Some Existence Theorems on All Fractional (<Emphasis Type="Italic">g</Emphasis>, <Emphasis Type="Italic">f</Emphasis>)-factors with Prescribed Properties

Let G be a graph, and g, f: V (G) → Z⁺ with g(x) ≤ f(x) for each x ∈ V (G). We say that G admits all fractional (g, f)-factors if G contains an fractional r-factor for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for any x ∈ V (G). Let H be a subgraph of G. We say that G has all fractional (g, f)-factors excluding H if for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for all x ∈ V (G), G has a fractional r-factor F _h such that E(H) ∩ E(F _h ) = θ, where h: E(G) → [0, 1] is a function. In this paper, we show a characterization for the existence of all fractional (g, f)-factors excluding H and obtain two sufficient conditions for a graph to have all fractional (g, f)-factors excluding H.     

Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.     

A New Class of Generalized Polynomials Associated with Hermite and Euler Polynomials

In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson’s polynomials \({\Phi_{n}^{(\alpha)}(x,\nu)}\) of degree n and order α introduced by Dere and Simsek. The concepts of Euler numbers E _n , Euler polynomials E _n (x), generalized Euler numbers E _n (a, b), generalized Euler polynomials E _n (x; a, b, c) of Luo et al., Hermite–Bernoulli polynomials \({{_HE}_n(x,y)}\) of Dattoli et al. and \({{_HE}_n^{(\alpha)} (x,y)}\) of Pathan are generalized to the one \({ {_HE}_n^{(\alpha)}(x,y,a,b,c)}\) which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between E _n , E _n (x), E _n (a, b), E _n (x; a, b, c) and \({{}_HE_n^{(\alpha)}(x,y;a,b,c)}\) are established. Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions.     

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

Connecting orbits near the adiabatic limit of Lagrangian systems with turning points

We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to action of a time-periodic force field with potential U(q, t, ε) = f(εt)V(q) depending slowly on time. It is assumed that the factor f(τ) is periodic and vanishes at least at one point on the period. Let X _c denote a set of isolated critical points of V(x) at which V(x) distinguishes its maximum or minimum. In the adiabatic limit ε → 0 we prove the existence of a set E _h such that the system possesses a rich class of doubly asymptotic trajectories connecting points of X _c for ε ∈ E _h .     

On the positive definiteness of some functions related to the Schoenberg problem

For a broad class of functions f: [0,+∞) → ?, we prove that the function f(ρ ^λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ? 0 and α(E, ρ) is the Schoenberg constant.     

Group analysis of the one-dimensional boltzmann equation: II. Equivalence groups and symmetry groups in the special case

We obtain relations that define the equivalence algebra of the family of one-dimensional Boltzmann equations f _t + cf _x + F(t, x, c)f _c = 0 and show that all equations of that form are locally equivalent. We carry out the group classification of the equation with respect to the function F in the special case where the function F and the transformations of the variables t and x are assumed to be independent of c. We show that, under such constraints for the transformation and the family of equations, the maximum possible symmetry algebra is eight-dimensional, which corresponds to an equation with a linear function F.     

On criteria of estimation of the errors of cubature formulas

The paper considers cubature formulas for calculating integrals of functions f(X), X = (x ₁, …, x _n ) which are defined on the n-dimensional unit hypercube K ⁿ = [0, 1] ⁿ and have integrable mixed derivatives of the kind \(\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)\), 0 ≤ α _j ≤ 2. We estimate the errors R[f] = \(\smallint _{K^n } \) f(X)dX ? Σ _{k = 1} ^N c _k f(X(k)) of cubature formulas (c _k > 0) as functions of the weights c _k of nodes X(k) and properties of integrable functions. The error is estimated in terms of the integrals of the derivatives of f over r-dimensional faces (r≤n) of the hypercube K ⁿ : |R(f)| ≤ \(\sum _{\alpha _j } \) G(α _j )\(\int_{K^r } {\left| {\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)} \right|} \) dX _r , where coefficients G(α _j ) are criteria which depend only on parameters c _k and X(k). We present an algorithm to calculate these criteria in the two- and n-dimensional cases. Examples are given. A particular case of the criteria is the discrepancy, and the algorithm proposed is a generalization of those used to compute the discrepancy. The results obtained can be used for optimization of cubature formulas as functions of c _k and X(k).     

Planar Graded Lattices and the <Emphasis Type="Bold">c</Emphasis><Subscript>1</Subscript>-Median Property

Let L be a lattice of finite length, ξ = (x ₁,…, x _k )∈L ^k , and y ∈ L. The remoteness r(y, ξ) of y from ξ is d(y, x ₁)+?+d(y, x _k ), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x ₁∨?∨x _k . In other words, L satisfies the so-called c ₁ -median property.     

On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power

A linear differential operator P(x, D) = P(x₁,... x _n , D₁,..., D _n ) = ∑_αγ_α(x)D^α with coefficients γ_α(x) defined in E ⁿ is called formally almost hypoelliptic in E ⁿ if all the derivatives D^ν_ξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in Eⁿ. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.     

Expectiles,Omega Ratios and Stochastic Ordering

In this paper we introduce the expectile order, defined by X ≤ _e Y if e _α (X) ≤e _α (Y) for each α ∈ (0, 1), where e _α denotes the α-expectile. We show that the expectile order is equivalent to the pointwise ordering of the Omega ratios, and we derive several necessary and sufficient conditions. In the case of equal means, the expectile order can be easily characterized by means of the stop-loss transform; in the more general case of different means we provide some sufficient conditions. In contrast with the more common stochastic orders such as ≤ _{s t} and ≤ _{c x} , the expectile order is not generated by a class of utility functions and is not closed with respect to convolutions. As an illustration, we compare the ≤ _{s t} , ≤ _{i c x} and ≤ _e orders in the family of Lomax distributions and compare Lomax distributions fitted to real world data of natural disasters in the U.S. caused by different sources of weather risk like storms or floods.     

Group-Theoretical Analysis of Variable Coefficient Nonlinear Telegraph Equations

Given a class \(\mathcal{F(\theta)}\) of differential equations with arbitrary element θ, the problems of symmetry group, nonclassical symmetry and conservation law classifications are to determine for each member \(f\in\mathcal{F(\theta)}\) the structure of its Lie symmetry group G _f , conditional symmetry Q _f and conservation law \(\mathop {\rm CL}\nolimits _{f}\) under some proper equivalence transformations groups.In this paper, an extensive investigation of these three aspects is carried out for the class of variable coefficient (1+1)-dimensional nonlinear telegraph equations with coefficients depending on the space variable f(x)u _tt =(g(x)H(u)u _x ) _x +h(x)K(u)u _x . The usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements are first constructed. Then using the technique of variable gauges of arbitrary elements under equivalence transformations, we restrict ourselves to the symmetry group classifications for the equations with two different gauges g=1 and g=h. In order to get the ultimate classification, the method of furcate split is also used and consequently a number of new interesting nonlinear invariant models which have non-trivial invariance algebra are obtained. As an application, exact solutions for some equations which are singled out from the classification results are constructed by the classical method of Lie reduction.The classification of nonclassical symmetries for the classes of differential equations with gauge g=1 is discussed within the framework of singular reduction operator. This enabled to obtain some exact solutions of the nonlinear telegraph equation which are invariant under certain conditional symmetries.Using the direct method, we also carry out two classifications of local conservation laws up to equivalence relations generated by both usual and extended equivalence groups. Equivalence with respect to these groups and correct choice of gauge coefficients of equations play the major role for simple and clear formulation of the final results.     

On some mean values for the divisor function and the Riemann zeta-function

Let Δ(x) and E(x) denote respectively the error terms in the summatory formula for the divisor function and in the mean square formula for ζ(s) on the critical line. We consider some general mean values for Δ(x) and E(x) and discover interesting differences between these two functions. In particular, this yields evidence that E(x) is more negative than Δ(x).     

Irreducibility criteria of Schur-type and Pólya-type

Let \({f(x)=(x-a_1)\cdots (x-a_m)}\), where a ₁, . . . , a _m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether \({(f(x))^{2^k}+1}\) is irreducible for every k ≥ 1. In 1919 Pólya proved that if \({P(x)\in\mathbb{Z}[x]}\) is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2^?N N! where \({N=\lceil m/2\rceil}\), then P(x) is irreducible. A great number of authors have published results of Schur-type or Pólya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Pólya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.     

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.     

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.     

Fujita type conditions to heat equation with variable source

This paper studies heat equation with variable exponent u _t = Δu + u^p(x) + u ^q in ? ^N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p_? = inf p(x) ≤ p(x) ≤ sup p(x) = p₊. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p₊, q} ≤ 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q ≤ 1 with p₊ > 1, or 1 < q < 1 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p_? ≤ p₊ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p_? > 1 + \(\frac{2}{N}\); and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p_? < 1 + \(\frac{2}{N}\) < p₊.     

Almost Empty Monochromatic Quadrilaterals in Planar Point Sets

For positive integers c, s ≥ 1, r ≥ 3, let W _r (c, s) be the least integer such that if a set of at least W _r (c, s) points in the plane, no three of which are collinear, is colored with c colors, then this set contains a monochromatic r-gon with at most s interior points. As is known, if r = 3, then W _r (c, s)=W _r (c, s). In this paper, we extend these results to the case r = 4. We prove that W4(2, 1) = 11, W4(3, 2) ≤ 120, and the least integer μ4(c) such that W4(c, μ4(c)) < ∞ is bounded by \(\left\lfloor {\frac{{c - 1}}{2}} \right\rfloor \cdot 2 \leqslant \mu 4\left( c \right) \leqslant 2c - 3\),where c ≥ 2.     

On the analytic and Cauchy capacities

We give new sufficient conditions for a compact set E ? C to satisfy γ(E) = γ_c(E), where γ is the analytic capacity and γ_c is the Cauchy capacity. As a consequence, we provide examples of compact plane sets such that the above equality holds but the Ahlfors function is not the Cauchy transform of any complex Borel measure supported on the set.     

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).