The number of strictly increasing mappings of fences

Let X and Y be fences of size n and m, respectively and n, m be either both even or both odd integers (i.e., |m-n| is an even integer). Let \(r = \left\lfloor {{{(n - 1)} \mathord{\left/ {\vphantom {{(n - 1)} 2}} \right. \kern-0em} 2}} \right\rfloor\) . If 1<n<-m then there are \(a_{n,m} = (m + 1)2^{n - 2} - 2(n - 1)(\begin{array}{*{20}c} {n - 2} \\ r \\ \end{array} )\) of strictly increasing mappings of X to Y. If 1<-m<-n<-2m and s=1/2(n?m) then there are a _n,m+b _n,m+c _n of such mappings, where $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$      

Diameter of the set of midpoints of chords of a bounded closed set

The estimate is obtained for the diameter d(S_n(a)) of the set S_n(a) of midpoints of chords of length ≥a(0n, namely $$d(S_n (a)) \leqslant \left\{ \begin{gathered} 1 - a^2 /2, n = 2, \hfill \\ \sqrt {1 - a^2 /2,} n \geqslant 3, \hfill \\ \end{gathered} \right.$$ and it is shown that the inequality cannot be improved.     

On the convergence rate of the partial sums of positive entire Dirichlet series

Пусть Λ=(λ_n) — возрастаю щая к+∞ последователь ность неотрицательных чис ел, λ₀=0, а S₊(Λ) — класс абсолют но сходящихся в С рядо в Дирихле вида $$F\left( z \right) = \mathop \sum \limits_{k = 0}^\infty a_k \exp \left\{ {z\lambda _k } \right\},$$ где a₀=1 и a_k>0 (k∈N). Положим $$\begin{gathered} S_n \left( z \right) = \mathop \sum \limits_{k = 1}^\infty a_k \exp \left\{ {z\lambda _k } \right\}, \hfill \\ \sigma _n \left( F \right) = \max \left\{ {\frac{1}{{S_n \left( x \right)}} - \frac{1}{{F\left( x \right)}}:x \in R} \right\}. \hfill \\ \end{gathered} $$ Доказано, что для того, чтобы для любой функц ии F∈S₊(Λ) выполнялось равенст во $$\mathop {\lim \sup }\limits_{n \to \infty } \frac{1}{{\ln n}}\ln \frac{1}{{\sigma _n \left( F \right)}} = + \infty ,$$ необходимо и достато чно, чтобы $$\mathop \sum \limits_{n = 1}^\infty \frac{1}{{n\lambda _n }}< + \infty .$$ Аналогичные результ ы получены для различ ных подклассов классаS ₊ (Λ), определяемых условиями на убывани е коэффициентова _n.     

Interpolation by polyhedral functions

A polyhedral functionlp(Δ_n) (f). interpolating a function f, defined on a polygon Φ, is defined by a set of interpolating nodes Δ_n ?Φ and a partition P(Δ_n) of the polygon Φ into triangles with vertices at the points of Δ_n. In this article we will compute for convex moduli of continuity the quatities $$\begin{gathered} E (H_\Phi ^\omega ; P (\Delta _n )) = sup || f - l_{p(\Delta _n )} (f)||, \hfill \\ f \in H_\Phi ^\omega \hfill \\ \end{gathered} $$ and also give an asymptotic estimate of the quantities $$\begin{gathered} E_n (H_\Phi ^\omega ) = infinf E (H_\Phi ^\omega ; P (\Delta _n )). \hfill \\ \Delta _n P(\Delta _n ) \hfill \\ \end{gathered} $$      

Some variations on Kac's formula for Brownian motion

We show that under suitable conditions $$\begin{gathered} E_x f\left\{ {a + \int_0^t \beta \left[ {b + \int_0^s {\alpha \left( {X_r } \right)dr, c + s, X_s } } \right]ds, b + \int_0^t {\alpha \left( {X_s } \right)ds, c + t, X_t } } \right\} \hfill \\ = e^{tG} f\left[ {a, b, c, x} \right] \hfill \\ \end{gathered} $$ whereX _t is a Brownian motion andG is the generator of a (C ₀) contraction semigroupe ^tG.     

Hyperbolic stochastic differential equations: Absolute continuity of the LKW of the solution at a fixed point

LetW be the Wiener process onT=[0, 1]². Consider the stochastic integral equation $$\begin{gathered} X_\zeta = x_0 + \int_{R_\zeta } {a_1 (\zeta \prime )X(s\prime ,dt\prime )ds\prime + } \int_{R_\zeta } {a_2 (\zeta \prime )X(ds\prime ,t\prime )dt\prime } \hfill \\ + \int_{R_\zeta } {a_3 (X_{\zeta \prime , } \zeta \prime )W(ds\prime ,dt\prime ) + } \int_{R_\zeta } {a_4 (X_{\zeta \prime , } \zeta \prime )ds\prime ,dt\prime ,} \hfill \\ \end{gathered} $$ whereR _ζ =(s, t) ∈ T, andx ₀ ∈ ?. Under some assumptions on the coefficients a_i, the existence and uniqueness of a solution for this stochastic integral equation is already known (see [6]). In this paper we present some sufficient conditions for the law ofX _ζ to have a density.     

Nontrivial solution of a quasilinear elliptic equation with critical growth in ℝn

Suppose Δ_n u = div (¦ ?u ¦^n-2?u) denotes then-Laplacian. We prove the existence of a nontrivial solution for the problem $$\left\{ \begin{gathered} - \Delta _n u + \left| u \right|^{n - 2} u = \int {(x,u)u^{n - 2} in \mathbb{R}^n } \hfill \\ u \in W^{1,n} (\mathbb{R}^n ) \hfill \\ \end{gathered} \right.$$ wheref(x, t) =o(t) ast → 0 and ¦f(x, t)¦ ≤C exp(α_n¦t¦^n/(n-1)) for some constantC > 0 and for allx∈?;t∈? with α_n =nω _n ^1/(n-1) , ω_n = surface measure ofS ^n-1.     

A new neighborhood union condition for Hamiltonian graphs

For a vertex set {u ₁,u ₂,...,u _k} of a graphG withn vertices, let $$\begin{gathered} s(G;\{ u_1 ,u_2 ,...,u_k \} ) = \sum {1 \leqslant i< j \leqslant k\left| {N(u_i ) \cup N(u_j )} \right|,} \hfill \\ NC_k = \min \{ s(G;\{ x_1 ,...,x_k \} )\} :\{ x_1 ,...,x_k \} is an independent set\} . \hfill \\ \end{gathered} $$ In this paper, we shall prove that ifG is 3-connected andNC ₄≥3n, thenG is either a hamiltonian or Petersen graph. This generalizes some results on the neighborhood union conditions for hamiltonian graphs.     

Inequalities for convex bodies and polar reciprocal lattices inR n II: Application ofK-convexity

The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inR ⁿ . The Minkowski functional? ⁿ ofU, the polar bodyU ⁰, the dual latticeL ^*, the covering radius μ(L, U), and the successive minima λ _i ,i=1, …,n, are defined in the usual way. Let $\mathcal{L}_n $ be the family of all lattices inR ⁿ . Given a convex bodyU, we define $$\begin{gathered} mh(U){\text{ }} = {\text{ }}\sup {\text{ }}\max \lambda _i (L,U)\lambda _{n - i + 1} (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n 1 \leqslant i \leqslant n \hfill \\ lh(U){\text{ }} = {\text{ }}\sup {\text{ }}\lambda _1 (L,U) \cdot \mu (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n \hfill \\ \end{gathered} $$ and kh(U) is defined as the smallest positive numbers for which, given arbitrary $L \in \mathcal{L}_n $ andx∈R ⁿ /(L+U), somey∈L ^* with ∥y∥ _U ⁰?sd(xy,Z) can be found. It is proved $$C_1 n \leqslant jh(U) \leqslant C_2 nK(R_U^n ) \leqslant C_3 n(1 + \log n),$$ , for j=k, l, m, whereC ₁,C ₂,C ₃ are some numerical constants andK(R _U ⁿ ) is theK-convexity constant of the normed space (R ⁿ , ∥∥U). This is an essential strengthening of the bounds obtained in [2]. The bounds for lh(U) are then applied to improve the results of Kannan and Lovász [5] estimating the lattice width of a convex bodyU by the number of lattice points inU.     

New iterations with errors for approximating common fixed points for two generalized asymptotically quasi-nonexpansive nonself-mappings

Let X be a real uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T ₁, T ₂: C → X be two uniformly L-Lipschitzian, generalized asymptotically quasi-nonexpansive non-self-mappings of C satisfying condition A′ with sequences {k _n ⁽ⁱ⁾ } and {δ _n ⁽ⁱ⁾ } ? [1, ∞),, i = 1, 2, respectively such that Σ _n=1 ^∞ (k _n ⁽ⁱ⁾ ? 1) < ∞, Σ _n=1 ⁽ⁱ⁾ δ _n ⁽ⁱ⁾ < ∞, and F = F(T ₁) ∩ F(T ₂) ≠ ?. For an arbitrary x ₁ ∈ C, let {x _n } be the sequence in C defined by $$ \begin{gathered} y_n = P\left( {\left( {1 - \beta _n - \gamma _n } \right)x_n + \beta _n T_2 \left( {PT_2 } \right)^{n - 1} x_n + \gamma _n v_n } \right), \hfill \\ x_{n + 1} = P\left( {\left( {1 - \alpha _n - \lambda _n } \right)y_n + \alpha _n T_1 \left( {PT_1 } \right)^{n - 1} x_n + \lambda _n u_n } \right), n \geqslant 1, \hfill \\ \end{gathered} $$ where {α _n }, {β _n }, {γ _n } and {λ _n } are appropriate real sequences in [0, 1) such that Σ _n=1 ^∞ ] γ _n < ∞, Σ _n=1 ^∞ λ _n < ∞, and {u _n }, }v _n } are bounded sequences in C. Then {x _n } and {y _n } converge strongly to a common fixed point of T ₁ and T ₂ under suitable conditions.     

Subalgebras of free products of algebras of the variety\mathfrak{u}_{m,n}

The variety \(\mathfrak{u}_{m,n} \) is defined by the system of n-ary operations ω_i,..., ω_m, the system of m-ary operations ?_i,..., ?_n, 1≤ m ≤ n, and the system of identities $$\begin{gathered} x_1 ...x_n \omega _1 ...x_1 ...x_n \omega _m \varphi _i = x_i (i = 1,...,n), \hfill \\ x_1 ...x_m \varphi _1 ...x_1 ...x_m \varphi _n \omega _j = x_j (i = 1,...,m), \hfill \\ \end{gathered} $$ It is proved in this paper that the subalgebra U of the free product \(\Pi _{i \in I}^* A_i \) of the algebras A_i (i ε I) can be expanded as the free product of nonempty intersections U ∩ A_i (i ε I) and a free algebra.     

Stability of the Faedo-Galerkin approximation of nonlinear wave-equations

Present investigation analyses the Ljapunov stability of the systems of ordinary differential equations arising in then-th step of the Faedo-Galerkin approximation for the nonlinear wave-equation $$\begin{gathered} u_{tt} - u_{xx} + M(u) = 0 \hfill \\ u(0,t) = u(1,t) = 0 \hfill \\ u(x,0) = \Phi (x); u_t (x,0) = \Psi (x). \hfill \\ \end{gathered}$$ For the nonlinearities of the classM (u)=u ^{2 p+1} ,p ∈N, ann-independent stability result is given. Thus also the stability of the original equation is shown.     

A decomposition for someU-type statistics

Define , $S_{k,n} = \Sigma _{1 \leqslant i_1< \cdot \cdot \cdot< l_k \leqslant n} X_{i_1 } \cdot \cdot \cdot X_{i_k } ,n \geqslant k \geqslant {\text{1}}$ where {X, X _n ,n≥1} are i.i.d. random variables withEX=0,EX ²=1 and letH _k (·) denote the Hermite polynomial of degreek. By establishing an LIL for products of correlated sums of i.i.d. random variables, the a.s. decomposition $$\begin{gathered} k!S_{k,n} = n^{k/2} H_k (S_{1n} /n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ) - \left( {\begin{array}{*{20}c} k \\ 2 \\ \end{array} } \right)S_{1.n}^{k - 2} \sum\limits_{i = 1}^n {(X_i^2 - 1)} \hfill \\ + O(n^{(k - 1)/2} (\log \log n)^{(k - 3/2} ) \hfill \\ \end{gathered} $$ valid whenEX ⁴<∞, elicits an LIL forη _k,n =k!S _k,n ?n ^k/2 H _k (S _1n /n ^1/2) under a reduced normalization. Moreover, whenE|X| ^p <∞ for somep in [2, 4], a Marcinkiewicz-Zygmund type strong law forη _k,n is obtained, likewise under a reduced normalization.     

On solvability of Dirichlet problem for second order elliptic equation

The paper gives some solvability conditions of the Dirichlet problem for the second order elliptic equation $$ - div(A(x)\nabla u) + (\bar b(x),\nabla u) - div(\bar c(x)u) + d(x)u = f(x) - divF(x),x \in Q,u|_{\partial Q} = u_0 \in L_2 (\partial Q) $$ in bounded domain Q ? R _n (n ≥ 2) with smooth boundary ?Q ∈ C ₁. In particular, it is proved that if the homogeneous problem has only the trivial solution, then for any u ₀ ∈L ₂(?Q) and f, F from the corresponding functional spaces the solution of the non-homogeneous problem exists, from Gushchin’s space $ C_{n - 1} (\bar Q) $ and the following inequality is true: $$ \begin{gathered} \left\| u \right\|_{C_{n - 1} (\bar Q)}^2 + \mathop \smallint \limits_Q r\left| {\nabla u} \right|^2 dx \leqslant \hfill \\ \leqslant C\left( {\left\| {u_0 } \right\|_{L_2 (\partial Q)}^2 + \mathop \smallint \limits_Q r^3 (1 + |\ln r|)^{3/2} f^2 dx + \mathop \smallint \limits_Q r(1 + |\ln r|)^{3/2} |F|^2 dx} \right) \hfill \\ \end{gathered} $$ where r(x) is the distance from a point x ∈ Q to the boundary ?Q and the constant C does not depend on u ₀, f and F.     

Some 3-adic congruences for binomial sums

We prove some 3-adic congruences for binomial sums,which were conjectured by Zhi-Wei Sun.For example,for any integer m≡1(mod 3)and any positive integer n,we have31n n.1Xk=01mk 2k kmin{3(n),3(m.1).1},where 3(n)denotes the 3-adic order of n.In our proofs,we use several auxiliary combinatorial identities and a series converging to 0 over the 3-adic field.     

On the convergence and divergence of multiple Fourier series with respect to the Walsh—Paley system in the spacesC andL

Работа касается вопр осов сходимости и рас ходимости кратных рядов Фурье п о системе Уолша-Пэли в метрикахС и L. Из доказанных тео рем следует, в частности, ч то еишf∈Е (Е=С или E=L) и существует н атуральноеi ₀ (1≦i ₀ ≦N) так ое, что и $$\begin{gathered} \omega (\delta _{i_0 } ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _{i_0 } }}}}} \right) (\delta _{i_0 } \to 0) \hfill \\ \omega (\delta _k ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _k }}}}} \right), k \ne i_0 (\delta _k \to 0), k = 1,2, ...,N, \hfill \\ \end{gathered}$$ тоN-кратный ряд Фурье функцииf по системе У олша-Пэли сходится по Прингсхе йму в смысле метрики пространств аЕ. Доказано также, что вы шеотмеченное утверж дение неусиляемо в метрикеL не только для системы Уолша, но и для некоторого класс а ОНС, ограниченных в совок упности.     

On some boundary value problems with integral conditions for functional differential equations

For the functional differential equationu ⁽ⁿ⁾ (t)=f(u)(t) we have established the sufficient conditions for solvability and unique solvability of the boundary value problems $$u^{(i)} (0) = c_i (i = 0,...,m - 1), \smallint _0^{ + \infty } |u^{(m)} (t)|^2 dt< + \infty $$ and $$\begin{gathered} u^{(i)} (0) = c_i (i = 0),...,m - 1, \hfill \\ \smallint _0^{ + \infty } t^{2j} |u^{(j)} (t)|^2 dt< + \infty (j = 0,...,m), \hfill \\ \end{gathered} $$ wheren≥2,m is the integer part of $\tfrac{n}{2}$ ,c _i ∈R, andf is the continuous operator acting from the space of (n?1)-times continuously differentiable functions given on an interval [0,+∞] into the space of locally Lebesgue integrable functions.     

Fractal relaxed Dirichlet problems

Given aself similar fractal K ? ? ⁿ of Hausdorff dimension α>n?2, andc ₁>0, we give an easy and explicit construction, using the self similarity properties ofK, of a sequence of closed sets? _h such that for every bounded open setΩ?? ⁿ and for everyf ∈ L²(Ω) the solutions to $$\left\{ \begin{gathered} - \Delta u_h = f in \Omega \backslash \varepsilon _h \hfill \\ u_h = 0 on \partial (\Omega \backslash \varepsilon _h ) \hfill \\ \end{gathered} \right.$$ converge to the solution of the relaxed Dirichlet boundary value problem $$\left\{ \begin{gathered} - \Delta u + uc_1 \mathcal{H}_{\left| K \right.}^\alpha = f in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$ (H _∣ ^α denotes the restriction of the α-dimensional Hausdorff measure toK). The condition α>n?2 is strict.     

Expressing a prime as difference between two numbers containing all the previous primes

We consider the simultaneous equations $$\begin{gathered} p_1^{\alpha _1 } p_2^{\alpha _2 } ...p_n^{\alpha _n } = P_1 P_2 \hfill \\ p_{n + 1} = P_1 - P_2 \hfill \\ \end{gathered} $$ wherep ₁,p ₂,...,p _n+1 are then+1 first primes and α₁,...,α _n ,P ₁,P ₂ are integers. The equations are solved completely forn≦3 and all solutions given under certain restrictions on the α _i 's forn≦9.     

Global structure of positive solutions for superlinear 2mth-boundary value problems

We consider boundary value problems for nonlinear 2mth-order eigenvalue problem $$ \begin{gathered} ( - 1)^m u^{(2m)} (t) = \lambda a(t)f(u(t)),0 < t < 1, \hfill \\ u^{(2i)} (0) = u^{(2i)} (1) = 0,i = 0,1,2,...,m - 1. \hfill \\ \end{gathered} $$ . where a ∈ C([0, 1], [0, ∞)) and a(t ₀) > 0 for some t ₀ ∈ [0, 1], f ∈ C([0, ∞), [0, ∞)) and f(s) > 0 for s > 0, and f ₀ = ∞, where $ \mathop {\lim }\limits_{s \to 0^ + } f(s)/s $ . We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.