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We show abstract versions for Banach couples of several limiting compact interpolation theorems established by Edmunds and Opic for couples of [ABSTRACT FROM AUTHOR]

Linear operators with off‐diagonal decay appear in many areas of mathematics including harmonic and numerical analysis, and their stability is one of the basic assumptions. In this paper, we consider a family of localized integral operators in the Beurling algebra with kernels having mild singularity near the diagonal and certain Hölder continuity property, and prove that their weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other on a space of homogeneous type with Ahlfors regular measure. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations, the results are new and generalize and improve some early results in this field. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the warped product pointwise semi‐slant submanifolds which are isometrically immersed into Sasakian manifolds. First, we prove some characterizations results in terms of the shape operator, under which influence a pointwise semi‐slant submanifold of a Sasakian manifold can be reduced to a warped product submanifold. Then, we determine a geometric inequality for the second fundamental form regarding to intrinsic invariant and extrinsic invariant using the Gauss equation instead of the Codazzi equation. Evenmore, we give some applications of this inequality into Sasakian space forms, and we will investigate the status of equalities in the inequality. As a particular case, we provide numerous applications of the Green lemma, the Laplacian of warped functions and some partial differential equations. Some triviality results for connected, compact warped product pointwise semi‐slant submanifolds of Sasakian space form by means of Hamiltonian and the kinetic energy of warped function involving boundary conditions are established. [ABSTRACT FROM AUTHOR]

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations. [ABSTRACT FROM AUTHOR]

In this paper we consider regularity coefficients of a discrete linear time‐varying systems. The main result presents a complete description of the relations between Lyapunov, Perron and Grobman regularity coefficients of mutually adjoint systems. [ABSTRACT FROM AUTHOR]

In this paper, helicoidal flat surfaces in the 3‐dimensional sphere S3 are considered. A complete classification of such surfaces, that generalizes a classification of rotational flat surfaces, is given in terms of the first and second fundamental forms for asymptotic parameters. The result consists in a relation between helicoidal flat surfaces and linear solutions of the corresponding homogeneous wave equation for the angle function. [ABSTRACT FROM AUTHOR]

This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time‐changed by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well‐posed Dirichlet problem for space‐time fractional diffusions in one or several variables. This solves an open problem in numerical analysis. [ABSTRACT FROM AUTHOR]

Let S=K[x1,...,xn] denote the polynomial ring in n variables over a field K with each degxi=1 and let I⊂S be a homogeneous ideal of S with dimS/I=d. The Hilbert series of S/I is of the form hS/I(λ)/(1−λ)d, where hS/I(λ)=h0+h1λ+h2λ2+⋯+hsλs with hs≠0 is the h‐polynomial of S/I. It is known that, when S/I is Cohen–Macaulay, one has reg(S/I)=deghS/I(λ), where reg(S/I) is the (Castelnuovo–Mumford) regularity of S/I. In the present paper, given arbitrary integers r and s with r≥1 and s≥1, a monomial ideal I of S=K[x1,...,xn] with n≫0 for which reg(S/I)=r and deghS/I(λ)=s will be constructed. Furthermore, we give a class of edge ideals I⊂S of Cameron–Walker graphs with reg(S/I)=deghS/I(λ) for which S/I is not Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of a generalized Camassa–Holm equation. We first derive two global existence results and two blow‐up results from the relationship between the Degasperis–Procesi equation and the generalized Camassa–Holm equation. We then prove the existence and uniqueness of global weak solutions under some certain sign conditions on the initial data. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals, given an arbitrary locally compact Abelian group, with the existence of integrable functions all whose convolution powers are essentially unbounded. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the essential spectrum of the operator L A u ( x ) = − ( A ( x ) u ′ ( x ) ) ′ + u ( x )where A ( x ) is a positive absolutely continuous function on (0, 1) that resembles x 2 α for some α ≥ 1. We prove that the essential spectrum of L A coincides with the essential spectrum of the operator L α u ( x ) : = − ( x 2 α u ′ ( x ) ) ′ + u ( x ). [ABSTRACT FROM AUTHOR]

In this paper, we study the following Kirchhoff type elliptic problem with critical growth: [ABSTRACT FROM AUTHOR]

In this paper we study local existence, uniqueness, and continuous dependence of an abstract integrodifferential equation. We also present a result on unique continuation and a blow-up alternative for mild solutions of the integrodifferential equation. Finally, we apply our results to an interesting strongly damped plate equation with memory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stationary Navier-Stokes equation in the whole plane and in the two-dimensional exterior domain invariant under the action of the cyclic group of order 4, and gives a condition on the potentials yielding the external force, and on the boundary value, sufficient for the unique existence of a small solution equivariant with respect to the aforementioned cyclic group. [ABSTRACT FROM AUTHOR]

The biharmonic Dirichlet boundary value problem on a bounded domain is the focus of the present paper. By Riesz' representation theorem the existence and uniqueness of a weak solution is quite direct. The problem that we are interested in appears when one is looking for constructive approximations of a solution. Numerical methods using for example finite elements, prefer systems of second equations to fourth order problems. Ciarlet and Raviart in and Monk in consider approaches through second order problems assuming that the domain is smooth. We will discuss what happens when the domain has corners. Moreover, we will suggest a setting, which is in some sense between Ciarlet-Raviart and Monk, that inherits the benefits of both settings and that will give the weak solution through a system type approach. [ABSTRACT FROM AUTHOR]

In this paper, we give the explicit bounds for the data of objects involved in some basic theorems of singularity theory: the inverse, implicit and rank theorems for Lipschitz mappings, the splitting lemma and the Morse lemma, the density and openness of Morse functions. We expect that the results will make singularities more applicable and will be useful for numerical analysis and some fields of computing. [ABSTRACT FROM AUTHOR]

Abstract: We study complex interpolation of Herz‐type Triebel–Lizorkin spaces by using the Calderón product method. Additionally we present complex interpolation between Herz‐type Triebel–Lizorkin spaces and Triebel–Lizorkin spaces F ∞ , β s. Moreover, we apply these results to obtain the complex interpolation of Triebel–Lizorkin spaces equipped with power weights and between b m o (or h p) spaces and Herz spaces. [ABSTRACT FROM AUTHOR]

In the paper, abstract interpolation of Hardy spaces on circular domains is studied. In particular, an extension of Peter Jones's result on the interpolation of Hardy spaces on the disc to multiply-connected domains is presented. Moreover, some applications in approximation theory are given. [ABSTRACT FROM AUTHOR]

We study the two‐dimensional stationary Navier–Stokes equations describing flows around a rotating disk. The existence of unique solutions is established for any rotating speed, and qualitative effects of a large rotation are described precisely by exhibiting a boundary layer structure and an axisymmetrization of the flow. [ABSTRACT FROM AUTHOR]

Let p(·) be a measurable function on Rn with 0

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L2 restrictions of Fourier transforms onto spheres in Rn which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass m≥0 in dimensions n≥2 is derived. [ABSTRACT FROM AUTHOR]

Let X1,⋯,Xn be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: fd,β(x)=const·1−∥x∥2β,∥x∥<1,(thebetacase)and f∼d,β(x)=const·1+∥x∥2−β,x∈Rd,(thebeta"case).We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of X1,⋯,Xn. Asymptotic formulae were obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when β↓−1, respectively β↑+∞, we can also cover the models in which X1,⋯,Xn are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of ±X1,⋯,±Xn and 0,X1,⋯,Xn. One of the main tools used in the proofs is the Blaschke–Petkantschin formula. [ABSTRACT FROM AUTHOR]

In , Huayi Chen introduces the notion of an approximable graded algebra, which he uses to prove a Fujita‐type theorem in the arithmetic setting, and asks if any such algebra is a sub‐algebra of the graded section ring of a big line bundle on an algebraic variety. We give a counter‐example showing that this is not the case. [ABSTRACT FROM AUTHOR]

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n-th order nonlinear differential equations, equations with a generalized ϕ-Laplacian, and nonlinear partial differential systems. [ABSTRACT FROM AUTHOR]

A connected, linearly ordered subset [ABSTRACT FROM AUTHOR]

The finite Euler product is known one of the classical zeta functions in number theory. In [1], [2] and [3], we have introduced some multivariable zeta functions and studied their definable probability distributions on [ABSTRACT FROM AUTHOR]

We classify solutions of the third Painlevé equation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$P^{\prime }_{\rm III}$\end{document}, which are meromorphic around the origin, and determine their linear monodromy. Combined with 4, 5, all of meromorphic solutions of the Painlevé equations around fixed singular points of the Briot-Bouquet type are completely determined. We characterize the linear monodromy of meromorphic solutions for the third, fifth and sixth Painlevé equations. [ABSTRACT FROM AUTHOR]

We prove that if the average Sylow number (ignoring the Sylow numbers that are one) of a finite group G is ⩽7, then G is solvable. [ABSTRACT FROM AUTHOR]