Hirzebruch–Milnor classes of complete intersections

We prove a new formula for the Hirzebruch–Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern–Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusiński and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern–Milnor classes of complete intersections with isolated singularities.     

Combinatorial Thom–Hirzebruch Characteristic Classes

With any flat combinatorial metric on an arbitrary oriented combinatorial manifold, one associates a family of signature operators S _t depending on a small real parameter t. These operators have support on a Const. t-neighborhood of the diagonal and inherit the local symmetry of the combinatorial structure. One shows that when t0, the measures ^d , introduced by Connes, Sullivan and the author, applied onto the operators S _t , converge toward a simplicial chain representing the Thom–Hirzebruch homology class of corresponding degree.     

Stability of Frobenius Pull–Backs of Tangent Bundles of Weighted Complete Intersections

The purpose of this paper is to show that every e–th Frobenius pull–back of the tangent bundle of a smooth weighted complete intersection defined over an algebraically closed field of positive characteristic is μ–stable if the Picard group is isomorphic to ℤ, in particular, if the dimension of the smooth weighted complete intersection is at least 3.     

A generalized Hirzebruch Riemann–Roch theorem

This short Note proves a generalization of the Hirzebruch Riemann–Roch theorem equivalent to the Cardy condition. This is done using an earlier result that explicitly describes what the Mukai pairing on Hochschild homology descends to in Hodge cohomology via the Hochschild–Kostant–Rosenberg map twisted by the root Todd genus. To cite this article: A.C. Ramadoss, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The Stanley–Reisner Ideals of Polygons as Set-Theoretic Complete Intersections

We show that the Stanley–Reisner ideal of the one-dimensional simplicial complex whose diagram is an n-gon is always a set-theoretic complete intersection in any positive characteristic.     

The Equivariant Hirzebruch–Riemann–Roch Theorem and the Geometry of Derived Loop Spaces

Mathematical Notes -     

Applications aux sommes elliptiques d'Apostol–Dedekind–Zagier

In this Note, we give two applications to our work [Bayad, C. R. Acad. Sci. Paris, Ser. I 339 (2004); DOI: 10.1016/j.crma.2004.07.018] concerning multiple elliptic Apostol–Dedekind–Zagier sums. These elliptic sums are defined by means of certain Jacobi modular forms of two variables $D_{\tau }(z\text{;}\phi )$. When $\text{Im}(\tau )\to \infty$, these elliptic sums give the classical Apostol–Dedekind–Zagier multiple sums [Apostol, Duke Math. J. 17 (1950) 147–157, Pacific. J. Math 2 (1952) 1–9; Zagier, Math. Ann, 202 (1973) 149–172]. We give a reciprocity law for these sums. To cite this article: A. Bayad, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Bifurcation of limit cycles near equivariant compound cycles

In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z3 equivariant system can have 13 limit cycles.     

On ordinary generalized geometric–arithmetic index

The ordinary generalized geometric–arithmetic index of graphs is introduced and some properties especially lower and upper bounds in terms of other graph invariants and topological indices are obtained.     

Hyperelliptic integrals and generalized arithmetic–geometric mean

We show how certain determinants of hyperelliptic periods can be computed using a generalized arithmetic-geometric mean iteration, whose initialisation parameters depend only on the position of the ramification points. Special attention is paid to the explicit form of this dependence and the signs occurring in the real domain.     

Robust cycles in Kolmogorov–Lotka–Volterra class of models with intraspecific co-operation

Using the Andronov–Hopf bifurcation theorem and the Poincaré–Bendixson Theorem, we explore robust cyclical possibilities in Kolmogorov–Lotka–Volterra class of models with positive intraspecific cooperation (in the form of social networks) in the prey population. We find that this additional feedback effect of intraspecific cooperation introduces nonlinearities which modify the cyclical outcomes of the model. We show that the cyclical outcomes are more robust than in the existing literature in this area due to introduction of such non-linearities. We also demonstrate the possibilities of multiple limit cycles under certain situations.     

Hamiltonian cycles in I–graphs

The I–graphs generalize the family of generalized Petersen graphs. We show that a connected I–graph which is not a generalized Petersen graph is Hamiltonian.