Resolutions of generic points lying on a smooth quadric

For a 0-dimensional schemeX on a smooth quadricQ we define a special type of resolution of its ideal sheaf as a locally freeO _Q. These resolutions allow to find, for schemes which are generic inQ, the minimal free resolution ofX as a subscheme of ℙ³. For almost all such schemes the graded Betti numbers in ℙ³ depend only on the Hilbert function ofX in ℙ³. Work done with financial support of M.U.R.S.T., while the authors were members of C.N.R.     

Tight sets,weighted <Emphasis Type="Italic">m</Emphasis>-covers,weighted <Emphasis Type="Italic">m</Emphasis>-ovoids,and minihypers

Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342–349, 2008; Govaerts and Storme 4:279–286, 2004; Govaerts et al. 28:659–672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q⁻(5,3), to 56-caps contained in this elliptic quadric Q⁻(5,3).      

On Macaulayfication of sheaves

Let X be a quasi-projective scheme and ℱ a coherent sheaf of modules over X such that its non-Cohen–Macaulay locus is at most one dimensional. We use and extend the techniques of Brodmann to construct proper birational morphisms of quasi-projective schemes f:Y→X and Cohen–Macaulay coherent sheaves of modules over Y that are isomorphic to the pull-back of ℱ away from the exceptional locus of f. Certain blow-ups of X at locally complete intersections subschemes which contain non-reduced scheme structures on the non-Cohen–Macaulay locus of ℱ are the main part of the construction. Received: 19 February 1998 / Revised version: 28 December 1998     

On γ-semi-open sets

Summary We introduce the concepts of γ-semi-open set, γ_s-set, γ^s-set, generalized γ_s-set, generalized γ^s-set, semi-T₁^γspace and semi-R₀^γspace by using γ-open-sets.     

Duality for Multiple Objective Fractional Subset Programming with Generalized (, ρ, σ, θ)-V-Type-I Functions

In this paper, we use a new class of generalized convex n-set functions, called (,ρ, σ, θ)-V-Type-I and related non-convex functions to establish appropriate duality theorems for three parametric and three semi-parametric dual models to the primal problem. This work extends an earlier work of Zalmai [Computer and Mathematics with Applications 43 (2002) 1489–1520] to a wider class of functions.This research is supported by the Department of Science and Technology, Ministry of Science and Technology, Government of India, under the Fast Track Scheme for Young Scientist through grant No. SR/FTP/MS-22/2001     

Cantor series constructions contrasting two notions of normality

Rényi (Mat Lapok 7:77–100, 1956) made a definition that gives a generalization of simple normality in the context of Q-Cantor series. In Mance ), a definition of Q-normality was given that generalizes the notion of normality in the context of Q-Cantor series. In this work, we examine both Q-normality and Q-distribution normality, treated in Lafer (Normal numbers with respect to Cantor series representation, 1974) and S̆alát (Czechoslovak Math J 18(93):476–488, 1968). Specifically, while the non-equivalence of these two notions is implicit in Lafer (Normal numbers with respect to Cantor series representation. Washington State University, 1974), in this paper, we give an explicit construction witnessing the nontrivial direction. That is, we construct a base Q as well as a real x that is Q-normal yet not Q-distribution normal. We next approach the topic of simultaneous normality by constructing an explicit example of a base Q as well as a real x that is both Q-normal and Q-distribution normal.     

On the boundary behavior of imbeddings of metric spaces into a Euclidean space

We investigate the boundary behavior of so-called Q-homeomorphisms with respect to a measure in some metric spaces. We formulate a series of conditions for the function Q(x) and the boundary of the domain under which any Q-homeomorphism with respect to a measure admits a continuous extension to a boundary point. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1068–1074, August, 2007.     

Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for <Emphasis Type="Italic">Q</Emphasis>-mappings

We prove that an open discrete Q-mapping has a continuous extension to an isolated boundary point if the function Q(x) has finite mean oscillation or logarithmic singularities of order at most n – 1 at this point. Moreover, the extended mapping is open and discrete and is a Q-mapping. As a corollary, we obtain an analog of the well-known Sokhotskii–Weierstrass theorem on Q-mappings. In particular, we prove that an open discrete Q-mapping takes any value infinitely many times in the neighborhood of an essential singularity, except, possibly, for a certain set of capacity zero. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 116–126, January, 2009.     

Homotopy Gerstenhaber Structures and Vertex Algebras

We provide a simple construction of a G _∞-algebra structure on an important class of vertex algebras V, which lifts the Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications to algebraic topology: the construction of a sheaf of G _∞ algebras on a Calabi–Yau manifold M, extending the operations of multiplication and bracket of functions and vector fields on M, and of a Lie_∞ structure related to the bracket of Courant (Trans Amer Math Soc 319:631–661, 1990).     

Extended finite operator calculus—an example of algebraization of analysis

“A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota—Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward's calculus of sequences in its afterwards elaborated form—a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota—Mullin or equivalently—of umbral calculus of Roman and Rota. At the same time this calculus is an example of the algebraization of the analysis—here restricted to the algebra of polynomials. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one. This is a review article based on the recent first author contributions [1]. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by Ward [2], Viskov [7, 8], Markowsky [12], Roman [28–32] on one side and the Rota-oriented notation on the other side [9–11, 1, 3, 4, 35] (see also [33]).     

Topology of the hyperspace of convex bodies of constant width

The hyperspace of all convex bodies of constant width in Euclidean spaceR ⁿ ,n≥2, is proved to be homeomorphic to a contractibleQ-manifold (Q denotes the Hilbert cube). The proof makes use of an explicitly constructed retraction of the entire hyperspace of convex bodies on the hyperspace of convex bodies of constant width. Translated fromMaternaticheskie Zametki, Vol. 62, No. 6, pp. 813–819, December, 1997 Translated by V. N. Dubrovsky     

Local behavior of <Emphasis Type="Italic">Q</Emphasis>-homeomorphisms in Loewner spaces

We study the problem of the elimination of isolated singularities for so-called Q-homeomorphisms in Loewner spaces. We formulate several conditions for a function Q(x) under which every Q-homeomorphism admits a continuous extension to an isolated singular point. We also consider the problem of the homeomorphicity of the extension obtained. The results are applied to Riemannian manifolds and Carnot groups. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1378–1388, October, 2008.     

Dynamics of neighborhoods of points under a continuous mapping of an interval

Let {I, f, Z ⁺} be a dynamical system induced by a continuous mapping f of a closed bounded interval I into itself. To describe the dynamics of neighborhoods of points unstable under the mapping f, we propose the concept of the εω-set ω ^{f, ε}(x) of a point x as the ω-limit set of the ε-neighborhood of the point x. We investigate the relationship between the εω-set and the domain of influence of a point. It is also shown that the domain of influence of an unstable point is always a cycle of intervals. The results obtained can be directly used in the theory of difference equations with continuous time and similar equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1534–1547, November, 2005.     

On some inequalities of probabilistic number theory

Let Q ₊ denote the set of positive rational numbers. We define discrete probability measures ν _x on the σ-algebra of subsets of Q ₊.We introduce additive functions ƒ: Q ₊ → G and obtain a bound for ν_x(ƒ (r) ∉ X+X−X) using a probability related to some independent random variables. This inequality is an analogue to that proved by I. Ruzsa for additive arithmetical functions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 256–266, April–June, 2006.     

On locally <Emphasis Type="Italic">A</Emphasis>-convex <Emphasis Type="Italic">H</Emphasis>*-algebras

We endow any proper A-convex H*-algebra (E, τ) with a locally pre-C*-topology. The latter is equivalent to that introduced by the pre C*-norm given by Ptàk function when (E, τ) is a Q-algebra. We also prove that the algebra of complex numbers is the unique proper locally A-convex H*-algebra which is barrelled and Q-algebra.      

Local rigidity of quasi-regular varieties

For a G-variety X with an open orbit, we define its boundary ∂ X as the complement of the open orbit. The action sheaf S _X is the subsheaf of the tangent sheaf made of vector fields tangent to ∂ X. We prove, for a large family of smooth spherical varieties, the vanishing of the cohomology groups H ⁱ (X, S _X ) for i > 0, extending results of Bien and Brion (Compos. Math. 104:1–26, 1996). We apply these results to study the local rigidity of the smooth projective varieties with Picard number one classified in Pasquier (Math. Ann., in press).     

Hopf Algebras of Type <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>, Twistings and the FRT-construction

We study pointed Hopf algebras of the form U(R _Q ), (Faddeev et al., Quantization of Lie groups and Lie algebras. Algebraic Analysis, vol. I, Academic, Boston, MA, pp. 129–139, 1988; Faddeev et al., Quantum groups. Braid group, knot theory and statistical mechanics. Adv. Ser. Math. Phys., vol. 9, World Science, Teaneck, NJ, pp. 97–110, 1989; Larson and Towber, Commun. Algebra 19(12):3295–3345, 1991), where R _Q is the Yang–Baxter operator associated with the multiparameter deformation of GL _n supplied in Artin et al. (Commun. Pure Appl. Math. 44:8–9, 879–895, 1991) and Sudbery (J. Phys. A, 23(15):697–704, 1990). We show that U(R _Q ) is of type A _n in the sense of Andruskiewitsch and Schneider (Adv. Math. 154:1–45, 2000; Pointed Hopf algebras. Recent developments in Hopf Algebras Theory, MSRI Series, Cambridge University Press, Cambridge, 2002). We consider the non-negative part of U(R _Q ) and show that for two sets of parameters, the corresponding Hopf sub-algebras can be obtained from each other by twisting the multiplication if and only if they possess the same groups of grouplike elements. We exhibit families of finite-dimensional Hopf algebras arising from U(R _Q ) with non-isomorphic groups of grouplike elements. We then discuss the case when the quantum determinant is central in A(R _Q ) and show that under some assumptions on the group of grouplike elements, two finite-dimensional Hopf algebras U(R _Q ), U(R _Q′) can be obtained from each other by twisting the comultiplication if and only if In the last part we show that U _Q is always a quotient of a double crossproduct. I wish to thank UIC, where some of the work was done, for hospitality.     

Generalized V-univexity type-I for multiobjective programming with n-set functions

In the last time important results in multiobjective programming involving type-I functions were obtained (Yuan et al. in: Konnov et al. (eds) Lecture notes in economics and mathematical systems, 2007; Mishra et al. An Univ Bucureşti Ser Mat, LII(2): 207–224, 2003). Following one of these ways, we study optimality conditions and generalized Mond-Weir duality for multiobjective programming involving n-set functions which satisfy appropriate generalized univexity V-type-I conditions. We introduce classes of functions called (ρ, ρ′)-V-univex type-I, (ρ, ρ′)-quasi V-univex type-I, (ρ, ρ′)-pseudo V-univex type-I, (ρ, ρ′)-quasi pseudo V-univex type-I, and (ρ, ρ′)-pseudo quasi V-univex type-I. Finally, a general frame for constructing functions of these classes is considered. This research was supported by Grant PN II code ID No. 112/01.10.2007, CEEX code 1/2006 No. 1531/2006, and CNCSIS A No. 105 GR/2006.     

L’application cotangente des surfaces de type général

We study surfaces of general type S whose cotangent sheaf is generated by its global sections. We define a map called the cotangent map of S that enables us to understand the obstructions to the ampleness of the cotangent sheaf of S. These obstructions are curves on S that we call “non-ample”. We classify the surfaces with an infinite number of non-ample curves and we partly classify the non-ample curves.