The goal of this paper is to prove the equivariant version of Bloch’s Riemann-Roch isomorphism between the higher algebraic K-theory and the higher Chow groups of smooth varieties. We show that for a linear algebraic group G acting on a smooth variety X , although there is no Chern character map from the equivariant K-groups to equivariant higher Chow groups, there is indeed such a map K i (X)⊗...

The cohomology of the classifying space BU(n) of the unitary groups can be identified with the the ring of symmetric polynomials on n variables by restricting to the cohomology of BT , where T ⊂ U(n) is a maximal torus. In this paper we explore the situation where BT = (CP) is replaced by a product of finite dimensional projective spaces (CP ), fitting into an associated bundle U(n)×T (S ) → (C...

We discuss rather systematically the principle, implicit in earlier works, that for a “random” element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the splitting field of the characteristic polynomial, computed using any faitfhful representation, has Galois group isomorphic to the Weyl group of the underlying algebraic group. Besides tools such as t...

Homological characteristics of digital objects can be obtained in a straightforward manner computing an algebraic map φ over a finite cell complex K (with coefficients in the finite field F2 = {0, 1}) which represents the digital object [9]. Computable homological information includes the Euler characteristic, homology generators and representative cycles, higher (co)homology operations, etc. T...

Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co-)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming. Our approach is based on formulating the memb...

1: After identification of the algebra of exponential generating series with the shuffle algebra of ordinary formal power series, the exponential map exp! : XK[[X]] −→ 1 +XK[[X]] for the associated Lie group with multiplication given by the shuffle product is well-defined over an arbitrary field K by a result going back to Hurwitz. The main result of this paper states that exp! (and its recipro...

The Hölder continuity of the truncated moment map a shade function in Euclidean space is established vicinity principal semi-algebraic set. proof combines volume bounds sets and convex optimization methods. main estimate applied to potential type transform specific two real variables, for perturbations quadrature domains.

A variant of the Hénon map is described in which the linear term is replaced by one that involves a much earlier iterate of the map. By varying the time delay, this map can be used to explore the transition from low-dimensional to high-dimensional dynamics in a chaotic system with minimal algebraic complexity, including a detailed comparison of the Kaplan-Yorke and correlation dimensions. The h...

Most problems in algebra, at some stage, must consider algebraic equations. Sometimes solving them, other times producing them – so their existence manifests a desired structure. Akin to doing differential equations, a researcher in algebraic equations must specialize to get somewhere. My speciality is algebraic equations in two variables, say w and z, where one variable, say z, interprets as a...

A complete minimal surface in R is said to be pseudo-algebraic, if its Weierstrass data are defined on a compact Riemann surface finitely many points removed M = M\{P1, . . . , Pn} and extend meromorphically across punctures ([KKM], see also [G]). The punctured Riemann surface M on which the Weierstrass data are defined is called the basic domain. The difference between a complete minimal surfa...