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In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.

In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).

In this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space and may open new avenues of research in hypercomplex analysis.

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.

This is the first in a series of three papers where we study the integral manifolds of the charged three-body problem. The integral manifolds are the fibers of the map of integrals. Their topological type may change at critical values of the map of integrals. Due to the non-compactness of the integral manifolds one has to take into account besides ‘ordinary’ critical points also critical points at infinity. In the present paper we concentrate on ‘ordinary’ critical points and in particular elucidate their connection to central configurations. In a second paper we will study critical points at infinity. The implications for the Hill regions, i.e. the projections of the integral manifolds to configuration space, are the subject of a third paper.

In this paper we extend the idea of weighted density of Balcerzak et al. (2015) by using a modulus function and introduce the idea f -density of weight g of subsets of ω ≔ { 0 , 1 , … } (at the same time extending the notion of f -density (Aizpuru et al., 2014)), which we name d g f where g : ω → [ 0 , ∞ ) satisfies g ( n ) → ∞ and n ∕ g ( n ) ↛ 0 and f is a modulus function. The aim of this paper is to show that we can get new ideals Z g ( f ) consisting of sets A ⊂ ω for which d g f ( A ) = 0 different from all the previously constructed ideals Z g of Balcerzak et al. (2015) and moreover they retain all the nice properties of the ideals Z g .

The first part of this paper considers the controllability for a functional semilinear differential inclusion governed by a family of operators { A ( t ) : t ∈ [ 0 , b ] } generating an evolution operator in a Banach space in the presence of noninstantaneous impulse effects. In the second part of this paper we study the controllability for a fractional noninstantaneous impulsive semilinear differential inclusion with delay, where the linear part is an infinitesimal generator of a C 0 − semigroup. Using a weakly convergent criterion in the space of piecewise continuous functions and weak topology theory (for weak sequentially closed graph operators) we establish sufficient conditions to guarantee controllability results. Examples are given to illustrate the abstract results.

Let M m be a closed smooth manifold equipped with a smooth involution having fixed point set of the form F n ∪ F 3 , where F n and F 3 are submanifolds with dimensions n and 3, respectively, where 3 n m and with the normal bundles over F n and F 3 being nonbounding. The authors of this paper, together with Patricia E. Desideri, previously showed that, when n is even, then m ≤ n + 4 , which we call a small codimension phenomenon. Further, they showed that this small bound is best possible. In this paper we study this problem for n odd, which is much more complicated, requiring more sophisticated techniques involving characteristic numbers. We show in this case that m ≤ M ( n − 3 ) + 6 , where M ( n ) is the Stong–Pergher number (see the definition of M ( n ) in Section 1). Further, we show that this bound is almost best possible, in the sense that there exists an example with m = M ( n − 3 ) + 5 , which means that for n odd the small codimension phenomenon does not occur and the bound in question is meaningful. The existence of these bounds is guaranteed by the famous Five Halves Theorem of J. Boardman, which establishes that, under the above hypotheses, m ≤ 5 2 n .

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert $A$-modules over locally $C^{*}$-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon--Nikodym theorem for this type of completely positive $n\times n$ matrices., 17 pages; The paper was revised and some material is added to it. To appear in Indag. Math

In this paper, the Taylor sequence space t p θ of non-absolute type is introduced which consists of all sequences whose Taylor transforms of order θ , ( 0 θ 1 ) , are in the space l p . It is shown that the space t p θ is a normed space which includes the space l p where 1 ≤ p ≤ ∞ . Moreover, in this paper a general upper estimate is obtained for the norm of Hausdorff matrices and Norlund matrices as operators from l p into the Taylor sequence space t p θ . Finally, the results are extended to the matrix domain of an arbitrary invertible matrix E in the sequence space l p .

In a recent paper, Byrnes et al. (2014) have developed some recurrence relations for the hypergeometric zeta functions. Moreover, the authors made two conjectures for arithmetical properties of the denominators of the reduced fraction of the hypergeometric Bernoulli numbers. In this paper, we prove these conjectures using some recurrence relations. Furthermore, we assert that the above properties hold for both Carlitz and Howard numbers.

In the first part of the paper we introduce some geometric tools needed to describe slow–fast Hamiltonian systems on smooth manifolds. We start with a smooth bundle p : M → B where ( M , ω ) is a C ∞ -smooth presymplectic manifold with a closed constant rank 2-form ω and ( B , λ ) is a smooth symplectic manifold. The 2-form ω is supposed to be compatible with the structure of the bundle, that is the bundle fibers are symplectic manifolds with respect to the 2-form ω and the distribution on M generated by kernels of ω is transverse to the tangent spaces of the leaves and the dimensions of the kernels and of the leaves are supplementary. This allows one to define a symplectic structure Ω e = ω + e − 1 p ∗ λ on M for any positive small e , where p ∗ λ is the lift of the 2-form λ to M . Given a smooth Hamiltonian H on M one gets a slow–fast Hamiltonian system with respect to Ω e . We define a slow manifold S M for this system. Assuming S M is a smooth submanifold, we define a slow Hamiltonian flow on S M . The second part of the paper deals with singularities of the restriction of p to S M . We show that if dim M = 4 , dim B = 2 and Hamilton function H is generic, then the behavior of the system near a singularity of fold type is described, to the main order, by the equation Painleve-I, and if this singularity is a cusp, then the related equation is Painleve-II.

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 − F1 does not contain any intervals but ℒeb(F2 − F1)> 0 almost surely, conditioned on non-extinction.

An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the theory of wavelets of compact support. We discuss or survey many different proofs of the identity, and also its relationship with Gauss hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. The paper concludes with a discussion of a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella hypergeometric functions and corresponding PDE's. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral., Comment: 20 pages; added in v3: more references to earlier occurrences of the identity and its multivariable analogue, combinatorial proof of the identity and extension to noninteger m,n, proof of multivariable identity by splitting up Dirichlet's multivariable beta integral

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

A partial flag variety P K of a Kac–Moody group G has a natural stratification into projected Richardson varieties. When G is a connected reductive group, a Bruhat atlas for P K was constructed in He et al. (2013): P K is locally modelled with Schubert varieties in some Kac–Moody flag variety as stratified spaces. The existence of Bruhat atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac–Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff–Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the J -Schubert varieties for a Birkhoff–Bruhat atlas. The notion of the J -Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac–Moody group). The main result of this paper is the construction of a Birkhoff–Bruhat atlas for any partial flag variety P K of a Kac–Moody group. We also construct a combinatorial atlas for the index set Q K of the projected Richardson varieties in P K . As a consequence, we show that Q K has some nice combinatorial properties. This gives a new proof and generalizes the work of Williams (2007) in the case where the group G is a connected reductive group.

A positive operator T : E → E on a Banach lattice E with an order continuous norm is said to be B -Volterra with respect to a Boolean algebra B of order projections of E if the bands canonically corresponding to elements of B are left fixed by T . A linearly ordered sequence ξ in B connecting 0 to 1 is called a forward filtration. A forward filtration can be used to lift the action of the B -Volterra operator T from the underlying Banach lattice E to an action of a new norm continuous operator T ˆ ξ : M r ( ξ ) → M r ( ξ ) on the Banach lattice M r ( ξ ) of regular bounded martingales on E corresponding to ξ . In the present paper, we study properties of these actions. The set of forward filtrations are left fixed by a function which erases the first order projection of a forward filtration and which shifts the remaining order projections towards 0. This function canonically induces a norm continuous shift operator s between two Banach lattices of regular bounded martingales. Moreover, the operators T ˆ ξ and s commute. Utilizing this fact with inductive limits, we construct a categorical limit space M T , ξ which is called the associated space of the pair ( T , ξ ) . We present new connections between theories of Boolean algebras, abstract martingales and Banach lattices.

In this paper we discuss two different existing algorithms for computing topological entropy and we implement one of them in order to compute the isentropes for cubic polynomials.

In this paper, the mixed L p -surface area measures are defined and the mixed L p Minkowski inequality is obtained consequently. Furthermore, the mixed L p projection inequality for mixed projection bodies is established.

Let f ( x ) be a non-zero polynomial with complex coefficients, and M p = ∫ 0 1 f ( x ) p d x for p a positive integer. In a recent paper, Muger and Tuset showed that lim sup p → ∞ | M p | 1 ∕ p > 0 , and conjectured that this limit is equal to the maximum amongst the critical values of f together with the values | f ( 0 ) | and | f ( 1 ) | . We give an example that shows that this conjecture is false. It also may be natural to guess that lim sup p → ∞ | M p | 1 ∕ p is equal to the maximum of | f ( x ) | on [ 0 , 1 ] . However, we give a counterexample to this as well. We also provide a few more guesses as to the behavior of the quantity lim sup p → ∞ | M p | 1 ∕ p .

Let E be the Sierpinski gasket, i.e., the self-similar set generated by the IFS f a ( x ) = x + a q : a ∈ { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) } . In this paper, we provide a description of the following set for 2 q 3 D q = { dim H ( E ∩ ( E + t ) ) : t ∈ T } , where T is the set of t = ( t 1 , t 2 ) with t ∈ E − E and t 1 , t 2 have unique q -expansions w.r.t − 1 , 0 , 1 .

The paper deals with the study of Banach spaces whose duals are injective Banach lattices. Davies in 1967 proved that an ordered Banach space is an L 1 -predual space if and only if it is a simplex space. In 2007 Duan and Lin proved that a real Banach space is an L 1 -predual space if and only if its every four-point subset is centerable. We prove the counterparts of these remarkable results for injectives by the new machinery of Boolean valued transfer from L 1 -spaces to injective Banach lattices.

The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) L p ( ⋅ ) ( Ω ) . In the second part strictly singular and disjointly strictly singular operators between spaces L p ( ⋅ ) ( Ω ) are studied. New results on the disjoint strict singularity of the inclusions L p ( ⋅ ) ( Ω ) ↪ L q ( ⋅ ) ( Ω ) are given.

The main purpose of this paper is to fully characterize continuous concave functions ψ such that the corresponding Marcinkiewicz Banach function space M ψ is a Grothendieck space.

The principal goal of this paper is to extend the classical problem of finding the values of α ∈ ℂ for which either f ˆ α ( z ) = ∫ 0 z ( f ( ζ ) ∕ ζ ) α d ζ or f α ( z ) = ∫ 0 z ( f ′ ( ζ ) ) α d ζ are univalent, whenever f belongs to some subclasses of univalent mappings in D , to the case of harmonic mappings, by considering the shear construction introduced by Clunie and Sheil-Small in [4] .

Ohno’s relation is a well-known family of relations among multiple zeta values, which can naturally be regarded as a type of duality for a certain power series which we call an Ohno sum. In this paper, we investigate Q -linear relations among Ohno sums which are not contained in Ohno’s relation. We prove two new families of such relations, and pose several further conjectural families of such relations.

The aim of this paper is to classify all invariant generalized complex structures on a partial flag manifold F Θ with at most four isotropy summands. To classify them all we proved that an invariant generalized almost complex structure on F Θ is ‘constant’ in each component of the isotropy representation.

This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the boundary of the domain. One-dimensional L 1 and L p inequalities and their multidimensional analogues are proved. We consider spatial inequalities in open convex domains with the finite inner radius. Constants in these inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases.

Given β ∈ ( 1 , 2 ] , the β -transformation T β : x ↦ β x ( mod 1 ) on the circle [ 0 , 1 ) with a hole [ 0 , t ) was investigated by Kalle et al. (2019). They described the set-valued bifurcation set E β ≔ { t ∈ [ 0 , 1 ) : K β ( t ′ ) ≠ K β ( t ) ∀ t ′ > t } , where K β ( t ) ≔ { x ∈ [ 0 , 1 ) : T β n ( x ) ≥ t ∀ n ≥ 0 } is the survivor set. In this paper we investigate the dimension bifurcation set ℬ β ≔ { t ∈ [ 0 , 1 ) : dim H K β ( t ′ ) ≠ dim H K β ( t ) ∀ t ′ > t } , where dim H denotes the Hausdorff dimension. We show that if β ∈ ( 1 , 2 ] is a multinacci number then the two bifurcation sets ℬ β and E β coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for β a multinacci number we have dim H ( E β ∩ [ t , 1 ] ) = dim H K β ( t ) for any t ∈ [ 0 , 1 ) . This confirms a conjecture of Kalle et al. for β a multinacci number.

By analogy with the classical definition, a Norm Hilbert space E is defined as a Banach space over a valued field K in which each closed subspace has an orthocomplement. In the rank one case (that is, the value group as well as the set of norms of the space are contained in [ 0 , ∞ ) ), they were described by van Rooij in his classical book of 1978, but the name itself was introduced in 1999 by Ochsenius and Schikhof for the case of spaces with an infinite rank valuation. Here we shall only consider spaces over fields with value groups contained in ( R + , ⋅ ) . Yet for the set of their norms we borrow, from the infinite rank case, the notion of a G -module. That structure allows for a greater complexity than what is offered by ordered subsets of R . In this paper we describe a new class of Norm Hilbert spaces, those in which the G -module has a convex base. Their characteristics will be the focus of our study.

Let $X$ be a Noetherian separated and finite dimensional scheme over a field $\mathbb{K}$ of characteristic zero. The goal of this paper is to study deformations of $X$ over a differential graded local Artin $\mathbb{K}$-algebra by using local Tate-Quillen resolutions, i.e., the algebraic analog of the Palamodov's resolvent of a complex space. The above goal is achieved by describing the DG-Lie algebra controlling deformation theory of a diagram of differential graded commutative algebras, indexed by a direct Reedy category., Final version. To appear in Indagationes Mathematicae

This paper establishes the mapping properties of integral operators on space of bounded mean oscillation and Campanato spaces. In particular, we have the Hardy’s inequality and the boundedness of the Hadamard fractional integrals on space of bounded mean oscillation and Campanato spaces.

In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, R -graph, central graph. Also, it generalizes the construction of overlay graph (Somodi et al., 2017), and consequently, Q -graph, total graph, and quasitotal graph. We denote this new graph by [ S ( G ) ] H 2 H 1 , where G is a graph and, H 1 and H 2 are suitable graphs corresponding to G . Further, we define several new unary graph operations which becomes particular cases of this construction. We determine the Adjacency and Laplacian spectra of [ S ( G ) ] H 2 H 1 for some classes of graphs G , H 1 and H 2 . From these results, we derive the L -spectrum of the graphs obtained by the unary graph operations mentioned above. As applications, these results enable us to compute the number of spanning trees and Kirchhoff index of these graphs.

This paper introduced the calculus of mappings on finite product of spaces, allowing different degrees of differentiability in the different factors. This enables us to prove an important feature in the infinite-dimensional Lie theory, the exponential law in generalized setting for locally convex spaces and for manifolds modelled on locally convex spaces.

Let n be a nonzero integer. A set of nonzero integers { a 1 , … , a m } such that a i a j + n is a perfect square for all 1 ≤ i j ≤ m is called a D ( n ) - m -tuple. In this paper, we consider the question, for a given integer n which is not a perfect square, how large and how small can be the largest element in a D ( n ) -quadruple. We construct families of D ( n ) -quadruples in which the largest element is of order of magnitude | n | 3 , resp. | n | 2 ∕ 5 .

One of the most influential versions of the classical Schwarz–Pick Lemma is probably that of Ahlfors. Pulling back a conformal semimetric on a Riemann surface under any holomorphic map from the open unit disk equipped with a Poincare metric, the curvature of which is assumed to bound from above the curvature of the Riemann surface, he successfully showed that a conformal semimetric to be compared with the Poincare metric is obtained. In the present paper, we give a comparison theorem between two conformal semimetrics of variable curvature in the same spirit. Our main theorem is a local one by its nature, but global results can be derived therefrom.

In this paper, we study the topological structure of the Hahn field F whose elements are functions from the additive abelian group of rational numbers Q to the real numbers field R , with well-ordered support. After reviewing the algebraic and order structures of the field F , we introduce different vector topologies on F that are induced by families of semi-norms, all of which are weaker than the order or valuation topology. We compare those vector topologies and we identify the weakest one which we denote by τ w and whose properties are similar to those of the weak topology on the Levi-Civita field (Shamseddine, 2010). In particular, we state and prove a convergence criterion for power series in F , τ w that is similar to that for power series on the Levi-Civita field in its weak topology (Shamseddine, 2013).

We study the class P of positive linear functionals φ on L ∞ ( [ 1 , ∞ ) ) for which φ ( f ) = α if lim x → ∞ 1 x ∫ 1 x f ( t ) d t = α . The semigroup of translations f ( x ) ↦ f ( r x ) on L ∞ ( [ 1 , ∞ ) ) , where r ∈ [ 1 , ∞ ) , plays a crucial role in the study of P . In particular, we give three different expressions of their extremal values, which can be considered main results of this paper. We also study linear functionals on l ∞ , the set of all real-valued bounded functions on natural numbers N , which extend Cesaro mean and give similar results about their extremal values, including a functional analytic proof of the classical result of Polya.

Let F Θ = G ∕ P Θ be a generalized flag manifold, where G is a real non-compact semi-simple Lie group and P Θ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow F Θ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in R n and use them to compute the boundary operator ∂ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of ∂ are 0 or ± 2 (so that Z 2 -homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse–Witten complex is solved.

In this paper, we construct a class of weighted Bergman spaces such that they can generate H ∞ in the sense of Mobius invariance. H ∞ is also equal to the intersection of these spaces. Applying the relation between H ∞ and these spaces, we show that the characterization via higher order derivatives does not hold for some of these weighted Bergman spaces.

In this paper, we will study the maximality of functions and a new Cegrell class N m ( Ω ) which is between the classes F m ( Ω ) and E m ( Ω ) . We study the Dirichlet problem, the subsolution theorem and the stability theorem for this class.

The paper is concerned with the scope of semi-intuitionistic set theories that relate to various foundational stances. It also provides a proof for a second conjecture of Feferman’s that relates the concepts for which the law of excluded middle obtains to those that are absolute with regard to the relevant test structures, or more precisely of Δ 1 complexity. The latter is then used to show that a plethora of statements is indeterminate with respect to various semi-intuitionistic set theories.

This paper sketches a way of supplementing classical mathematics with a motivation for a Brouwerian theory of free choice sequences. The idea is that time is unending, i.e. that one can never come to an end of it, but also indeterminate, so that in a branching time model only one branch represents the ‘actual’ one. The branching can be random or subject to various restrictions imposed by the creating subject. The fact that the underlying mathematics is classical makes such perhaps delicate issues as the fan theorem no longer problematic. On this model, only intuitionistic logic applies to the Brouwerian free choice sequences, and there it applies not because of any skepticism about classical mathematics, but because there is no ‘end of time’ from the standpoint of which everything about the sequences can be decided.

Brouwer’s views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent branch of logic with connections to lattice theory, topology, modal logic, and other areas. This paper aims to present a modern account of semantics for intuitionistic propositional systems. The guiding idea is that of a hierarchy of semantics, organized by increasing generality: from the least general Kripke semantics on through Beth semantics, topological semantics, Dragalin semantics, and finally to the most general algebraic semantics. While the Kripke, topological, and algebraic semantics have been extensively studied, the Beth and Dragalin semantics have received less attention. We bring Beth and Dragalin semantics to the fore, relating them to the concept of a nucleus from pointfree topology, which provides a unifying perspective on the semantic hierarchy.

The aim of this paper is to initiate a systematic study of the Orlicz Figa – Talamanca Herz algebras on locally compact groups. We also introduce and study invariant means on the dual of these algebras.

Let U n be a complex unitary group and D n be its subgroup that consists of n × n diagonal matrices whose diagonal elements are complex numbers of modulus 1. In this paper we present some general results on Jordan triple product maps between U n and U m . As a main result we present the general form of all (continuous) Jordan triple product maps from the group U n to the group D m . Also, we characterize the general form of all (continuous) maps between the groups D n and D m that preserve the Jordan triple product. These are the (continuous) maps φ : D n → D m which satisfy φ ( V W V ) = φ ( V ) φ ( W ) φ ( V ) , V , W ∈ D n .

In this paper, we study the Heinz type inequalities for mappings satisfying Poisson’s equation. Some results generalize the ones obtained by Partyka and Sakan.

Let G be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group H ( G ) to be frames and Riesz bases in terms of the group Fourier transform.

Let X be a real normed vector space and f : ( 0 , ∞ ) → X . In this paper, we prove the hyperstability of the logarithmic functional equation f x y − y f ( x ) = 0 on Γ of Lebesgue measure zero. More precisely, we prove that if f : ( 0 , ∞ ) → X satisfies ‖ f x y − y f ( x ) ‖ ≤ ϕ ( x , y ) for all ( x , y ) ∈ Γ + ⊂ { ( x , y ) : y > α ( x ) } [ resp. Γ − ⊂ { ( x , y ) : 0 y α ( x ) } ] of Lebesgue measure zero, where α : ( 0 , ∞ ) → ( 0 , ∞ ) is an arbitrary given function and ϕ : ( 0 , ∞ ) × ( 0 , ∞ ) → [ 0 , ∞ ) satisfies the condition ϕ ( x , y ) ∕ y → 0 as y → ∞ [resp. y → 0 ], then f satisfies the functional equation f ( x y ) = y f ( x ) for all x > 0 and y > 0 .