cycle independent of ?. We also prove some related results for quasi-finite correspondences. The proofs are based on the work of Cisinski and Deglise on triangulated categories of motives.     

Stability of basic sequences via nonlinear <Emphasis Type="Italic">ε</Emphasis>-isometries

In this paper, Let X, Y be two real Banach spaces and ε ≥ 0. A mapping f: X → Y is said to be a standard ε-isometry provided f(0) = 0 and

$$\parallel f\left( x \right) - f\left( y \right)\parallel - \parallel x - y\parallel | \leqslant \varepsilon $$

(1)

for all x, y ∈ X. If ε = 0, then it is simply called a standard isometry. We prove a sufficient and necessary condition for which {f(x_n)}n≥1 is a basic sequence of Y equivalent to {x_n}n≥1 whenever {x_n}n≥1 is a basic sequence in X and f: X → Y is a nonlinear standard isometry. As a corollary we obtain the stability of basic sequences under the perturbation by nonlinear and non-surjective standard ε-isometries.