New versions of the Colombeau algebras

We construct some versions of the Colombeau theory. In particular, we construct the Colombeau algebra generated by harmonic (or polyharmonic) regularizations of distributions connected with a half‐space and by analytic regularizations of distributions connected with an octant. Unlike the standard Colombeau's scheme, our theory has new generalized functions that can be easily represented as weak asymptotics whose coefficients are distributions, i.e., in form of asymptotic distributions . The algebra of asymptotic distributions generated by the linear span of associated homogeneous distributions (in the one‐dimensional case) which we constructed earlier [9] can be embedded as a subalgebra into our version of Colombeau algebra. The representation of distributional products in the form of weak asymptotic series proved very useful in solving problems which arise in the theory of discontinuous solutions of hyperbolic systems of conservation laws [10]–[16], [49] and [50]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Archimedean-based Marshall-Olkin Distributions and Related Dependence Structures

In this paper we study the dependence properties of a family of bivariate distributions (that we call Archimedean-based Marshall-Olkin distributions) that extends the class of the Generalized Marshall-Olkin distributions of Li and Pellerey, J Multivar Anal, 102, (10), 1399–1409, 2011 in order to allow for an Archimedean type of dependence among the underlying shocks’ arrival times. The associated family of copulas (that we call Archimedean-based Marshall-Olkin copulas) includes several well known copula functions as specific cases for which we provide a different costruction and represents a particular case of implementation of Morillas, Metrika, 61, (2), 169–184, 2005 construction. It is shown that Archimedean-based copulas are obtained through suitable transformations of bivariate Archimedean copulas: this induces asymmetry, and the corresponding Kendall’s function and Kendall’s tau as well as the tail dependence parameters are studied. The type of dependence so modeled is wide and illustrated through examples and the validity of the weak Lack of memory property (characterizing the Marshall-Olkin distribution) is also investigated and the sub-family of distributions satisfying it identified. Moreover, the main theoretical results are extended to the multidimensional version of the considered distributions and estimation issues discussed.     

Weakly Compact and Absolutely Summing Polynomials

This paper shows that, contrary to the case of linear operators, absolutely summing homogeneous polynomials are not always weakly compact. It is also shown that, regardless of the infinite dimensional Banach space E and the positive integer n, there exists an n-homogeneous polynomial P from E to E that plays the role of the identity operator in the sense that P is neither compact nor absolutely r-summing for any r, and P is weakly compact if and only if E is reflexive.     

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.     

Cartan coherent configurations

The Cartan scheme \(\mathcal{X}\) of a finite group G with a (B, N)-pair is defined to be the coherent configuration associated with the action of G on the right cosets of the Cartan subgroup \(B\cap N\) by right multiplication. It is proved that if G is a simple group of Lie type, then asymptotically the coherent configuration \(\mathcal{X}\) is 2-separable, i.e., the array of 2-dimensional intersection numbers determines \(\mathcal{X}\) up to isomorphism. It is also proved that in this case, the base number of \(\mathcal{X}\) equals 2. This enables us to construct a polynomial-time algorithm for recognizing Cartan schemes when the rank of G and the order of the underlying field are sufficiently large. One of the key points in the proof is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.     

Density of multivariate homogeneous polynomials on star like domains

The famous Weierstrass theorem asserts that every continuous function on a compact set in ${\mathbb{R}}^d$ can be uniformly approximated by algebraic polynomials. A related interesting problem consists in studying the same question for the important subclass of homogeneous polynomials containing only monomials of the same degree. The corresponding conjecture claims that every continuous function on the boundary of convex 0-symmetric bodies can be uniformly approximated by pairs of homogeneous polynomials. The main objective of the present paper is to review the recent progress on this conjecture and provide a new unified treatment of the same problem on non convex star like domains. It will be shown that the boundary of every 0-symmetric non convex star like domain contains an exceptional zero set so that a continuous function can be uniformly approximated on the boundary of the domain by a sum of two homogeneous polynomials if and only if the function vanishes on this zero set. Thus the Weierstrass type approximation problem for homogeneous polynomials on non convex star like domains amounts to the study of these exceptional zero sets. We will also present an extension of a theorem of Varjú which describes the exceptional zero sets for intersections of star like domains. These results combined with certain transformations of the underlying region will lead to the discovery of some new classes of convex and non convex domains for which the Weierstrass type approximation result holds for homogeneous polynomials.     

Quasi Co-Hopfian Modules and Applications

We carry out an extensive study of modules M _R with the property that M/f(M) is singular for all injective endomorphisms f of M. Such modules called “quasi co-Hopfian”, generalize co-Hopfian modules. It is shown that a ring R is semisimple if and only if every quasi co-Hopfian R-module is co-Hopfian. Every module contains a unique largest fully invariant quasi co-Hopfian submodule. This submodule is determined for some modules including the semisimple ones. Over right nonsingular rings several equivalent conditions to being quasi co-Hopfian are given. Modules with all submodules quasi co-Hopfian are called “completely quasi co-Hopfian” (cqcH). Over right nonsingular rings and over certain right Noetherian rings, it is proved that every finite reduced rank module is cqcH. For a right nonsingular ring which is right semi-Artinian (resp. right FBN) the class of cqcH modules is the same as the class of finite reduced rank modules if and only if there are only finitely many isomorphism classes of nonsingular R-modules which are simple (resp. indecomposable injective).     

Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type

Processes of Ornstein-Uhlenbeck type on $\text{R}$^d are analogues of the Ornstein-Uhlenbeck process on $\text{R}$^d with the Brownian motion part replaced by general processes with homogeneous independent increments. The class of operator-selfdecomposable distributions of Urbanik is characterized as the class of limit distributions of such processes. Continuity of the correspondence is proved. Integro-differential equations for operator-selfdecomposable distributions are established. Examples are given for null recurrence and transience of processes of Ornstein-Uhlenbeck type on $\text{R}$¹.     

Second-order invex functions in nonlinear programming

We introduce a notion of a second-order invex function. A Fréchet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Fréchet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type I invex data are obtained. An answer to the question when a kernel, which is not identically equal to zero, exists is given.     

ON A SET OF h-HARMONIC HOMOGENEOUS POLYNOMIALS

In this paper, we study the homogeneous polynomials orthogonal with the weight function h(x(d))=x1^2k1…xd^2kd on S^(d-1).We obtain the explicit formula on a basis of the orthogonal homogeneous polynomials of degree n. It is simpler than the formula in,and can be regarded as an extension of under the weighted case.     

Invariant <Emphasis Type="Italic">f</Emphasis>-structures on naturally reductive homogeneous spaces

We study invariant metric f-structures on naturally reductive homogeneous spaces and establish their relation to generalized Hermitian geometry. We prove a series of criteria characterizing geometric and algebraic properties of important classes of metric f-structures: nearly Kähler, Hermitian, Kähler, and Killing structures. It is shown that canonical f-structures on homogeneous Φ-spaces of order k (homogeneous k-symmetric spaces) play remarkable part in this line of investigation. In particular, we present the final results concerning canonical f-structures on naturally reductive homogeneous Φ-spaces of order 4 and 5.     

On <Emphasis Type="Italic">δ</Emphasis>-homogeneous Riemannian manifolds. II

We continue the study of the δ-homogeneous Riemannian manifolds defined in a more general case by V. N. Berestovski? and C. P. Plaut. Each of these manifolds has nonnegative sectional curvature. We prove in particular that every naturally reductive compact homogeneous Riemannian manifold of positive Euler characteristic is δ-homogeneous.     

On factorial states of operator algebras

Results for the factorial state space of a C¹-algebra A which are analogous to results of 11., 12., 572–612),Tomiyama and Takesaki (Tohôku Math. J. (2) 13 (1961), 498–523) for the pure state space. It is shown that A is prime if and only if the (type I) factorial states are dense in the state space. It follows that every factorial state is a w¹-limit of type I factorial states. The factorial state space of a von Neumann algebra is determined, and it is shown that if A is unital and acts non-degenerately on a Hilbert space then the factorial state space of the generated von Neumann algebra restricts precisely to the factorial state space of A. It is shown that the set of factorial states is w¹-compact if and only if A is unital, liminal and has Hausdorff primitive ideal space.     

Fractional Abstract Cauchy Problems

This paper is concerned with fractional abstract Cauchy problems with order \({\alpha\in(1,2)}\). The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (FACP ₀) is well-posed if and only if its coefficient operator A generates an α-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (FACP _f ).     

Any 2-asummable bipartite function is weighted threshold

Possible characterizations of which positive boolean functions are weighted threshold were studied in the 60s and 70s. It is known that a boolean function is weighted threshold if and only if it is k-asummable for every value of k. Furthermore, for some particular subfamilies of functions (those with up to eight variables, and graph functions), it is known that a function is weighted threshold if and only if it is 2-asummable.In this work we prove that bipartite functions also satisfy this property: a bipartite function is weighted threshold if and only if it is 2-asummable. In a bipartite function the set of variables can be partitioned in two classes, such that all the variables in the same class play exactly the same role in the function.     

Modular classes of skew algebroid relations

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E ^* which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e., mod(E)?=?0. Further, the relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as the modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and of a Poisson map.     

Invertibility in the Flag Kernels Algebra on the Heisenberg Group

Flag kernels are tempered distributions which generalize these of Calderón–Zygmund type. For any homogeneous group \(\mathbb {G}\) the class of operators which acts on \(L^{2}(\mathbb {G})\) by convolution with a flag kernel is closed under composition. In the case of the Heisenberg group we prove the inverse-closed property for this algebra. It means that if an operator from this algebra is invertible on \(L^{2}(\mathbb {G})\), then its inversion remains in the class.     

Injectivity relative to closed submodules

Let R be a ring. An R-module X is called c-injective if, for every closed submodule L of every R-module M, every homomorphism from L to X lifts to M. It is proved that if R is a Dedekind domain then an R-module X is c-injective if and only if X is isomorphic to a direct product of homogeneous semisimple R-modules and injective R-modules. It is also proved that a commutative Noetherian domain R is Dedekind if and only if every simple R-module is c-injective.     

Homological Behavior of Auslander’s k-Gorenstein Rings

In this paper we mainly study the homological properties of dual modules over k-Gorenstein rings. For a right quasi k-Gorenstein ring Λ, we show that the right self-injective dimension of Λ is at most k if and only if each M?∈ mod Λ satisfying the condition that Ext $_{\Lambda}^i(M, \Lambda)=0$ for any 1?≤?i?≤?k is reflexive. For an ∞-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of Λ are identical. In addition, we show that if Λ is a left quasi ∞-Gorenstein ring and M?∈ mod Λ with gradeM finite, then Ext $_{\Lambda}^i($ Ext $_{\Lambda ^{op}}^i($ Ext $_{\Lambda}^{{\rm grade}M}(M, \Lambda), \Lambda), \Lambda)=0$ if and only if i?≠gradeM. For a 2-Gorenstein ring Λ, we show that a non-zero proper left ideal I of Λ is reflexive if and only if Λ/I has no non-zero pseudo-null submodule.