On the distribution of integer random variables related by a certain linear inequality: III

We consider tuples {N _jk }, j = 1, 2, ..., k = 1, ..., q _j , of nonnegative integers such that $$ \sum\limits_{j = 1}^\infty {\sum\limits_{k = 1}^{q_j } {jN_{jk} } } \leqslant M. $$ Assuming that q _j ～ j ^d?1, 1 < d < 2, we study how the probabilities of deviations of the sums $ \sum\nolimits_{j = j_1 }^{j_2 } {\sum\nolimits_{k = 1}^{q_j } {N_{jk} } } $ N _jk from the corresponding integrals of the Bose-Einstein distribution depend on the choice of the interval [j ₁,j ₂].     

An extremal problem for algebraic polynomials in the symmetric discrete Gegenbauer-Sobolev space

We study discrete Sobolev spaces with symmetric inner product $$\left\langle {f,g} \right\rangle _\alpha = \int_{ - 1}^1 {f g d\mu _\alpha } + M[f(1)g(1) + f( - 1)g( - 1)] + K[f'(1)g'(1) + f'( - 1)g'( - 1)]$$ , where M ≥ 0, k ≥ 0, and $$d\mu _\alpha (x) = \frac{{\Gamma (2\alpha + 2)}}{{2^{2\alpha + 1} \Gamma ^2 (\alpha + 1)}}(1 - x^2 )^\alpha dx, \alpha > - 1$$ , is the Gegenbauer probability measure. We obtain the solution of the following extremal problem: Calculate $$\mathop {\inf }\limits_{a_0 ,a_1 ,...,a_{N - r} } \left\{ {\langle P_N^{(r)} ,P_N^{(r)} \rangle _\alpha ,1 \leqslant r \leqslant N - 1, P_N^{(r)} (x) = \sum\limits_{j = N - r + 1}^N {a_j^0 x^j } + \sum\limits_{j = 0}^{N - r} {a_j x^j } } \right\}$$ , where the a _j ⁰ , j = N ? r + 1, N ? r + 2, ..., N ? 1, N, a _N ⁰ > 0, are fixed numbers, and find the extremal polynomial.     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

Oscillatory Properties of Solutions of Impulsive Differential Equations With Several Retarded Arguments

The impulsive differential equation $\begin{gathered} x\prime (t) + \sum\limits_{i = 1}^m {p_i (t)x(t - \tau _i ) = 0,} {\text{ }}t \ne \xi _k , \\ \Delta x(\xi _k ) = b_k x(\xi _k ) \\ \end{gathered} $ with several retarded arguments is considered, where p _i(t) ≥ 0, 1 + b _k > 0 for i = 1, ..., m, t ≥ 0, $k \in \mathbb{N}$ . Sufficient conditions for the oscillation of all solutions of this equation are found.     

Оценки отклонений оп тимальных сеток вL p -норме и теория квад ратурных формул

Let E^s=[0, 1]^s be then-dimensional unit cube, 1<p<∞, anda=(a ₁, ...,a _s ) some set of natural numbers. Denote byL _p ^(a) , (E ^s ) the class of functionsf: E ^s → C for which $$\left\| {\frac{{\partial ^{b_1 + \cdots + b_s } f}}{{\partial x_1^{b_1 } \cdots \partial x_s^{b_s } }}} \right\|_p \leqslant 1,$$ where $$0< b_1< a_1 , ..., 0< b_s< a_s .$$ Set $$R_p^{\left( a \right)} \left( N \right) = \mathop {\inf }\limits_{card \mathfrak{S} = N} R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right),$$ where $R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right)$ is the error of the quadrature formulas on the mesh $\mathfrak{S}$ (for the classL _p ^(a) (E ^s )), consisting of N nodes and weights, and the infimum is taken with respect to all possibleN nodes and weights. In this paper, the two-sided estimate $$\frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }} \ll _{p, a} R^{\left( a \right)} \left( N \right) \ll _{p, a} \frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }}$$ is proved for every natural numberN > 1, whered=min{a ₁, ...,a _s }, whilel is the number of those components of a which coincide withd. An analogous result is proved for theL _p -norm of the deviation of meshes.     

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let ${\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)}$ with the non-negative potential V belonging to reverse H?lder class with respect to the measure ??(x)dx, where ??(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying ${\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. }$ We obtain some estimates for ${V^{\alpha}\mathcal{L}^{-\alpha}}$ on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator ${[b, V^{\alpha} \mathcal{L}^{-\alpha}]}$ when ${b\in BMO_\omega}$ and 0?<??? ?? 1.     

A moment problem associated to rational Szegő functions

Leta ₁,...,a _p be distinct points in the finite complex plane ?, such that |a _j|>1,j=1,..., p and let \(b_j = 1/\bar \alpha _j ,\) j=1,..., p. Let μ₀, μ _π ^(j) , ν _π ^(j) j=1,..., p;n=1, 2,... be given complex numbers. We consider the following moment problem. Find a distribution ψ on [?π, π], with infinitely many points of increase, such that $$\begin{array}{l} \int_{ - \pi }^\pi {d\psi (\theta ) = \mu _0 ,} \\ \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - a_j )^n }} = \mu _n^{(j)} ,} \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - b_j )^n }} = v_n^{(j)} ,} j = 1,...,p;n = 1,2,.... \\ \end{array}$$ It will be shown that this problem has a unique solution if the moments generate a positive-definite Hermitian inner product on the linear space of rational functions with no poles in the extended complex plane ?^* outside {a ₁,...,a _p,b ₁,...,b _p}.     

Structural and geometric characteristics of sets of convergence and divergence of multiple Fourier series with J k -lacunary sequence of rectangular partial sums

We study multiple trigonometric Fourier series of functions f in the classes $L_p \left( {\mathbb{T}^N } \right)$ , p > 1, which equal zero on some set $\mathfrak{A}, \mathfrak{A} \subset \mathbb{T}^N , \mu \mathfrak{A} > 0$ (µ is the Lebesgue measure), $\mathbb{T}^N = \left[ { - \pi ,\pi } \right]^N$ , N ≥ 3. We consider the case when rectangular partial sums of the indicated Fourier series S _n (x; f) have index n = (n ₁, ..., n _N ) ∈ ? ^N , in which k (k ≥ 1) components on the places {j ₁, ..., j _k } = J _k ? {1, ..., N} are elements of (single) lacunary sequences (i.e., we consider multiple Fourier series with J _k -lacunary sequence of partial sums). A correlation is found of the number k and location (the “sample” J _k ) of lacunary sequences in the index n with the structural and geometric characteristics of the set $\mathfrak{A}$ , which determines possibility of convergence almost everywhere of the considered series on some subset of positive measure $\mathfrak{A}_1$ of the set $\mathfrak{A}$ .     

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.     

On the convergence to infinity of Fourier series along dense subsequences of numbers

В статье доказываетс я Теорема.Какова бы ни была возрастающая последовательность натуральных чисел {H _k } _{k = 1} ^∞ c $$\mathop {\lim }\limits_{k \to \infty } \frac{{H_k }}{k} = + \infty$$ , существует функцияf∈L(0, 2π) такая, что для почт и всех x∈(0, 2π) можно найти возраст ающую последовательность номеров {n_k(x)} _k=1 ^∞ ,удовлетворяющую усл овиям 1) $$n_k (x) \leqq H_k , k = 1,2, ...,$$ 2) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t} (x)} (x,f) = + \infty ,$$ 3) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t - 1} (x)} (x,f) = - \infty$$ .     

On the cosets of the 2q-power group in the unit group modulop

Letp be a prime number ≡ 3 mod 4,G ^p the unit group of ?/p?, andg a generator ofG _p. Letq be an odd divisor ofp - 1 andG _p ^2q = {t ^2q;t ∈G _pthe subgroup of index2q inG _p. The groupG _p ² / _p ^2q consists of the classes \(\bar g^{2j} \) ,j = 0,...,q – 1. In this paper we study the ’excesses’ of the classes \(\bar g^{2j} \) in {l,...,(_p–l)/2}, i.e., the numbers \(\Phi _j = \left| {\left\{ {k;1 \leqslant k \leqslant \left( {p - 1} \right)/2,\bar k \in \bar g^{2j} } \right\}} \right| - \left| {\left\{ {k;\left( {p - 1} \right)/2 \leqslant k \leqslant p - 1,\bar k \in \bar g^{2j} } \right\}} \right|\) ,j = 0.....q — 1. First we express therelative class number h _2q of the subfieldK _2q? ?(e^2#x03C0;i/p ) of degree [K _2q: ?] =2q in terms of these excesses. We use this formula to establish certaincongruences for the Ф^j. E.g., ifq ∈ {3,5,11}, each number Фj is congruent modulo 4 to each other iff 2 dividesh _2q ^- . Finally we study thevariance of the excesses, i.e., the number \(\sigma ^2 = ((\Phi _0 - \hat \Phi )^2 + \ldots + (\Phi _{q - 1} - \hat \Phi )^2 )/(q - 1)\) , where \(\hat \Phi \) is the mean value of the numbers Ф_j. We obtain an explicit lower bound for σ² in terms ofh _2q ^- /h ₂ ^- . Moreover, we show that log σ² is asymptotically equal to 21og(h _2q ^- h ₂ ^- )/(q - 1) forp→∞. Three tables illustrate the results.     

Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

On orthogonal polynomials

Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р _n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa _j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μ_j+1-μ_j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pv_j(x)} последовател ьности {р_n(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).     

Some extremal properties of positive trigonometric polynomials

A class P_n of even positive trigonometric polynomials tn(?)=a₀ + a₁ cos ?+ ... + a_n cos · n?, satisfying the conditions: a_k ≥0 (k = 0,1, ..., n), a₀ < a₁ is considered. The behavior of the sequence of functionals $$v_n = _{t_n \mathop { \in P_n }\limits^{\inf } } \frac{{t_n \left( 0 \right) - a_o }}{{\left( {\sqrt {a_1 } - \sqrt {a_o } } \right)}}$$ , is studied; two-sided estimations are given for V_n and \(V_\infty = \mathop {\lim }\limits_{n \to \infty } V_n \) .     

Density of primes in l-th power residues

Given a prime number l, a finite set of integers S?=?{a ₁, ...,a _m } and m many l-th roots of unity $\zeta_l^{r_i}, i=1, \ldots ,m$ we study the distribution of primes p in ?(ζ _l ) such that the l-th residue symbol of a _i with respect to p is $\zeta_l^{r_i}, \mbox{ for all } i$ . We find out that this is related to the degree of the extension $\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$ . We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from S?=?{a ₁, ...,a _m }. This latter argument enables one to describe the degree $\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$ in much simpler terms.     

Moduli of continuity defined by zero continuation of functions andK-functionals with restrictions

We consider the followingK-functional: $$K(\delta ,f)_p : = \mathop {\sup }\limits_{g \in W_{p U}^r } \left\{ {\left\| {f - g} \right\|_{L_p } + \delta \sum\limits_{j = 0}^r {\left\| {g^{(j)} } \right\|_{L_p } } } \right\}, \delta \geqslant 0,$$ where ? ∈L _p :=L _p [0, 1] andW _p,U ^r is a subspace of the Sobolev spaceW _p ^r [0, 1], 1≤p≤∞, which consists of functionsg such that $\int_0^1 {g^{(l_j )} (\tau ) d\sigma _j (\tau ) = 0, j = 1, ... , n} $ . Assume that 0≤l _l ≤...≤l _n ≤r-1 and there is at least one point τ _j of jump for each function σ _j , and if τ _j =τ _s forj ≠s, thenl _j ≠l _s . Let $\hat f(t) = f(t)$ , 0≤t≤1, let $\hat f(t) = 0$ ,t<0, and let the modulus of continuity of the functionf be given by the equality $$\hat \omega _0^{[l]} (\delta ,f)_p : = \mathop {\sup }\limits_{0 \leqslant h \leqslant \delta } \left\| {\sum\limits_{j = 0}^l {( - 1)^j \left( \begin{gathered} l \hfill \\ j \hfill \\ \end{gathered} \right)\hat f( - hj)} } \right\|_{L_p } , \delta \geqslant 0.$$ We obtain the estimates $K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 ]} (\delta ,f)_p $ and $K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 + 1]} (\delta ^\beta ,f)_p $ , where β=(pl _l + 1)/p(l ₁ + 1), and the constantc>0 does not depend on δ>0 and ? ∈L _p . We also establish some other estimates for the consideredK-functional.     

Some new estimates of the Fourier-Bessel transform in the space \mathbb{L}_2 \left( {\mathbb{R}_ + } \right)

The Fourier-Bessel integral transform $$g\left( x \right) = F\left[ f \right]\left( x \right) = \frac{1} {{2^p \Gamma \left( {p + 1} \right)}}\int\limits_0^{ + \infty } {t^{2p + 1} f\left( x \right)j_p \left( {xt} \right)dt}$$ is considered in the space $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ . Here, j _p (u) = ((2 ^p Γ(p+1))/(u ^p ))J _p (u) and J _p (u) is a Bessel function of the first kind. New estimates are proved for the integral $$\delta _N^2 \left( f \right) = \int\limits_N^{ + \infty } {x^{2p + 1} g^2 \left( x \right)dx, N > 0,}$$ in $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ for some classes of functions characterized by a generalized modulus of continuity.     

Decay estimates for nonlinear nonlocal diffusion problems in the whole space

In this paper, we obtain bounds for the decay rate in the L ^r (? ^d )-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, $$u_t \left( {x,t} \right) = \int_{\mathbb{R}^d } {K\left( {x,y} \right)\left| {u\left( {y,t} \right) - u\left( {x,t} \right)} \right|^{p - 2} \left( {u\left( {y,t} \right) - u\left( {x,t} \right)} \right)dy, x \in \mathbb{R}^d , t > 0.}$$ . We consider a kernel of the form K(x, y) = ψ(y?a(x)) + ψ(x?a(y)), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a(x) = Ax. To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $$T\left( u \right) = - \int_{\mathbb{R}^d } {K\left( {x,y} \right)\left| {u\left( y \right) - u\left( x \right)} \right|^{p - 2} \left( {u\left( y \right) - u\left( x \right)} \right)dy, 1 \leqslant p < \infty .}$$ . The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space ? ^d : $$\lambda _{1,p} \left( {\mathbb{R}^d } \right) = 2\left( {\int_{\mathbb{R}^d } {\psi \left( z \right)dz} } \right)\left| {\frac{1} {{\left| {\det A} \right|^{1/p} }} - 1} \right|^p .$$ Moreover, we deal with the p = ∞ eigenvalue problem, studying the limit of λ _1,p ^1/p as p→∞.     

Estimates of capacities associated with Besov spaces

Definition

Let A??ⁿ, 0<β≤∞. Define $$h_{\varphi ,\beta } (A) = \inf \left( {\sum\limits_{i = 0}^{ + \infty } {\left( {m_j \varphi (2^{ - i} } \right)^\beta } } \right)^{1/\beta } $$ where the infinum is taken over all coverings of A by a countable number of balls, whose radii r_j do not exceed 1, while m_i is the number of balls from this covering whose radii r_j belong to the set (2^?i?1, 2^?i], i∈N₀.

Theorem 1

Let p≤1, θ=∞, and let the function ?(t)t^lp?n increase. Then the following conditions are 2quivalent;

1. for any compact set K, K??ⁿ, if $\overline {cap} (K, X) = 0$ , then h_?,∞(K)=0;

Theorem 2

Let θ<1. Then for any set A the inequalities $c_1 \overline {cap} (A,X) \leqslant h_{t^{n - lp} ,\theta /p} (A) \leqslant c_2 \overline {cap} (A,X)$ hold. Bibliography:6 titles.