Equal sums of four seventh powers

In this paper, the method used to find the smallest, nontrivial, positive integer solution of is discussed. The solution is

Factors enabling this discovery are advances in computing power, available workstation memory, and the appropriate choice of optimized algorithms.

     

On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

     

On standardized models of isogenous elliptic curves

Let be isogenous elliptic curves over given by standardized Weierstrass models. We show that (in the obvious notation)

and, moreover, that there are integers such that

where .

     

Closed form summation of -finite sequences

We consider sums of the form

in which each is a sequence that satisfies a linear recurrence of degree , with constant coefficients. We assume further that the 's and the 's are all nonnegative integers. We prove that such a sum always has a closed form, in the sense that it evaluates to a linear combination of a finite set of monomials in the values of the sequences with coefficients that are polynomials in . We explicitly describe two different sets of monomials that will form such a linear combination, and give an algorithm for finding these closed forms, thereby completely automating the solution of this class of summation problems. We exhibit tools for determining when these explicit evaluations are unique of their type, and prove that in a number of interesting cases they are indeed unique. We also discuss some special features of the case of ``indefinite summation", in which .

     

Stability of wavelet frames and Riesz bases, with respect to dilations and translations

We consider the perturbation problem of wavelet frame (Riesz basis) about dilation and translation parameters and . For wavelet functions whose Fourier transforms have small supports, we give a method to determine whether the perturbation system is a frame (Riesz basis) and prove the stability about dilation parameter on Paley-Wiener space. For a great deal of wavelet functions, we give a definite answer to the stability about translation . Moreover, if the Fourier transform has small support, we can estimate the frame bounds about the perturbation of translation parameter . Our methods can be used to handle nonhomogeneous frames (Riesz basis).

     

On random algebraic polynomials

This paper provides asymptotic estimates for the expected number of real zeros and -level crossings of a random algebraic polynomial of the form , where are independent standard normal random variables and is a constant independent of . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form .

     

Monic integer Chebyshev problem

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let denote the monic polynomials of degree with integer coefficients. A monic integer Chebyshev polynomial satisfies

and the monic integer Chebyshev constant is then defined by

This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied.

We compute for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases.

Conjecture. Suppose is an interval whose endpoints are consecutive Farey fractions. This is characterized by Then

This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater.

     

A new statistic for the problem

A finite -trajectory is a sequence of positive integers such that if is odd, if is even, 1$"> if and . For such a sequence we define . We prove that if is odd and . Histograms suggest that may have an interesting limiting distribution.     

Parametric decomposition of powers of ideals versus regularity of sequences

Let be an ideal in a Noetherian local ring . Then the sequence is -regular if every is a non-zerodivisor in and if for all integers , where runs over the elements of the set .

     

Uniqueness for non-harmonic trigonometric series

When , and

if

then

More generalized results are given.

     

Adjoint restriction estimates and scaling on spheres

We test the restriction conjecture, in its adjoint form, against a class of measures on the sphere . The densities are smoothed out characteristic functions of rectangular caps on , where are fixed nonnegative numbers.

     

A strengthening of the Carleman-Hardy-Pólya inequality

As a consequence of a more general statement proved in the paper, it is deduced that, if , and , then

with equality if and only if . This is a new refinement of Carleman's classic inequality.

     

An infinite family of bounds on zeros of analytic functions and relationship to Smale's bound

Smale's analysis of Newton's iteration function induce a lower bound on the gap between two distinct zeros of a given complex-valued analytic function . In this paper we make use of a fundamental family of iteration functions , , to derive an infinite family of lower bounds on the above gap. However, even for , where coincides with Newton's, our lower bound is more than twice as good as Smale's bound or its improved version given by Blum, Cucker, Shub, and Smale. When is a complex polynomial of degree , for small the corresponding bound is computable in arithmetic operations. For quadratic polynomials, as increases the lower bounds converge to the actual gap. We show how to use these bounds to compute lower bounds on the distance between an arbitrary point and the nearest root of . In particular, using the latter result, we show that, given a complex polynomial , , for each we can compute upper and lower bounds and such that the roots of lie in the annulus . In particular, , ; and , , where . An application of the latter bounds is within Weyl's classical quad-tree algorithm for computing all roots of a given complex polynomial.

     

Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions

Any product of real powers of Jacobian elliptic functions can be written in the form . If all three 's are even integers, the indefinite integral of this product with respect to is a constant times a multivariate hypergeometric function with half-odd-integral 's and , showing it to be an incomplete elliptic integral of the second kind unless all three 's are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables }, allowing as many as six integrals to take a unified form. Thirty -functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting , , and in , which is symmetric in its first two variables and has an efficient algorithm for numerical computation.

     

一类上三角矩阵环$W_{n}(p, q)$的斜Armendariz性质

设α是环R的一个自同态,称环R是α-斜Armendariz环,如果在R[x;α]中,(∑_(i=0)~ma_ix~i)(∑_(j=0)~nb_jx~j)=0,那么a_ia~i(b_j)=0,其中0≤i≤m,0≤j≤n.设R是α-rigid环,则R上的上三角矩阵环的子环W_n(p,q)是α~—-斜Armendariz环.     

Asymptotic Behavior in a Quasilinear Fully Parabolic Chemotaxis System with Indirect Signal Production and Logistic Source

In this paper, we study the asymptotic behavior of solutions to a quasilinear fully parabolic chemotaxis system with indirect signal production and logistic sourceunder homogeneous Neumann boundary conditions in a smooth bounded domain $Ω⊂\mathbb{R}^n$ $(n ≥1)$, where $b ≥0$, $γ ≥1$, $a_i ≥1$, $µ$, $b_i >0$ $(i =1,2)$, $D$, $S∈ C^2([0,∞))$ fulfilling $D(s) ≥ a_0(s+1)^{−α}$, $0 ≤ S(s) ≤ b_0(s+1)^β$ for all $s ≥ 0,$ where $a_0,b_0 > 0$ and $α,β ∈ \mathbb{R}$ are constants. The purpose of this paper is to prove that if $b ≥ 0$ and $µ > 0$ sufficiently large, the globally bounded solution $(u,v,w)$ with nonnegative initial data $(u_0,v_0,w_0)$ satisfies $$\Big\| u(·,t)− \Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\|_{L^∞(Ω)}+\Big\| v(·,t)−\frac{b_1b_2}{a_1a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)} +\Big\| w(·,t)−\frac{b_2}{a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)}→0$$ as $t→∞$.     

一类线性差分方程的亚纯解与一个亚纯函数分担3个值的唯一性

主要研究差分方程a_1(z)f(x+1)+a_0(z)f(z)=F(z)的一个有穷级超越亚纯解f(z)与亚纯函数g(z)分担0,1,∞CM时的唯一性问题(其中a_(z),a0(z),F(z)为非零多项式,且满足a_1(z)+a_0(z)■0),得到f(x)≡g(z),或f(z)+g(z)≡f(z)g(z),或存在一个多项式β(z)=az+b_0和一个常数a_0满足e~(a_0)≠e~(b_0),使得f(z)=(1-e~(β(x)))/(e~(β(x))(e~(a_o-b_0)-1))与g(z)=(1-e~(β(x)))/(1-e~(b_o-a_0)),其中a(≠0),b_0为常数.     

-actions whose fixed data has a section

Given a collection of real vector bundles over a closed manifold , suppose that, for some is of the form , where is the trivial one-dimensional bundle. In this paper we prove that if is the fixed data of a -action, then the same is true for the Whitney sum obtained from by replacing by . This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible -actions fixing the disjoint union of an even projective space and a point.     

Dimension of families of determinantal schemes

A scheme of codimension is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous matrix and is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers and we denote by (resp. ) the locus of good (resp. standard) determinantal schemes of codimension defined by the maximal minors of a matrix where is a homogeneous polynomial of degree .

In this paper we address the following three fundamental problems: To determine (1) the dimension of (resp. ) in terms of and , (2) whether the closure of is an irreducible component of , and (3) when is generically smooth along . Concerning question (1) we give an upper bound for the dimension of (resp. ) which works for all integers and , and we conjecture that this bound is sharp. The conjecture is proved for , and for under some restriction on and . For questions (2) and (3) we have an affirmative answer for and , and for under certain numerical assumptions.

     

Small prime solutions of quadratic equations II

Let be non-zero integers and any integer. Suppose that and for . In this paper we prove that (i) if the are not all of the same sign, then the above quadratic equation has prime solutions satisfying and (ii) if all the are positive and , then the quadratic equation is soluble in primes Our previous results are and in place of and above, respectively.