On a class of strongly contractive quadratic recurrent systems

We consider a class of nonlinear recurrent systems of the form \( {\Lambda_p} = \frac{1}{p}\sum\limits_{{p_1} = 1}^{p - 1} {f\left( {\frac{{{p_1}}}{p}} \right){\Lambda_{{p_1}}}{\Lambda_{p - {p_1}}}} \), p > 1, where f is a given function on the interval [0, 1] and Λ₁ = x is an adjustable real-valued parameter. Under some suitable assumptions on the function f, we show that there exists an initial value x ^* for which Λ _p = Λ _p (x ^* ) → const as p → ∞. More precise asymptotics of Λ _p is also derived.     

Powerful digraphs

We introduce the concept of a powerful digraph and establish that a powerful digraph structure is included into the saturated structure of each nonprincipal powerful type p possessing the global pairwise intersection property and the similarity property for the theories of graph structures of type p and some of its first-order definable restrictions (all powerful types in the available theories with finitely many (> 1) pairwise nonisomorphic countable models have this property). We describe the structures of the transitive closures of the saturated powerful digraphs that occur in the models of theories with nonprincipal powerful 1-types provided that the number of nonprincipal 1-types is finite. We prove that a powerful digraph structure, considered in a model of a simple theory, induces an infinite weight, which implies that the powerful digraphs do not occur in the structures of the available classes of the simple theories (like the supersimple or finitely based theories) that do not contain theories with finitely many (> 1) countable models.     

On the Waring-Goldbach Problem for Six Cubes and Two Biquadrates

Let P_r denote an almost prime with at most r prime factors, counted according to multiplicity. In the present paper, it is proved that for any sufficiently large even integer n, the equation

$$n = {x^3} + p_1^3 + p_2^3 + p_3^3 + p_4^3 + p_5^3 + p_6^4 + p_7^4$$

has solutions in primes p_i with x being a P₆. This result constitutes a refinement upon that of Hooley C.

     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

Sums of Values of Nonprincipal Characters over a Sequence of Shifted Primes

For a nonprincipal character χ modulo D, we prove a nontrivial estimate of the form Σ_n≤x Λ(n)χ(n ? l) \( \ll x\exp \{ - 0.6\sqrt {\ln D} \} \) for the sum of values of χ over a sequence of shifted primes in the case when x ≥ D^1/2+ε, (l,D) = 1, and the modulus of the primitive character generated by χ is a cube-free number.     

Riesz basis property of system of root functions of second-order differential operator with involution

The properties of the root functions are studied for an arbitrary operator generated in L ₂(?1, 1) by the operation with involution of the form Lu = ?u″(x)+αu″(?x)+q(x)u(x)+ qν(x)u(ν(x)), where α ∈ (?1, 1), ν(x) is an absolutely continuous involution of the segment [?1, 1] and the coefficients q(x) and qν(x) are summable functions on (?1, 1). Necessary and sufficient conditions are obtained for the unconditional basis property in L ₂(?1, 1) for the system of the root functions of the operator.     

Propagation of waves in a randomly stratified medium: An inverse problem

We pose and solve an inverse problem of finding a coefficient in the wave equation in the inhomogeneous semispace on the scattering data of a plane wave incident from the homogeneous semispace. The unknown coefficient is a sum of a deterministic summand of one variable (the “depth” z) and a small random summand α(x, z). We look for the deterministic summand, the expectation E(α(x, z)) =: m(z), and the second moment r(x ₁ t - x ₂, z ₁, z ₂):= E(α(x ₁, z ₁)α(x ₂, z ₂)). Here the symbol E(·) stands for expectation. The stratification property of a medium means that (i) the deterministic summand depends only on z, (ii) m(z) depends only on z, and (iii) the second moment for fixed z ₁ and z ₂ depends only on x ₁ ? x ₂.     

Integral averaging technique for oscillation of damped half-linear oscillators

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?_p(x′))′ + b(t)?_p(x′) + c(t)?_p(x) = 0, where ?p(x) = |x|^p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.     

Regularized traces of singular differential operators with canonical boundary conditions

A self-adjoint differential operator \(\mathbb{L}\) of order 2m is considered in L ₂[0,∞) with the classic boundary conditions \(y^{(k_1 )} (0) = y^{(k_2 )} (0) = y^{(k_3 )} (0) = \ldots = y^{(k_m )} (0) = 0\), where 0 ≤ k ₁ < k ₂ < ... < k _m ≤ 2m ? 1 and {k _s } _s=1 ^m ∪ {2m ? 1 ? k _s } _s=1 ^m = {0, 1, 2, ..., 2m ? 1}. The operator \(\mathbb{L}\) is perturbed by the operator of multiplication by a real measurable bounded function q(x) with a compact support: \(\mathbb{P}\) f(x) = q(x)f(x), f ∈ L ₂[0,∞). The regularized trace of the operator \(\mathbb{L} + \mathbb{P}\) is calculated.     

Some results on the annihilator graph of a commutative ring

Let R be a commutative ring. The annihilator graph of R, denoted by AG(R), is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann _R (xy) ≠ ann _R (x) ∪ ann _R (y), where for z ∈ R, ann _R (z) = {r ∈ R: rz = 0}. In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1, 2 or 3. Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG(R) and AG(T(R)) are isomorphic, where T(R) is the total quotient ring of R. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n, where n ? 1.     

Output stabilization for a class of uncertain systems

The system

$$\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u,{\kern 1pt} \frac{{dy}}{{dt}} = A\left( \cdot \right)y + B\left( \cdot \right)u + D\left( {C*y - v} \right)$$

where v = C*x is an output, u = S*y is a control, A(·) ∈ R ^{n × n} , B(·) ∈ R ^{n × (n–p)}, C ∈ R ^{n × (n–p)}, and D ∈ R ^{n × (n–p)}, is considered. The elements α_ij(·) and β_ij(·) of the matrices A(·) and B(·) are arbitrary functionals satisfying the conditions

$$\mathop {\sup }\limits_{\left( \cdot \right)} |{\alpha _{ij}}\left( \cdot \right)| < \infty \left( {i,j \in 1,n} \right),\mathop {\sup }\limits_{\left( \cdot \right)} |{\beta _{ij}}\left( \cdot \right)| < \infty \left( {i \in 1,n,j \in 1,n - p} \right).$$

It is assumed that A(·) ∈ Z ₁ ∪ Z ₃ and A*(·) ∈ Z ₁ ∪ Z ₃, where Z ₁ is the class of matrices in which the first p elements of the kth superdiagonal are sign-definite and the elements above them are sufficiently small. The class Z ₃ differs from Z _t1 in that the elements between this superdiagonal and the (k + 1)th row are sufficiently small. If k > p, then the elements of the p × p square in the upper left corner of the matrix are sufficiently small as well. By using special quadratic Lyapunov functions, a matrix D for which y(t)–x(t) → 0 exponentially as t → ∞ is first found, and then a matrix S for which the vectors x(t) and y(t) have the same property is constructed.

     

An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).     

Global existence of real roots and random Newton flow algorithm for nonlinear system of equations

To solve nonlinear system of equation, F(x) = 0, a continuous Newton flow x _t (t) = V (x) = ?(DF(x))^?1 F(x), x(0) = x ⁰ and its mathematical properties, such as the central field, global existence and uniqueness of real roots and the structure of the singular surface, are studied. We concisely introduce random Newton flow algorithm (NFA) for finding all roots, based on discrete Newton flow x ^j+1 = x ^j + hV (x ^j ) with random initial value x ⁰ and h ∈ (0, 1], and three computable quantities, g _j , d _j and K _j . The numerical experiments with dimension n = 300 are provided.     

Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Let {Q _n ^(α,β) (x)} _n=0 ^∞ denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product

$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$

where λ>0 and d μ _α,β(x)=(x?a)(1?x)^α?1(1+x)^β?1 dx, d ν _α,β(x)=(1?x) ^α (1+x) ^β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q _n ^(α,β) are obtained.

     

Hyperquasipolynomials for the Theta-Function

Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ? → ? satisfying f(x+y)g(x?y) = α₁(x)β₁(y)+· · ·+α_r(x)β_r(y) for some r ∈ ? and α_j, β_j: ? → ? are described.     

Invariance of the order and type of a sequence of operators

In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖A_n(x)‖_p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖A_n(x)‖_p even if an explicit form of seminorms is unknown.     

Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

It is well-known that Morgan-Voyce polynomials B _n(x) and b _n(x) satisfy both a Sturm-Liouville equation of second order and a three-term recurrence equation ([SWAMY, M.: Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968), 167–175]). We study Diophantine equations involving these polynomials as well as other modified classical orthogonal polynomials with this property. Let A, B, C ∈ ? and {pk(x)} be a sequence of polynomials defined by

$\begin{gathered} p_0 (x) = 1 \hfill \\ p_1 (x) = x - c_0 \hfill \\ p_{n + 1} (x) = (x - c_n )p_n (x) - d_n p_{n - 1} (x), n = 1,2,..., \hfill \\ \end{gathered} $

with

$(c_0 ,c_n ,d_n ) \in \{ (A,A,B),(A + B,A,B^2 ),(A,Bn + A,\tfrac{1}{4}B^2 n^2 + Cn)\} $

with A ≠ 0, B > 0 in the first, B ≠ 0 in the second and C > ?¼B ² in the third case. We show that the Diophantine equation

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with m > n ≥ 4,

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≠ 0 has at most finitely many solutions in rational integers x, y.

     

New algorithm for computing the Hermite interpolation polynomial

Let x ₀, x ₁,? , x _n , be a set of n + 1 distinct real numbers (i.e., x _i ≠ x _j , for i ≠ j) and y _{i, k} , for i = 0,1,? , n, and k = 0 ,1 ,? , n _i , with n _i ≥ 1, be given of real numbers, we know that there exists a unique polynomial p _{N ? 1}(x) of degree N ? 1 where \(N={\sum }_{i=0}^{n}(n_{i}+1)\), such that \(p_{N-1}^{(k)}(x_{i})=y_{i,k}\), for i = 0,1,? , n and k = 0,1,? , n _i . P _N?1(x) is the Hermite interpolation polynomial for the set {(x _i , y _{i, k} ), i = 0,1,? , n, k = 0,1,? , n _i }. The polynomial p _N?1(x) can be computed by using the Lagrange polynomials. This paper presents a new method for computing Hermite interpolation polynomials, for a particular case n _i = 1. We will reformulate the Hermite interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA). Some properties of this algorithm will be studied and some examples will also be given.     

Remarks on D-integral complete multipartite graphs

A graph is called distance integral (or D-integral) if all eigenvalues of its distance matrix are integers. In their study of D-integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on D-integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs \({K_{{p_1},{p_2},{p_3}}}\) with p₁ < p₂ < p₃, and \({K_{{p_1},{p_2},{p_3},{p_4}}}\) with p₁ < p₂ < p₃ < p₄, as well as the infinite classes of distance integral complete multipartite graphs \({K_{{a_1}{p_1},{a_2}{p_2},...,{a_s}{p_s}}}\) with s = 5, 6.