Cyclotomic torsion on Fermat Jacobians

We prove that the torsion part of the Mordell–Weil group of the Jacobian of a Fermat curve over a cyclotomic field is contained in the kernel of a certain isogeny. This provides a natural analogue of a similar result on Jacobians of Fermat quotient curves.     

On “bent” functions

Let P(x) be a function from GF(2ⁿ) to GF(2). P(x) is called “bent” if all Fourier coefficients of (−1)^P(x) are ±1. The polynomial degree of a bent function P(x) is studied, as are the properties of the Fourier transform of (−1)^P(x), and a connection with Hadamard matrices.     

K2 of a Fermat quotient and the value of its L-function

Two linearly independent elements of k ₂ of a certain quotient of a Fermat curve is exhibited in an explicit manner. The covolume of the regulator and the value of the L-function of the curve is numerically computed, and their ratio is nearly equal to a simple rational number.     

Average-case complexity and decision problems in group theory

We investigate the average-case complexity of decision problems for finitely generated groups, in particular, the word and membership problems. Using our recent results on “generic-case complexity”, we show that if a finitely generated group G has word problem solvable in subexponential time and has a subgroup of finite index which possesses a non-elementary word-hyperbolic quotient group, then the average-case complexity of the word problem of G is linear time, uniformly with respect to the collection of all length-invariant measures on G. This results applies to many of the groups usually studied in geometric group theory: for example, all braid groups B_n, all groups of hyperbolic knots, many Coxeter groups and all Artin groups of extra-large type.     

Approximation of unbounded functions via compactification

Approximants to functions f(s) that are allowed to possess infinite limits on their interval of definition, are constructed.To this end a compactification of Rⁿ is developed which is based on the projection of Rⁿ on a bowl-shaped subset of a parabolic surface. This compactification induces a bijection and a metric with several desirable properties that make it a useful tool for rational approximation of unbounded functions.Roughly speaking this compactification enables us to show that unbounded functions can be approximated by rational functions on a closed interval; thus we also obtain an extension to Weierstrass’ celebrated theorem. An extension to a Fourier-type theorem is also obtained. Roughly speaking, our result states that unbounded periodic functions can be approximated by quotients of certain trigonometric sums. The characteristics of the main results are the following. The approximations do not require the original approximated function to possess a restricted rate of growth. Neither do they require that the approximated function possess any amount of smoothness. Moreover, the numerator and denominator, in an approximating quotient are guaranteed not to vanish simultaneously. Furthermore, some of the proposed approximations are guaranteed to be bounded at every point at which the original approximated function is bounded. Beside the tool of compactification we also employ Bernstein polynomials and Cesaro means of “trigonometric sums”.     

Polynomial identity rings as rings of functions

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a PGL_n-action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the “coordinate ring” of a “variety” in this setting. For n=1 our definitions and results reduce to those of classical affine algebraic geometry.     

Some Asymptotic Formulae for Gaussian Distributions

This paper considers asymptotic expansions of certain expectations which appear in the theory of large deviation for Gaussian random vectors with values in a separable real Hilbert space. A typical application is to calculation of the “tails” of distributions of smooth functionals,p(r)=P{Φ(r⁻¹ξ)0},r→∞, e.g., the probability that a centered Gaussian random vector hits the exterior of a large sphere surrounding the origin. The method provides asymptotic formulae for the probability itself and not for its logarithm in a situation, where it is natural to expect thatp(r)=c′r^D exp{−c″r²}. Calculations are based on a combination of the method of characteristic functionals with the Laplace method used to find asymptotics of integrals containing a fast decaying function with “small” support.     

An application of the Turán theorem to domination in graphs

A function f:V(G)→{+1,−1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least 1. The signed domination number γ_s(G) of G is the minimum weight of a signed dominating function on G. By simply changing “{+1,−1}” in the above definition to “{+1,0,−1}”, we can define the minus dominating function and the minus domination number of G. In this note, by applying the Turán theorem, we present sharp lower bounds on the signed domination number for a graph containing no (k+1)-cliques. As a result, we generalize a previous result due to Kang et al. on the minus domination number of k-partite graphs to graphs containing no (k+1)-cliques and characterize the extremal graphs.     

The b-transform

The b-transform is used to convert entire functions into “primary b-functions” by replacing the powers and factorials in the Taylor series of the entire function with corresponding “generalized powers” (which arise from a polynomial function with combinatorial applications) and “generalized factorials.” The b-transform of the exponential function turns out to be a generalization of the Euler partition generating function, and partition generating functions play a key role in obtaining results for the b-transforms of the elementary entire transcendental functions. A variety of normal-looking results arise, including generalizations of Euler's formula and De Moivre's theorem. Applications to discrete probability and applied mathematics (i.e., damped harmonic motion) are indicated. Also, generalized derivatives are obtained by extending the concept of a b-transform.     

The fermat equation over quadratic fields

Kummer's method of proof is applied to the Fermat equation over quadratic fields. The concept of an m-regular prime, p, is introduced and it is shown that for certain values of m, the Fermat equation with exponent p has no nontrivial solutions over the field Q(√m).     

Which posets have a scattered MacNeille completion?

A poset is order-scattered if it does not embed the chain η of the rational numbers. We prove that there are eleven posets such that N(P), the MacNeille completion of P, is order-scattered if and only if P embeds none of these posets. Moreover these posets are pairwise non-embeddable in each other. This result completes a previous characterisation due to Duffus, Pouzet, Rival [4]. The proof is based on the “bracket relation”: a famous result of F. Galvin.Dedicated to the memory of Ivan Rival.Received June 16, 2004; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Two-sided bounds and leading term for rates of convergence in the multivariate central limit theorem

For a sequence of independent and identically distributed random vectors, upper and lower bounds are obtained for the discrepancy between the probability measure P_n, induced by their normalized sum, and the Normal measure Φ. The upper and lower bounds are of the same order of magnitude. These results may be derived by a “leading term” approach, in which a signed measure Q_n is introduced as a first order approximation to P_n − Φ. The purpose of this paper is to investigate properties of the leading term.     

Reverse mathematics and well-ordering principles: A pilot study

The larger project broached here is to look at the generally sentence “if X is well-ordered then f(X) is well-ordered”, where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory T_f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we prove in this paper that the statement “if X is well-ordered then ε_X is well-ordered” is equivalent to . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalbán, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte’s method of proof search (deduction chains) [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a (small) fragment of .     

Diagnosability of regular systems

A novel approach aimed at evaluating the diagnosability of regular systems under the PMC model is introduced. The diagnosability is defined as the ability to provide a correct diagnosis, although possibly incomplete. This concept is somehow intermediate between one-step diagnosability and sequential diagnosability. A lower bound to diagnosability is determined by lower bounding the minimum of a “syndrome-dependent” bound t_σ over the set of all the admissible syndromes. In turn, t_σ is determined by evaluating the cardinality of the smallest consistent fault set containing an aggregate of maximum cardinality. The new approach, which applies to any regular system, relies on the “edge-isoperimetric inequalities” of connected components of units declaring each other non-faulty. This approach has been used to derive tight lower bounds to the diagnosability of toroidal grids and hypercubes, which improve the existing bounds for the same structures.     

Twisted loop groups and their affine flag varieties

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a “twisted case”; a consequence of our results is that our construction also includes the flag varieties for Kac–Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t)) and some applications to local models of Shimura varieties.     

Algebraic K-theory of non-linear projective spaces

In the spirit of “The Fundamental Theorem for the algebraic K-theory of spaces: I” (J. Pure Appl. Algebra 160 (2001) 21–52) we introduce a category of sheaves of topological spaces on n-dimensional projective space and present a calculation of its K-theory, a “non-linear” analogue of Quillen's isomorphism K_i(P_Rⁿ)₀ⁿK_i(R).     

Complete p-Descent for Jacobians of Hermitian Curves

Let X be the Fermat curve of degree q+1 over the field k of q² elements, where q is some prime power. Considering the Jacobian J of X as a constant abelian variety over the function field k(X), we calculate the multiplicities, in subfactors of the Shafarevich–Tate group, of representations associated with the action on X of a finite unitary group. J is isogenous to a power of a supersingular elliptic curve E, the structure of whose Shafarevich–Tate group is also described.     

Induced disjoint paths problem in a planar digraph

As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}. The objective is to find k paths P₁,…,P_k such that P_i is a path from s_i to t_i and P_i and P_j have neither common vertices nor adjacent vertices for any distinct i,j.The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We investigate the computational complexity of several variants of the induced disjoint paths problem. We show that the induced disjoint paths problem is (i) solvable in polynomial time when k is fixed and G is a directed (or undirected) planar graph, (ii) NP-hard when k=2 and G is an acyclic directed graph, (iii) NP-hard when k=2 and G is an undirected general graph.As an application of our first result, we show that we can find in polynomial time certain structures called a “hole” and a “theta” in a planar graph.     

Fermat–Reyes method in the ring of Fermat reals

To discover derivatives, Pierre de Fermat used to assume a non-zero increment h in the incremental ratio and, after some calculations, to set h=0 in the final result. This method, which sounds as inconsistent, can be perfectly formalized with the Fermat–Reyes theorem about existence and uniqueness of a smooth incremental ratio. In the present work, we will introduce the cartesian closed category where to study and prove this theorem and describe in general the Fermat method. The framework is the theory of Fermat reals, an extension of the real field containing nilpotent infinitesimals which does not need any knowledge of mathematical logic. This key theorem will be essential in the development of differential and integral calculus for smooth functions defined on the ring of Fermat reals and also for infinite-dimensional operators like derivatives and integrals.