Solution of the Ulam Stability Problem for an Euler Type Quadratic Functional Equation

In 1968 S.M. Ulam proposed the problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?’’. In 1978 according to P.M. Gruber this kind of problems is of particular interest in probability theory and in the case of functional equations of different types. In 1997 W. Schuster established a new vector quadratic identity on the basis of the well-known Euler type theorem on quadrilaterals: If ABCD is a quadrilateral and M, N are the mid-points of the diagonals AC, BD as well as A′, B′, C′, D′ are the mid-points of the sides AB, BC, CD, DA, then |AB|² + |BC|² + |CD|² + |DA|² = 2|A′C′|² + 2|B′D′|² + 4|MN|². Employing in this paper the above geometric identity we introduce the new Euler type quadratic functional equation

$\begin{array}{l}2{[}Q(x_{0} - x_{1}+Q(x_{1}-x_{2})+Q(x_{2}- x_{3})+Q(x_{3}-x_{0}){]}\\\qquad = Q(x_{0}-x_{1}-x_{2}+x_{3})+Q(x_{0} + x_{1}-x_{2}-x_{3})+2Q(x_{0}-x_{1}+ x_{2}-x_{3})\end{array}$

for all vectors (x₀, x₁, x₂, x₃) X⁴, with X and Y linear spaces. For every x ∈ R set Q(x) = x². Then the mapping Q : X → Y is quadratic. Note also that if Q : R → R is quadratic, then we have Q(x) = Q(1)x². Besides note that the geometric interpretation of the special example

$\begin{array}{l}2{[}(x_{0} - x_{1})^{2}+ (x_{1}-x_{2})^{2}+ (x_{2}-x_{3})^{2}+(x_{3}-x_{0})^{2}{]}\\\qquad = (x_{0}-x_{1}-x_{2} + x_{3})^{2}+(x_{0} + x_{1}-x_{2}-x_{3})^{2} + 2(x_{0}-x_{1}+ x_{2}-x_{3})^{2}\end{array}$

leads to the above-mentioned Euler type theorem on quadrilaterals ABCD with position vectors x₀, x₁, x₂, x₃ of vertices A, B, C, D, respectively. Then we solve the Ulam stability problem for the afore-mentioned equation, with non-linear Euler type quadratic mappings Q : X → Y.

     

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.     

On the deficiency index of the vector-valued Sturm–Liouville operator

Let R₊:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R₊ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R₊ and the elements of the matrix functions P^?1, Q, and R are measurable on R₊ and summable on each of its closed finite subintervals. We study the operators generated in the space L_n²(R₊) by formal expressions of the form l[f] = ?(P(f' ? Rf))' ? R*P(f' ? Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = ?(P₀f')' + i((Q₀f)' + Q₀f') + P'₁f, where everywhere the derivatives are understood in the sense of distributions and P₀, Q₀, and P₁ are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P₀^?1 exists and ∥P0∥, ∥P₀^?1∥, ∥P₀^?1∥∥P₁∥², ∥P₀^?1∥∥Q₀∥² ∈ L_loc¹(R₊). Themain goal in this paper is to study of the deficiency index of the minimal operator L₀ generated by expression l[f] in L_n²(R₊) in terms of the matrix functions P, Q, and R (P₀, Q₀, and P₁). The obtained results are applied to differential operators generated by expressions of the form \(l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f\), where x_k, k = 1, 2,..., is an increasing sequence of positive numbers, with lim_k→+∞x_k = +∞, H_k is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.     

Boolean Lattices: Ramsey Properties and Embeddings

A subposet Q ^′ of a poset Q is a copy of a poset P if there is a bijection f between elements of P and Q ^′ such that x ≤ y in P iff f(x) ≤ f(y) in Q. For posets P, P ^′, let the poset Ramsey number R(P, P ^′) be the smallest N such that no matter how the elements of the Boolean lattice Q _N are colored red and blue, there is a copy of P with all red elements or a copy of P ^′ with all blue elements. We provide some general bounds on R(P, P ^′) and focus on the situation when P and P ^′ are both Boolean lattices. In addition, we give asymptotically tight bounds for the number of copies of Q _n in Q _N and for a multicolor version of a poset Ramsey number.     

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.

     

On a generalization of the Hadwiger–Nelson problem

For a field F and a quadratic form Q defined on an n-dimensional vector space V over F, let QG _Q , called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u,w ∈ V form an edge if and only if Q(v ? w) = 1. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger–Nelson problem. In the present paper, we will prove that for a local field F of characteristic zero, the Borel chromatic number of QG _Q is infinite if and only if Q represents zero non-trivially over F. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009 [6].     

Invariance of Picard Dimensions Under Basic Perturbations

The Picard dimension \(\dim \mu\) of a signed Radon measure μ on the punctured closed unit ball 0?x|?≦?1 in the d-dimensional euclidean space with d?≧?2 is the cardinal number of the set of extremal rays of the cone of positive continuous distributional solutions u of the Schrödinger equation (???Δ?+?μ)u?=?0 on the punctured open unit ball 0?x|?x|?=?1. If the Green function of the above equation on 0?x|?Δ?+?μ)u?=?δ _y , the Dirac measure supported by the point y, exists for every y in 0?x|?μ is referred to as being hyperbolic on 0?x|?γ is a radial Radon measure which is both positive and absolutely continuous with respect to the d-dimensional Lebesgue measure dx whose Radon–Nikodym density dγ(x)/dx is bounded by a positive constant multiple of |x|^???2. The purpose of this paper is to show that the Picard dimensions of hyperbolic radial Radon measures μ are invariant under basic perturbations \(\gamma: \dim(\mu+\gamma)=\dim\mu\). Three applications of this invariance are also given.     

On the tree structure of the power digraphs modulo n

For any two positive integers n and k ? 2, let G(n, k) be a digraph whose set of vertices is {0, 1, …, n ? 1} and such that there is a directed edge from a vertex a to a vertex b if a ^k ≡ b (mod n). Let \(n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} \) be the prime factorization of n. Let P be the set of all primes dividing n and let P ₁, P ₂ ? P be such that P ₁ ∪ P ₂ = P and P ₁ ∩ P ₂ = ?. A fundamental constituent of G(n, k), denoted by \(G_{{P_2}}^*(n,k)\), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \(\prod\nolimits_{{p_i} \in {P_2}} {{p_i}} \) and are relatively prime to all primes q ∈ P ₁. L. Somer and M. K?i?ek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic.     

Stochastic Hamiltonian flows with singular coefficients

In this paper, we study the following stochastic Hamiltonian system in ?^2d (a second order stochastic differential equation):

$$d{\dot X_t} = b({X_t},{\dot X_t})dt + \sigma ({X_t},{\dot X_t})d{W_t},({X_0},{\dot X_0}) = (x,v) \in \mathbb{R}^{2d},$$

where b(x; v) : ?^2d → ?^d and σ(x; v): ?^2d → ?^d ? ?^d are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and b ∈ H _p ^2/3,0 and ?σ ∈ L^p for some p > 2(2d+1), where H _p ^{α, β} is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that (x, v) ? Z_t(x, v) := (X_t, ?_t)(x, v) forms a stochastic homeomorphism flow, and (x, v) ? Z_t(x, v) is weakly differentiable with ess.sup_{x, v} E(sup_{t∈[0, T]} |?Z_t(x, v)|^q) < ∞ for all q ? 1 and T ? 0. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli (2008) and Trevisan (2016).

     

Existence and concentration behavior of sign-changing solutions for quasilinear Schrödinger equations

We consider the quasilinear Schrödinger equations of the form ?ε²Δu + V(x)u ? ε²Δ(u²)u = g(u), x∈ R^N, where ε > 0 is a small parameter, the nonlinearity g(u) ∈ C¹(R) is an odd function with subcritical growth and V(x) is a positive Hölder continuous function which is bounded from below, away from zero, and inf_ΛV(x) < inf?_ΛV(x) for some open bounded subset Λ of R^N. We prove that there is an ε₀ > 0 such that for all ε ∈ (0, ε₀], the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε → 0⁺.     

Generic hypersurface singularities

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x₁,...,x_r be closed points in general position in projective spacePn, then the linear subspaceV ofH⁰ (?ⁿ,O(d)) (the space of homogeneous polynomials of degreed on ?ⁿ) formed by those polynomials which are singular at eachx_i, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x₁,...,x_r. As such, the “expected” value for the dimension ofV is max(0,h⁰(O(d))?r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.     

Ramsey numbers of cubes versus cliques

The cube graph Q _n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2 ⁿ vertices. The Ramsey number r(Q _n ;K _s ) is the minimum N such that every graph of order N contains the cube graph Q _n or an independent set of order s. In 1983, Burr and Erd?s asked whether the simple lower bound r(Q _n ;K _s )≥(s?1)(2 ⁿ ?1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.     

Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

Let O ? R ^d be a bounded domain of class C ^1,1. Let 0 < ε - 1. In L ₂(O;C ⁿ ) we consider a positive definite strongly elliptic second-order operator B _D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B _D,ε ? ζQ ₀(·/ε))^?1 as ε → 0. Here the matrix-valued function Q ₀ is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L ₂(O;C ⁿ )-operator norm and in the norm of operators acting from L ₂(O;C ⁿ ) to the Sobolev space H ¹(O;C ⁿ ) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q ₀(x/ε)? _t v _ε (x, t) = ?(B _D,ε v _ε )(x, t).     

Dirac Operator with a Potential of Special Form and with the Periodic Boundary Conditions

We consider the Dirac operator on the interval [0, 1] with the periodic boundary conditions and with a continuous potential Q(x) whose diagonal is zero and which satisfies the condition Q(x) = Q^T(1?x), x ∈ [0, 1]. We establish a relationship between the spectrum of this operator and the spectra of related functional-differential operators with involution. We prove that the system of eigenfunctions of this Dirac operator has the Riesz basis property in the space L ₂ ² [0, 1].     

A partition theorem for a large dense linear order

Let κ be a cardinal which is measurable after generically adding ?_κ+ω many Cohen subsets to κ, and let ?_κ = (Q, ≤ _Q ) be the strongly κ-dense linear order of size κ. We prove, for 2 ≤ m < ω, that there is a finite value t _m ⁺ such that the set [Q] ^m of m-tuples from Q can be partitioned into classes 〈C _i : i < t _m ⁺ }〉 such that for any coloring a class C _i in fewer than κ colors, there is a copy ?* of ?_κ such that [?*] ^m ? C _i is monochromatic. It follows that \(\mathbb{Q}_\kappa \to (\mathbb{Q}_\kappa )_{ < \kappa /t_m^ + }^m \), that is, for any coloring of [?_κ] ^m with fewer than κ colors there is a copy Q′ ? Q of ?_κ such that [Q′] ^m has at most t _m ⁺ colors. On the other hand, we show that there are colorings of ?_κ such that if Q′ ? Q is any copy of ?_κ then C _i ? [Q′] ≠ ø; for all i < t _m ⁺ , and hence \(\mathbb{Q}_\kappa \nrightarrow [\mathbb{Q}_\kappa ]_{t_m^ + }^m \).We characterize t _m ⁺ as the cardinality of a certain finite set of ordered trees and obtain an upper and a lower bound on its value. In particular, t ₂ ⁺ = 2 and for m > 2 we have t _m ⁺ > t _m , the m-th tangent number.The stated condition on κ is the hypothesis for a result of Shelah on which our work relies. A model in which this condition holds simultaneously for all m can be obtained by forcing from a model with a κ^+ω-strong cardinal, but it follows from earlier results of Hajnal and Komjáth that our result, and hence Shelah’s theorem, is not directly implied by any large cardinal assumption.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

On idempotent <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated to a von Neumann algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let L_p(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P ? Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ L_p(M, τ), then A² ∈ L_p(M, τ). Let n be a positive integer, n > 2, and A = Aⁿ ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μ_t(A) belong to the set {0} ∪ [‖A^n?2‖^?1, ‖A‖] for all t > 0; 2) either μ_t(A) ≥ 1 for all t > 0 or there is a t₀ > 0 such that μ_t(A) = 0 for all t > t₀. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z² = 0, ZP = 0, and PZ = Z. Here, if Q ∈ L_p(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L₁(M, τ) and if A = A³ and A ? A² ∈ M, then τ(A) ∈ R.     

Generalized competition index of primitive digraphs

For any positive integers k and m, the k-step m-competition graph C _m ^k (D) of a digraph D has the same set of vertices as D and there is an edge between vertices x and y if and only if there are distinct m vertices v₁, v₂, · · ·, v _m in D such that there are directed walks of length k from x to v _i and from y to v _i for all 1 ≤ i ≤ m. The m-competition index of a primitive digraph D is the smallest positive integer k such that C _m ^k (D) is a complete graph. In this paper, we obtained some sharp upper bounds for the m-competition indices of various classes of primitive digraphs.     

On the Ramsey number of the triangle and the cube

The Ramsey number r(K ₃,Q _n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K _N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erd?s conjectured that r(K ₃,Q _n )=2 ⁿ⁺¹?1 for every n∈?, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K ₃,Q _n )?7000·2 ⁿ . Here we show that r(K ₃,Q _n )=(1+o(1))2 ⁿ⁺¹ as n→∞.