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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 Lp spaces.

We investigate the Chow groups of projective determinantal varieties and those of their strata of matrices of fixed rank, using Chern class computations (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Let c e IN. For d e ℤ with gcd (d, c) = 1 let δ(d, c) be defined by d · δ(d, c) 1 mod c, 1 ⩽ δ(d, c) ⩽ c. Let s, t e IN with 1 ⩽ s ⩽ c, 1 ⩽ t ⩽ c. The main result is that for arbitrary fixed e > 0, but uniformly over c, s and t Equivalently, the points (d, δ(c,d)) are approximately uniformly distributed in [0, c] × [0, c]: The two-dimensional discrepancy of {(d,δ (c, d)): 1 ⩽ d ⩽ c and (c, d)= 1} in [0,c] × [0,c] is Oe(ce-1/2).

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1-dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.

In this paper, we study time-asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space-time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time-asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support., Comment: This paper is an improved and extended version of arXiv:1507.00883. Moreover this second version contains supplementary information on wave packets to motivate our results and comments on the applicability of our method to the study of certain hyperbolic equations

In this paper we consider the following question. When does there exist a square root of a probability measure supported on R+? This question is naturally related to subnormality of weighted shifts. The main result of this paper is that if μ is a finitely atomic probability measure having at most 4 atoms, then μ has a square root, i.e., there exists a measure ν such that μ=ν*ν (* means the convolution) if and only if the Aluthge transform of a subnormal weighted shift with Berger measure μ is subnormal. As an application of them, we give non-trivial, large classes of probability measures having a square root. We also prove that there are 6 and 7-atomic probability measures which don't have any square root. Our results have a connection to the following long-open problem in Operator Theory: characterize the subnormal operators having a square root.

In this paper we prove a sharp distortion property of the Cassinian metric under Mobius transformations of a punctured ball onto another punctured ball. The paper also deals with discussion on local convexity properties of the Cassinian metric balls in proper subdomains of Rn. Inclusion properties of the Cassinian metric balls with other hyperbolic-type metric balls are also investigated. In particular, several conjectures are also stated in response to sharpness of the inclusion relations.

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.

The paper is devoted to the existence and uniqueness of local solutions for the density-dependent non-Newtonian compressible fluids with vacuum in one-dimensional bounded intervals. The important points in this paper are that the initial density may vanish in an open subset and the viscosity coefficient is nonlinearly dependent of density and shear rate.

The paper introduces a class of functions where the resolving operator for a system of Kolmogorov–Feller-type equations with a small parameter is well posed in forward and backward times. The introduced class of functions is invariant under the resolving operator if the solution is understood in the weak sense with an exponential weight. The paper continues the study of [6].

In this paper we consider the system of the non-steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y, respectively, to be the space of “possible” solutions of this problem and the space of its data. We define the operator and formulate our problem in terms of operator equations. Let and be the Frechet derivative of at . We prove that is one-to-one and onto Y. Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. The next result proved in the Appendix of this paper is W2, 2-regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.

Tessellations of that use convex polyhedral cells to fill the space can be extremely complicated, especially if they are not “facet-to-facet”, that is, if the facets of a cell do not necessarily coincide with the facets of that cell's neighbours. In a recent paper [15], we have developed a theory which covers these complicated cases, at least with respect to their combinatorial topology. The theory required seven parameters, three of which suffice for facet-to-facet cases; the remaining four parameters are needed for the awkward adjacency concepts that arise in the general case. This current paper establishes constraints that apply to these seven parameters and so defines a permissible region within their seven-dimensional space, a region which we discover is not bounded. Our constraints in the relatively simple facet-to-facet case are also new.

In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. In the present paper this affine method is refined and adapted to the pseudo-Riemannian case. Using this method and elementary topology it is proved that any homogeneous Lorentzian manifold of even dimension admits a light-like homogeneous geodesic. The method is illustrated in detail with an example of the Lie group of dimension 3 with an invariant metric, which does not admit any light-like homogeneous geodesic.

In this paper we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equation, where uniform trace estimates on the half-line are shown., Comment: Some typos corrected. Accepted for publication in Mathematische Nachrichten

The estimate of integral points in right-angled simplices has many applications in number theory, complex geometry, toric variety and tropical geometry. In [24], [25], [27], the second author and other coworkers gave a sharp upper estimate that counts the number of positive integral points in n dimensional () real right-angled simplices with vertices whose distance to the origin are at least . A natural problem is how to form a new sharp estimate without the minimal distance assumption. In this paper, we formulate the Number Theoretic Conjecture which is a direct correspondence of the Yau Geometry conjecture. We have proved this conjecture for . This paper gives hope to prove the new conjecture in general. As an application, we give a sharp estimate of the Dickman-de Bruijn function for .

We consider a nonlinear wave equation with a quasilinear quadratic nonlinearity. Slow spatial and temporal modulations of the envelope of an underlying carrier wave can be described formally by an NLS equation. It is the purpose of this paper to present a method which allows to prove error estimates between this formal approximation and true solutions of the quasilinear wave equation in case . The paper contains the first validity proof of the NLS approximation for a nonlinear wave equation with quasilinear quadratic terms.

The paper is the second in a set of two papers, which are devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non-selfadjoint unbounded matrix differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary-value problem (IBVP) for a non-homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with a one-parameter family of dissipative boundary conditions at one end and with a heavy load at the other end; this filament problem is treated for two cases of the boundary parameter: non-singular and singular; (c) IBVP for a three-dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping; (d) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (e) IBVP for a coupled Euler-Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending-torsion vibration model); (f) IBVP for two coupled Timoshenko beams model, which is currently accepted as an appropriate model describing vibrational behavior of a longer double-walled carbon nanotube. Problems have been discussed in the first paper of the aforementioned set. Problems are discussed in the present paper.

Let be a regular local smoothing of a nodal curve. In this paper, we find a modular description of the Neron maps associated to Abel-Jacobi maps with values in Esteves's fine compactified Jacobian and with source the B-smooth locus of either the double product of over B or the degree-2 Hilbert scheme of the family f. This is the first of a series of two papers dedicated to the construction of a resolution of the degree-2 Abel-Jacobi map for a regular smoothing of a nodal curve.

In this paper, we study the minimal free resolution of a nondegenerate projective variety when X is contained in a variety Y of minimal degree as a divisor. Such a variety is of interest because of its extremal behavior with respect to various properties. The graded Betti diagram of X has been completely known only when X is arithmetically Cohen-Macaulay. Our main result in the present paper provides a detailed description of the graded Betti diagram of X for the case where X is not arithmetically Cohen-Macaulay.

In this paper we deal with the hyponormality of Toeplitz operators with matrix-valued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded-type Toeplitz operators (i.e., the symbol is a matrix-valued function such that Φ and are of bounded type). In particular, we get a much simpler criterion for the hyponormality of when the co-analytic part of the symbol Φ is a left divisor of the analytic part.

Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood--Paley square function for Dunkl heat flows in $\mathbb{R}^d$ is introduced by employing the full "gradient" induced by the corresponding carr\'{e} du champ operator and then the $L^p$ boundedness is studied for all $p\in(1,\infty)$. For $p\in(1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\in[2,\infty)$, we restrict to a particular case when the corresponding Weyl group is isomorphic to $\mathbb{Z}_2^d$ and apply a probabilistic method to prove the $L^p$ boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry--Emery, which may be of independent interest, plays a crucial role. The results are dimension-free., Comment: Comments welcome!

In this paper, we obtain difference analogues of the second main theorem for meromorphic functions in several complex variables from which difference analogues of Picard-type theorems are also obtained. Our results are improvements or extensions of some recent results of papers [Proc. Royal Soc. Edinburgh Section A Math. 137, 457–474 (2007); Comput. Meth. Funct. Theor. 12, No. 1, 343–361 (2012)]. The method we used is very different from theirs.

I give an explicitly verifiable necessary and sufficient condition for the uniqueness of the eigenform on finitely ramified fractals, once an eigenform is known. This improves the results of my previous paper [16], where I gave some necessary and some sufficient conditions, and with a relatively mild additional requirement on the known eigenform. The result of this paper can be interpreted as a uniqueness result for self-similar energies on finitely ramified fractals.

In this paper, we discuss hybrid point sets built from two of the most prominent classes of sequences used in quasi-Monte Carlo methods. We derive an existence result on point sets with low diaphony, where the components of the points involved stem from a digital (t, s)-sequence on the one hand, and from a lattice point set on the other. Moreover, we outline how the hybrid diaphony of the point sets considered in this paper relates to the worst-case integration error in suitable function spaces.

We investigate in the paper general (not necessarily definite) first-order symmetric systems of differential equations in the framework of extension theory of symmetric linear relations. For this aim we first introduce the new notion of a boundary pair for A*, where A is a symmetric linear relation, is a boundary Hilbert space, is a subspace in and is a (possibly multivalued) linear mapping satisfying the abstract Green's identity. Unlike known concept of a boundary pair for A* our definition is suitable for relations A with possibly unequal deficiency indices n±(A). Next, the general symmetric system Jy′(t) − B(t)y(t) = Δ(t)f(t) on an interval is considered in the paper. We characterize explicitly the corresponding minimal relation , which enables us to construct a special (so-called decomposing) boundary pair for the maximal relation . It turns out that the system is definite if and only if the mapping Γ is single-valued, in which case a decomposing boundary pair turns into the decomposing boundary triplet for . By using such a triplet we describe in terms of boundary conditions proper extensions of in the case of the regular endpoint a and arbitrary (possibly unequal) deficiency indices . We also show that self-adjoint decomposing boundary conditions exist only for Hamiltonian systems; moreover, we describe all such conditions in the compact form. These results are generalizations of the known results by Rofe-Beketov on regular differential operators. Finally, we characterize all maximal dissipative and accumulative separated boundary conditions for arbitrary (not necessarily Hamiltonian) definite symmetric systems

Rychkov defined weighted Besov spaces and weighted Triebel-Lizorkin spaces coming with a weight in the class , which is even wider than the class Ap due to Muckenhoupt. In the present paper we are concerned with the atomic decomposition of these spaces. As an application, we obtain some key theorems in the theory of function spaces. Finally we conclude this paper with some helpful examples showing the difference between weighted spaces and unweighted spaces.

This paper studies a three-species predator-prey model with cross-diffusion. In a previous paper 11 of Pang and Wang, it was proved that the system admits global classical solutions if the exponents in cross diffusion term satisfy m, l ≥ 1. In the present paper, we continue considering the asymptotic behavior of the solutions, and obtain that the system possesses a global attractor in fact, if the cross diffusion exponents are even larger with m, l ≥ 2.

odinger operators. 2000 Mathematics Subject Classification Primary 35J10, 45P05; Secondary 35P05, 47B10. The Evans function is a Wronskian type determinant used to detect point spectrum of differential operators obtained by linearizing PDEs about special solutions such as traveling waves, etc. This work is a sequel to the paper “Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves”, published by F. Gesztesy, K. Zumbrun and the second author in J. Math. Pures Appl. 90, 160‐200 (2008), where the Evans and Jost functions for the Schr¨ odinger equations have been considered. In the current work, we study the Evans function for the general case of linear ODE systems, and choose it to agree with the modified Fredholm determinant of the respective Birman-Schwinger type integral operator. The Evans function is thus the determinant of the matrix composed of the so-called generalized Jost solutions. These are the solutions of the homogeneous perturbed differential equation which are asymptotic to some reference solutions of the unperturbed equation. One of the main results of the current paper is a formula for the derivative of the Evans function for the first order systems. Its proof uses a matrix composed of the newly introduced modified Jost solutions. These are the solutions of an inhomogeneous perturbed differential equation with the inhomogeneous term constructed by means of the above-mentioned generalized Jost solutions. Copyright line will be provided by the publisher

Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third-order multi-point boundary value problem at resonance where f: [0, 1] × R3 R is a continuous function, 0 < ξ1 < ⋅⋅⋅ < ξm < 1, αi ∈ R, i = 1, …, m, m ≥ 1 and 0 < η1 < η2 < ⋅⋅⋅ < ηn < 1, βj ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′′′) is equal to 2. Since all the existence results for third-order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.

The present paper is the first one in a series of two papers devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non-selfadjoint unbounded differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary-value problem (IBVP) for a non-homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with dissipative boundary condition at one end and with a heavy load at the other end; (c) IBVP for a three-dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping. In the second paper of the series the following problems will be considered: (a) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (b) IBVP for a coupled Euler–Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending-torsion vibration model); (c) IBVP for a system of two Timoshenko beams coupled through linear Van der Waals forces. The model of (c) describes vibrational motion of a double-walled carbon nanotube. In all of the above cases, the result has been obtained by using Krien’s Theorem on completeness of root vectors of a dissipative operator with a nuclear imaginary part. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In this paper we study a class of elliptic second-order differential operators on finite dimensional convex compact sets whose principal part degenerates on a subset of the boundary of the domain. We show that the closures of these operators generate Feller semigroups. Moreover, we approximate these semigroups by iterates of suitable positive linear operators which we also introduce and study in this paper for the first time, and which we refer to as modified Bernstein-Schnabl operators. As a consequence of this approximation we investigate some regularity properties preserved by the semigroup. Finally, we consider the special case of the finite dimensional simplex and the well-known Wright-Fisher diffusion model of gene frequency used in population genetics. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim