Some non-archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non-spherically complete base fields, to the following problem, posed in 13, p. 95: Does there exist an absolutely convex edged set B in a non-archimedean locally convex space such that its closure is not edged?

We study limit K-spaces for general Banach couples, not necessarily ordered. They correspond to the extreme choice θ = 0, 1 in the realization of the real method as a K-space. We also show the connection of these limit spaces with interpolation methods defined by the unit square.

Beginning in 2006, G. Gentili and D. C. Struppa developed a theory of regular quaternionic functions with properties that recall classical results in complex analysis. For instance, in each Euclidean ball B(0, R) centered at 0 the set of regular functions coincides with that of quaternionic power series converging in B(0, R). In 2009 the author proposed a classification of singularities of regular functions as removable, essential or as poles and studied poles by constructing the ring of quotients. In that article, not only the statements, but also the proving techniques were confined to the special case of balls B(0, R). Quite recently, F. Colombo, G. Gentili and I. Sabadini (2010) and the same authors in collaboration with D. C. Struppa (2009) identified a larger class of domains, on which the theory of regular functions is natural and not limited to quaternionic power series. The present article studies singularities in this new context, beginning with the construction of the ring of quotients and of Laurent-type expansions at points p other than the origin. These expansions, which differ significantly from their complex analogs, allow a classification of singularities that is consistent with the one given in 2009. Poles are studied, as well as essential singularities, for which a version of the Casorati-Weierstrass Theorem is proven.

In this paper we study the Cauchy-Kowalewski extension of real analytic functions satisfying a system of differential equations connected to bicomplex analysis, and we use this extension to study the product in the space of bicomplex holomorphic functions. We also show how these ideas can be used to define a Fourier transform for bicomplex holomorphic functions.

To each irreducible infinite dimensional representation of a C*-algebra , we associate a collection of irreducible norm-continuous unitary representations of its unitary group , whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group are. These are precisely the representations arising in the decomposition of the tensor products under . We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which acts transitively and that the corresponding norm-closed momentum sets distinguish inequivalent representations of this type.

We consider spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with classes of locally convex topologies, which include and generalize various known ones such as weighted space- or inductive limit-type topologies. The main result states that every continuous linear functional on such a function space can be expressed as an integral with respect to some canonical (dual space-valued) vector measure.

We consider the whole-line inverse scattering problem for Sturm-Liouville equations which have constant coefficients on a half-line. Since in this case the reflection coefficient determines a Weyl-Titchmarsh m-function, it determines the coefficients up to some simple Liouville transformations. Given inverse spectral theory, proofs are fairly simple but provide extensions of known results as we require less smoothness and less decay than is customary.

Let X be an open subset of and ν the restriction of the usual Lebesgue measure of to X. In this paper, we investigate properties of the range of positive integral operators on L²(X, ν), in connection with the reproducing kernel Hilbert space of the generating kernel. Assuming differentiability assumptions on the kernel, we deduce smoothness properties for the functions in the range of the operator and also properties of the so-called inclusion map. The results are deduced when the assumptions are defined via both, weak and partial derivatives. Further, assuming the generating kernel has a Mercer-like expansion based on sufficiently smooth functions, we deduce results on the term-by-term differentiability of the series and reproducing properties for the derivatives of the functions in the reproducing kernel Hilbert space.

In this paper we study three dimensional homogeneous Finsler manifolds. We first obtain a complete list of the three-dimensional homogeneous manifolds which admit invariant Finsler metrics. Then we consider invariant Randers metrics and present the classification of three dimensional homogeneous Randers spaces under isometrics.

We develop a maximal regularity approach in temporally weighted L_p-spaces for vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions, both of static and of relaxation type. Normal ellipticity and conditions of Lopatinskii-Shapiro type are the basic structural assumptions. The weighted framework allows to reduce the initial regularity and to avoid compatibility conditions at the boundary, and it provides an inherent smoothing effect of the solutions. Our main tools are interpolation and trace theory for anisotropic Slobodetskii spaces with temporal weights, operator-valued functional calculus, as well as localization and perturbation arguments.

We consider a constrained minimal energy problem with an external field over noncompact classes of vector measures (μⁱ)_{i ∈ I} of finite or infinite dimensions on a locally compact space. The components μⁱ are nonnegative Radon measures satisfying normalizing conditions, supported by given A_i and such that σⁱ − μⁱ ≥ 0, the constraints σⁱ, i ∈ I, being given. For a particular matrix of interaction between μⁱ, i ∈ I, and a rather general class of positive definite kernels, sufficient conditions for the existence of minimizers are established and their uniqueness and vague compactness are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also analyze continuity properties of minimizers in the vague and strong topologies when A_i and σⁱ are varied. Almost all results are valid also for classical kernels in , which is important in applications.

Dedicated to Professor V. I. Burenkov on the occasion of his 70th birthday We characterize the traces of vector-valued Besov and Lizorkin-Triebel spaces. Therefrom we derive the corresponding assertions for the vector-valued Sobolev spaces . Here we do not assume the UMD property for the Banach space E.

In this work we consider a viscoelastic problem with a kernel whose derivative may be positive on some subsets. Starting from a “generalized” necessary condition we show that it is in fact also sufficient to prove a decay of arbitrary type provided that the non-decreasingness rate or/and zone of the kernel is small enough.

We study Esteves's fine compactified Jacobians for nodal curves. We give a proof of the fact that, for a one-parameter regular local smoothing of a nodal curve X, the relative smooth locus of a relative fine compactified Jacobian is isomorphic to the Néron model of the Jacobian of the general fiber, and thus it provides a modular compactification of it. We show that each fine compactified Jacobian of X admits a stratification in terms of certain fine compactified Jacobians of partial normalizations of X and, moreover, that it can be realized as a quotient of the smooth locus of a suitable fine compactified Jacobian of the total blowup of X. Finally, we determine when a fine compactified Jacobian is isomorphic to the corresponding Oda-Seshadri's coarse compactified Jacobian.

We give an intrinsic estimate of the number of connected components of the complementary set to the amoeba of an exponential sum with real spectrum improving the result of Forsberg, Passare and Tsikh in the polynomial case and that of Ronkin in the exponential one.

The positivity preserving property for the biharmonic operator with Dirichlet boundary condition is investigated. We discuss here the case where the domain is an ellipse (that may degenerate to a strip) and the data is a polynomial function. We provide various conditions for which the positivity is preserved.

We consider the space of complete rank two collineations. Starting from its description as a limit of GIT-quotients of a Grassmanian G(2, n) by a certain -action, we determine the Cox ring by means of toric ambient modifications.

Slice regular functions have been introduced in 20 as solutions of a special partial differential operator with variable coefficients. As such they do not naturally form a sheaf. In this paper we use a modified definition of slice regularity, see 21, to introduce the sheaf of slice regular functions with values in in the algebra of quaternions and, more in general, in a Clifford algebra and we study its cohomological properties. We show that the first cohomology group with coefficients in the sheaf of slice regular functions vanishes for any open set in the space of quaternions (resp. the space of paravectors in ). However, we prove that not all the open sets are domains of slice regularity but only those special sets which are axially symmetric, i.e., invariant with respect to rotations that fix the real axis.

We give sufficient conditions on a class of R-modules in order for the class of complexes of -modules, , to be covering in the category of complexes of R-modules. More precisely, we prove that if is precovering in R − Mod and if is closed under direct limits, direct products, and extensions, then the class is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module C_n is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.

We classify the compatible left-symmetric algebraic structures on the Witt algebra satisfying certain non-graded conditions. It is unexpected that they are Novikov algebras. Furthermore, as applications, we study the induced non-graded modules of the Witt algebra and the induced Lie algebras by Novikov-Poisson algebras’ approach and Balinskii-Novikov's construction.

We explore the interlacing between model category structures attained to classes of modules of finite -dimension, for certain classes of modules . As an application we give a model structure approach to the finitistic dimension conjectures and present a new conceptual framework in which these conjectures can be studied.

For any , we determine the optimal constant such that the following holds. If is the Haar system on [0,1], then for any vectors a_k from a separable Hilbert space and , we have

This is generalized to the sharp weak-type inequality

where X, Y stand for -valued martingales such that Y is differentially subordinate to X.

In this paper, we study the inversion formula for recovering a function from its windowed Fourier transform. We give a rigorous proof for an inversion formula which is known in engineering. We show that the integral involved in the formula is convergent almost everywhere on as well as in L^p for all 1 < p < ∞ if the function to be reconstructed is.

Let F/E be an abelian Galois extension of function fields over an algebraic closed field K of characteristic p > 0. Denote by G the Galois group of the extension F/E. In this paper, we study Ω(m), the space of holomorphic m-(poly)differentials of the function field of F when G is cyclic or a certain elementary abelian group of order pⁿ; we give bases for each case when the base field is rational, introduce the Boseck invariants and give an elementary approach to the G module structure of Ω(m) in terms of Boseck invariants. The last computation is achieved without any restriction on the base field in the cyclic case, while in the elementary abelian case it is assumed that the base field is rational. Finally, an application to the computation of the tangent space of the deformation functor of curves with automorphisms is given.

We prove that there exists a rank 1 perturbation of a unitary operator on a complex separable infinite dimensionai Hilbert space which is hypercyclic.

The purpose of this paper is to introduce five martingale Orlicz-Hardy spaces and to establish the atomic decomposition theorem. As applications we show the relation among five martingale Orlicz-Hardy spaces and the duality, namely, the dual of martingale Orlicz-Hardy spaces are generalized martingale Campanato spaces. Further, we prove a John-Nirenberg type inequality for generalized martingale Campanato spaces when the stochastic basis is regular.

We reconsider the fundamental work of Fichtner 2 and exhibit the permanental structure of the ideal Bose gas again, using a new approach which combines a characterization of infinitely divisible random measures (due to Kerstan, Kummer and Matthes 4, 6 and Mecke 9, 10) with a decomposition of the moment measures into its factorial measures due to Krickeberg 5. To be more precise, we exhibit the moment measures of all orders of the general ideal Bose gas in terms of certain “loop” integrals. This representation can be considered as a point process analogue of the old idea of Symanzik 15 that local times and self-crossings of the Brownian motion can be used as a tool in quantum field theory.

Behind the notion of a general ideal Bose gas there is a class of infinitely divisible point processes of all orders with a Lévy-measure belonging to some large class of measures containing that of the classical ideal Bose gas considered by Fichtner.

It is well-known that the calculation of moments of higher order of point processes is notoriously complicated. See for instance Krickeberg’s calculations for the Poisson or the Cox process in 5.

Relations to the work of Shirai, Takahashi 12 and Soshnikov 14 on permanental and determinantal processes are outlined.

We consider the following Keller-Segel system of degenerate type: (KS): where m > 1, γ > 0, q ⩾ 2m. We shall first construct a weak solution u(x, t) of (KS) such that u^{m − 1} is Lipschitz continuous and such that for δ > 0 is of class C¹ with respect to the space variable x. As a by-product, we prove the property of finite speed of propagation of a weak solution u(x, t) of (KS), i.e., that a weak solution u(x, t) of (KS) has a compact support in x for all t > 0 if the initial data u₀(x) has a compact support in . We also give both upper and lower bounds of the interface of the weak solution u of (KS).

In this paper, we study anisotropic Bessel potential and Besov spaces, where the anisotropy measures the extra amount of regularity in certain directions. Some basic properties of these spaces are established along with applications to elliptic boundary value problems.

By using the variant version of Mountain Pass Theorem, the existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems is obtained. The result obtained generalizes and improves some known works.

We develop a precise analysis of J. O’Hara’s knot functionals E^(α), α ∈ [2, 3), that serve as self-repulsive potentials on (knotted) closed curves. First we derive continuity of E^(α) on injective and regular H² curves and then we establish Fréchet differentiability of E^(α) and state several first variation formulae. Motivated by ideas of Z.-X. He in his work on the specific functional E⁽²⁾, the so-called Möbius Energy, we prove C^∞-smoothness of critical points of the appropriately rescaled functionals by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.

Recently, importance of the Besov space has been acknowledged by analysts studing such subsets with lower dimension than the whole space as fractals in the Euclidean space. On the other hand, by taking an extension K of local field K′, K′ is contained in K as a subset with lower dimension than the whole space K and the present author showed that it can be viewed as a d-set of K in terms of fractal analysis. In this article, Besov spaces on separable extension of the field of p-adic numbers and on its d-set will be consistently introduced. After that, a trace theorem showing a relationship between those Besov spaces will be presented and finally a penetrating stochastic process into K′ will be addressed as an application of the trace theorem.

In this paper, we introduce the compositions of smooth Wiener-Poisson functional and Schwartz distribution by using Malliavin calculus jump type. We also prove Watanabe's theorem in the jump-diffusion case, that is, the asymptotic expansion formula for the compositions of a smooth Wiener-Poisson functional with Schwartz distributions.

Let G be a locally compact unimodular group. Let dμ = ϕ dλ be a probability measure with continuous density ϕ w.r.t. the Haar measure λ. One of the important characteristics of the random walk on G driven by μ is the probability of return at time n to a small neighborhood of the starting point, a quantity controlled by ϕ⁽ⁿ⁾(e). This paper develops the idea of discrete subordination in this context. To any Bernstein function ψ, we associate a new measure μ_ψ, the ψ-subordinate of μ, and we discuss the moment properties of μ_ψ as well as the behavior of .

This paper is concerned with the following problem

where , p > 1, q₁, q₂ > 0 and Ω is a bounded domain in with smooth boundary. We will give a rather complete classification of the parameters (m, q₁, q₂) based on the global existence and blow up features of solutions. In particular, it is shown that there is a threshold m_c, such that if m < m_c, then there exists a critical Fujita exponent pair (q₁, q₂) for the nonlinear sources, while if m ≥ m_c, such exponent pair does not exist.

In this paper, we give sufficient conditions for the uniform boundedness and uniform ultimate boundedness of solutions of a class of retarded functional differential equations with impulse effects acting on variable times. We employ the theory of generalized ordinary differential equations to obtain our results. As an example, we investigate the boundedness of the solution of a circulating fuel nuclear reactor model.

Let k be an algebraically closed field of prime characteristic p > 2, and let S(n) be the special Lie superalgebra over k. The iso-classes of simple restricted modules of these algebras are classified, and the character formulas of restricted simple modules are given.

We obtain new conditions under which a system of linear functional differential equations with singular coefficients has a unique solution possessing a given initial value and satisfying a certain growth restriction.

We offer a groupoid-theoretic approach to computing invariants. We illustrate this approach by describing the Gel'fand-MacPherson correspondence and the Gale transform. We also provide Zariski-local descriptions of the moduli space of ordered points in . We give an explicit description of the moduli space over . In characteristic 2, the singularity at the totally ramified cover is isomorphic to the affine cone over the Veronese embedding .

We consider the anisotropic mean curvature flow for parametrized hypersurfaces and provide a new definition of gradient flow that takes into account the anisotropic nature of space. With this new approach we succeed in identifying the natural candidate for the anisotropic mean curvature vector using a variational method. Under the obtained flow Wulff shapes shrink self similarly. The new definition of gradient flow relies on understanding which Banach structure plays a fundamental role and performing a consistent choice of function spaces. The geometric setting we introduce also allows us to provide and justify a natural formulation for the anisotropic Willmore functional. Its first variation is computed.

We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle πα, α ∈ (1, 2], at infinity. In this paper, we show that every such function is close-to-convex of order (α − 1) and is included in the set of univalent functions of bounded boundary rotation. Many interesting consequences of this result are obtained. We also determine the extreme points of the set of concave functions with respect to the linear structure of the Hornich space.

We investigate the problem of existence of free alternating group actions on products of equidimensional spheres. Most notably, we prove that for a prime number p ≥ 7, the alternating group cannot act freely on a finite-dimensional CW-complex with the integral cohomology ring isomorphic to that of (Sⁿ)^p for any positive integer n. This extends previous results due to L. P. Plakhta.

In this paper, we study the fixed-point theorems for a class of mixed monotone operators. Using Thompson's metric techniques and iterative methods, we construct monotone sequences which converge to the unique fixed point of the operator. Our results extend previous works of Guo 5 and Huang-Huang-Tsai 8. Applications to nonlinear second/fourth order boundary-value problems are considered.

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems depending on a parameter which include as particular cases the Dirichlet problem and the Robin problem. By using Schauder's fixed point theorem and the Leray–Schauder degree, we derive lower bounds on the number of solutions of our problem. The results here extend earlier theorems due to Kazdan–Warner and also Amann–Hess to the degenerate case.

We study the possibility of using Krylov-Bogolyubov averaging to find solutions of some class of differential equations that tend to constant vectors as t → ∞. We also construct the asymptotics of solutions of nonautonomous Van der Pol equation as t → ∞.

We prove C^{0, α} regularity for local minimizers u of functionals with p(x)-growth of the type

in the class , where the exponent function p : Ω → (1, ∞) is assumed to be continuous with a modulus of continuity satisfying

and 1 < γ₁ ⩽ p(x) ≤ γ₂ < +∞. Moreover, is a given obstacle function, whose gradient Dψ belongs to a Morrey space with n − γ₁ < λ < n and q > γ₂. We do not assume any quantitative continuity of the integrand function f.

We give a K-theory proof of the invariance under cobordism of the family index. We consider elliptic pseudodifferential families on a continuous fibre bundle with smooth fibres , and define a notion of cobordant families using K¹-groups on fibrations with boundary. We show that the index of two such families is the same using properties of the push-forward map in K-theory to reduce it to families on .

We explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrödinger operators (also known as Bessel operators). We also investigate the connections with the generalized Bäcklund–Darboux transformation.

We prove the following inclusion

where WF_* denotes the non-quasianalytic Beurling or Roumieu wave front set, Ω is an open subset of , P is a linear partial differential operator with suitable ultradifferentiable coefficients, and Σ is the characteristic set of P. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.

Using an idea going back to Madelung, we construct global in time solutions to the transport equation corresponding to the asymptotic solution of the Kolmogorov-Feller equation describing a system with diffusion, potential and jump terms. To do that we use the construction of a generalized delta-shock solution of the continuity equation for a discontinuous velocity field. We also discuss corresponding problem of asymptotic solution construction (Maslov tunnel asymptotics).

Sieg and Wegner showed that the stable exact sequences define a maximal exact structure (in the sense of Quillen) in any pre-abelian category 41. We generalize this result to weakly idempotent complete additive categories.

For a self-affine tile in generated by an expanding matrix and an integral consecutive collinear digit set , Leung and Lau [Trans. Amer. Math. Soc. 359, 3337–3355 (2007).] provided a necessary and sufficient algebraic condition for it to be disklike. They also characterized the neighborhood structure of all disklike tiles in terms of the algebraic data A and . In this paper, we completely characterize the neighborhood structure of those non-disklike tiles. While disklike tiles can only have either six or eight edge or vertex neighbors, non-disklike tiles have much richer neighborhood structure. In particular, other than a finite set, a Cantor set, or a set containing a nontrivial continuum, neighbors can intersect in a union of a Cantor set and a countable set.

Let G be an abelian topological group. The symbol denotes the group of all continuous characters endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that χ(E)⊆φ([− 1/4, 1/4]) holds only for the trivial character , where is the canonical homomorphism. A super-sequence is a non-empty compact Hausdorff space S with at most one non-isolated point (to which Sconverges). We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super-sequence converging to 0 that is qc-dense in G. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group G, the restriction homomorphism defined by for , is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component G_a contains a super-sequence converging to the identity that is qc-dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group.

Let L = −Δ + V be a Schrödinger operator on (n ≥ 3), where is a nonnegative potential belonging to certain reverse Hölder class B_s for . In this article, we prove the boundedness of some integral operators related to L, such as L⁻¹∇², L⁻¹V and L⁻¹( − Δ) on the space .

Let Z_α be a positive α-stable random variable and We show the existence of an unbounded open domain D in [1/2, 1] × ( − ∞, −1/2] with a cusp at (1/2, −1/2), characterized by the complete monotonicity of the function such that is unimodal if and only if (α, r)∉D.

The notion of semigroups of Lipschitz operators associated with abstract quasilinear evolution equations is introduced and a product formula for such semigroups is established. The product formula obtained in the paper is applied to the solvability of the Cauchy problem for a first order quasilinear system through a finite difference scheme of the Lax-Friedrichs type.