The aim of this paper is to establish the isomorphic classification of Besov spaces over [0, 1]^{d}. Using the identification of the Besov space with the -infinite direct sum of finite-dimensional spaces (which holds independently of the dimension and of the smoothness degree of the space ) we show that , , is a family of mutually non-isomorphic spaces. The only exception is the isomorphism between the spaces and , which follows from Pełczyński's isomorphism between and . We also tell apart the isomorphic classes of spaces from the isomorphic classes of Besov spaces over the Euclidean space .

We provide existence results for semilinear differential inclusions involving measures:

- (0.1)

where *A* is the infinitesimal generator of a *C*_{0}-semigroup of contractions on a separable Banach space *X* and is a right-continuous non-decreasing function. The existence of mild solutions, as well as the compactness of the solution set are obtained via Kakutani–Ky Fan's fixed point theorem in the space of regulated functions endowed with weak, respectively strong topologies. Some examples of special cases covered by our existence results have been included.

This paper studies the following nonhomogeneous elliptic system involving Hardy–Sobolev critical exponents

where , Ω is a *C*^{1} open bounded domain in containing the origin, and . The existence result of positive ground state solution is established.

Let be a closed, connected, strictly pseudoconvex CR manifold with dimension . We define the second CR Yamabe invariant in terms of the second eigenvalue of the Yamabe operator and the volume of *M* over the pseudo-convex pseudo-hermitian structures conformal to θ. Then we study when it is attained and classify the CR-sphere by its second CR Yamabe invariant. This work is motivated by the work of B. Ammann and E. Humbert on the Riemannian context.

The harmonic Neumann function is constructed for a class of hyperbolic strips inside the unit disc via the parqueting-reflection principle. As it turns out this Neumann function is related to the respective Green function, see [the second author, Green function for a hyperbolic strip and a class of related plane domains, Appl. Anal. **93**(2014), 2370–2385], in the same way as for the case of e.g. the unit disc or half planes, etc. On this basis the Neumann problem for the Poisson equation is solved in an explicit way. Such explicit solutions serve for applications as in engineering or mathematical physics.

From certain triangle functors, called nonnegative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of nonnegative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. In particular, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. Our results can also be applied to provide shorter proofs of some known results on homological conjectures.

]]>We prove an optimal theorem for a weak solution of an elliptic system in divergence form with measurable coefficients in a homogenization problem. Our theorem is sharp with respect to the assumption on the coefficients. Indeed, we allow the very rapidly oscillating coefficients to be merely measurable in one variable.

]]>The present paper establishes a duality relation for the spectra of self-affine measures. This is done under the condition of compatible pair and is motivated by a duality conjecture of Dutkay and Jorgensen on the spectrality of self-affine measures. For the spectral self-affine measure, we first obtain a structural property of spectra which indicates that one can get new spectra from old ones. We then establish a duality property for the spectra which confirms the conjecture in a certain case.

]]>By means of a new change of variable we prove the existence of a positive 2π-periodic solution for the Mathieu–Duffing type equations having its nonlinearity a super-linear growth. As result we can guarantee the existence of 2π-periodic solutions even assuming that the parameter of the associated Mathieu equation is in the contentious zone of resonance.

]]>We study the twisted Novikov homology of the complement of a complex hypersurface in general position at infinity. We give a self-contained topological proof of the vanishing (except possibly in the middle degree) of the twisted Novikov homology groups associated to positive cohomology classes of degree one defined on the complement.

]]>In this paper, we consider the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the subcritical case when the velocity dissipation dominates. More precisely, we establish the global regularity result of the 2D Boussinesq equations in a new range of fractional powers of the Laplacian, namely with . Therefore, this result significantly improves the previous work which obtained the global regularity result for with , where is an explicit function.

]]>In this paper we introduce two Bishop–Phelps–Bollobás type properties for bounded linear operators between two Banach spaces *X* and *Y*: property 1 and property 2. These properties are motivated by a Kim–Lee result which states, under our notation, that a Banach space *X* is uniformly convex if and only if the pair satisfies property 2. Positive results of pairs of Banach spaces satisfying property 1 are given and concrete pairs of Banach spaces failing both properties are exhibited. A complete characterization of property 1 for the pairs is also provided.

By Jahnke–Peternell–Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exist 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.

]]>For a locally compact quantum group we define its center, , and its quantum group of inner automorphisms, . We show that one obtains a natural isomorphism between and , we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.

]]>In this work, we introduce the concept of μ-pseudo almost automorphic processes in distribution. We use the μ-ergodic process to define the spaces of μ-pseudo almost automorphic processes in the square mean sense. We establish many interesting results on the functional space of such processes like a composition theorem. Under some appropriate assumptions, we establish the existence, the uniqueness and the stability of the square-mean μ-pseudo almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise. We provide an example to illustrate our results.

]]>We show that a Schrödinger operator with a δ-interaction of strength α supported on a bounded or unbounded *C*^{2}-hypersurface , can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator with a singular interaction is regarded as a self-adjoint realization of the formal differential expression , where is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.

We prove unique existence of mild solutions on for the Navier–Stokes equations in an exterior domain in , subject to the non-slip boundary condition.

]]>Efficient sufficient conditions are established for the solvability of the mixed problem

where in the case where the homogeneous linear problem has nontrivial solutions.

]]>The volume of the unit ball of the Lebesgue sequence space is very well known since the times of Dirichlet. We calculate the volume of the unit ball of the mixed norm , whose special cases are nowadays popular in machine learning under the name of group Lasso. We give two proofs of the main results, one in the spirit of Dirichlet, the other one using polarization identities. The result is given by a closed formula involving the gamma function, only slightly more complicated than the one of Dirichlet. We consider the real as well as the complex case. We also consider the anisotropic unit balls. We close by an overview of open problems.

]]>We construct Calabi–Yau 3-fold orbifolds embedded in weighted projective space in codimension 4. Each Hilbert series we consider is realised by at least two deformation families of Calabi–Yau 3-folds, distinguished by their topology, echoing a similar phenomenon for Fano 3-folds in high codimension.

]]>For the following singularly perturbed problem

we construct a solution which concentrates at several given isolated positive local minimum components of *V* as . Here, the nonlinearity *f* is of critical growth. Moreover, the monotonicity of and the so-called Ambrosetti–Rabinowitz condition are not required.

We investigate the relationship among several numerical invariants associated to a (free) projective hypersurface *V*: the sequence of mixed multiplicities of its Jacobian ideal, the Hilbert polynomial of its Milnor algebra, and the sequence of exponents when *V* is free. As a byproduct, we obtain explicit equations for some of the homaloidal surfaces in the projective 3-dimensional space constructed by C. Ciliberto, F. Russo and A. Simis.

In this paper, we investigate the locally uniformly non-square point of Orlicz–Bochner sequence spaces endowed with Luxemburg norm. Analysing and combining the generating function *M* and properties of the real Banach space *X*, we get sufficient and necessary conditions of locally uniformly non-square point, which contributes to criteria for locally uniform non-squareness in Orlicz–Bochner sequence spaces. The results generalize the corresponding results in the classical Orlicz sequence spaces.

A sufficient condition for higher-order Sobolev-type embeddings on bounded domains of Carnot–Carathéodory spaces is established for the class of rearrangement-invariant function spaces. The condition takes form of a one-dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.

]]>We consider rational varieties with a torus action of complexity one and extend the combinatorial approach via the Cox ring developed for the complete case in earlier work to the non-complete, e.g. affine, case. This includes in particular a description of all factorially graded affine algebras of complexity one with only constant homogeneous invertible elements in terms of canonical generators and relations.

]]>In this paper, we study the topology of real analytic map-germs with isolated critical value , with . We compare the topology of *f* with the topology of the compositions , where are the projections , for . As a main result, we give necessary and sufficient conditions for *f* to have a Lê–Milnor fibration in the tube.

In this paper we first introduce the concept of a double modified analytic function space Fourier–Feynman transform using the double modified analytic function space integral. We then proceed to establish the existence of the modified analytic function space Fourier–Feynman transform for all functionals in the Banach algebra. Finally we use this double modified analytic function space transform to explain various physical phenomenon.

]]>We consider the general Hardy type operator where is a positive and measurable kernel. To characterize the weights *u* and *v* so that is still an open problem for any parameters *p* and *q*. However, for special cases the solution is known for some parameters *p* and *q*. In this paper the current status of this problem is described and discussed mainly for the case In particular, some new scales of characterizations in classical situations are described, some new proofs and results are given and open questions are raised.

The centroid of a subset of with positive volume is a well-known characteristic. An interesting task is to generalize its definition to at least some sets of zero volume. In the presented paper we propose two possible ways how to do that. The first is based on the Hausdorff measure of an appropriate dimension. The second is given by the limit of centroids of ε-neighbourhoods of the particular set when ε goes to 0. For both generalizations we discuss their existence and basic properties. Then we focus on sufficient conditions of existence of the second generalization and on conditions when both generalizations coincide. It turns out that they can be formulated with the help of the Minkowski content, rectifiability, and self-similarity. Since the centroid is often used in stochastic geometry as a centre function for certain particle processes, we present properties that are needed for both generalizations to be valid centre functions. Finally, we also show their continuity on compact convex *m*-sets with respect to the Hausdorff metric topology.

In this paper, we discuss the invariance property of Property under holomorphic maps on any compact subset in . An obstruction to Property of any compact subset regarding the fine topology is also given, which generalizes a classical result of N. Sibony on the complex plane.

]]>Given non-negative measurable functions on we study the high dimensional Hardy operator between Orlicz–Lorentz spaces , where *f* is a measurable function of and is the ball of radius *t* in . We give sufficient conditions of boundedness of and . We investigate also boundedness and compactness of between weighted and classical Lorentz spaces. The function spaces considered here do not need to be Banach spaces. Specifying the weights and the Orlicz functions we recover the existing results as well as we obtain new results in the new and old settings.

For a bounded sequence of matrices defining a nonautonomous dynamics with discrete time, we obtain all possible relations between the regularity coefficients introduced by Lyapunov, Perron and Grobman. This includes considering general inequalities between the coefficients and showing that these inequalities are the best possible, in the sense that for any three nonnegative numbers satisfying them, and for no others, there exists a bounded sequence of matrices having the numbers respectively as Lyapunov, Perron and Grobman coefficients. Moreover, we establish inequalities between the three coefficients and some other regularity coefficients.

]]>Precise descriptions of the spaces associated with weighted Sobolev spaces on the real line are given.

]]>We consider grand Lebesgue spaces on sets of infinite measure and study the dependence of these spaces on the choice of the so-called. We also consider Mikhlin and Marcinkiewicz theorems on Fourier multipliers in the setting of grand spaces.

]]>We define Atkinson's semi-definite *p*-Laplacian eigenvalue problems, which include the regular *p*-Laplacian eigenvalue problems with *L*^{1} coefficient functions. Then we show that the Sturm oscillation theorem also holds for this eigenvalue problem.

A hypersurface of the space form has a canonical principal direction (CPD) relative to the closed and conformal vector field *Z* of if the projection of *Z* to *M* is a principal direction of *M*. We show that CPD hypersurfaces with constant mean curvature are foliated by isoparametric hypersurfaces. In particular, we show that a CPD surface with constant mean curvature of space form is invariant by the flow of a Killing vector field whose action is polar on . As consequence we show that a compact CPD minimal surface of the sphere is a Clifford torus. Finally, we consider the case when a CPD Euclidean hypersurface has zero Gauss–Kronecker curvature.

In this paper, we study the well-posedness of the degenerate differential equations with fractional derivative in Lebesgue–Bochner spaces , periodic Besov spaces and periodic Triebel–Lizorkin spaces , where *A* and *M* are closed linear operators in a complex Banach space *X* satisfying , and is the fractional derivative in the sense of Weyl. Using known operator-valued Fourier multiplier results, we completely characterize the well-posedness of this problem in the above three function spaces by the *R*-bounedness (or the norm boundedness) of the *M*-resolvent of *A*.

We prove that the locus of Hilbert schemes of *n* points on a projective *K*3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well. Along the way we prove an integral constraint on the monodromy group of generalized Kummer manifolds.

We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand–Shilov spaces. Moreover, we deduce composition and certain invariance properties of these classes.

]]>As outlined below, this paper is devoted to a Carleson-type-measure-based study of the holomorphic Campanato 2-space on the open unit ball of , comprising all Hardy 2-functions whose oscillations in non-isotropic metric balls on the compact unit sphere are proportional to some power of the radius other than the dimension .

]]>The contribution of this paper is two-fold. The first one is to derive a simple formula of the mean curvature form for a hypersurface in the Randers space with a Killing vector field, by considering the Busemann–Hausdorff measure and Holmes–Thompson measure simultaneously. The second one is to obtain the explicit local expressions of two types of nontrivial rotational BH-minimal surfaces in a Randers domain of constant flag curvature , which are the first examples of BH-minimal surfaces in the hyperbolic Randers space.

]]>The article deals with the class consisting of non-vanishing functions *f* that are analytic and univalent in such that the complement is a convex set, and the angle at ∞ is less than or equal to for some . Related to this class is the class of concave univalent mappings in , but this differs from with the standard normalization A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for settled by Avkhadiev et al. . Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for is also presented.

We consider the initial value problem of the 3D incompressible Boussinesq equations for rotating stratified fluids. We establish the dispersive and the Strichartz estimates for the linear propagator associated with both the rotation and the stable stratification. As an application, we give a simple proof of a unique existence of global in time solutions to our system using just a contraction mapping principle.

]]>In this note, we first prove the non-degeneracy property of extremals for the optimal Hardy–Littlewood–Sobolev inequality on the Heisenberg group, as an application, a perturbation result for the CR fractional Yamabe problem is obtained, this generalizes a classical result of Malchiodi and Uguzzoni .

]]>The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on , in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.

]]>In this paper, we develop the theory for a family of neural network (NN) operators of the Kantorovich type, in the general setting of Orlicz spaces. In particular, a modular convergence theorem is established. In this way, we study the above family of operators in many instances of useful spaces by a unique general approach. The above NN operators provide a constructive approximation process, in which the coefficients, the weights, and the thresholds of the networks needed in order to approximate a given function *f*, are known. At the end of the paper, several examples of Orlicz spaces, and of sigmoidal activation functions for which the present theory can be applied, are studied in details.

Let be a connected reductive complex algebraic group with split real form . Consider a strict wonderful -variety **X** equipped with its σ-equivariant real structure, and let *X* be the corresponding real locus. Further, let *E* be a real differentiable *G*-vector bundle over *X*. In this paper, we introduce a distribution character for the regular representation of *G* on the space of smooth sections of *E* given in terms of the spherical roots of , and show that on a certain open subset of *G* of transversal elements it is locally integrable and given by a sum over fixed points.

A geodesic in a homogeneous Finsler space is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of *G*. A homogeneous Finsler space is called Finsler g.o. space if its all geodesics are homogeneous. Recently, the author studied Finsler g.o. spaces and generalized some geometric results on Riemannian g.o. spaces to the Finslerian setting. In the present paper, we investigate homogeneous geodesics in homogeneous spaces, and obtain the sufficient and necessary condition for an space to be a g.o. space. As an application, we get a series of new examples of Finsler g.o. spaces.

Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of 2*n*-dimensional nondegenerate hypersurfaces ruled by *n*-planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with respect to which the ruling is Lagrangian.

In this paper we consider the *k*-plane Nikodym maximal estimates in the variable Lebesgue spaces . We first formulate the problem about the boundedness of the *k*-plane Nikodym maximal and show that the maximal estimate in is equivalent to that in for . So, the optimal Nikodym maximal estimate in follows from Cordoba's estimate.

We consider a planar Riemann surface *R* made of a non-compact simply connected plane domain from which an infinite discrete set of points is removed. We give several conditions for the collars of the cusps in *R* caused by these points to be uniformly distributed in *R* in terms of Euclidean geometry. Then we associate a graph *G* with *R* by taking the Voronoi diagram for the uniformly distributed cusps and show that *G* represents certain geometric and analytic properties of *R*.

We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous *m*-homogeneous non-analytic polynomials on *c*_{0} contains an isomorphic copy of ℓ_{1}. Moreover, we can have this copy of ℓ_{1} in such a way that every non-zero element of it fails to be analytic at precisely the same point.

This paper addresses the problem of well-posedness of non-autonomous linear evolution equations in uniformly convex Banach spaces. We assume that for each *t* is the generator of a quasi-contractive, strongly continuous group, where the domain *D* and the growth exponent are independent of *t*. Well-posedness holds provided that is Lipschitz for all . Hölder continuity of degree is not sufficient and the assumption of uniform convexity cannot be dropped.

We study the maximal immediate extensions of valued fields whose residue fields are perfect and whose value groups are divisible by the residue characteristic if it is positive. In the case where there is such an extension which has finite transcendence degree we derive strong properties of the field and the extension and show that the maximal immediate extension is unique up to isomorphism, although these fields need not be Kaplansky fields. If the maximal immediate extension is an algebraic extension, we show that it is equal to the perfect hull and the completion of the field.

]]>It is known that applying an -homothetic deformation to a complex contact manifold whose vertical space is annihilated by the curvature yields a condition which is invariant under -homothetic deformations. A complex contact manifold satisfying this condition is said to be a complex -space.

In this paper, we deal with the questions of Bochner, conformal and conharmonic flatness of complex -spaces when , and prove that such kind of spaces cannot be Bochner flat, conformally flat or conharmonically flat.

We show the existence of a nodal solution with two nodal domains for a generalized Kirchhoff equation of the type

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

We show a picture of the relations among different types of summability of series in the space of integrable functions with respect to a vector measure *m* of relatively norm compact range. In order to do that, we study the class of the so-called *m*-1-summing operators. We give several applications regarding the existence of copies of *c*_{0} in , as well as on *m*-1-summing operators which are weakly compact, Asplund or weakly precompact.

In this paper, we give a new characterization for the boundedness of weighted differentiation composition operator from logarithmic Bloch spaces to Bloch-type spaces and calculate its essential norm in terms of the *n*-th power of induced analytic self-map on the unit disk. From which a sufficient and necessary condition of compactness of this operator follows immediately.

We present existence theorems for coupled system of quadratic integral equations of generalized Chandrasekhar type which has numerous application (cf. , , and ). Also, the asymptotic stability of solutions will be considered.

]]>It is well-known that for a general operator *T* on Hilbert space, if *T* is subnormal, then is subnormal for all natural numbers . It is also well-known that if *T* is hyponormal, then *T*^{2} need not be hyponormal. However, for a unilateral weighted shift , the hyponormality of (detected by the condition for all ) does imply the hyponormality of every power . Conversely, we easily see that for a weighted shift is not hyponormal, therefore not subnormal, but is subnormal for all . Hence, it is interesting to note when for some , the subnormality of implies the subnormality of *T*. In this article, we construct a non trivial large class of weighted shifts such that for some , the subnormality of guarantees the subnormality of . We also prove that there are weighted shifts with non-constant tail such that hyponormality of a power or powers does not guarantee hyponormality of the original one. Our results have a partial connection to the following two long-open problems in Operator Theory: (i) characterize the subnormal operators having a square root; (ii) classify all subnormal operators whose square roots are also subnormal. Our results partially depend on new formulas for the determinant of generalized Cauchy–Hankel matrices and on criteria for their positive semi-definiteness.

We give for generalized Durrmeyer type series and their linear combinations quantitative Voronosvskaja formulae in terms of the classical Peetre K-functional. Finally we apply the general theory to various kernels

]]>In this paper necessary and sufficient conditions are deduced for the close-to-convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag–Leffler expansions for Bessel functions of the first kind.

]]>Three spectral problems generated by the same Sturm–Liouville equation are considered: Neumann–Dirichlet problem (the Neumann condition at the left end and the Dirichlet condition at the right end) on the whole interval [0, *a*], Neumann–Dirichlet problem on and Dirichlet–Dirichlet problem on . The three spectra inverse problem, i.e. the problem of recovering the Sturm–Liuville equation using the three spectra of these boundary value problems is completely solved.

This paper deals with dimension-controllable (tractable) embeddings of Besov spaces on *n*-dimensional torus into small Lebesgue spaces. Our techniques rely on the approximation structure of Besov spaces, extrapolation properties of small Lebesgue spaces and interpolation.

Let be the ring of (continuous) semialgebraic functions on a semialgebraic set *M* and its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps and induced by the inclusion of a semialgebraic subset *N* of *M*. The ring can be understood as the localization of at the multiplicative subset of those bounded semialgebraic functions on *M* with empty zero set. This provides a natural inclusion that reduces both problems above to an analysis of the fibers of the spectral map . If we denote , it holds that the restriction map is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of at the points of *Z*. The size of the fibers of prime ideals “close” to the complement provides valuable information concerning how *N* is immersed inside *M*. If *N* is dense in *M*, the map is surjective and the generic fiber of a prime ideal contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber is a finite set for . If such is the case, our procedure allows us to compute the size *s* of . If in addition *N* is locally compact and *M* is pure dimensional, *s* coincides with the number of minimal prime ideals contained in .

In this paper we study direct and inverse problems for discrete and continuous skew-selfadjoint Dirac systems with rectangular (possibly non-square) pseudo-exponential potentials. For such a system the Weyl function is a strictly proper rational matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a nonlinear generalized discrete Heisenberg magnet model, extending earlier results for the isotropic case. A large part of the paper is devoted to the related discrete systems of which the pseudo-exponential potential depends on an additional continuous time parameter. Our technique allows us to obtain explicit solutions for the generalized discrete Heisenberg magnet model and evolution of the Weyl functions.

]]>In the first part of the paper we study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fréchet space . In the second one non homogeneous microlocal properties are introduced and propagation of Sobolev singularities for solutions to (pseudo)differential equations is given. For both the arguments actual examples are provided.

]]>In this work we show that if is a linear differential operator of order ν with smooth complex coefficients in from a complex vector space *E* to a complex vector space *F*, the Sobolev a priori estimate

holds locally at any point if and only if is elliptic and the constant coefficient homogeneous operator is canceling in the sense of Van Schaftingen for every which means that

Here is the homogeneous part of order ν of and is the principal symbol of . This result implies and unifies the proofs of several estimates for complexes and pseudo-complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.

]]>In this paper, necessary conditions and sufficient conditions for the irregular shearlet systems to be frames are studied. We show that the irregular shearlet systems to possess upper frame bounds, the space-scale-shear parameters must be relatively separated. We prove that if the irregular shearlet systems possess the lower frame bound and the space-scale-shear parameters satisfy certain condition, then the lower shearlet density is strictly positive. We apply these results to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems. We prove that for a feasible class of shearlet generators introduced by P. Kittipoom et al., each relatively separated sequence with sufficiently hight density will generate a frame. Explicit frame bounds are given. We also study the stability of shearlet frames and show that a frame generated by certain shearlet function remains a frame when the space-scale-shear parameters and the generating function undergo small perturbations. Explicit stability bounds are given. Using pseudo-spline functions of type I and II, we construct a family of irregular shearlet frames consisting of compactly supported shearlets to illustrate our results.

]]>In this paper, we introduce the notion of a left-symmetric algebroid, which is a generalization of a left-symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie algebroid. We construct left-symmetric algebroids from -operators on Lie algebroids. We study phase spaces of Lie algebroids in terms of left-symmetric algebroids. Representations of left-symmetric algebroids are studied in detail. At last, we study deformations of left-symmetric algebroids, which could be controlled by the second cohomology class in the deformation cohomology.

]]>By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups with respect to the standard -Wasserstein distance for all . In particular, we show that for the Itô stochastic differential equation

if the drift term *b* is such that for any ,

holds with some positive constants *K*_{1}, *K*_{2} and , then there is a constant such that for all , and ,

where is a positive constant. This improves the main result in where the exponential convergence is only proved for the *L*^{1}-Wasserstein distance.

We study the capitulation problem of the 2-class group of some cyclic number fields *M* with large degree and 2-class group isomorphic to . Precisely, we give the structure of the Galois group of the maximal unramified 2-extension over *M*.

We consider the equation which is called Holling–Tanner population model

where is a bifurcation parameter and are unknown constants. In this paper, we determine the unknown constants from the asymptotic behavior of the bifurcation curve , where .

]]>We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We show that an operator of higher order Schrödinger type is a Calderón–Zygmund operator. We also show that there exist potentials which satisfy our assumptions but are not nonnegative polynomials.

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