There are eight‐element orthogonal exponentials on the spatial Sierpinski gasket

The self‐affine measure corresponding to an expanding matrix and the digit set in the space is supported on the spatial Sierpinski gasket, where are the standard basis of unit column vectors in and . In the case and , it is conjectured that the cardinality of orthogonal exponentials in the Hilbert space is at most “4”, where the number 4 is the best upper bound. That is, all the four‐element sets of orthogonal exponentials are maximal. This conjecture has been proved to be false by giving a class of the five‐element orthogonal exponentials in . In the present paper, we construct a class of the eight‐element orthogonal exponentials in the corresponding Hilbert space to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.     

Localized peaking regimes for quasilinear parabolic equations

This paper deals with the asymptotic behavior as of all weak (energy) solutions of a class of equations with the following model representative: with prescribed global energy function Here , , , Ω is a bounded smooth domain, . Particularly, in the case it is proved that the solution u remains uniformly bounded as in an arbitrary subdomain and the sharp upper estimate of when has been obtained depending on and . In the case for all , sharp sufficient conditions on degeneration of near that guarantee the above mentioned boundedness for an arbitrary (even large) solution have been found and the sharp upper estimate of a final profile of the solution when has been obtained.     

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

We construct a bounded C¹ domain Ω in for which the regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in such that the solution of in Ω and either on or on is contained in but not in for any . An analogous result holds for Sobolev spaces with .     

‐Theory for Schrödinger systems

In this article, we study for the ‐realization of the vector‐valued Schrödinger operator . Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, we prove that the ‐realization of , defined on the intersection of the natural domains of the differential and multiplication operators which form , generates a strongly continuous contraction semigroup on . We also study additional properties of the semigroup such as extension to L¹, positivity, ultracontractivity and prove that the generator has compact resolvent.     

Semilinear subdiffusion with memory in multidimensional domains

For , we investigate the nonlinear integro‐differential equation on a multidimensional space domain where is the Caputo fractional derivative and and are uniform elliptic operators with smooth coefficients depending on time. Under suitable conditions on the nonlinearity, the global existence and uniqueness of the classical solution to the related initial and boundary value problems are established.     

The Tarski irredundant basis theorem and the finite soluble groups

Denote by and , respectively, the smallest and the largest cardinality of a minimal generating set of a finite group G. The Tarski irredundant basis theorem implies that for every k with there exist a minimal generating set , an index and in G such that is again a minimal generating set of G. In this case we say that is an immediate descendant of ω. There are several examples of minimal generating sets of cardinality smaller than which have no immediate descendant and so it appears an interesting problem to investigate under which conditions an immediate descendant exists. In this paper we discuss this problem in the case of finite soluble groups.     

A geometric property in and its applications

In this work, we initiate the study of the geometry of the variable exponent sequence space when . In 1931 Orlicz introduced the variable exponent sequence spaces while studying lacunary Fourier series. Since then, much progress has been made in the understanding of these spaces and of their continuous counterpart. In particular, it is well known that is uniformly convex if and only if the exponent is bounded away from 1 and infinity. The geometry of when either or remains largely ill‐understood. We state and prove a modular version of the geometric property of when , known as uniform convexity in every direction. We present specific applications to fixed point theory. In particular we obtain an analogue to the classical Kirk's fixed point theorem in when .     

A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

In this article, we study the eigenvalues of p‐fractional Hardy operator where , , , and Ω is an unbounded domain in with Lipschitz boundary containing 0. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is uniquely associated to a nonnegative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues as .     

Variable Hardy–Lorentz spaces

Let be a measurable function on with . We introduce the variable Hardy–Lorentz space for via the radial grand maximal function. Under the assumption that satisfies the log‐Hölder condition, we establish a version of Fefferman–Stein vector‐valued inequality in variable Lorentz space by interpolation. We also construct atomic decompositions for , and develop a theory of real interpolation and formulate the dual space of the variable Hardy–Lorentz space with and . As a byproduct, we obtain a new John–Nirenberg theorem. Furthermore, we get equivalent characterizations of the variable Hardy–Lorentz space by means of the Lusin area function, the Littlewood–Paley g‐function and the Littlewood–Paley ‐function. Finally, we investigate the boundedness of singular operators on for and .     

The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system

In this paper, we study the following critical fractional Schrödinger–Poisson system where is a small parameter, and , is the fractional critical exponent for 3‐dimension, has a positive global minimum, and are positive and have global maximums. We obtain the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials and Q as the concentration position of these ground state solutions as . Moreover, we consider some properties of these ground state solutions, such as convergence and decay estimate.     

Expected intrinsic volumes and facet numbers of random beta‐polytopes

Let be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: and We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of . Asymptotic formulae were obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when , respectively , we can also cover the models in which are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of and . One of the main tools used in the proofs is the Blaschke–Petkantschin formula.     

A Szegő limit theorem for translation‐invariant operators on polygons

We prove Szeg?‐type trace asymptotics for translation‐invariant operators on polygons. More precisely, consider a Fourier multiplier on with a sufficiently decaying, smooth symbol . Let be the interior of a polygon and, for , define its scaled version . Then we study the spectral asymptotics for the operator , the spatial restriction of A onto : for entire functions h with we provide a complete asymptotic expansion of as . These trace asymptotics consist of three terms that reflect the geometry of the polygon. If P is replaced by a domain with smooth boundary, a complete asymptotic expansion of the trace has been known for more than 30 years. However, for polygons the formula for the constant order term in the asymptotics is new. In particular, we show that each corner of the polygon produces an extra contribution; as a consequence, the constant order term exhibits an anomaly similar to the heat trace asymptotics for the Dirichlet Laplacian.     

Cox rings and algebraic maps

Given a morphism from a Mori Dream Space X to a smooth Mori Dream Space Y and quasicoherent sheaves on X and on Y, we describe the inverse image of by F and the direct image of by F in terms of the corresponding modules over the Cox rings graded in the class groups.     

On heteroclinic solutions for BVPs involving ‐Laplacian operators without asymptotic or growth assumptions

In this paper we consider the second order discontinuous equation in the real line, with ? an increasing homeomorphism such that and , with , for , a L¹‐Carathéodory function and verifying an adequate relation. We remark that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities ? and f. Moreover, as far as we know, our main result is even new when , that is, for the equation      

Existence and regularity of positive solutions of a degenerate elliptic problem

Existence and regularity of positive solutions of a degenerate elliptic Dirichlet problem of the form in Ω, on , where Ω is a bounded smooth domain in , , are obtained via new embeddings of some weighted Sobolev spaces with singular weights and . It is seen that and admit many singular points in Ω. The main embedding results in this paper provide some generalizations of the well‐known Caffarelli–Kohn–Nirenberg inequality.     

Embedding Banach spaces into the space of bounded functions with countable support

We prove that a WLD subspace of the space consisting of all bounded, countably supported functions on a set Γ embeds isomorphically into if and only if it does not contain isometric copies of . Moreover, a subspace of is constructed that has an unconditional basis, does not embed into , and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of ).     

Smoothable zero dimensional schemes and special projections of algebraic varieties

We study the degrees of generators of the ideal of a projected Veronese variety to depending on the center of projection. This is related to the geometry of zero dimensional schemes of length 8 in , Cremona transforms of , and the geometry of Tonoli Calabi‐Yau threefolds of degree 17 in .     

Noncommutative topology and Jordan operator algebras

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a “full” noncommutative topology in the ‐algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of ‐algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about for a left ideal L in a ‐algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in ‐algebras and various subspaces of ‐algebras, related to open and closed projections and technical ‐algebraic results of Brown.