Modification systems and integration in their Chow groups

We introduce a notion of integration on the category of proper birational maps to a given variety X, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers of nonsingular birational varieties; ‘stringy’ Chern classes of singular varieties; and a zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern–Schwartz–MacPherson classes of possibly singular varieties, placing them into a context in which a ‘change-of-variable’ formula holds.     

On Ginzburg's Bivariant Chern Classes, II

For a morphism whose target variety is nonsingular, the Chern–Schwartz–MacPherson class homomorphism followed by capping with the pullback of the Segre class of the target variety is called the Ginzburg–Chern class. In this paper, using the Verdier–Riemann–Roch for Chern Class, we show that the correspondence assigning to a bivariant constructible function on any morphism with nonsingular target variety the Ginzburg–Chern class of it is the unique Grothendieck transformation satisfying the 'normalization condition' that for morphisms to a point it becomes the Chern–Schwartz–MacPherson class homomorphism, except for that the bivariant homology pullback is considered only for a smooth morphism.     

Cyclic permutations and evolutionary trees

Given a tree with leaf set X, there are certain ways of arranging the elements of X in a circular order so that can be embedded in the plane and ‘preserve’ this ordering. We investigate some new combinatorial properties of these ‘circular orderings.’ We then use these properties to establish two results concerning dissimilarity maps on X that are induced by edge-weighted trees with leaf set X.     

The coarse Baum–Connes conjecture via coarse geometry

The C₀ coarse structure on a metric space is a refinement of the bounded structure and is closely related to the topology of the space. In this paper we will prove the C₀ version of the coarse Baum–Connes conjecture and show that K_*(C^*X₀) is a topological invariant for a broad class of metric spaces. Using this result we construct a ‘geometric’ obstruction group to the coarse Baum–Connes conjecture for the bounded coarse structure. We then show under the assumption of finite asymptotic dimension that the obstructions vanish, and hence we obtain a new proof of the coarse Baum–Connes conjecture in this context.     

Elementarity and Dimensions

An alternative proof of Fedorchuk’s recent result that dim X ≤ Dg X for compact Hausdorff spaces X is given. The problem is reduced to the metric case by using the Lowenheim-Skolem theorem.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 292–298.Original Russian Text Copyright © 2005 by K. P. Hart.     

Differentiation of Singular Integrals with Piecewise-Continuous Density and Boundary Values of Derivatives of a Cauchy-Type Integral

We establish sufficient conditions for the differentiability of a singular Cauchy integral with piecewise-continuous density. Formulas for the nth-order derivatives of a singular Cauchy integral and for the boundary values of the nth-order derivatives of a Cauchy-type integral are obtained.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 222–229, February, 2005.     

Projective Completions of Jordan Pairs,Part II: Manifold Structures and Symmetric Spaces

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields , and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ‘compact-like’ duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as ‘standard models’ – they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra.Mathematics Subject Classiffications (2000). primary: 17C36, 46H70, 17C65; secondary: 17C30, 17C90     

Optimal reinsurance under VaR and CTE risk measures

Let X denote the loss initially assumed by an insurer. In a reinsurance design, the insurer cedes part of its loss, say f(X), to a reinsurer, and thus the insurer retains a loss I_f(X)=X−f(X). In return, the insurer is obligated to compensate the reinsurer for undertaking the risk by paying the reinsurance premium. Hence, the sum of the retained loss and the reinsurance premium can be interpreted as the total cost of managing the risk in the presence of reinsurance. Based on a technique used in [Müller, A., Stoyan, D., 2002. Comparison Methods for Stochastic Models and Risks. In: Willey Series in Probability and Statistics] and motivated by [Cai J., Tan K.S., 2007. Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measure. Astin Bull. 37 (1), 93–112] on using the value-at-risk (VaR) and the conditional tail expectation (CTE) of an insurer’s total cost as the criteria for determining the optimal reinsurance, this paper derives the optimal ceded loss functions in a class of increasing convex ceded loss functions. The results indicate that depending on the risk measure’s level of confidence and the safety loading for the reinsurance premium, the optimal reinsurance can be in the forms of stop-loss, quota-share, or change-loss.     

One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

We investigate the existence of a separately continuous function f: X × Y → ℝ with a one-point set of discontinuity points in the case where the topological spaces X and Y satisfy conditions of compactness type. In particular, it is shown that, for compact spaces X and Y and nonisolated points x ₀ ∈ X and y ₀ ∈ Y, a separately continuous function f: X × Y → ℝ with the set of discontinuity points {(x ₀, y ₀)} exists if and only if there exist sequences of nonempty functionally open sets in X and Y that converge to x ₀ and y ₀, respectively.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 94–101, January, 2005.     

A fuzzy adaptive network approach to parameter estimation in cases where independent variables come from an exponential distribution

In a regression analysis, it is assumed that the observations come from a single class in a data cluster and the simple functional relationship between the dependent and independent variables can be expressed using the general model; Y=f(X)+ε. However; a data cluster may consist of a combination of observations that have different distributions that are derived from different clusters. When faced with issues of estimating a regression model for fuzzy inputs that have been derived from different distributions, this regression model has been termed the ‘switching regression model’ and it is expressed with . Here l_i indicates the class number of each independent variable and p is indicative of the number of independent variables [J.R. Jang, ANFIS: Adaptive-network-based fuzzy inference system, IEEE Transaction on Systems, Man and Cybernetics 23 (3) (1993) 665–685; M. Michel, Fuzzy clustering and switching regression models using ambiguity and distance rejects, Fuzzy Sets and Systems 122 (2001) 363–399; E.Q. Richard, A new approach to estimating switching regressions, Journal of the American Statistical Association 67 (338) (1972) 306–310].In this study, adaptive networks have been used to construct a model that has been formed by gathering obtained models. There are methods that suggest the class numbers of independent variables heuristically. Alternatively, in defining the optimal class number of independent variables, the use of suggested validity criterion for fuzzy clustering has been aimed. In the case that independent variables have an exponential distribution, an algorithm has been suggested for defining the unknown parameter of the switching regression model and for obtaining the estimated values after obtaining an optimal membership function, which is suitable for exponential distribution.     

On spaces which are linearly D

We introduce a generalization of D-spaces, which we call linearly D-spaces. The following results are obtained for a T₁-space X.

– X is linearly Lindelöf if, and only if, X is a linearly D-space of countable extent.

– X is linearly D provided that X is submetaLindelöf.

– X is linearly D provided that X is the union of finitely many linearly D-subspaces.

– X is compact provided that X is countably compact and X is the union of countably many linearly D-subspaces.

Dually Degenerate Varieties and the Generalization of a Theorem of Griffiths–Harris

The dual variety X^* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X^* is a hypersurface in the dual space (P^N)^*. If dimX^*<N–1, then the variety X is called dually degenerate. The authors refine these definitions for a variety XP^N with a degenerate Gauss map of rankr. For such a variety, in the general case, the dimension of its dual variety X^* is N–l–1, where l=n–r, and X is dually degenerate if dimX^*<N–l–1. In 1979 Griffiths and Harris proved that a smooth variety XP^N is dually degenerate if and only if all its second fundamental forms are singular. The authors generalize this theorem for a variety XP^N with a degenerate Gauss map of rankr. Mathematics Subject Classification (2000) 53A20.     

On the tautological ring of ℳ g

We prove that any product of tautological classes of _g of degreed (in the Chow ring of _g with rational coefficients) vanishes ford>g–2 and is proportional to the class of the hyperelliptic locus in degreeg–2.Oblatum 28-I-1195 & 15-IV-1995In memory of Nicolaas H. Kuiper (1920–1994)     

Classes de Hirzebruch et classes de Chern motiviques

Let X be a complex algebraic variety. We define and study new theories of characteristic classes, defined on the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. One of them, $T_y$ is a homology class version of the motivic measure and generalizes the corresponding Hirzebruch characteristic. It unifies the Chern class transformation of Schwartz and MacPherson, the Todd class transformation of Baum–Fulton–MacPherson and the L-class transformation of Cappell–Shaneson. To cite this article: J.-P. Brasselet et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Subtractive Categories

We introduce a notion of a subtractive category. It generalizes the notion of a pointed subtractive variety of universal algebras in the sense of A. Ursini. Subtractive categories are closely related to Mal’tsev and additive categories: (i) a category C with finite limits is a Mal’tsev category if and only if for every object X in C the category Pt(X)=((X,1_X)↓(C↓X)) of “points over X” is subtractive; (ii) a pointed category C with finite limits is additive if and only if C is subtractive and half-additive.Mathematics Subject Classifications (2000) 18C99, 18E05, 08B05.     

Pseudo-Index of Fano Manifolds and Smooth Blow-Ups

Suppose π: X → Y is a smooth blow-up along a submanifold Z of Y between complex Fano manifolds X and Y of pseudo-indices i_X and i_Y respectively (recall that i_X is defined by i_X :=min {−K_X·C | C is a rational curve of X}). We prove that if 2 dim (Z) < dim (Y)+i_Y −1 and show that this result is optimal by classifying the ‘boundary’ cases. As expected, these results are obtained by studying rational curves on X and Y.     

Separately Continuous Functions on Products and Their Dependence on ℵ Coordinates

We investigate necessary and sufficient conditions on topological products X = ∏_{s ∈ s} X _s and Y = ∏_{t ∈ T} Y _t for every separately continuous function f: X × Y → ℝ to be dependent on at most ℵ coordinates with respect to a certain coordinate.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1357 – 1368, October, 2004.     

A Microlocal Riemann–Hilbert Correspondence

We present a microlocal version of the Riemann–Hilbert correspondence for regular holonomic D-modules. We show that a regular holonomic system of microdifferential equations is associated to a perverse sheaf concentrated in degree 0. Moreover, we show that this perverse sheaf can be recovered from the local system it determines on the complementary of its singular locus. We characterize the classes of perverse sheaves and local systems associated to regular holonomic systems of microdifferential equations.     

Some existence and construction results of polygonal designs

This paper revisits the existence and construction problems for polygonal designs (a special class of partially balanced incomplete block designs associated with regular polygons). We present new polygonal designs with various parameter sets by explicit construction. In doing so we employ several construction methods — some conventional and some new. We also establish a link between a class of polygonal designs of block size 3 and the cyclically generated ‘λ-fold triple systems’. Finally, we show that the existence question for a certain class of polygonal designs is equivalent to the existence question for ‘perfect grouping systems’ which we introduce.