We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extending the classical results from smooth manifolds and conical spaces. The estimates, together with strong continuity of the heat operator, yield short-time existence of solutions to certain semilinear parabolic equations. Our discussion reviews and generalizes earlier work by Jeffres and Loya.

]]>We re-examine Shatalov-Sternin's proof of existence of resurgent solutions of a linear ODE. In particular, we take a closer look at the “Riemann surface” (actually, a two-dimensional complex manifold) whose existence, endless continuability and other properties are claimed by those authors. We present a detailed argument for a part of the “Riemann surface” relevant for the exact WKB method.

]]>For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of *L*^{2} harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of the foliation) is a Witt space and if the metric considered is a foliated cusp metric, then no such resolution is required.

We solve the univalence problem in the class of α-project starlike function. The α-project close-to-convex function, α-project Bazilevič and α-project Φ-like functions are also considered here.

]]>The purpose of the present paper is to discuss the role of second order elliptic operators of the type on the existence of a positive solution for the problem involving critical exponent

where Ω is a smooth bounded domain in , , and λ is a real parameter. In particular, we show that if the function has an interior global minimum point *x*_{0} such that is comparable to , where and is the identity matrix of order *n*, then the range of values of λ for which the problem above has a positive solution can change drastically from to .

We show abstract versions for Banach couples of several limiting compact interpolation theorems established by Edmunds and Opic for couples of spaces.

]]>We study general (not necessarily Hamiltonian) first-order symmetric systems on an interval with both singular endpoints *a* and *b*. For such a system we give a criterion of existence and description of self-adjoint separated boundary conditions. We prove the Titchmarsh type formula for the characteristic matrix of the self-adjoint linear relation in generated by separated boundary conditions as well as by a certain type of mixed boundary conditions. This formula enables one to express in terms of the *m*-functions at the endpoints of . By using the Titchmarsh type formula we parametrize all spectral functions corresponding to boundary problems with the mentioned boundary conditions immediately in terms of self-adjoint boundary parameters at the end points *a* and *b*.

In this paper we give explicit modular maps for the family of abelian surfaces and that of the abelian surfaces whose endomorphism algebra contains .

We obtain a description of the Shimura variety for the latter family, also. The notion of the family of *K*3 surfaces with a fixed marking plays a central role. As the basement of our study we use the expressions of those families given by Clingher–Doran, A. Nagano and the work of A. Kumar as well.

The paper is the second in a set of two papers, which are devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non-selfadjoint unbounded matrix differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (*a*) initial boundary-value problem (IBVP) for a non-homogeneous string with both distributed and boundary damping; (*b*) IBVP for small vibrations of an ideal filament with a one-parameter family of dissipative boundary conditions at one end and with a heavy load at the other end; this filament problem is treated for two cases of the boundary parameter: non-singular and singular; (*c*) IBVP for a three-dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping; (*d*) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (*e*) IBVP for a coupled Euler-Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending-torsion vibration model); (*f*) IBVP for two coupled Timoshenko beams model, which is currently accepted as an appropriate model describing vibrational behavior of a longer double-walled carbon nanotube. Problems have been discussed in the first paper of the aforementioned set. Problems are discussed in the present paper.

In this paper we determine the radius of convexity and the radius of starlikeness of the functions Γ and The basic tools of our work are the developments of the functions in function series.

]]>In the limit we analyse the generators of families of reversible jump processes in associated with a class of symmetric non-local Dirichlet-forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of a certain eikonal equation. Fine results are sensitive to the rate function being or just Lipschitz. Our estimates are analogous to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice . Although our final interest is in the (sub)stochastic jump process, technically this is a pure analysis paper, inspired by PDE techniques.

]]>Let be a strictly stationary sequence of negatively associated random variables with zero mean and finite variance. We set and , . If , then for any , we show the precise rates of the first moment convergence in the law of the iterated logarithm for a kind of weighted infinite series of and as , and as .

]]>For an open set we study the algebra of continuous linear operators on admitting the monomials as eigenvectors. We give a concrete representation of these operators and evaluate it explicitly for the unit ball and the whole of . We also study the topology of and the algebra of eigenvalue sequences.

]]>In this paper, we obtain new results for the weak-AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong-AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak-AFPP in such frames. In connection, we provide a simple criterion for the containement of ℓ_{1}-sequences in terms of strongly-equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak-AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non-locally convex) Hausdorff vector topology which is complete, non-metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the -AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if *X* is a non-locally convex *F*-space with an absolute basis, then the weak-AFPP is equivalent to the fact that every bounded convex subset of *X* is compact.

In this article we give an upper bound for the number of cusps on a cuspidal curve on a Hirzebruch surface. We adapt the results that have been found for a similar question asked for cuspidal curves on the projective plane, and restate the results in this new setting.

]]>Let be a regular local smoothing of a nodal curve. In this paper, we find a modular description of the Néron maps associated to Abel-Jacobi maps with values in Esteves's fine compactified Jacobian and with source the *B*-smooth locus of either the double product of over *B* or the degree-2 Hilbert scheme of the family *f*. This is the first of a series of two papers dedicated to the construction of a resolution of the degree-2 Abel-Jacobi map for a regular smoothing of a nodal curve.

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between Jacobi forms and skew-holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on and the space of skew-holomorphic Jacobi cusp forms on with the same half-integral weight to the Eichler cohomology group of with a coefficient module coming from polynomials.

]]>We consider real interpolation methods defined by means of slowly varying functions and rearrangement invariant spaces, for which we present a collection of reiteration theorems for interpolation and extrapolation spaces. As an application we obtain interpolation formulas for Lorentz-Karamata type spaces, for Zygmund spaces , and for the grand and small Lebesgue spaces.

]]>Extensions of the multidimensional Heisenberg group by one-parameter groups of matrix dilations are introduced and classified up to isomorphism. Each group is isomorphic to both, a subgroup of the symplectic group and a subgroup of the affine group, and its metaplectic representation splits into two irreducible subrepresentations, each of which is equivalent to a subrepresentation of the wavelet representation.

]]>We show the existence of a nodal solution (sign-changing solution) for a Kirchhoff equation of the type

where Ω is a bounded domain in , *M* is a general *C*^{1} class function and *f* is a superlinear *C*^{1} class function with subcritical growth. The proof is based on a minimization argument and a quantitative deformation lemma.

Let *X* be an *n*-dimensional smooth projective variety with an *n*-block collection , with , of coherent sheaves on *X* that generate the bounded derived category . We give a cohomological characterisation of torsion-free sheaves on *X* that are the cohomology of monads of the form

where . We apply the result to get a cohomological characterisation when *X* is the projective space, the smooth hyperquadric or the Fano threefold *V*_{5}. We construct a family of monads on a Segre variety and apply our main result to this family.

We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three-dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.

]]>In this paper, we will consider the higher-order functional dynamic equations of the form

on an above-unbounded time scale , where and , . The function is a rd-continuous function such that . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.

]]>We show that a compact surface of genus greater than one, without focal points and a finite number of bubbles (“good” shaped regions of positive curvature) is in the closure of Anosov metrics. Compact surfaces of nonpositive curvature and genus greater than one are in the closure of Anosov metrics, by Hamilton's work about the Ricci flow. We generalize this fact to the above surfaces without focal points admitting regions of positive curvature using a “magnetic” version of the Ricci flow, the so-called Ricci Yang-Mills flow.

]]>Let be a holomorphic foliation with ample canonical bundle on a smooth projective surface *X*. We obtain an upper bound on the order of its automorphism group which depends only on and provided this group is finite. Here, and are the canonical bundles of and *X*, respectively.

In this article, Riemann-type boundary-value problem of single-periodic polyanalytic functions has been investigated. By the decomposition of single-periodic polyanalytic functions, the problem is transformed into *n* equivalent and independent Riemann boundary-value problems of single-periodic analytic functions, which has been discussed in details according to two growth orders of functions. Finally, we obtain the explicit expression of the solution and the conditions of solvability for Riemann problem of the single-periodic polyanalytic functions.

This article presents a family of nonlinear differential identities for the spatially periodic function , which is essentially the Jacobian elliptic function with one non-trivial parameter . More precisely, we show that this function fulfills equations of the form

for all . We give explicit expressions for the coefficients and for given *s*.

Moreover, we show that for any *s* the set of functions constitutes a basis for . By virtue of our formulas the problem of finding a periodic solution to any nonlinear wave equation reduces to a problem in the coefficients. A finite ansatz exactly solves the KdV equation (giving the well-known cnoidal wave solution) and the Kawahara equation. An infinite ansatz is expected to be especially efficient if the equation to be solved can be considered a perturbation of the KdV equation.

We give a theory of idèles with coefficients for smooth surfaces over a field. It is an analogue of Beilinson/Huber's theory of higher adèles, but handling cycle module sheaves instead of quasi-coherent ones. We prove that they give a flasque resolution of the cycle module sheaves in the Zariski topology. As a technical ingredient we show the Gersten property for cycle modules on equicharacteristic complete regular local rings, which might be of independent interest.

]]>Let be bounded with a smooth boundary Γ and let *S* be the symmetric operator in given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self-adjoint extensions of *S* by providing an explicit one-to-one correspondence between such extensions and the class of Dirichlet forms in which are additively decomposable by the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Kreĭn's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell-type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.

We study the regularity of binomial edge ideals. For a closed graph *G* we show that the regularity of the binomial edge ideal coincides with the regularity of and can be expressed in terms of the combinatorial data of *G*. In addition, we give positive answers to Matsuda-Murai conjecture for some classes of graphs.

We consider incompressible generalized Newtonian fluids in two space dimensions perturbed by an additive Gaussian noise. The velocity field of such a fluid obeys a stochastic partial differential equation with fully nonlinear drift due to the dependence of viscosity on the shear rate. In particular, we assume that the extra stress tensor is of power law type, i.e. a polynomial of degree , , i.e. the shear thinning case. We prove that the associated Kolmogorov operator *K* admits at least one infinitesimally invariant measure μ satisfying certain exponential moment estimates. Moreover, *K* is *L*^{2}-unique w.r.t. μ provided , where is the second root of , approximately .

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form

As an application we consider the nonlinear Schrödinger system

for and exponents *q* which satisfy in case and in case . Generalizing the results of Wei and Yao for we find new sufficient conditions and necessary conditions on such that precisely one positive solution exists. Our results dealing with the special case are optimal. Finally, an application to a multi-component nonlinear Schrödinger system is given.

Let *Y* be a projective variety over a field *k* (of arbitrary characteristic). Assume that the normalization *X* of *Y* is such that is normal, being the algebraic closure of *k*. We define a notion of strong semistability for vector bundles on *Y*. We show that a vector bundle on *Y* is strongly semistable if and only if its pull back to *X* is strongly semistable and hence it is a tensor category. In case , we show that strongly semistable vector bundles on *Y* form a neutral Tannakian category. We define the holonomy group scheme of *Y* to be the Tannakian group scheme for this category. For a strongly semistable principal *G*-bundle , we construct a holonomy group scheme. We show that if *Y* is an integral complex nodal curve, then the holonomy group of a strongly semistable vector bundle on *Y* is the Zariski closure of the (topological) fundamental group of *Y*.

We show that for a compact hypergroup *K*, the hypergroup algebra is amenable as a Banach algebra if the set of hyperdimensions of irreducible representations of *K* is bounded above. Conversely if is amenable, the set of ratios of the hyperdimension to the dimension of irreducible representations of *K* is bounded above. These are equivalent for compact commutative hypergroups.

In this note we discuss an abstract framework for standard boundary value problems in divergence form with maximal monotone relations as “coefficients”. A reformulation of the respective problems is constructed such that they turn out to be unitarily equivalent to inverting a maximal monotone relation in a Hilbert space. The method is based on the idea of “tailor-made” distributions as provided by the construction of extrapolation spaces, see e.g. [Picard, McGhee: Partial Differential Equations: A unified Hilbert Space Approach (De Gruyter, 2011)]. The abstract framework is illustrated by various examples.

]]>Four classes of closed subspaces of an inner product space *S* that can naturally replace the lattice of projections in a Hilbert space are: the complete/cocomplete subspaces , the splitting subspaces , the quasi-splitting subspaces and the orthogonally-closed subspaces . It is well-known that in general the algebraic structure of these families differ remarkably and they coalesce if and only if *S* is a Hilbert space. It is also known that when *S* is a hyperplane in its completion i.e. then and . On the other extreme, when i.e. then and . Motivated by this and in contrast to it, we show that in general the codimension of *S* in bears very little relation to the properties of these families. In particular, we show that the equalities and can hold for inner product spaces with arbitrary codimension in . At the end we also contribute to the study of the algebraic structure of by testing it for the Riesz interpolation property. We show that may fail to enjoy the Riesz interpolation property in both extreme situations when *S* is “very small” (i.e. and when *S* is ‘very big’ (i.e. .

We compute the *K*-groups of the -algebra of bounded operators generated by the Boutet de Monvel operators with classical SG-symbols of order (0,0) and type 0 on , as defined by Schrohe, Kapanadze and Schulze. In order to adapt the techniques used in Melo, Nest, Schick and Schrohe's work on the *K*-theory of Boutet de Monvel's algebra on compact manifolds, we regard the symbols as functions defined on the radial compactifications of and . This allows us to give useful descriptions of the kernel and the image of the continuous extension of the boundary principal symbol map, which defines a -algebra homomorphism. We are then able to compute the *K*-groups of the algebra using the standard *K*-theory six-term cyclic exact sequence associated to that homomorphism.

Continuing the study of bounded geometry for Riemannian foliations, begun by Sanguiao, we introduce a chart-free definition of this concept. Our main theorem states that it is equivalent to a condition involving certain normal foliation charts. For this type of charts, it is also shown that the derivatives of the changes of coordinates are uniformly bounded, and there are nice partitions of unity. Applications to a trace formula for foliated flows will be given in a forthcoming paper.

]]>We study the dynamical boundary value problem for Hamilton-Jacobi equations of the eikonal type with a small parameter. We establish two results concerning the asymptotic behavior of solutions of the Hamilton-Jacobi equations: one concerns with the convergence of solutions as the parameter goes to zero and the other with the large-time asymptotics of solutions of the limit equation.

]]>We prove a multiplier theorem for certain Laplacians with drift on Damek–Ricci spaces, which are a class of Lie groups of exponential growth. Our theorem generalizes previous results obtained by W. Hebisch, G. Mauceri and S. Meda on Lie groups of polynomial growth.

]]>This note is a companion to the article *On the mutually non isomorphic* *spaces* published in this journal, in which P. Cembranos and J. Mendoza showed that is a collection of mutually non isomorphic Banach spaces [5]. We now complete the picture by allowing the non-locally convex relatives to be part of their natural family and see that, in fact, no two members of the extended class are isomorphic. Our approach is novel in the sense that we reach the isomorphism obstructions from the perspective of bases techniques and the different convexities of the spaces, both methods being intrinsic to quasi-Banach spaces.

An extension of conformal equivalence for Finsler metrics is introduced and called weakly conformal equivalence and is used to define the weakly conformal transformations. The conformal Lichnerowicz-Obata conjecture is refined to weakly conformal Finsler geometry. It is proved that: *If X is a weakly conformal complete vector field on a connected Finsler space* (*M, F*) *of dimension* , *then, at least one of the following statements holds*: (a) *There exists a Finsler metric F*_{1} *weakly conformally equivalent to F such that X is a Killing vector field of the Finsler metric*, (b) *M is diffeomorphic to the sphere* *and the Finsler metric is weakly conformally equivalent to the standard Riemannian metric on* , *and* (c) *M is diffeomorphic to the Euclidean space* *and the Finsler metric F is weakly conformally equivalent to a Minkowski metric on* . The considerations invite further dynamics on Finsler manifolds.

In this paper we prove that there does not exist any Hopf real hypersurface in complex hyperbolic two-plane Grassmannians with parallel Ricci tensor.

]]>The knowledge about the stability properties of spatially localized structures in linear periodic media with and without defects is fundamental for many fields in nature. Its importance for the design of photonic crystals is, for example, described in and . Against this background, we consider a one-dimensional linear Klein-Gordon equation to which both a spatially periodic Lamé potential and a spatially localized perturbation are added. Given the dispersive character of the underlying equation, it is the purpose of this paper to deduce time-decay rates for its solutions. We show that, generically, the part of the solution which is orthogonal to possible eigenfunctions of the perturbed Hill operator associated to the problem decays with a rate of w.r.t. the norm. In weighted *L*^{2} norms, we even get a time decay of . Furthermore, we consider the situation of a perturbing potential that is only made up of a spatially localized part which, now, can be slightly more general. It is well-known that, in general, it is not possible to obtain the endpoint estimate in one space dimension by means of the wave operators drawn from scattering theory. For this reason, we proceed directly and prove, along the lines of , the expected decay rate of .

A generalized bounded variation characterization of Banach spaces possessing the Radon-Nikodym property is given in terms of the average range. We prove that a Banach space *X* has the Radon-Nikodym property if and only if for each function of generalized bounded variation on [0, 1], the average range is a nonempty set at almost all .

A Sobolev type embedding for Triebel-Lizorkin-Morrey-Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy-Morrey spaces and Hardy-Lorentz spaces are obtained.

]]>In this paper, we consider hypersurfaces in the unit lightlike sphere. The unit sphere can be canonically embedded in the lightcone and de Sitter space in Minkowski space. We investigate these hypersurfaces in the framework of the theory of Legendrian dualities between pseudo-spheres in Minkowski space.

]]>For , the symmetric functions are defined by

where , and are non-negative integers. In this paper, the Schur convexity, geometric Schur convexity and harmonic Schur convexity of are investigated. As applications, Schur convexity for the other symmetric functions is obtained by a bijective transformation of independent variable for a Schur convex function, some analytic and geometric inequalities are established by using the theory of majorization, in particular, we derive from our results a generalization of Sharpiro's inequality, and give a new generalization of Safta's conjecture in the *n*-dimensional space and others.

In this paper, we study the random dynamical system generated by a stochastic reaction-diffusion equation with multiplicative noise and prove the existence of an -random attractor for such a random dynamical system. The nonlinearity *f* is supposed to satisfy some growth of arbitrary order .

Partially ample divisors are defined by relaxing the different conditions that characterize the ample divisors. We prove that for nef and big divisors such notions coincide. We also prove that the partial ampleness of big divisors are preserved in the positive parts of the Fujita-Zariski decomposition.

]]>In this paper we introduce the notion of decomposability in the space of Henstock-Kurzweil-Pettis integrable (for short *HKP*-integrable) functions. We show representations theorems for decomposable sets of *HKP*-integrable or Henstock integrable functions, in terms of the family of selections of suitable multifunctions.

In 1996, H. Volkmer observed that the inequality

is satisfied with some positive constant for a certain class of functions *f* on [ − 1, 1] if the eigenfunctions of the problem

form a Riesz basis of the Hilbert space . Here the weight is assumed to satisfy a.e. on ( − 1, 1).

We present two criteria in terms of Weyl–Titchmarsh *m*-functions for the Volkmer inequality to be valid. Note that one of these criteria is new even for the classical HELP inequality. Using these results we improve the result of Volkmer by showing that this inequality is valid if the operator associated with the spectral problem satisfies the linear resolvent growth condition. In particular, we show that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if *r* is odd.

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4-dimensional half conformally flat manifolds *M*^{4}. In fact, we shall show that for a nontrivial must be isometric to a sphere and *f* is some height function on

We construct relative PEL type embeddings in mixed characteristic (0, 2) between hermitian orthogonal Shimura varieties of PEL type. We use this to prove the existence of integral canonical models in unramified mixed characteristic (0, 2) of hermitian orthogonal Shimura varieties of PEL type.

]]>We describe a method for doing computations with Orlov's equivalence between the bounded derived category of certain hypersurfaces and the stable category of graded matrix factorisations of the polynomials describing these hypersurfaces. In the case of a smooth elliptic curve over an algebraically closed field we describe the indecomposable graded matrix factorisations of rank one. Since every indecomposable maximal Cohen-Macaulay module over the completion of a smooth cubic curve is gradable, we obtain explicit descriptions of all indecomposable rank one matrix factorisations of smooth cubic potentials. Finally, we explain how to compute all indecomposable matrix factorisations of higher rank with the help of a computer algebra system.

]]>In this paper we deal with the hyponormality of Toeplitz operators with matrix-valued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded-type Toeplitz operators (i.e., the symbol is a matrix-valued function such that Φ and are of bounded type). In particular, we get a much simpler criterion for the hyponormality of when the co-analytic part of the symbol Φ is a left divisor of the analytic part.

]]>We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds endowed with a timelike conformal vector field *V*. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally umbilical hypersurfaces in terms of their higher order mean curvatures. For instance, supposing an appropriated restriction on the norm of the tangential component of the vector field *V*, we are able to show that such hypersurfaces must be totally umbilical provided that either some of their higher order mean curvatures are linearly related or one of them is constant. Applications to the so-called generalized Robertson-Walker spacetimes are given. In particular, we extend to the Lorentzian context a classical result due to Jellett .

This paper addresses the analysis of the weak solution of in a bounded domain Ω subject to the boundary condition on , when the data *f* belongs to and . We prove existence and uniqueness of solution for this problem in the Nikolskii space . Moreover, we obtain energy estimates regarding the Nikolskii norm of ω in terms of the norm of *f*.

We establish a Trudinger–Moser type inequality in a weighted Sobolev space. The inequality is applied in the study of the elliptic equation

where , *f* has exponential critical growth and *h* belongs to the dual of an appropriate function space. We prove that the problem has at least two weak solutions provided is small.

In the first part of this paper we introduce a method for computing Hilbert decompositions (and consequently the Hilbert depth) of a finitely generated multigraded module *M* over the polynomial ring by reducing the problem to the computation of the finite set of the new defined Hilbert partitions. In the second part we show how Hilbert partitions may be used for computing the Stanley depth of the module *M*. In particular, we answer two open questions posed by Herzog in .

In this paper, we improve a recent result by Li and Peng on products of functions in and , where is a Schrödinger operator with *V* satisfying an appropriate reverse Hölder inequality. More precisely, we prove that such products may be written as the sum of two continuous bilinear operators, one from into , the other one from into , where the space is the set of distributions *f* whose grand maximal function satisfies

We discuss the boundedness and compactness of some integral-type operators acting from spaces to mixed-norm spaces on the unit ball of .

]]>In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the solid cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is zero, we give sufficient conditions to conclude that the surface is part of a plane or a catenoid.

]]>In this paper, we study the Plancherel measure of a class of non-connected nilpotent groups which is of special interest in Gabor theory. Let *G* be a time-frequency group. That is where , are translations and modulations operators acting in and *B* is a non-singular matrix. We compute the Plancherel measure of the left regular representation of *G* which is denoted by *L*. The action of *G* on induces a representation which we call a Gabor representation. Motivated by the admissibility of this representation, we compute the decomposition of *L* into direct integral of irreducible representations by providing a precise description of the unitary dual and its Plancherel measure. As a result, we generalize Hartmut Führ's results which are only obtained for the restricted case where , and Even in the case where *G* is not type I, we are able to obtain a decomposition of the left regular representation of *G* into a direct integral decomposition of irreducible representations when . Some interesting applications to Gabor theory are given as well. For example, when *B* is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of *G*.

We show that under some appropriate assumptions, every weak solution (e.g. energetic solution) to a given rate-independent system is of class SBV, or has finite jumps, or is even piecewise *C*^{1}. Our assumption is essentially imposed on the energy functional, but not convexity is required.

We use the theory of generalized linear ordinary differential equations in Banach spaces to study linear measure functional differential equations with infinite delay. We obtain new results concerning the existence, uniqueness, and continuous dependence of solutions. Even for equations with a finite delay, our results are stronger than the existing ones. Finally, we present an application to functional differential equations with impulses.

]]>In this paper, we study the minimal free resolution of a nondegenerate projective variety when *X* is contained in a variety *Y* of minimal degree as a divisor. Such a variety is of interest because of its extremal behavior with respect to various properties. The graded Betti diagram of *X* has been completely known only when *X* is arithmetically Cohen-Macaulay. Our main result in the present paper provides a detailed description of the graded Betti diagram of *X* for the case where *X* is not arithmetically Cohen-Macaulay.

We provide a structural generalization of a theorem by Kleiman–Piene, concerning the enumerative geometry of nodal curves in a complete linear system on a smooth projective surface *S*. Provided that *r*, the number of nodes, is sufficiently small compared to the ampleness of the linear system, we show that, under certain assumptions, the number of *r*-nodal curves passing through points in general position on *S* is given by a Bell polynomial in universally defined integers which we identify, using classical intersection theory, as linear, integral polynomials evaluated in four basic Chern numbers. Furthermore, we provide a decomposition of the as a sum of three terms with distinct geometric interpretations, and discuss the relationship between these polynomials and Kazarian's Thom polynomials for multisingularities of maps.

Hudzik, Kamińska and Mastyło obtained some geometric properties of Calderón–Lozanovskiĭ function spaces which are defined on a nonatomic σ-measure space in Houston. J. Math. **22** (1996), but left the case of atomic measure unsolved. We studied the relevant problems for the sequence spaces and obtained the following main results:

*For the Calderón–Lozanovskiĭ sequence spaces*,*is order continuous if and only if**and e is order continuous*.*Let*Φ*be strictly convex on*,*then the convex characteristic**whenever e is not order continuous or*;*if e is uniformly monotone and*,*then*.*For the Orlicz-Lorentz sequence space*,*if**or*,*or*ω*is not regular*;*if*Φ*is strictly convex on*,*and*ω*is regular*.