An estimate of an incomplete linear form in several algebraic numbers

Letμ>m?1, letν be a rational number, and letω _k=b _k ^v , where b_k ≠ 0 are distinct numbers of an imaginary quadratic field K, which satisfy some additional conditions. Then $$\begin{gathered} |{}_1x_1 \omega _1 + ... + x_m \omega _m | > X^{ - \mu } , \hfill \\ X = \max |x_k | \geqslant X, > 0, \hfill \\ 1 \leqslant k \leqslant m \hfill \\ \end{gathered}$$ where x₁, ..., x_m are integers of the field K, and X₀ is an effective constant.     

Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f(x, y) from the class L ₂ = L ₂((a, b) × (c, d); p(x)q(y)) with the weight p(x)q(y) and the norm $$\left\| f \right\| = \sqrt {\int\limits_a^b {\int\limits_c^d {p(x)q(x)f^2 (x,y)dxdy} } }$$ are approximated by an orthonormal system of orthogonal P _n (x)Q _n (y), n, m = 0, 1, ..., with weights p(x) and q(y). Let $$E_N (f) = \mathop {\inf }\limits_{P_N } \left\| {f - P_N } \right\|$$ denote the best approximation of f ?? L ₂ by algebraic polynomials of the form $$\begin{array}{*{20}c} {P_N (x,y) = \sum\limits_{0 < n,m < N} {a_{m,n} x^n y^m ,} } \\ {P_1 (x,y) = const.} \\ \end{array}$$ . Consider a double Fourier series of f ?? L ₂ in the polynomials P _n (x)Q _m (y), n, m = 0, 1, ..., and its ??hyperbolic?? partial sums $$\begin{array}{*{20}c} {S_1 (f;x,y) = c_{0,0} (f)P_o (x)Q_o (y),} \\ {S_N (f;x,y) = \sum\limits_{0 < n,m < N} {c_{n,m} (f)P_n (x)Q_m (y), N = 2,3, \ldots .} } \\ \end{array}$$ A generalized shift operator Fh and a kth-order generalized modulus of continuity ?? _k (A, h) of a function f ?? L ₂ are used to prove the following sharp estimate for the convergence rate of the approximation: $\begin{gathered} E_N (f) \leqslant (1 - (1 - h)^{2\sqrt N } )^{ - k} \Omega _k (f;h),h \in (0,1), \hfill \\ N = 4,5,...;k = 1,2,... \hfill \\ \end{gathered} $ . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.     

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.     

Dichotomies for Lorentz spaces

Assume that L ^p,q , $L^{p_1 ,q_1 } ,...,L^{p_n ,q_n } $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \{ (f_1 ,...,f_n ) \in L^{p_{1,} q_1 } \times \cdots \times L^{p_n ,q_n } :f_1 \cdots f_n \in L^{p,q} \} $ . We prove the following dichotomy: either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is σ-porous in $L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ , provided 1/p ≠ 1/p ₁ + … + 1/p _n . In general case we obtain that either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is meager. This is a generalization of the results for classical L ^p spaces.     

Spectral properties of some regular boundary value problems for fourth order differential operators

In this paper we consider the problem $\begin{gathered} y^{iv} + p_2 (x)y'' + p_1 (x)y' + p_0 (x)y = \lambda y,0 < x < 1, \hfill \\ y^{(s)} (1) - ( - 1)^\sigma y^{(s)} (0) + \sum\limits_{l = 0}^{s - 1} {\alpha _{s,l} y^{(l)} (0) = 0,} s = 1,2,3, \hfill \\ y(1) - ( - 1)^\sigma y(0) = 0, \hfill \\ \end{gathered} $ where λ is a spectral parameter; p _j (x) ∈ L ₁(0, 1), j = 0, 1, 2, are complex-valued functions; α _s;l , s = 1, 2, 3, $l = \overline {0,s - 1} $ , are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α _3,2 + α _1,0 ≠ α _2,1. It is proved that the system of root functions of this spectral problem forms a basis in the space L _p (0, 1), 1 < p < ∞, when α _3,2+α _1,0 ≠ α _2,1, p _j (x) ∈ W ₁ ^j (0, 1), j = 1, 2, and p ₀(x) ∈ L ₁(0, 1); moreover, this basis is unconditional for p = 2.     

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.     

Some extremal properties of positive trigonometric polynomials

For n=8 an upper bound is given for the functional $$V_n = \mathop {\inf }\limits_{t_n } \frac{{\alpha _1 + \alpha _2 + \cdots + \alpha _n }}{{\left( {\sqrt {\alpha _1 } - \sqrt {\alpha _0 } } \right)^2 }}$$ , which is defined on the class of even, nonnegative, trigonometric polynomials \(t_n (\phi ) = \sum\nolimits_{k = 0}^n {\alpha _k } cos k\phi \) , such that α _k ? 0 (k=0, ...,n) α₁>α₀ :V _s ? 34.54461566.     

The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic stability

We study discretization in classes of integro-differential equations where the functions aj(t),1≤j≤n,are completely monotonic on(0,∞) and locally integrable,but not constant.The equations are discretized using the backward Euler method in combination with order one convolution quadrature for the memory term.The stability properties of the discretization are derived in the weighted l1(ρ;0,∞) norm,where ρ is a given weight function.Applications to the weighted l1 stability of the numerical solutions of a related equation in Hilbert space are given.     

Heat equation with a singular potential on the boundary and the Kato inequality

This paper is concerned with the heat equation in the half-space ? ₊ ^N with the singular potential function on the boundary, (P) $\left\{ \begin{gathered} \frac{\partial } {{\partial t}}u - \Delta u = 0\operatorname{in} \mathbb{R}_ + ^N \times (0,T), \hfill \\ \frac{\partial } {{\partial x_N }}u + \frac{\omega } {{|x|}}u = 0on\partial \mathbb{R}_ + ^N \times (0,T), \hfill \\ u(x,0) = u_0 (x) \geqslant ()0in\mathbb{R}_ + ^N , \hfill \\ \end{gathered} \right. $ where N ?? 3, ?? > 0, 0 < T ?? ??, and u ₀ ?? C ₀(? ₊ ^N ). We prove the existence of a threshold number ?? _N for the existence and the nonexistence of positive solutions of (P), which is characterized as the best constant of the Kato inequality in ? ₊ ^N .     

The lattice point discrepancy of a torus in ℝ3

This article provides an asymptotic formula for the number of integer points in the three-dimensional body $$ \left( \begin{gathered} x \hfill \\ y \hfill \\ z \hfill \\ \end{gathered} \right) = t\left( \begin{gathered} (a + r\cos \alpha )\cos \beta \hfill \\ (a + r\cos \alpha )\sin \beta \hfill \\ r\sin \alpha \hfill \\ \end{gathered} \right),0 \leqq \alpha ,\beta < 2\pi ,0 \leqq r \leqq b, $$ for fixed a > b > 0 and large t.     

Approximation by rational functions with polynomials of positive coefficients as the denominators

The present paper proves that if f(x) ∈ C[0,1], changes its sign exactly l times at 0 ＜ y1 ＜ y2 ＜y1＜1 in(0,1),then there exists a pn (x)пn( ),such that |f(x)-p(x)/pn(x)|≤ Cωφ(f,n-1/2),where ρ(x) is defined by ρ(x)=l∏i=1(x - yi), if f(x) ≥ 0 for x ∈ (yl, 1),-1∏i=1(x-yi), iff(x) ＜ 0 for x ∈ (y1,1).which improves and generalizes the result of [7].     

Finite difference method of first boundary problem for quasilinear parabolic systems (IV)

More work is done to study the explicit, weak and strong implicit difference solution for the first boundary problem of quasilinear parabolic system: $$\begin{gathered} u_t = ( - 1)^{M + 1} A(x,t,u, \cdots ,u_x M - 1)u_x 2M + f(x,t,u, \cdots u_x 2M - 1), \hfill \\ (x,t) \in Q_T = \left| {0< x< l,0< t \leqslant T} \right|, \hfill \\ u_x ^k (0,t) = u_x ^k (l,t) = 0 (k = 0,1, \cdots ,M - 1),0< t \leqslant T, \hfill \\ u(x,0) = \varphi (x),0 \leqslant x \leqslant l, \hfill \\ \end{gathered} $$ whereu, ?, andf arem-dimensional vector valued functions, A is anm×m positively definite matrix, and $u_t = \frac{{\partial u}}{{\partial t}},u_x ^k = \frac{{\partial ^k u}}{{\partial x^k }}$ . For this problem, the convergence of iteration for the general difference schemes is proved.     

L p inequalities for polynomials with restricted zeros

LetP(z) be a polynomial of degreen which does not vanish in the disk |z|<k. It has been proved that for eachp>0 andk≥1, $$\begin{gathered} \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P^{(s)} (e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} \leqslant n(n - 1) \cdots (n - s + 1) B_p \hfill \\ \times \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P(e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} , \hfill \\ \end{gathered} $$ where $B_p = \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {k^s + e^{i\alpha } } \right|^p d\alpha } } \right\}^{ - 1/p} $ andP ^(s)(z) is thesth derivative ofP(z). This result generalizes well-known inequality due to De Bruijn. Asp→∞, it gives an inequality due to Govil and Rahman which as a special case gives a result conjectured by Erdös and first proved by Lax.     

Another approach to characterizations of generalized triangle inequalities in normed spaces

In this paper, we consider a generalized triangle inequality of the following type: $$\left\| {x_1 + \cdots + x_n } \right\|^p \leqslant \frac{{\left\| {x_1 } \right\|^p }} {{\mu _1 }} + \cdots + \frac{{\left\| {x_2 } \right\|^p }} {{\mu _n }}\left( {for all x_1 , \ldots ,x_n \in X} \right),$$ where (X, ‖·‖) is a normed space, (µ₁, ..., µ _n ) ∈ ? ⁿ and p > 0. By using ψ-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which is given by [Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Nonlinear Anal., 2012, 75(2), 735–741].     

An infinite algebraic system in the irregular case

We obtain sufficient conditions for the nontrivial solvability of systems of the form $$ \phi _i = b_i + \lambda _i \sum\limits_{j = 0}^\infty {a_{i - j} \phi _j ,i \in \mathbb{Z}_ + \underline{\underline {def}} \{ 0,1,2...,n,...\} ,} $$ and of the corresponding homogeneous systems. It is assumed that the sequences b = (b ₀, b ₁, b ₂, …) and λ = (λ ₀, λ ₁, λ ₂, …) and the Toeplitz matrix A = (a _i?j ) satisfy the conditions $$ \begin{gathered} a_j \geqslant 0,j \in \mathbb{Z},\sum\limits_{j = - \infty }^\infty {a_j = 1,} \sum\limits_{j = - \infty }^\infty {|j|a_j < \infty ,\sum\limits_{j = - \infty }^\infty {ja_j < 0,} } \hfill \\ b_j \geqslant 0,j \in \mathbb{Z},\sum\limits_{j = 0}^\infty {b_j = \infty ,} 1 \leqslant \lambda _i \leqslant \left( {\sum\limits_{j = - \infty }^i {a_j } } \right)^{ - 1} ,i \in \mathbb{Z}_ + . \hfill \\ \end{gathered} $$ . Under these conditions, we construct bounded solutions of homogeneous and inhomogeneous systems of the form indicated above.     

Bell Numbers, Log-Concavity, and Log-Convexity

Let {b _k (n)} _n=0 ^∞ be the Bell numbers of order k. It is proved that the sequence {b _k (n)/n!} _n=0 ^∞ is log-concave and the sequence {b _k (n)} _n=0 ^∞ is log-convex, or equivalently, the following inequalities hold for all n?0, $$1 \leqslant \frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} \leqslant \frac{{n + 2}}{{n + 1}}$$ . Let {α(n)} _n=0 ^∞ be a sequence of positive numbers with α(0)=1. We show that if {α(n)} _n=0 ^∞ is log-convex, then α(n)α(m)?α(n+m), ?n,m?0. On the other hand, if {α(n)/n!} _n=0 ^∞ is log-concave, then $$\alpha (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)\alpha (n)\alpha (m),{\text{ }}\forall n,m \geqslant 0$$ . In particular, we have the following inequalities for the Bell numbers $$b_k (n)b_k (m) \leqslant b_k (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)b_k (n)b_k (m),{\text{ }}\forall n,m \geqslant 0$$ . Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.     

Boundary parametrization of planar self-affine tiles with collinear digit set

We consider a class of planar self-affine tiles T = M-1 a∈D(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:M =(0-B 1-A),D = {(00),...,(|B|0-1)}.We give a parametrization S1 →T of the boundary of T with the following standard properties.It is H¨older continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on T and have algebraic preimages.We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.     

Symmetry results for systems involving fractional Laplacian

In this paper we investigate symmetry results for positive solutions of systems involving the fractional Laplacian (1) $\left\{ \begin{gathered} ( - \Delta )^{\alpha _1 } u_1 (x) = f_1 (u_2 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ ( - \Delta )^{\alpha _2 } u_2 (x) = f_2 (u_1 (x)),x \in \mathbb{R}^\mathbb{N} , \hfill \\ \lim _{|x| \to \infty } u_1 (x) = \lim _{|x| \to \infty } u_2 (x) = 0 \hfill \\ \end{gathered} \right. $ where N ≥ 2 and α ₁, α ₂ ∈ (0, 1). We prove symmetry properties by the method of moving planes.     

On a second-order matrix differential operator

We consider the second-order matrix differential operator $$N = \left( {\begin{array}{*{20}c} { - \frac{d}{{dx}}\left( {p_0 \frac{d}{{dx}}} \right) + p_1 } \\ r \\ \end{array} \begin{array}{*{20}c} r \\ { - \frac{d}{{dx}}\left( {q_0 \frac{d}{{dx}}} \right) + q_1 } \\ \end{array} } \right)$$ determined by the expression Nφ, [0 ?x < ∞), where \(\phi = \left( {\begin{array}{*{20}c} U \\ V \\ \end{array} } \right)\) . It has been proved that if p₀, q₀, p₁, q₁,r satisfy certain conditions, then N is in the limit point case at ∞. It has been also shown that certain differential operators in the Hilbert space L² of vectors, generated by the operator N, are symmetric and self-adjoint.     

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 $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$