Explicit merit factor formulae for Fekete and Turyn polynomials

We give explicit formulas for the norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials

where is the Legendre symbol. For example for an odd prime,

where is the class number of . Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of . This is the sequence that has the largest known asymptotic merit factor. Explicitly,

where denotes the nearest integer, satisfies

where

Indeed we derive a closed form for the norm of all shifted Fekete polynomials

Namely

and if .

     

On the Diophantine equation : Higher-order recurrences

Let be a field of characteristic and let be a linear recurring sequence of degree in defined by the initial terms and by the difference equation

with . Finally, let be an element of . In this paper we are giving fairly general conditions depending only on on , and on under which the Diophantine equation

has only finitely many solutions . Moreover, we are giving an upper bound for the number of solutions, which depends only on . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

     

Sums of unit fractions

Erdos and Straus conjectured that for any positive integer the equation has a solution in positive integers , and . Let k \geq 3$"> and

We show that parametric solutions can be used to find upper bounds on where the number of parameters increases exponentially with . This enables us to prove

0. \end{displaymath}">

This improves upon earlier work by Viola (1973) and Shen (1986), and is an ``exponential generalization' of the work of Vaughan (1970), who considered the case .

     

Poincaré series of resolutions of surface singularities

Let be a resolution of singularities of a normal surface singularity , with integral exceptional divisors . We consider the Poincaré series

where

We show that if has characteristic zero and is a semi-abelian variety, then the Poincaré series is rational. However, we give examples to show that this series can be irrational if either of these conditions fails.

     

Nonlinear Cauchy-Riemann operators in

This paper has arisen from an effort to provide a comprehensive and unifying development of the -theory of quasiconformal mappings in . The governing equations for these mappings form nonlinear differential systems of the first order, analogous in many respects to the Cauchy-Riemann equations in the complex plane. This approach demands that one must work out certain variational integrals involving the Jacobian determinant. Guided by such integrals, we introduce two nonlinear differential operators, denoted by and , which act on weakly differentiable deformations of a domain .

Solutions to the so-called Cauchy-Riemann equations and are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental -estimate

In quest of the best constant , we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.

     

Damped wave equation with a critical nonlinearity

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity

where 0,$"> and space dimensions . Assume that the initial data

where \frac{n}{2},$"> weighted Sobolev spaces are

Also we suppose that

0,\int u_{0}\left( x\right) dx>0, \end{displaymath}">

where

Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property

for all 0,$"> where

     

Geometry of chain complexes and outer automorphisms under derived equivalence

The two main theorems proved here are as follows: If is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization of the family of finite -module complexes with fixed sequence of dimensions and an ``almost projective' complex , there exists a canonical vector space embedding

where is the pertinent product of general linear groups acting on , tangent spaces at are denoted by , and is identified with its image in the derived category .

     

The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

The definite integral

is related to the Laplace transform of the digamma function

by when . Certain analytic expressions for in the complementary range, , are also provided.

     

On Littlewood's boundedness problem for sublinear Duffing equations

In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations

where the 1-periodic function is a smooth function and satisfies sublinearity:

     

Weierstrass functions with random phases

Consider the function

where 1$">, , and is a non-constant 1-periodic Lipschitz function. The phases are chosen independently with respect to the uniform probability measure on . We prove that with probability one, we can choose a sequence of scales such that for every interval of length , the oscillation of satisfies . Moreover, the inequality is almost surely true at every scale. When is a transcendental number, these results can be improved: the minoration is true for every choice of the phases and at every scale.

     

On the correlations of directions in the Euclidean plane

Let denote the repartition of the -level correlation measure of the finite set of directions , where is the fixed point and is an integer lattice point in the square . We show that the average of the pair correlation repartition over in a fixed disc converges as . More precisely we prove, for every and , the estimate

We also prove that for each individual point , the -level correlation diverges at any point as , and we give an explicit lower bound for the rate of divergence.

     

A global approach to fully nonlinear parabolic problems

We consider the general initial-boundary value problem

(1)        
(2)        
(3)        
where is a bounded open set in with sufficiently smooth boundary.  The problem (1)-(3) is first reduced to the analogous problem in the space with zero initial condition and

The resulting problem is then reduced to the problem where the operator satisfies Condition  This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces.  The local and global solvability of the operator equation are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.

     

Whitney's extension problem for multivariate -functions

We prove that the trace of the space to an arbitrary closed subset is characterized by the following ``finiteness' property. A function belongs to the trace space if and only if the restriction to an arbitrary subset consisting of at most can be extended to a function such that

The constant is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.

     

Spectral asymptotics for Sturm-Liouville equations with indefinite weight

The Sturm-Liouville equation

is considered subject to the boundary conditions

We assume that is positive and that is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to for , or equivalently up to for , the eigenvalues of the above boundary value problem.

     

The classical problem of the calculus of variations in the autonomous case: Relaxation and Lipschitzianity of solutions

We consider the problem of minimizing

Under the assumption that the Lagrangian is continuous and satisfies a growth assumption that does not imply superlinear growth, we provide a result on the relaxation of the functional and show that a solution to the minimum problem is Lipschitzian.

     

One-dimensional asymptotic classes of finite structures

A collection of finite -structures is a 1-dimensional asymptotic class if for every and every formula , where :

(i)

There is a positive constant and a finite set such that for every and , either , or for some ,

(ii)

For every , there is an -formula , such that is precisely the set of with

One-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.

     

Torsion subgroups of elliptic curves in short Weierstrass form

In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any 0$">, all but finitely many curves

where and are integers satisfying \vert B\vert^{1+\varepsilon}>0$">, have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to \vert B\vert^{2+\varepsilon}>0$">, then the now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterlé.

     

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction.

Let the continued fraction expansion of any irrational number be denoted by and let the -th convergent of this continued fraction expansion be denoted by . Let

where . Let . It is shown that if , then the Rogers-Ramanujan continued fraction diverges at . is an uncountable set of measure zero. It is also shown that there is an uncountable set of points such that if , then does not converge generally.

It is further shown that does not converge generally for 1$">. However we show that does converge generally if is a primitive -th root of unity, for some . Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.

     

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

We consider purely inseparable extensions of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group and a vector space decomposition such that and , where denotes the integral closure. Moreover, is Cohen-Macaulay if and only if is Cohen-Macaulay. Furthermore, is polynomial if and only if is polynomial, and if and only if

where and .

     

Boundedness and oscillation for nonlinear dynamic equations on a time scale

We obtain some boundedness and oscillation criteria for solutions to the nonlinear dynamic equation

on time scales. In particular, no explicit sign assumptions are made with respect to the coefficient . We illustrate the results by several examples, including a nonlinear Emden-Fowler dynamic equation.