On the 2<Emphasis Type="Italic">m</Emphasis>-th power mean of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions with the weight of trigonometric sums

Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a ₀ + a ₁ x + ... + a _k x ^k is a k-degree polynomial with integral coefficients such that (p, a ₀, a ₁, ..., a _k ) = 1, for any integer m, we study the asymptotic property of

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

Estimation of the norm of the derivative of a monotone rational function in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We show that the derivative of an arbitrary rational function R of degree n that increases on the segment [−1, 1] satisfies the following equality for all 0 < ε < 1 and p, q > 1:

Multivariate Refinement Equations and Convergence of Cascade Algorithms in Lp（0〈p〈1）Spaces

We consider the solutions of refinement equations written in the form

On expansions of numbers in alternating <Emphasis Type="Italic">s</Emphasis>-adic series and Ostrogradskii series of the first and second kind

We present expansions of real numbers in alternating s-adic series (1 < s ∈ N), in particular, s-adic Ostrogradskii series of the first and second kind. We study the “geometry” of this representation of numbers and solve metric and probability problems, including the problem of structure and metric-topological and fractal properties of the distribution of the random variable

Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

The <Emphasis Type="Italic">p</Emphasis>-adic order of some fibonomial coefficients whose entries are powers of <Emphasis Type="Italic">p</Emphasis>

Let (F _n ) _n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${\left[ {\begin{array}{*{20}{c}} m \\ k \end{array}} \right]_F} = \frac{{{F_{m - k + 1}} \cdots {F_{m - 1}}{F_m}}}{{{F_1} \cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then \(p{\left| {\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]} \right._F}\) for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a, b ≥ 1 and for all prime number p.

     

On the Distribution of Values of <Emphasis Type="Italic">L</Emphasis>(1, <Emphasis Type="Italic">f</Emphasis>)

Let S_k(N)⁺ be the set of newforms of weight k for Γ₀(N), and let L (s, f), f ∈ S_k(N)⁺, be the Hecke L-function of the form f. It is proved that for every integer m ≥ 1, k = 2, and N = p → ∞,

$\mathop \sum \limits_{f \in S_2 (N)^ + } L^m (1,f) = \frac{1}{{12}}B_m N + O(N^{1 - \alpha } ),$

where B_m is a constant defined in the paper, and α = α(m) > 0 is a certain constant. This result implies the existence of the distribution function of the sequence

$\{ L(1,f),f \in S_2 (N)^ + \} ,\quad N = p \to \infty ,$

and also yields an explicit expression for the corresponding characteristic function. Bibliography: 11 titles.

     

Approximate <Emphasis Type="Italic">C</Emphasis>*-ternary ring homomorphisms

In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation

On the phase transitions of (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-SAT

Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T _k (V) denote the set of all 2 ^k n ^k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T _k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log₂ n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that

$$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$

     

The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).     

An Example of a Fast-Decreasing (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">A</Emphasis>)-Lacunary Function

A function analytic in the unit disk is called (p, A)-lacunary if the inequalities n _k Ak ^p hold for all k 0 with some 1 < p < and A > 0. In this paper, for 1 < p < 2 and A > 0, we construct a (p, A)-lacunary function f _1,p,A (z) decreasing as x 1 – 0 at a rate close to the optimal rate for (p, A)-lacunary functions. Bibliography: 6 titles.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> boundedness for multilinear fractional integral on product spaces

For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation

Distance-Regular Shilla Graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript> = <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript>

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ₁ equal to a₃. For a Shilla graph, let us put a = a₃ and b = k/a. It is proved in this paper that a Shilla graph with b₂ = c₂ and noninteger eigenvalues has the following intersection array:

$$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$

If Γ is a Q-polynomial Shilla graph with b₂ = c₂ and b = 2r, then the graph Γ has intersection array

$$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$

and, for any vertex u in Γ, the subgraph Γ₃(u) is an antipodal distance-regular graph with intersection array

$$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$

The Shilla graphs with b₂ = c₂ and b = 4 are also classified in the paper.

     

A remark on the existence of entire large and bounded solutions to a (<Emphasis Type="Italic">k</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">k</Emphasis><Subscript>2</Subscript>)-Hessian system with gradient term

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system

$$\left\{ {\begin{array}{*{20}c}{S_{k_1 } \left( {\lambda \left( {D^2 u_1 } \right)} \right) + a_1 \left( {\left| x \right|} \right)\left| {\nabla u_1 } \right|^{k_1 } = p_1 \left( {\left| x \right|} \right)f_1 \left( {u_2 } \right)} & {for x \in \mathbb{R}^N ,} \\{S_{k_2 } \left( {\lambda \left( {D^2 u_2 } \right)} \right) + a_2 \left( {\left| x \right|} \right)\left| {\nabla u_2 } \right|^{k_2 } = p_2 \left( {\left| x \right|} \right)f_2 \left( {u_1 } \right)} & {for x \in \mathbb{R}^N .} \\\end{array} } \right.$$

Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the k _i -Hessian operator, a ₁, p ₁, f ₁, a ₂, p ₂ and f ₂ are continuous functions.

     

An optimal<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bound on the Krein spectral shift function

Let ξ_A,B be the Krein spectral shift function for a pair of operatorsA, B, with C =A-B trace class. We establish the bound

Univalent Functions, <Emphasis Type="Italic">VMOA</Emphasis> and Related Spaces

This paper is concerned mainly with the logarithmic Bloch space ℬ_log which consists of those functions f which are analytic in the unit disc \mathbbD{\mathbb{D}} and satisfy sup_{|z| < 1}(1-|z|)log\frac11-|z||f^￠(z)| < ￥\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty , and the analytic Besov spaces B ^p , 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of:

On approximation of functions by trigonometric polynomials with incomplete spectrum in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, 0 < <Emphasis Type="Italic">p</Emphasis> < 1

Let B be a set of integers with certain arithmetic properties. We obtain estimates of the best approximation of functions in the space L _p , 0 < p <1, by trigonometric polynomials that are constructed by the system {e^ikx}_{k ? \mathbbZ\B} \{e^{ikx}\}_{k\in \mathbb{Z}\backslash B} . Bibliography: 13 titles.     

The Dual Spaces of Sets of Difference Sequences of Order <Emphasis Type="Italic">m</Emphasis> and Matrix Transformations

Let p = （pk）k=0^∞ be a bounded sequence of positive reals, m C N and u be s sequence of nonzero terms. If x = （xk）k=0^∞ is any sequence of complex numbers we write Δ（m）x for the sequence of the m th order differences of x and Δu^（m）X = {x=（x）k=0^∞ uΔ（m）x ∈ X} for any set X of sequences. We determine the α-, β- and γ-duals of the sets Δμ^（m）X for X=co（p）,c（p）,l∞（p） and characterize some matrix transformations between these spaces Δ^（m）X.