The de-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of the de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute the cohomologies of the Lie algebra g of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as the Shapiro lemma.     

Reelle algebraische Funktionen mit vorgegebenen Null- und Polstellen

This note continues the investigations of Knebusch on algebraic curves over real closed fields and was initiated by reading [3]. Especially we ask for the existence of real algebraic functions with given zeroes and poles, a question going back to Witt [4]. We study the real nature of coverings of real algebraic curves, and if the covering has degree two, we get algebraic proofs for results, which in the classical case have been obtained by topological methods in [2].     

Some Algebraic Properties of Cerf Diagrams of One-Parameter Function Families

We obtain results concerning Arnold's problem about a generalization of the Pontryagin-Thom construction in cobordism theory to real algebraic functions. The Pontryagin-Thom construction in the Wells form is transferred to the space of real functions. The relation of the problem with algebraic K-theory and the h-principle due to Eliashberg and Mishachev is revealed.     

Study of generalized eigenfunctions of a perturbed isotropic elastic half‐space

We consider the self‐adjoint operator governing the propagation of elastic waves in a perturbed isotropic half‐space (perturbation with compact support of a homogeneous isotropic half‐space) with a free boundary condition. We propose a method to obtain, numerical values included, a complete set of generalized eigenfunctions that diagonalize this operator. The first step gives an explicit representation of these functions using a perturbative method. The unbounded boundary is a new difficulty compared with the method used by Wilcox [25], who set the problem in the complement of bounded open set. The second step is based on a boundary integral equations method which allows us to compute these functions. For this, we need to determine explicitly the Green's function of (A₀ – ω²), where A₀ is the self‐adjoint operator describing elastic waves in a homogeneous isotropic half‐space. Copyright © 2000 John Wiley & Sons, Ltd.     

Quantitative Generalized Bertini-Sard Theorem for Smooth Affine Varieties

Let X ⊂ ℂⁿ be a smooth affine variety of dimension n – r and let f = (f₁,..., f_m): X → ℂ_m be a polynomial dominant mapping. We prove that the set K(f) of generalized critical values of f (which always contains the bifurcation set B(f) of f) is a proper algebraic subset of ℂ_m. We give an explicit upper bound for the degree of a hypersurface containing K(f). If I(X)—the ideal of X—is generated by polynomials of degree at most D and deg f_i ≤ d, then K(f) is contained in an algebraic hypersurface of degree at most (d + (m – 1)(d – 1)+(D – 1)r)_n-rD_r. In particular if X is a hypersurface of degree D and f: X → ℂ is a polynomial of degree d, then f has at most (d + D – 1)_n-1D generalized critical values. This bound is asymptotically optimal for f linear. We give an algorithm to compute the set K(f) effectively. Moreover, we obtain similar results in the real case.     

Explicit Universal Deformations of Even Galois Representations

We investigate the case of deformations of even Galois representations. Our methods are the group theoretic ones mainly developed by Nigel Boston to study odd representations. We present conditions for Borel and tame cases under which the universal deformation ring is isomorphic to ?_p[[T]] and where we compute the universal deformation explicitly. Furthermore we produce a family of examples of totally real S₃ extensions which satisfy the above conditions in the tame case and we give examples in the Borel case. Finally we study the change of the deformation space under enlarging the ramification and thus give an example of an even representation that is not twist-finite.     

On the Distributional Index 2F1-transform

In this paper the index ₂F₁-transform is defined on a space of generalized functions by using the method of adjoints. We prove that smooth functions having suitable asymptotic behaviors around zero and infinite are ₂F₁-transforms of generalized functions.     

Non-special divisors on real algebraic curves and embeddings into real projective spaces

We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.) of the real algebraic curve into the real projective space ℙ ^r for r≥3 (r=2, resp.). We study these embeddings in quite some detail. Received: October 17, 2001?Published online: February 20, 2003     

The enumeration of bipartite graphs

The standard construction of graphs with n connected components is modified here for bicolored graphs by letting S_n × H act on the function space $\text{Y}{}^1\text{where}\text{X={1,2,…,n}, Y}$ is the set of connected bicolored graphs, and H is the group that interchanges the vertex colors. Then DeBruijn's Generalization of Polya's Theorem is applied to arrive at a direct algebraic relationship between the generating functions for bicolored and connected bicolored graphs. As the former generating function is easily computable, this relationship gives us the latter generating function which is precisely the generating function for connected bipartite graphs.     

Monodromy of certain Painlevé–VI transcendents and reflection groups

We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ= and 2α=(2μ-1)² with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space. Oblatum 19-III-1999 & 25-XI-1999?Published online: 21 February 2000     

Real Quotient Singularities and Nonsingular Real Algebraic Curves in the Boundary of the Moduli Space

The quotient of a real analytic manifold by a properly discontinuous group action is, in general, only a semianalytic variety. We study the boundary of such a quotient, i.e., the set of points at which the quotient is not analytic. We apply the results to the moduli space M_g/ of nonsingular real algebraic curves of genus g (g2). This moduli space has a natural structure of a semianalytic variety. We determine the dimension of the boundary of any connected component of M_g/. It turns out that every connected component has a nonempty boundary. In particular, no connected component of M_g/ is real analytic. We conclude that M_g/ is not a real analytic variety.     

On measure-preserving functions over ?3

This paper is devoted to (discrete) p-adic dynamical systems, an important domain of algebraic and arithmetic dynamics [31]?C[41], [5]?C[8]. In this note we study properties of measurepreserving dynamical systems in the case p = 3. This case differs crucially from the case p = 2. The latter was studied in the very detail in [43]. We state results on all compatible functions which preserve measure on the space of 3-adic integers, using previous work of A. Khrennikov and author of present paper, see [24]. To illustrate one of the obtained theorems we describe conditions for the 3-adic generalized polynomial to be measure-preserving on ?₃. The generalized polynomials with integral coefficients were studied in [17, 33] and represent an important class of T-functions. In turn, it is well known that T-functions are well-used to create secure and efficient stream ciphers, pseudorandom number generators.     

Generating functions for generalized Hermite polynomials associated with parabolic cylinder functions

ABSTRACT

In this article, we first give some basic properties of generalized Hermite polynomials associated with parabolic cylinder functions. We next use Weisner? group theoretic method and operational rules method to establish new generating functions for these generalized Hermite polynomials. The operational methods we use allow us to obtain unilateral, bilinear and bilateral generating functions by using the same procedure. Applications of generating functions obtained by Weisner? group theoretic method are discussed.     

Real and Rational Forms of Certain O<Subscript>2</Subscript>(ℂ)-actions,and a Solution to the Weak Complexification Problem

The main result of this paper is that there is a non-linearizable real algebraic action of the circle S¹ on ⁴, an action which becomes linearizable over . This solves the Weak Complexification Problem. We also show that for any field k of characteristic zero, there are non-linearizable algebraic actions of the group O₂(k) on four-dimensional affine k-space, and if k contains a square root of 3, then this action restricts to a non-linearizable action of the symmetric group S₃ on four-dimensional affine k-space.     

Commutative Sequences of Integrable Functions and Best Approximation With Respect to the Weighted Vector Measure Distance

Let λ be a countably additive vector measure with values in a separable real Hilbert space H. We define and study a pseudo metric on a Banach lattice of integrable functions related to λ that we call a λ-weighted distance. We compute the best approximation with respect to this distance to elements of the function space by the use of sequences with special geometric properties. The requirements on the sequence of functions are given in terms of a commutation relation between these functions that involves integration with respect to λ. We also compare the approximation that is obtained in this way with the corresponding projection on a particular Hilbert space.     

Local functionals on C∞(G) and probability

A characterization of local functionals on C_∞(G), the space of real continuous functions with compact supports on a locally compact space G, is given. Such functionals were defined by Gel'fand and Vilenkin, as they occur in the analysis of generalized random processes with independent values. The preceding characterization is then used in a representation of the characteristic functionals of the above processes on C_∞(G), and results analogous to those of Gel'fand and Vilenkin on a Schwartz space are obtained. Since C_∞(G) is not nuclear, this study presents new problems and it largely complements the earlier work.     

q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

Algebraic density property of homogeneous spaces

Let X be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that X is equipped with several fixed point free nondegenerate SL₂-actions satisfying some mild additional assumption. Then we prove that the Lie algebra generated by completely integrable algebraic vector fields on X coincides with the space of all algebraic vector fields. In particular, we show that apart from a few exceptions this fact is true for any homogeneous space of form G/R where G is a linear algebraic group and R is a closed proper reductive subgroup of G.