Nontrivial Estimates Research Articles

Large sieve inequalities for special characters to prime square moduli

In this paper, we develop a large sieve type inequality for some special characters whose moduli are squares of primes. Our result gives non-trivial estimate in certain ranges.

Additive representation in thin sequences, V: Mixed problems of Waring's type

In the first part of this series of papers (see Brudern, Kawada and Wooley [2]), we introduced an approach to additive problems in which one seeks to establish that almost all natural numbers in some fixed polynomial sequence are represented in a prescribed manner, thereby deriving non-trivial estimates for exceptional sets in thin sequences. We illustrated our methods by obtaining upper bounds for the exceptional sets associated with the representation of integers from quadratic, or cubic, polynomial sequences by sums of six cubes of positive integers. In subsequent parts of the series (see Brudern, Kawada and Wooley [3], [4], [5]), we adapted our core methods so as to tackle problems associated with the binary Goldbach problem, the expected asymptotic formula for the number of representations, and lower bounds for the number of integers represented in some prescribed manner. As is apparent from the opening part of this series, our methods are of great flexibility. The aim of the present paper is to provide variants of the ideas developed in the preceding opera, and here we will be concerned solely with methods which provide estimates for the size of exceptional sets in representation problems. The discerning reader will recognise that in several of the more exotic problems mentioned below, it is the existence of a non-trivial estimate for the exceptional set in question which is of interest. The investigation of the sharpest attainable estimate for this exceptional set should be politely deferred beyond any future occasion. We begin by exploring exceptional sets in polynomial sequences for additive problems involving mixed powers. Here one finds that sharp mean value estimates for mixed sums of powers, familiar to aficionados of the circle method, lead to surprisingly strong conclusions. Our first results, which we establish in §2, involve problems containing a block of four cubes. Here and elsewhere, ∗ Packard Fellow, and supported in part by NSF grant DMS-9622773. This paper benefitted from visits of various of the authors to Ann Arbor, Kyoto, Oberwolfach and Stuttgart, and the authors collectively thank these institutions for their hospitality and excellent working conditions. Received August 16, 2000.

Factorizations of large cycles in the symmetric group

The factorizations of an n-cycle of the symmetric group S n into m permutations with prescribed cycle types α 1,…, α m describe topological equivalence classes of one pole meromorphic functions on Riemann surfaces. This is one of the motivations for a vast literature on counting such factorizations. Their number, denoted by c α 1,…, α m ( n) , is also known as a connection coefficient of the center of the algebra of the symmetric group, whose multiplicative structure it describes. The relation to Riemann surfaces induces the definition of a genus for factorizations. It turns out that this genus is fully determined by the cycle types α 1,…, α m , and that it has a determinant influence on the complexity of computing connection coefficients. In this article, a new formula for c α 1,…, α m ( n) is given, that makes this influence of the genus explicit. Moreover, our formula is cancellation-free, thus contrasting with known formulae in terms of characters of the symmetric group. This feature allows us to derive non-trivial asymptotic estimates. Our results rely on combining classical methods of the theory of characters of the symmetric group with a combinatorial approach that was first introduced in the much simpler case m=2 by Goupil and Schaeffer.

On Arresting the Linear Growth Rate for the Magnetohydrodynamic Thermal Stability Problem in Completely Confined Fluids

This paper carries forward Sherman and Ostrach's ( J. Fluid Mech. 24 (1966) 661–671) analysis for the magnetohydrodynamic thermal stability problem in completely confined fluids and derives a bound for the linear complex growth rate of an arbitrary oscillatory perturbation which may be neutral or unstable through the use of some nontrivial integral estimates obtained from the coupled system of perturbation equations governing the problem.

On Waring's Problem for Cubes and Smooth Weyl Sums

Non-trivial estimates for fractional moments of smooth cubic Weyl sums are developed. Complemented by bounds for such sums of use on both the major and minor arcs in a Hardy-Littlewood dissection, these estimates are applied to derive an upper bound for the sth moment of the smooth cubic Weyl sum of the expected order of magnitude as soon as s> 7.691. Related arguments demonstrate that all large integers n are represented as the sum of eight cubes of natural numbers, all of whose prime divisors are at most exp (c(log nlog log n)1/2}, for a suitable positive number c. This conclusion improves a previous result of G. Harcos in which nine cubes are required. 1991 Mathematics Subject Classification: 11P05, 11L15, 11P55.

Incomplete Kloosterman sums and their applications

We obtain a non-trivial estimate for the upper bound of the absolute value of incomplete Kloosterman sums in which the number of terms is much less than the modulus.

Estimates for the Jung constant in Banach lattices

Sharp lower estimates for the Jung constantJ(E) in Banach latticesE satisfying an upperp-estimate and a lowerq-estimate are given. Moreover, the minimal value ofJ(E) with respect to equivalent renormings ofE is calculated inE=Lp,q for finitep andq, as well as in more general spacesE. Finally, a nontrivial estimate for the radiusrLp,∞(A) is obtained forA being a bounded sequence of disjointly supported functions inLp,∞.

Reproductive characteristics of Ethiopian highland sheep: II. Genetic parameters of semen characteristics and their relationships with testicular measurements in ram lambs

A study was conducted to investigate the effect of season on, and to estimate heritabilities of, and genetic correlations among, semen and spermatozoa characteristics and the relationship of these characteristics with measures of testicular or scrotal size in 6-, 9- and 12-month old ram lambs of Menz and Horro sheep breeds indigenous to Ethiopian highlands. A total of 278 ram lambs with substantial pedigree information were involved in the study. There was improvement in all semen and spermatozoa traits with age, the means at 12 months being consistently superior to values at 6 and 9 months of age. There were no significant breed differences in any of the traits studied, except semen volume at 9 months (0.67, S.E. 0.07 for Horro versus 0.39, S.E. 0.05 ml for Menz) and proportion of dead spermatozoa at 12 months (0.18, S.E. 0.03 versus 0.23, S.E. 0.02, respectively). Season was significant ( p<0.05) for most of the traits studied, and differences were attributed primarily to nutrition. Semen collected in the wet season had higher spermatozoa concentration while samples collected in the dry season had higher proportion of abnormalities. After correcting for differences in total spermatozoa abnormalities, the wet season (of collection) had the highest spermatozoa output (volume×concentration) in 12-month old ram lambs. Heritability estimates varied substantially in magnitude, ranging from zero to over 0.4. Traits with non-trivial heritability estimates were mass motility at 9 months (0.32, S.E. 0.11), individual motility at 9 months (0.32, S.E. 0.12) and at 12 months (0.16, S.E. 0.12) and proportion of abnormal spermatozoa at 9 months (0.35, S.E. 0.13). Genetic correlations among semen and spermatozoa characteristics, as well as correlations with testicular measurements, were medium to high and generally favourable, but were associated with large standard errors. The genetic correlations of scrotal circumference with semen volume (0.55, S.E. 0.11), mass motility (0.62, S.E. 0.20), individual motility (0.54, S.E. 0.12), concentration (0.25, S.E. 0.04) and proportion of abnormal spermatozoa (−0.75, S.E. 0.24) in 12-month old rams indicated that selection based on this trait, which is highly heritable and easy to measure and can be measured early in life, should have appreciable favourable correlated response in semen quality and spermatozoa production.

A generalization of the Jensen inequality

A new approach for estimating functions and functionals based on the concept of function superconvexity and a generalization of the Jensen inequality is proposed. A new concept is introduced and the associated analytical results are substantiated. Some nontrivial estimates of functions are derived.

On a Question of Igusa: Towards a Theory of Several Variable Asymptotic Expansions: I

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. A question of Igusa (from 1978) inquires about the singular behavior of the singular series, determined by a polynomial mapping ${\mathbf P}$ :K$^n$→K$^m$, m[les ]n, where K is a local field of characteristic zero. This paper describes in geometric terms the singularities of the singular series for two classes of polynomial maps ${\mathbf P}$=(P$_1$, P$_2$):K$^n$→K$^2$. The main result, which makes possible this description, is a type of uniformization of ${\mathbf P}$ by finitely many monomial maps μ(${\mathbf x}$)=(${\mathbf x}$$^$$_1$, ${\mathbf x}$$^ $$_2$), such that rank ($^$$_1$$_$$_2$)= 2. This is proved using resolution of singularities. Using this result, nontrivial estimates of oscillatory integrals with phase λ$_1$P$_1$+ λ$_2$P$_2$ are possible. These will be described elsewhere.

A Kuznetsov formula for Kloosterman sums on GLn

We prove a Kuznetsov sum formula for Kloosterman sums on GLn corresponding to the big Bruhat cell. Using this formula, a weighted sum of Kloosterman sums can be expressed in spectral decomposition. A non-trivial estimate of the spectral side might lead to a proof of cancellations in sums of Kloosterman sums on GLn.

Analytic Continuation of Representations and Estimates of Automorphic Forms

0.1. Analytic vectors and their analytic continuation. Let G be a Lie group and (�,G,V ) a continuous representation of G in a topological vector space V. A vector v ∈ V is called analytic if the functionv : g 7→ �(g)v is a real analytic function on G with values in V. This means that there exists a neighborhood U of G in its complexification GC such thatv extends to a holomorphic function on U. In other words, for each element g ∈ U we can unambiguously define the vector �(g)v asv(g), i.e., we can extend the action of G to a somewhat larger set. In this paper we will show that the possibility of such an extension sometimes allows one to prove some highly nontrivial estimates. Unless otherwise stated, G = SL(2, R), so GC = SL(2, C). We consider a typical representation ofG, i.e., a representation of the principal series. Namely, fix� ∈ C and consider the space Dof smooth homogeneous functions of degree � − 1 on R 2 0, i.e., D� = {� ∈ C ∞ (R 2 0) : �(ax,ay) = |a| �−1 �(x,y)}; we

Estimation of Exponential Sums over Primes in Short Intervals I

Let \(\) and \(\) be the von Mangoldt function and Mobius function, respectively, x real and y“small” compared with x. This paper gives, for the first time, a non-trivial estimate of the sum $$$$

A Geometric Inequality with Applications to the Kakeya Problem in Three Dimensions

It was shown by Besicovitch [F] that there are sets in Rd with measure zero that contain unit line segments in every direction. In [Fe2], C. Fefferman used Besicovitch sets to show that the ball-multiplier is bounded only on L2. Moreover, ideas originating in Fefferman’s work lead to alternate proofs, cf. [Fe3], [Co1,2], of the optimal boundedness result for Bochner– Riesz means, established originally by Carleson–Sjolin [CS], as well as for the restriction problem in R2, which had been solved earlier by Fefferman and Stein [Fe1]. It turns out that the crucial property of planar Besicovitch sets in this context is that they have maximal Hausdorff dimension. It is conjectured that Besicovitch sets E ⊂ Rd with d ≥ 3 have dimension equal to d. It is easy to show that dim(E) ≥ (d+ 1)/2. This was first improved by Bourgain [Bo], who showed, e.g., for d = 3 that dim(E) ≥ 7/3. A further improvement was then achieved by Wolff [W1], who proved that dim(E) ≥ (d + 2)/2 in all dimensions. Both these results where based in part on “bush-type” arguments. More precisely, Bourgain’s argument used the observation that tubes of thickness δ with 10δ-separated directions which intersect at some point x0 have to be disjoint outside a ball of radius 1/2 centered at x0. The improvement in [W1] is obtained by considering families of tubes intersecting a line. In this paper we present a different geometric approach that leads to a nontrivial estimate for Besicovitch sets in R3 – in fact Bourgain’s 7/3 bound. Our method is analogous to [KW] and will combine a geometric

Representations of the Character Groups of Fq

We give non-trivial unitary representations of the character groups of a finite field Fq. With the representation properties we deduce some results on character sums, and with the unitarity we obtain non-trivial estimates for the incomplete sums \(\sum_{\nu =1}^k\ G(\chi_i\chi _\nu^{- 1},\psi);1\leq k <q\).

Some new results about Jacobi forms

THE one-dimensional Jacobi forms were first systematically investigated by Eichler and Zagier[ll, and a similar theory in a higher-dimensional case was developed by ~ i e ~ l e r " ] , who gave the precise definition and some basic properties of higher-dimensional Jacobi forms. This note tries to give a satisfactory classification for a large kind of spaces of Jacobi forms. To be a little more explicit, we get a rank formula for Jacobi forms and then show that either all the nonzero elements are singular or none are. For cusp forms we also obtain some nontrivial estimates which can be considered as a generalization of Kohnen's result recorded in the preprint "On growth estimates for Fourier coefficients of Jacobi forms and an application".

Two applications of Weil-Serre's explicit formula

We apply Weil-Serre's explicit formula to produce non-trivial estimates of exponential sums along a curve and of the dual distance of subfield subcodes of algebraic-geometric (AG-)codes. These bounds work while the number of rational points of a curve is large when compared to its genus thus the Weil-Bombieri bound for exponential sums being trivial.

Analogues of Kloosterman sums

Non-trivial estimates are obtained for the upper bound of absolute values of analogues of Kloosterman sums in which the number of terms is much less that the modulus.

An Extension of the Weil-Carlitz-Uchiyama Bound

In this paper, we provide a new bound for exponential sums in one variable. This new bound gives non-trivial estimates for sets of polynomials where the classical bound of Weil-Carlitz-Uchiyama does not. Furthermore, infinite classes of examples are provided where our bound is tight.

On Zagier’s cusp form and the Ramanujan τ function

Zagier constructed a cusp form for each weightk of the full modular group. We use this construction to estimate Fourier coefficients of cusp forms. In particular, we get a non-trivial estimate, by elementary methods and indicate a relationship with the Lindelof hypothesis for classical Dirichlet L-functions.