Transformations z(t) = L(t)y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of an order n + 1 (n ≥ 1) which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $f\left( {s,w_{00} \upsilon _0 ,...,\sum\limits_{j = 0}^n {w_{nj\upsilon _j } } } \right) = \sum\limits_{j = 0}^n {w_{n + 1j\upsilon j} + w_{n + 1n + 1} f\left( {x,\upsilon ,\upsilon _1 ,...,\upsilon _n } \right),}$ where $w_{n + 10} = h\left( {s,x,x_1 ,u,u_1 ,...,u_n } \right),w_{n + 11} = g\left( {s,x,x_1 ,...,x_n ,u,u_1 ,...,u_n } \right){\text{ and }}w_{ij} = a_{ij} \left( {x_i ,...,x_{i - j + 1} ,u,u_1 ,...,u_{i - j} } \right)$ for the given functions a _ij is solved on $\mathbb{R},u \ne {\text{0}}$ .     

On L 1-convergence of Fourier series of complex valued functions under the GM7 condition

Let f??L _2?? be a real-valued even function with its Fourier series $\frac {a_{0}}{2}+\sum\limits_{n=1}^{\infty}{a_{n}}\cos nx$ , and let S _n (f,x) be the nth partial sum of the Fourier series, n?R1. The classical result says that if the nonnegative sequence {a _n } is decreasing and $\lim\limits_{n\to\infty} a_{n} =0$ , then $\lim\limits_{n\to\infty} \|{f-S_{n}(f)}\|_{L}=0$ if and only if $\lim\limits_{n\to\infty} a_{n}\log n=0$ . Later, the monotonicity condition set on {a _n } is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM₇ condition in real space. In this paper, we give a complete generalization to the complex space.     

Orlicz norm estimates for eigenvalues of matrices

Let φ be a supermultiplicative Orlicz function such that the function $t \mapsto \varphi \left( {\sqrt t } \right)$ is equivalent to a convex function. Then each complexn×n matrixT=(τ _ij ) _{i, j} satisfies the following eigenvalue estimate: $\left\| {\left( {\lambda _i \left( T \right)} \right)_{i = 1}^n } \right\|_{\ell _\varphi } \leqslant C\left\| ( \right\|\left( {\tau _{ij} } \right)_{i = 1}^n \left\| {_{_{\ell _{\varphi *} } } )_{j = 1}^n } \right\|\ell _{\bar \varphi } $ . Here, ?_* stands for Young’s conjugate function of φ, ?, $\bar \varphi $ is the minimal submultiplicative function dominating φ andC>0 a constant depending only on φ. For the power function φ(t)=t ^p ,p≥2 this is a celebrated result of Johnson, König, Maurey and Retherford from 1979. In this paper we prove the above result within a more general theory of related estimates.     

Fixed Points and Quartic Functional Equations in β-Banach Modules

Let M?=?{ 1, 2, . . . ,?n?} and let ${\mathcal {V}=\{\,I \subseteq M: 1 \in I\,\}}$ , where n is an integer greater than 1. Denote ${M{\setminus}{I}}$ by I ^c for ${I \in \mathcal {V}.}$ We investigate the solution of the following generalized quartic functional equation $$\begin{array}{ll} \sum\limits_{I \in\mathcal {V}}f\, \left({\sum\limits_{i \in I}}a_ix_i-\sum\limits_{i \in I^c}a_ix_i\right) \, = \,2^{n-2} \sum\limits_{1\leq i < j \leq n}a^2_{i}a^2_{j} \left[f(x_{i}+x_{j})+f(x_{i}-x_{j})\right] \\ \qquad \qquad \qquad \quad\quad\quad \quad\quad\quad\quad +\,2^{n-1} \sum\limits^{n}_{i=1}a^2_{i} \left(a^2_{i}-\sum\limits^{n}_{\substack{{j=1}\\{j\neq i}}}a^2_{j}\right)f(x_{i}) \end{array}$$ in β-Banach modules on a Banach algebra, where ${a_{1},\ldots, a_{n}\in \mathbb{Z}{\setminus}\{0\}}$ with a _? ?≠ ±1 for all ${\ell \in \{1 , 2, \ldots ,\,n-1\}}$ and a _n ?=?1. Moreover, using the fixed point method, we prove the generalized Hyers–Ulam stability of the above generalized quartic functional equation. Finally, we give an example that the generalized Hyers–Ulam stability does not work.     

The functional law of the iterated logarithm for truncated sums

The functional law of the iterated logarithm (FLIL) is obtained for truncated sums $S_n = \sum _{j = l}^n X_j I\{ X_j^{\text{2}} \leqslant b_n \} $ of independent symmetric random variables X_j, 1<-j≤n, b_n≤∞. Considering the random normalization $T_n^{1/{\text{2}}} = \left( {\sum\limits_{j = 1}^n {X_j^{\text{2}} } I\{ X_j^{\text{2}} \leqslant b_n \} } \right)^{1/{\text{2}}} ,$ we obtain an upper estimate in the FLIL, using only the condition that T_n→∞ almost surely. These results are useful in studying trimmed sums. Bibliography: 9 titles.     

Fixed Points and Stability for Quadratic Mappings in β–Normed Left Banach Modules on Banach Algebras

The goal of the present paper is to investigate some new stability results by applying the alternative fixed point of generalized quadratic functional equation $$\begin{array}{ll}f\left(\sum\limits_{i=1}^{n}a_ix_i\right)+{\sum\limits_{i=1}^{n-1}}{\sum\limits_{j=i+1}^{n}}\left[f(a_ix_i+a_jx_j)+2f(a_ix_i-a_jx_j)\right]\\ \qquad \quad = (3n-2){\sum\limits_{i=1}^{n}}a^2_{i}f(x_{i})\end{array}$$ in β–Banach modules on Banach algebras, where ${a_{1},\dots,a_{n}\in \mathbb{Z}{\setminus}\{0\}}$ and some ${\ell\in\{1 , 2 ,\dots, n-1\},}$ a _? ?≠ ±1 and a _n ?=?1, where n is a positive integer greater or at least equal to two.     

Logarithmic growth of theL 1-norm of the majorant of partial sums of an orthogonal series

It is proved that for anyN ×N orthogonal matrixA = {a _ij} we have $$\sum\limits_{i = 1}^N {\mathop {\max }\limits_{1 \leqslant n \leqslant N} |\left| {\sum\limits_{j = 1}^n {a_{ij} } } \right|} \geqslant \frac{1}{{30}}N^{1/2} \log N.$$ A multidimensional analog of this result is also established.     

Interior W 1,p Regularity and Hölder Continuity of Weak Solutions to a Class of Divergence Kolmogorov Equations with Discontinuous Coefficients

We consider a class of Kolmogorov equation $$Lu={\sum^{p_0}_{i,j=1}{\partial_{x_i}}(a_{ij}(z){\partial_{x_j}}u)}+{\sum^{N}_{i,j=1}b_{ij}x_{i}{\partial_{x_j}}u-{\partial_t}u}={\sum^{p_0}_{j=1}{\partial_{x_j}}F_{j}(z)}$$ in a bounded open domain ${\Omega \subset \mathbb{R}^{N+1}}$ , where the coefficients matrix (a _ij (z)) is symmetric uniformly positive definite on ${\mathbb{R}^{p_0} (1 \leq p_0 < N)}$ . We obtain interior W ^1,p (1 < p < ∞) regularity and Hölder continuity of weak solutions to the equation under the assumption that coefficients a _ij (z) belong to the ${VMO_L\cap L^\infty}$ and ${({b_{ij}})_{N \times N}}$ is a constant matrix such that the frozen operator ${L_{z_0}}$ is hypoelliptic.     

Алгебраические дроби Чебышева—Маркова на нескольких отрезках

The Chebyshev-Markov problem about real algebraic functions of the form $A_n (x) = \frac{{x^n + c_1 x^{n - 1} + ... + c_n }}{{\left( {\prod {_{i = 1}^{2n} } \left( {1 - a_{i,n} x} \right)} \right)^{1/2} }}$ deviated least from zero on a system of intervals $\left[ {b_1 ;b_2 } \right] \cup ... \cup \left[ {b_{2p - 1} ;b_{2p} } \right], - \infty< b_1 \leqslant b_2< ...< b_{2p - 1} \leqslant b_{2p}< + \infty $ is considered. The expression under the square root above is a real polynomial of degree less than 2n, which is positive on [b ₁;b _2p ]. The solution of this problem is given in a parametric form in terms of automorphic Schottky-Burside functions. Similar functions were first used by N. I. Akhyeser in the approximation theory.     

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

Estimates in Morrey Spaces and H?lder Continuity for Weak Solutions to Degenerate Elliptic Systems

In this paper, we establish gradient estimates in Morrey spaces and H?lder continuity for weak solutions of the following degenerate elliptic system $$-X_{\alpha}^{\ast}(a_{ij}^{\alpha\beta}(x)X_{\beta}u^{j})=g_{i}-X_{\alpha}^{\ast}f_{i}^{\alpha}(x),$$ where X ₁, . . . , X _q are real smooth vector fields satisfying H?rmander’s condition, coefficients ${a_{ij}^{\alpha \beta }\in VMO_X \cap L^\infty (\Omega ), \alpha,\beta=1,2, \,.\,.\,.\, ,q, i,j=1,2, \,.\,.\,.\, ,N, X_{\alpha}^{\ast}}$ is the transposed vector field of X _α.     

The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Transcendence of the values of infinite products in several variables

The aim of this paper is to prove the transcendence of certain infinite products. As applications, we get necessary and sufficient conditions for transcendence of the value of $\Pi_{k=0}^{\infty}(1+a_{k}^{(1)}{z_{1}r^{k}}+\cdot\cdot\cdot+a_{k}^{(m)}{z_{m}r^{k}})$ at appropriate algebraic points, where r ≥ 2 is an integer and {a_n ⁽ⁱ⁾}_{n≥ 0} (1 ≤ i ≤ m) are suitable sequences of algebraic numbers.     

Global Morrey estimates for a class of Ornstein-Uhlenbeck operators

In this paper, we consider a class of hypoelliptic Ornstein-Uhlenbeck operators in ? ^N given by $\mathcal{A} = \sum\limits_{i,j = 1}^{p_0 } {a_{ij} \partial _{x_i x_j }^2 + } \sum\limits_{i,j = 1}^N {b_{ij} x_i \partial _x } ,$ where (a _ij ), (b _ij ) are N × N constant matrices, and (a _ij ) is symmetric and positive semidefinite. We deduce global Morrey estimates forA from similar estimates of its evolution operator L on a strip domain S = ? ^N × [?1, 1].     

The Stability of Systems of Neutral Type with Small Time Lag

In this paper, we have obtained the equivalence theorems of stability between the system of differential equations $[{\dot x_i}(t) = \sum\limits_{j = 1}^n {{a_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{b_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{c_{ij}}{{\dot x}_j}(t)} (i = 1,2, \cdots ,n)\]$ and the system of differential-difference equations of neutral type $[{\dot x_i}(t) = \sum\limits_{j = 1}^n {{a_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{b_{ij}}{x_j}(t - {\Delta _{ij}})} + \sum\limits_{j = 1}^n {{c_{ij}}{{\dot x}_j}(t - {\Delta _{ij}})} (i = 1,2, \cdots ,n)\]$ where a_ij, b_ij, c_ij are given constants, and \Delta_ij are non-negative real constants.     

The analytic spread of codimension two monomial varieties

Let ${\rm} A=k[{u_{1}^{a_{1}}},{u_{2}^{a_{2}}},\dots,{u_{n}^{a_{n}}},{u_{1}^{c_{1}}} \dots {u_{n}^{c_{n}}},{u_{1}^{b_{1}}} \dots {u_{n}^{b_{n}}}]\ \subset k[{u_{1}}, \dots {u_{n}}],$ where, a_j, b_j, C_j ∈ ?, a_j > 0, (b_j, C_j) ≠ (0,0) for 1 ≤ j ≤ n, and, further ${\underline b}:=\ ({b_{1}}, \dots,{b_{n}})\ \not=\ 0 $ and ${\underline c}:=\ ({c_{1}}, \dots,{c_{n}})\ \not=\ 0 $ . The main result says that the defining ideal I ? m = (x₁,…, x_n, y, z) ? k[x₁,…, x_n, y, z] of the semigroup ring A has analytic spread ?(I_m) at most three.     

Subspace Arrangements of Type B n and D n

Let ${\mathcal{D}}_{n,k} $ be the family of linear subspaces of ?ⁿ given by all equations of the form $\varepsilon _1 x_{i_1 } = \varepsilon _2 x_{i_2 } = \cdot \cdot \cdot \varepsilon _k x_{i_k } ,$ for 1 ≤ < ? ? ? < i _k ≤ i and $\left( {\varepsilon _1 ,...,\varepsilon _k } \right)\varepsilon \left\{ { + 1, - 1} \right\}^k $ Also let ${\mathcal{B}}_{n,k,h} $ be ${\mathcal{D}}_{n,k} $ enlarged by the subspaces $x_{j_1 } = x_{j_2 } = \cdot \cdot \cdot x_{j_h } = 0,$ for 1 ≤. The special cases ${\mathcal{B}}_{n,2,1} $ and ${\mathcal{D}}_{n,2} $ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B _nand D _n respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of ${\mathcal{B}}_{n,k,h,} 1 \leqslant h < k$ , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold $\begin{gathered} {\mathcal{D}}_{n,2} \\ M_{n,k,h,} = {\mathbb{R}}^n \backslash \bigcup {{\mathcal{B}}_{n,k,h,} } \\ \end{gathered} $ . For instance, it is shown that $H^d \left( {M_{n,k,k - 1} } \right)$ is torsion-free and is nonzero if and only if d = t(k ? 2) for some $t,0 \leqslant t \leqslant \left[ {{n \mathord{\left/ {\vphantom {n k}} \right. \kern-0em} k}} \right]$ . Torsion-free cohomology follows also for the complement in ?ⁿof the complexification ${\mathcal{B}}_{n,k,h}^C ,1 \leqslant h < k$ .     

Gaussian fluctuation for linear eigenvalue statistics of large dilute Wigner matrices

This paper focuses on the dilute real symmetric Wigner matrix Mn=1√n(aij)n×n,whose offdiagonal entries aij(1 i=j n)have mean zero and unit variance,Ea4ij=θnα(θ0)and the fifth moments of aij satisfy a Lindeberg type condition.When the dilute parameter 0α13and the test function satisfies some regular conditions,it proves that the centered linear eigenvalue statistics of Mn obey the central limit theorem.     

Morrey-type regularity of solutions to parabolic problems with discontinuous data

We consider regular oblique derivative problem in cylinder Q _T ?=????× (0, T), ${\Omega\subset {\mathbb R}^n}$ for uniformly parabolic operator ${{{\mathfrak P}}=D_t- \sum_{i,j=1}^n a^{ij}(x)D_{ij}}$ with VMO principal coefficients. Its unique strong solvability is proved in Manuscr. Math. 203?C220 (2000), when ${{{\mathfrak P}}u\in L^p(Q_T)}$ , ${p\in(1,\infty)}$ . Our aim is to show that the solution belongs to the generalized Sobolev?CMorrey space ${W^{2,1}_{p,\omega}(Q_T)}$ , when ${{{\mathfrak P}}u\in L^{p,\omega} (Q_T)}$ , ${p\in (1, \infty)}$ , ${\omega(x,r):\,{\mathbb R}^{n+1}_+\to {\mathbb R}_+}$ . For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy?CDirichlet problem.