Differential operators for Siegel-Jacobi forms

For any positive integers n and m, H_(n,m):= H_n× C~(m,n) is called the Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are defined on this space. We compute the Chern connection of the Siegel-Jacobi space and use it to obtain derivations of Jacobi forms. Using these results, we construct a series of invariant differential operators for Siegel-Jacobi forms. Also two kinds of Maass-Shimura type differential operators for H_(n,m) are obtained.     

Estimation Problems for Periodically Correlated Isotropic Random Fields

Spectral theory of isotropic random fields in Euclidean space developed by M. I. Yadrenko is exploited to find a solution to the problem of optimal linear estimation of the functional

$$ A\zeta ={\sum\limits_{t=0}^{\infty}}\,\,\,{\int_{S_n}} \,\,a(t,x)\zeta (t,x)\,m_n(dx) $$

which depends on unknown values of a periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere S _n in Euclidean space E ⁿ random field ζ(t, x), t?∈?Z, x?∈?S _n . Estimates are based on observations of the field ζ(t, x)?+?θ(t, x) at points (t, x), t?=???1,???2, ..., x?∈?S _n , where θ(t, x) is an uncorrelated with ζ(t, x) periodically correlated with respect to time isotropic on the sphere S _n random field. Formulas for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional Aζ are obtained. The least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functional Aζ are determined for some special classes of spectral densities.

     

Spectral (isotropic) manifolds and their dimension

A set of n × n symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in ?ⁿ is called spectral or isotropic. In this paper, we establish that every locally symmetric C^k submanifoldMof ?ⁿ gives rise to a C^k spectral manifold for k ∈ {2, 3, …,∞,ω}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived. This work builds upon the results of Sylvester and ?ilhavý and uses characteristic properties of locally symmetric submanifolds established in recent works by the authors.     

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].     

Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions

In this paper we prove that for an arbitrary pair {T ₁, T ₀} of contractions on Hilbert space with trace class difference, there exists a function ξ in L ¹(T) (called a spectral shift function for the pair {T ₁, T ₀}) such that the trace formula trace(f(T ₁) ? f(T ₀)) = ∫_T f′(ζ)ξ(ζ)dζ holds for an arbitrary operator Lipschitz function f analytic in the unit disk.     

The tail of a quantum spin network

The tail of a sequence \(\{P_n(q)\}_{n \in \mathbb {N}}\) of formal power series in \(\mathbb {Z}[[q]]\) is the formal power series whose first n coefficients agree up to a common sign with the first n coefficients of \(P_n\). This paper studies the tail of a sequence of admissible trivalent graphs with edges colored n or 2n. We use local skein relations to understand and compute the tail of these graphs. We also give product formulas for the tail of such trivalent graphs. Furthermore, we show that our skein theoretic techniques naturally lead to a proof for the Andrews–Gordon identities for the two-variable Ramanujan theta function as well to corresponding identities for the false theta function.     

A note on the perturbations of compact quantum metric spaces

In this short note,we consider the perturbation of compact quantum metric spaces.We first show that for two compact quantum metric spaces(A,P) and(B,Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively,the quantum Gromov–Hausdorff distance between(A,P) and(B,Q) is small under certain conditions.Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.     

Kreĭn space representation and Lorentz groups of analytic Hilbert modules

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H²(D²). A closed subspace M in H²(D²) is called a submodule if z _i M ? M (i = 1, 2). An associated integral operator (defect operator) C _M captures much information about M. Using a Kre?n space indefinite metric on the range of C _M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.     

Generalizing the Krein–Rutman theorem,measures of noncompactness and the fixed point index

If L : Y → Y is a bounded linear map on a Banach space Y, the “radius of the essential spectrum” or “essential spectral radius” ρ(L) of L is well-defined and there are well-known formulas for ρ(L) in terms of measures of noncompactness. Now let \({C \subset D}\) be complete cones in a normed linear space (X, || · ||) and f : C → C a continuous map which is homogeneous of degree one and preserves the partial ordering induced by D. We prove (see Section 2) that various obvious analogs of the formulas for the essential spectral radius for the case f : C → C have serious defects, even when f is linear on C. We propose (see (3.5)) a definition for ρ _C (f), the “cone essential spectral radius of f,” which avoids these difficulties. If \({{\tilde r}_{C}(f)}\) denotes the (Bonsall) cone spectral radius of f, we conjecture (see Conjecture 4.1) that if \({\rho_{C}(f) < {\tilde r}_{C}(f)}\), then there exists \({u \in C {\backslash} \, \{0\}}\) with f(u) = ru where r ? r _C (f). If f satisfies certain additional conditions (for example, if f is a compact perturbation of a map which is linear on C), we obtain the conclusion of the conjecture; but in general we observe (Remark 4.7) that the conjecture is intimately related to old and difficult conjectures in asymptotic fixed point theory. In Section 5 we briefly discuss extensions of generalized max-plus operators which were our original motivation and for which Conjecture 4.1 is already nontrivial.     

Continuity of the Spectrum of a Field of Self-Adjoint Operators

Given a family of self-adjoint operators \({(A_t)_{t \in T}}\) indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum \({\sigma(A_t)}\) to be Vietoris continuous with respect to t. Equivalently the boundaries and the gap edges are continuous in t. If (T, d) is a complete metric space with metric d, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.     

Unicyclic nonintegral sum graphs

A graph G = (V,E) is an integral sum graph if there exists a labeling S(G) ? Z such that V = S(G) and every two distinct vertices u, υ ∈ V are adjacent if and only if u + υ ∈ V. A connected graph G = (V,E) is called unicyclic if |V| = |E|. In this paper two infinite series are constructed of unicyclic graphs that are not integral sum graphs.     

Regular variation of a random length sequence of random variables and application to risk assessment

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence C(N) = (C ₁,…,C _N ) of random length N, where C(N) comes from the product of a matrix A(N) of random size N × N and a random sequence X(N) of random length N. Our aim is to build a regular variation framework for such random sequences of random length, to study their spectral properties and, subsequently, to develop risk measures. In several applications, many risk indicators can be expressed from the extremal behavior of ∥C(N)∥, for some norm ∥?∥. We propose a generalization of Breiman’s Lemma that gives way to a tail estimate of ∥C(N)∥ and provides risk indicators such as the ruin probability and the tail index for Shot Noise Processes on a finite-time horizon. Lastly, we apply our main result to a model used in dietary risk assessment and in non-life insurance mathematics to illustrate the applicability of our method.     

Positive-fraction intersection results and variations of weak epsilon-nets

A connected Finsler space (M, F) is said to be homogeneous if it admits a transitive connected Lie group G of isometries. A geodesic in a homogeneous Finsler space (G / H, F) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of G. In this paper, we study the problem of the existence of homogeneous geodesics on a homogeneous Finsler space, and prove that any homogeneous Finsler space of odd dimension admits at least one homogeneous geodesic through each point.     

Strict Embeddings of Rearrangement Invariant Spaces

A Banach space E of measurable functions on [0,1] is called rearrangement invariant if E is a Banach lattice and equimeasurable functions have identical norms. The canonical inclusion E ? F of two rearrangement invariant spaces is said to be strict if functions from the unit ball of E have absolutely equicontinuous norms in F. Necessary and sufficient conditions for the strictness of canonical inclusion for Orlicz, Lorentz, and Marcinkiewicz spaces are obtained, and the relations of this concept to the disjoint strict singularity are studied.     

Ordering uniform supertrees by their spectral radii

A supertree is a connected and acyclic hypergraph. For a hypergraph H, the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H. By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on n vertices, which extends the known result.     

Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation

Using the method of spectral analysis, for the mixed type equation u_xx + (sgny)u_yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.     

<Emphasis Type="Italic">LJ</Emphasis>-spaces

In this paper LJ-spaces are introduced and studied. They are a common generalization of Lindelöf spaces and J-spaces researched by E. Michael. A space X is called an LJ-space if, whenever {A, B} is a closed cover of X with A ∩ B compact, then A or B is Lindelöf. Semi-strong LJ-spaces and strong LJ-spaces are also defined and investigated. It is demonstrated that the three spaces are different and have interesting properties and behaviors.     

Uniform,on the entire axis,convergence of the spectral expansion for Schrödinger operator with a potential-distribution

A uniform, on ?, estimate for the increment of the spectral function θ(λ; x, y) at x = y is proved for the self-adjoint Schrödinger operator A defined on the entire axis ? by the differential operation (?d/dx)² + q(x) with a potential-distribution q(x) that uniformly locally belongs to the space W ₂ ^?1. As a consequence, it is shown that for any function f(x) from the domain of power Aα of the operator with α > 1/4, the spectral expansion of the function that corresponds to the operator A is convergent absolutely and uniformly on the entire axis ?.     

Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential

For the self-adjoint Schrödinger operator ? defined on ? by the differential operation ?(d/dx)² + q(x) with a distribution potential q(x) uniformly locally belonging to the space W ₂ ^?1, we describe classes of functions whose spectral expansions corresponding to the operator ? absolutely and uniformly converge on the entire line ?. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.     

Random difference equations with subexponential innovations

We consider the random difference equations S = _d (X + S)Y and T = _d X + TY, where = _d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.