Numerous refinements of Pólya and Heinz operator inequalities

In this article, we present an infinite number of refinements of the Heinz inequality for real numbers and operators. Making use of them, infinitely many refinements of the classical Pólya inequality and their operator versions are deduced.     

Spectral structure of the Neumann–Poincaré operator on tori

We address the question whether there is a three-dimensional bounded domain such that the Neumann–Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann–Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values.     

一类椭圆型半变分不等式解的存在性(英文)

本文研究了一类Dirichlet边界的椭圆型半变分不等式问题.利用非光滑形式的环绕定理和非光滑形式的对称山路定理,得到了在相应假设条件下此不等式问题至少有一个非平凡解和无穷多解.本文中非光滑势能在原点处关于算子+V(x)的第一正特征值λ是不完全共振的.     

Jackson inequality in L2( \mathbbTN \mathbb{T}^N ) with generalized modulus of continuity

The sharp Jackson inequality is proved for the space of periodic functions of many variables with a mean-square norm for an arbitrary modulus of continuity generated by a difference operator with constant coefficients.     

Jackson inequality in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>) with generalized modulus of continuity

The sharp Jackson inequality is proved for the space of entire functions of many variables with mean-square norm for an arbitrary modulus of continuity generated by a difference operator with constant coefficients.     

Jackson inequality in L2($$ \mathbb{T}^N $$) with generalized modulus of continuity

The sharp Jackson inequality is proved for the space of periodic functions of many variables with a mean-square norm for an arbitrary modulus of continuity generated by a difference operator with constant coefficients.     

Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach

We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the essential spectrum of the discrete Schrödinger operator. For periodic Jacobi operators we relax the rational dependence conditions on the values of the quasi-momenta from this previous work. We then explore conditions that permit not just the existence of infinitely many subordinate solutions to the formal spectral equation but also the embedding of infinitely many eigenvalues.     

The Ornstein-Uhlenbeck semigroup in an infinite dimensional L2 setting

We study the number operator, N, of quantum field theory as a partial differential operator in infinitely many variables. Informally Nu(x) = ?Δu(x) + x · grad u(x). A large core for N is constructed which is invariant under e^?tN and on which this informal expression may be given a precise and natural meaning.     

Operator part of infinite system of differential equations

Let T be a differential operator in a Hilbert space generated by a first-order infinite system of an ordinary linear differential expression, which is subject to infinitely many boundary conditions. We solve completely for y from Ty = g. The main tools involved are operator parts of closed linear manifolds, which are closely related to generalized inverses.     

Coupled maps and analytic function spaces

We consider ergodic properties of weakly coupled analytic and expanding circle maps. For weak enough coupling a natural ergodic measure exists and exhibits exponent decay of time correlations. The marginal densities of the natural measure are analytic. A spatial decay of correlations (e.g. polynomial) in these densities may arise from a similar spatial decay of the couplings. The space of couplings and observables is a Banach algebra of analytic functions of infinitely many variables. This algebra acts upon a Banach module of complex measures with analytic marginal densities. A Perron-Frobenius type operator acts on the Banach module and we apply a re-summation technique to derive uniform bounds for this operator. Explicit bounds are calculated for some examples.     

On a boundary value problem of N. I. Ionkin type

We study the spectral properties of a second-order differential operator with regular but not strongly regular boundary conditions. We show that the system of root functions of this operator contains infinitely many associated functions. We prove that a specially chosen system of root functions of this operator forms a basis in the space L _p (0, 1), 1 < p < ∞, which is unconditional for p = 2.     

A hybrid approximation method for equilibrium and fixed point problems for a family of infinitely nonexpansive mappings and a monotone mapping

In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mapping in the framework of a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. Additionally, we utilize our results to study the optimization problem and find a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. Our results improve and extend the results announced by many others.     

On an operator inequality

Jensen's operator inequality characterizes operator convex functions of two variables (F. Hansen, Proc. Amer. Math. Soc. 125 (1997) 2093–2102). We give a simplified proof of this theorem formulated for matrices.     

Recursion Operator of the Narita–Itoh–Bogoyavlensky Lattice

We construct a recursion operator for the family of Narita–Itoh–Bogoyavlensky infinite lattice equations using its Lax presentation and present their mastersymmetries and bi‐Hamiltonian structures. We show that this highly nonlocal recursion operator generates infinitely many local symmetries.     

On the central limit theorem in Banach spaces

It is shown, with the use of a concentration inequality of LeCam, that associated with an infinitely divisible random variable with values in a separable Banach space there is a Lévy-Khintchine formula. A partial converse of this fact is also proved. Relations between the continuity of the compound Poisson and the Gaussian variables associated with a Lévy measure are studied. A central limit theorem is obtained and examples are given.     

On the spectral instability of the Sturm-Liouville operator with a complex potential

Using the Sturm-Liouville operator with a complex potential as an example, we analyze the spectral instability effect for operators that are far from being self-adjoint. We show that the addition of an arbitrarily small compactly supported function with an arbitrarily small support to the potential can substantially change the asymptotics of the spectrum. This fact justifies, in a sense, the necessity of well-known sufficient conditions for the potential under which the spectrum of the operator is localized around some ray. For an operator with a logarithmic growth, we construct a perturbation that preserves the asymptotics of the spectrum but has infinitely many poles inside the main sector.     

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

We consider a general Euler-Korteweg-Poisson system in R ³, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density - the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum. In this paper we show that, even in presence of a dispersive tensor, we have the same phenomena found by De Lellis and Székelyhidi.     

Local reduction based SQP-type method for semi-infinite programs with an infinite number of second-order cone constraints

The second-order cone program (SOCP) is an optimization problem with second-order cone (SOC) constraints and has achieved notable developments in the last decade. The classical semi-infinite program (SIP) is represented with infinitely many inequality constraints, and has been studied extensively so far. In this paper, we consider the SIP with infinitely many SOC constraints, called the SISOCP for short. Compared with the standard SIP and SOCP, the studies on the SISOCP are scarce, even though it has important applications such as Chebychev approximation for vector-valued functions. For solving the SISOCP, we develop an algorithm that combines a local reduction method with an SQP-type method. In this method, we reduce the SISOCP to an SOCP with finitely many SOC constraints by means of implicit functions and apply an SQP-type method to the latter problem. We study the global and local convergence properties of the proposed algorithm. Finally, we observe the effectiveness of the algorithm through some numerical experiments.     

Estimates for eigenvalues of quasilinear elliptic systems

In this paper we introduce the generalized eigenvalues of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many continuous eigencurves, which are obtained by variational methods. For the one-dimensional problem, we obtain an hyperbolic type function defining a region which contains all the generalized eigenvalues (variational or not), and the proof is based on a suitable generalization of Lyapunov's inequality for systems of ordinary differential equations. We also obtain a family of curves bounding by above the kth variational eigencurve.     

A remark on the space of metrics having nontrivial harmonic spinors

Let M be a closed spin manifold of dimension n ≡ 3 mod 4. We give a simple proof of the fact that the space of metrics on M with invertible Dirac operator is either empty or it has infinitely many path components.