Weighted Hardy's inequality in a limiting case and the perturbed Kolmogorov equation

In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant on the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein–Uhlenbeck operator perturbed by a critical singular potential in a two-dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the limiting case of results which are established in the work of Goldstein et al. [Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential. Appl Anal. 2012;91(11):2057–2071] and Hauer and Rhandi [A weighted Hardy inequality and nonexistence of positive solutions. Arch Math. 2013;100:273–287] in the non-critical cases, and are also considered as extensions of a result in Cabré and Martel [Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potential singulier. C R Acad Sci Paris Sér I Math. 1999;329:973–978] to the Kolmogorov operator case perturbed by a critical singular potential.     

Stiefel–Whitney invariants for bilinear forms

We examine potential extensions of the Stiefel–Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt?s Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)     

Fermion bound states in the Aharonov-Bohm field in 2+1 dimensions

We find exact solutions of the Dirac equation that describe fermion bound states in the Aharonov-Bohm potential in 2+1 dimensions with the particle spin taken into account. For this, we construct self-adjoint extensions of the Hamiltonian of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions. The self-adjoint extensions depend on a single parameter. We select the range of this parameter in which quantum fermion states are bound. We demonstrate that the energy levels of particles and antiparticles intersect. Because solutions of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions describe the behavior of relativistic fermions in the field of the cosmic string in 3+1 dimensions, our results can presumably be used to describe fermions in the cosmic string field.     

On Relations Between Bessel Potential Spaces and Riesz Potential Spaces

We present a relation between the Bessel potential spaces and the Riesz potential spaces. The ideas of the proof are to characterize each potential spaces and to give a correspondence between individual Bessel potentials and Riesz potentials.     

Static and dynamic best-response potential functions for the non-linear Cournot game

We show that the Cournot oligopoly game with non-linear market demand can be reformulated as a best-response potential game where the best-response potential function is linear-quadratic in the special case where marginal cost is normalized to zero. We also propose a new approach to show that the open-loop differential game with Ramsey dynamics admits a best-response Hamiltonian potential corresponding to the sum of the best-response potential function of the static game plus the scalar product of transition functions multiplied by the fictitious costate variables. Unlike the original differential game, its best-response representation yields the map of the instantaneous best reply functions.     

Lagrange’s identity and its generalizations

The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in coordinates and 3) of continuum of interacting particles governed by the well-known Vlasov kinetic equation.      

Choices and kernels in bipolar valued digraphs

We explore the extension of the notion of kernel (independent, dominant or absorbent, non-empty subset) of a digraph to valued graphs (or valued relations). We define various natural extensions and show the relationship between them. This work has potential interest for applications in choice decision problems.     

Anneaux polynomiaux a deux variables

Some types of extensions of skew fields are now known: Galois quadratic extensions ([7]), cyclic extensions ([1]), general quadratic extensions ([4]), binomial extensions ([5]). All these extensions belong to the class of pseudolinear extensions ([8]). A new class of extensions, the so-called «hexaphic extensions», which extends the class of pseudo-linear extensions, will be studied in the other papers in future. For this study, we introduce in this paper skew polynomials rings in two variables over a ring, which extend the well-known skew polynomial rings in one variable (pseudolinear) first studied by O. Ore ([10]).     

Convex-Concave Extensions

In this paper we introduce a new notion which we call convex-concave extensions. Convex-concave extensions provide for given nonlinear functions convex lower bound functions and concave upper bound functions, and can be viewed as a generalization of interval extensions. Convex-concave extensions can approximate the shape of a given function in a better way than interval extensions which deliver only constant lower and upper bounds for the range. Therefore, convex-concave extensions can be applied in a more flexible manner. For example, they can be used to construct convex relaxations. Moreover, it is demonstrated that in many cases the overestimation which is due to interval extensions can be drastically reduced. Applications and some numerical examples, including constrained global optimization problems of large scale, are presented.     

Global existence and decay estimates of the Boltzmann equation with frictional force: The general results

In this paper, we give the existence theory and the optimal time convergence rates of the solutions to the Boltzmann equation with frictional force near a global Maxwellian. We generalize our previous results on the same problem for hard sphere model into both hard potential and soft potential case. The main method used in this paper is the classic energy method combined with some new time–velocity weight functions to control the large velocity growth in the nonlinear term for the case of interactions with hard potentials and to deal with the singularity of the cross-section at zero relative velocity for the soft potential case.     

Note on Morita Equivalence in Ring Extensions

It seems that Morita invariance judges of the importance of classes of ring extensions concerned. Miyashita introduced the notion of Morita equivalence in ring extensions, and he showed that the classes of G-Galois extensions and Frobenius extensions are Morita invariant. After that, Ikehata showed that the classes of separable extensions, Hirata separable extensions, symmetric extensions, and QF-extensions are Morita invariant. In this article, we shall prove that the classes of several extensions are Morita invariant. Further, we will give an example of the class of ring extensions which is not Morita invariant.     

A Callias-Type Index Theorem with Degenerate Potentials

A generalization of Callias’ index theorem for self adjoint Dirac operators with skew adjoint potentials on asymptotically conic manifolds is presented in which the potential term may have constant rank nullspace at infinity. The index obtained depends on the choice of a family of Fredholm extensions, though as in the classical version it depends only on the data at infinity.     

Poisson double extensions

A double Ore extension was introduced by Zhang and Zhang(2008) to study a class of ArtinShelter regular algebras. Here we give a definition of Poisson double extension which may be considered as an analogue of the double Ore extension, and show that algebras in a class of double Ore extensions are deformation quantizations of Poisson double extensions. We also investigate the modular derivations of Poisson double extensions and the relationship between Poisson double extensions and iterated Poisson polynomial extensions.Results are illustrated by examples.     

Armendariz modules over skew PBW extensions

The aim of this article is to develop the theory of skew Armendariz and quasi-Armendariz modules over skew PBW extensions. We generalize the results of several works in the literature concerning these topics for the context of Ore extensions to another non-commutative rings which can not be expressed as iterated Ore extensions. As a consequence of our treatment, we extend and unify different results about the Armendariz, Baer, p.p., and p.q.-Baer properties for Ore extensions and skew PBW extensions.     

Extensions of commutative rings

In studying the minimal prime spectra of commutative rings with identity we have been able to identify several interesting types of extensions of rings. In particular, we determine what kind of ring extensions will result in a homeomorphisms of the hull-kernel and inverse topologies on the minimal prime spectra. We relate these types of extensions to other known types of extensions.     

Ore Extensions of Multiplier Hopf Algebras

The goal of this article is to generalize the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. First the concept of a Hopf–Ore extension of a multiplier Hopf algebra is introduced. We give a necessary and sufficient condition for Ore extensions to become a multiplier Hopf algebra. Finally, *-structures are constructed on Hopf–Ore extensions, and certain isomorphisms between Hopf–Ore extensions are discussed.     

Momentum Operators in the Unit Square

We investigate the skew-adjoint extensions of a partial derivative operator acting in the direction of one of the sides a unit square. We investigate the unitary equivalence of such extensions and the spectra of such extensions. It follows from our results, that such extensions need not have discete spectrum. We apply our techniques to the problem of finding commuting skew-adjoint extensions of the partial derivative operators acting in the directions of the sides of the unit square. While our results are most easily stated for the unit square, they are established for a larger class of domains, including certain fractal domains.     

A Criterion for Spectrum Decomposition

We obtain a necessary and sufficient condition for the decomposition of the spectrum of an arbitrary nonsymmetric potential whose least value is attained at finitely many points.     

Some characterizations of almost sure bounds for weighted multidimensional empirical distributions and a Glivenko-Cantelli theorem for sample quantiles

Summary Characterizations of almost sure bounds and a Glivenko-Cantelli theorem are obtained for certain weighted m-dimensional empirical distributions. These results constitute generalizations and extensions of the work of Shorack and Wellner (1978) and Wellner (1977, 1978). Also as an example of the potential use of the techniques developed in this paper a Glivenko-Cantelli type theorem is proven for sample quantiles.