Multiparameter perturbation theory of Fredholm operators applied to bloch functions

In the present paper, a family of linear Fredholm operators depending on several parameters is considered. We implement a general approach, which allows us to reduce the problem of finding the set Λ of parameters t = (t ₁, ..., t _n ) for which the equation A(t)u = 0 has a nonzero solution to a finite-dimensional case. This allows us to obtain perturbation theory formulas for simple and conic points of the set Λ by using the ordinary implicit function theorems. These formulas are applied to the existence problem for the conic points of the eigenvalue set E(k) in the space of Bloch functions of the two-dimensional Schrödinger operator with a periodic potential with respect to a hexagonal lattice.     

One Criterion for Nuclearity of Linear Operators

We prove a criterion for nuclearity of a linear operator and establish the form of the greatest two-sided ideal of the set of all compact Akhiezer integral operators in L₂ and the set of all Akhiezer integral operators in L₂.     

An upper estimate for the product linear nonhomogeneous forms for lattices with a small homogeneous minimum

One obtains the following transfer theorem, related to Minkowski's nonhomogeneous conjecture. Let Λ ? ?ⁿ be a point lattice, det Λ=1. We consider the nonhomogeneous Π(Λ) and the homogeneous L(Λ)=бⁿ. б ? 0, arithmetical minima of the lattice Λ. Then for sufficiently largen, if , then .     

Lattice sub-tilings and frames in LCA groups

Given a lattice Λ in a locally compact Abelian group G and a measurable subset Ω with finite and positive measure, then the set of characters associated with the dual lattice form a frame for $L^2(\mathrm{\Omega })$ if and only if the distinct translates by Λ of Ω have almost empty intersections. Some consequences of this results are the well-known Fuglede theorem for lattices, as well as a simple characterization for frames of modulates.     

A generalized Shimura correspondence for newforms

We associate a set of half integral weight forms to an integral weight newform of odd level. We prove an explicit identity relating the central values of the twist L-functions of the newform to the Fourier coefficients of the half integral weight forms.     

DIFFEOMORPHISMS WITH VARIOUS C1 STABLE PROPERTIES

Let M be a smooth compact manifold and Λ be a compact invariant set.In this article,we prove that,for every robustly transitive set Λ,f|Λ satisfies a C1-genericstable shadowable property (resp.,C1-gene...     

Prefinitely axiomatizable modal and intermediate logics

A logic Λ bounds a property P if all proper extensions of Λ have P while Λ itself does not. We construct logics bounding finite axiomatizability and logics bounding finite model property in the lattice of intermediate logics and in the lattice of normal extensions of K4.3. MSC: 03B45, 03B55.     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.     

Properties of solutions of cooperative games with transferable utilities

We understand a solution of a cooperative TU-game as the α-prenucleoli set, α ∈ R, which is a generalization of the notion of the [0, 1]-prenucleolus. We show that the set of all α-nucleoli takes into account the constructive power with the weight α and the blocking power with the weight (1 ? α) for all possible values of the parameter α. The further generalization of the solution by introducing two independent parameters makes no sense. We prove that the set of all α-prenucleoli satisfies properties of duality and independence with respect to the excess arrangement. For the considered solution we extend the covariance propertywith respect to strategically equivalent transformations.     

On coefficients of Kapteyn-type series

Quite recently Jankov and Pogány [JANKOV, D.—POGÁNY, T. K.: Integral representation of Schlömilch series, J. Classical Anal. 1 (2012) 75–84] derived a double integral representation of the Kapteyn-type series of Bessel functions. Here we completely describe the class of functions Λ = {α}, which generate the mentioned integral representation in the sense that the restrictions $\alpha |_\mathbb{N} = (\alpha _n )_{n \in \mathbb{N}} $ is the sequence of coefficients of the input Kapteyn-type series.     

The torsion-free part of the Ziegler spectrum of orders over Dedekind domains

We study the R-torsion-free part of the Ziegler spectrum of an order Λ over a Dedekind domain R. We underline and comment on the role of lattices over Λ. We describe the torsion-free part of the spectrum when Λ is of finite lattice representation type.     

On the coefficients of Neumann series of Bessel functions

Recently Pogány and Süli (Proc. Amer. Math. Soc. 137 (7) (2009) 2363-2368) derived a closed-form integral expression for Neumann series of Bessel functions. In this note we precisely characterize the class of functions α that generate the integral representation of a Neumann series of Bessel functions in the sense that the restriction αN_|=(αn) of a function α to the set N of all positive integers is the sequence of coefficients of the initial Neumann series.     

A refinement of the estimates of the arithmetic minimum of the product of nonhomogeneous linear forms for large dimensions

Let M _n be the nonhomogeneous arithmetic minimum of the lattice Λ ? ?ⁿ of determinant Δ>0. It is proved that forn?1,1·n⁵, we have . One obtains other estimates of a similar type in the nonhomogeneous Minkowski conjecture, obtained for the first time by N. G. Chebotarev. These estimates strengthen the results of B. P. Skubenko.     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

Nagata-like theorems for almost Prüfer v-multiplication domains

Let D be an integral domain and X an indeterminate over D . We show that if S is an almost splitting set of an integral domain D , then D is an APVMD if and only if both DS and DN(S) are APVMDs. We also prove that if {Dα}α∈I is a collection of quotient rings of D such that D=∩α∈IDα has finite character (that is, each nonzero d∈D is a unit in almost all Dα) and each of Dα is an APVMD, then D is an APVMD. Using these results, we give several Nagata-like theorems for APVMDs.     

On Integral Well-rounded Lattices in the Plane

We investigate distribution of integral well-rounded lattices in the plane, parameterizing the set of their similarity classes by solutions of the family of Pell-type Diophantine equations of the form x ²+Dy ²=z ² where D>0 is squarefree. We apply this parameterization to the study of the greatest minimal norm and the highest signal-to-noise ratio on the set of such lattices with fixed determinant, also estimating cardinality of these sets (up to rotation and reflection) for each determinant value. This investigation extends previous work of the first author in the specific cases of integer and hexagonal lattices and is motivated by the importance of integral well-rounded lattices for discrete optimization problems. We briefly discuss an application of our results to planar lattice transmitter networks.     

Random weights, robust lattice rules and the geometry of the cbcrc algorithm

In this paper we study lattice rules which are cubature formulae to approximate integrands over the unit cube [0,1] ^s from a weighted reproducing kernel Hilbert space. We assume that the weights are independent random variables with a given mean and variance for two reasons stemming from practical applications: (i) It is usually not known in practice how to choose the weights. Thus by assuming that the weights are random variables, we obtain robust constructions (with respect to the weights) of lattice rules. This, to some extend, removes the necessity to carefully choose the weights. (ii) In practice it is convenient to use the same lattice rule for many different integrands. The best choice of weights for each integrand may vary to some degree, hence considering the weights random variables does justice to how lattice rules are used in applications. In this paper the worst-case error is therefore a random variable depending on random weights. We show how one can construct lattice rules which perform well for weights taken from a set with large measure. Such lattice rules are therefore robust with respect to certain changes in the weights. The construction algorithm uses the component-by-component (cbc) idea based on two criteria, one using the mean of the worst case error and the second criterion using a bound on the variance of the worst-case error. We call the new algorithm the cbc2c (component-by-component with 2 constraints) algorithm. We also study a generalized version which uses r constraints which we call the cbcrc (component-by-component with r constraints) algorithm. We show that lattice rules generated by the cbcrc algorithm simultaneously work well for all weights in a subspace spanned by the chosen weights ?? ⁽¹⁾, . . . , ?? ^(r). Thus, in applications, instead of finding one set of weights, it is enough to find a convex polytope in which the optimal weights lie. The price for this method is a factor r in the upper bound on the error and in the construction cost of the lattice rule. Thus the burden of determining one set of weights very precisely can be shifted to the construction of good lattice rules. Numerical results indicate the benefit of using the cbc2c algorithm for certain choices of weights.     

Modular forms of orthogonal type and Jacobi theta-series

In this paper, we study Jacobi forms of half-integral index for any even integral positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A ₁=〈2〉). We construct Jacobi forms of singular (respectively, critical) weight in all dimensions n≥8 (respectively, n≥9). We give the Jacobi lifting for Jacobi forms of half-integral indices and we obtain an additive lifting construction of new reflective modular forms which are natural generalizations to O(2,n) (n=4, 5 and 6) of the Igusa modular form Δ ₅.     

Slopes of integral lattices

We use the dyadic trace to define the concept of slope for integral lattices. We present an introduction to the theory of the slope invariant. The main theorem states that a Siegel modular cusp form f of slope strictly less than the slope of an integral lattice with Gram matrix s satisfies f(sτ)=0 for all τ in the upper half plane. We compute the dyadic trace and the slope of each root lattice and we give applications to Siegel modular cusp forms.