Estimates For Sums Research Articles

Double exponential sums with exponential functions

We obtain several estimates for double rational exponential sums modulo a prime [Formula: see text] with products [Formula: see text] where both [Formula: see text] and [Formula: see text] run through short intervals and [Formula: see text] is fixed integer. We also obtain some new estimates for the number of points on exponential modular curves [Formula: see text] and similar.

Decay rate of iterated integrals of branched rough paths

Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals.

The analog of the Erdös distance problem in finite fields

In this paper, we give a proof of the result of Iosevich and Rudnev [Erdös distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007) 6127–6142] on the analog of the Erdös–Falconer distance problem in the case of a finite field of characteristic [Formula: see text], where [Formula: see text] is an odd prime, without using estimates for Kloosterman sums. We also address the case of characteristic 2.

Do children understand fraction addition?

Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, 2017) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments examining fourth to eighth graders' estimates of fraction sums. We found that roughly half of estimates of sums were smaller than the same child's estimate of one of the two addends in the problem. Moreover, children's estimates of fraction sums were no more accurate than if they had estimated each sum as the average of the smallest and largest possible response. This weak performance could not be attributed to poor mastery of arithmetic procedures, poor knowledge of individual fraction magnitudes, or general inability to estimate sums. These results suggest that a major source of difficulty in this domain is that many children's learning of fraction arithmetic procedures develops unconstrained by conceptual understanding of the procedures. Implications for education are discussed.

Reconstruction of precipitation variability in Estonia since the eighteenth century, inferred from oak and spruce tree rings

There is plenty of evidence for intensification of the global hydrological cycle. In Europe, the northern areas are predicted to receive more precipitation in the future and observational evidence suggests a parallel trend over the past decades. As a consequence, it would be essential to place the recent trend in precipitation in the context of proxy-based estimates of reconstructed precipitation variability over the past centuries. Tree rings are frequently used as proxy data for palaeoclimate reconstructions. Here we use deciduous (Quercus robur) and coniferous (Picea abies) tree-ring width chronologies from western Estonia to deduce past early-summer (June) precipitation variability since 1771. Statistical model transforming our tree-ring data into estimates of precipitation sums explains 42% of the variance in instrumental variability. Comparisons with products of gridded reconstructions of soil moisture and summer precipitation illustrate robust correlations with soil moisture (Palmer Drought Severity Index), but lowered correlation with summer precipitation estimates prior to mid-nineteenth century, these instabilities possibly reflecting the general uncertainties inherent to early meteorological and proxy data. Reconstructed precipitation variability was negatively correlated to the teleconnection indices of the North Atlantic Oscillation and the Scandinavia pattern, on annual to decadal and longer scales. These relationships demonstrate the positive precipitation anomalies to result from increase in zonal inflow and cyclonic activity, the negative anomalies being linked with the high pressure conditions enhanced during the atmospheric blocking episodes. Recently, the instrumental data have demonstrated a remarkable increase in summer (June) precipitation in the study region. Our tree-ring based reconstruction reproduces this trend in the context of precipitation history since eighteenth century and quantifies the unprecedented abundance of June precipitation over the recent years.

Validity of sum rules for the polarization transfer in electron bremsstrahlung

The polarization correlations, which describe the spin transfer from a high-energy spin-polarized electron to the photon in the elementary process of bremsstrahlung induced by strong potentials, obey a strict sum rule in the case of coplanar emission. The proof is carried out within the relativistic partial-wave approach. Further sum estimates are obtained by means of a variational principle under the sole condition that the transition amplitude is linear in the electron wavefunctions and in the photon field. Numerical sum rule results are given for light and heavy bare target atoms in the energy range 1–10 MeV.

On the Golomb’s conjecture and Lehmer’s numbers

Abstract Let p be an odd prime. For each integer a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one ā with 1 ≤ ā ≤ p − 1 such that a · ā ≡ 1 mod p. Let N(p) denote the set of all primitive roots a mod p with 1 ≤ a ≤ p − 1 in which a and ā are of opposite parity. The main purpose of this paper is using the analytic method and the estimate for the hybrid exponential sums to study the solvability of the congruence a + b ≡ 1 mod p with a, b ∈ N(p), and give a sharper asymptotic formula for the number of the solutions of the congruence equation.

Addition to V. P. Zastavnyi’s paper “Estimates for sums of moduli of blocks in trigonometric Fourier series”

Results on the integrability and integrability with power weight for sums of moduli of blocks from the series Ʃ1/k sin kx are extended to the class of series Ʃb k sin kx with coefficients b k of a more general form.

МЕТОДЫ ОЦЕНОК КОРОТКИХ СУММ КЛООСТЕРМАНА

This survey contains enlarged version of a mini-course which was read by the author in November 2015 during “Chinese - Russian workshop of exponential sums and sumsets”. This workshop was organized by professors Chaohua Jia (Institute of Mathematics, Academia Sinica) and Ke Gong (Henan University) in Academy of Mathematics and System Science, CAS (Beijing). The author is warmly grateful to them for the support and hospitality. The survey contains the Introduction, three parts and Conclusion. The basic definitions and results concerning the complete Kloosterman sums are given in the Introduction. The method of estimating of incomplete Kloosterman sums to moduli equal to the raising power of a fixed prime is described in the first part. This method is based on one idea of A. G. Postnikov which reduces the estimate of such sums to the estimate of the exponential sums with polynomial by I. M. Vinogradov’s mean value theorem. A. A. Karatsuba’s method of estimating of incomplete sums to an arbitrary moduli is described in the second part. This method is based on a very precise estimate of the number of solutions of one symmetric congruence involving inverse residues to a given modulus. This estimate plays the same role in thie problems under considering as Vinogradov’s mean value theorem in the estimating of corresponding exponential sums. The method of J. Bourgain and M. Z. Garaev is described in the third part. This method is based on very deep “sum-product estimate” and on the improvement of A. A. Karatsuba’s bound for the number of solutions of symmetric congruence. The Conclusion contains a series of recent results concerning the estimates of short Kloosterman sums.

Superconductivity and its enhancement under high pressure in “F-free” single crystals of CeOBiS2

“F-free” CeOBiS2 single crystals have successfully grown, thoroughly eliminating a concern about F-contamination by using a high-purity CsCl flux. The obtained crystals have a plate-like shape with a size of 1.0–3.0 mm in the well-developed plane. Single crystal X-ray structural analysis clearly revealed that the CeOBiS2 crystallizes with a space group P4/nmm (with lattice parameters of a = 4.0189 (6) Å, c = 13.573 (2) Å). The bond valence sum estimation and X-ray photoelectron spectroscopy analysis showed that the chemical state of Ce was in the mixed valence of Ce3+ and Ce4+. The single crystals show superconductivity with zero resistivity at ∼1.3 K. This is the first conclusive evidence of superconductivity driven by Ce valence fluctuation in surely non-doped CeOBiS2. The superconducting transition temperature was enhanced up to ∼3.8 K by applying hydrostatic pressure.

The vibrator-ground model and the vibroseis source wavelet

Miller and Pursey (1954) have shown that for an isotropic-homogeneous-elastic half-space, the far-field particle displacement is proportional to the surface stress if the surface stress is uniformly applied over a small disc. This relationship of force and far-field particle displacement establishes the theory of the vibroseis method. It implies that the vibrator ground force is proportional to the far field particle displacement if the ground is assumed to be an isotropic-homogeneous-elastic body and the vibrator baseplate is small enough comparing with the wavelength of interest. The true ground force is the integration of the pressures beneath the baseplate. However, this true ground force is not directly measured, and it is often estimated by a weighted sum of outputs from two accelerometers placed on the reaction mass and baseplate assemblies (Sallas and Weber, 1982; Sallas, 1984). The weighted sum estimation of the ground force, often referred as the weighted-sum ground force, has been widely used as the phase locking feedback signal on the vibrator force control since 1984. Initially, the vibroseis source wavelet embedded in correlated vibroseis data is assumed to be the autocorrelation of the pilot sweep (zero-phase with a flat spectrum).

One kind sixth power mean of the three-term exponential sums

Abstract In this paper, we use the estimate for trigonometric sums and the properties of the congruence equations to study the computational problem of one kind sixth power mean of the three-term exponential sums. As a conclusion, we give an exact computational formula for it.

CHOOSING AN APPROPRIATE CONTINGENCY SUM ESTIMATING METHODS FOR HIGHWAY CONSTRUCTION PROJECTS IN NIGERIA: A LITERATURE REVIEW

In order to establish a gap to be filled in a particular field, there is a need to gather information that allows the researcher to identify and analyse the current state of researches on the topic. Contingency sum estimation for projects plays an important role for inclusion for risk taken by the contractor on the bid amount. This is good in order to meet project objectives (cost, time, quality, health and safety, and environment). The main function of contingency sum is to cater forvariations that arise during the construction phase of projects. The aim of the study is to identifying and examine the various methods for estimating contingency sum for projects with a view to propose an appropriate method which will be suitable for a specific project in order to increase accuracy and reduce cost overrun. A literature review of selected papers (journals, conference proceedings and theses) on contingency estimation was carried out. The major findings from this study is that most studies were carried out on building projects while few were carried out on highway projects. From the analysis carried out, no single method of estimating has been effective for all types of projects. These methods of estimating contingency sums have developed over the years as a result of shortcomings of one method after the other. It is therefore recommended that peculiarity of each project, ease of using each of the methods and accuracy of the estimates needed should be the basis of choosing method of estimating contingency sum.

CHOOSING AN APPROPRIATE CONTINGENCY SUM ESTIMATING METHODS FOR HIGHWAY CONSTRUCTION PROJECTS IN NIGERIA: A LITERATURE REVIEW

In order to establish a gap to be filled in a particular field, there is a need to gather information that allows the researcher to identify and analyse the current state of researches on the topic. Contingency sum estimation for projects plays an important role for inclusion for risk taken by the contractor on the bid amount. This is good in order to meet project objectives (cost, time, quality, health and safety, and environment). The main function of contingency sum is to cater forvariations that arise during the construction phase of projects. The aim of the study is to identifying and examine the various methods for estimating contingency sum for projects with a view to propose an appropriate method which will be suitable for a specific project in order to increase accuracy and reduce cost overrun. A literature review of selected papers (journals, conference proceedings and theses) on contingency estimation was carried out. The major findings from this study is that most studies were carried out on building projects while few were carried out on highway projects. From the analysis carried out, no single method of estimating has been effective for all types of projects. These methods of estimating contingency sums have developed over the years as a result of shortcomings of one method after the other. It is therefore recommended that peculiarity of each project, ease of using each of the methods and accuracy of the estimates needed should be the basis of choosing method of estimating contingency sum.

A Three-Dimensional Localization Method for Multistatic SAR Based on Numerical Range-Doppler Algorithm and Entropy Minimization

In traditional localization methods for synthetic aperture radar (SAR), the range sum estimation and Doppler centroid estimation (DCE) are required. The DCE error can influence the localization accuracy greatly. In addition, the target height information cannot be obtained by these methods. In this paper, a three-dimensional localization method for multistatic SAR based on the numerical range-Doppler (RD) algorithm and entropy minimization principle is proposed. In this method, the raw data from each transmitter and receiver (T/R) pair are focused by the numerical RD algorithm with the initial location value of the reference target. Then, Newton iteration is used to solve the target location value with the information of the bistatic range sum (BRS) in different SAR images with respect to different T/R pairs. Generally, the initial location value of the reference target is not accurate, and it can influence the imaging quality and accuracy of other target locations. We use entropy to measure image quality and iterate imaging with the new location value of the reference target, until the entropy gets the minimum value. Therefore, we can get the optimal location value of the reference target, which can make image entropy reach the minimum. Finally, all targets can be located by the Newton iteration method with their BRS in each T/R pair that are obtained from the images with minimum entropy. Compared with traditional localization methods for monostatic SAR, the proposed method not only effectively eliminates the influences of DCE errors, but also can get the target height information. Therefore, it improves the localization accuracy and can achieve three-dimensional localization. The effectiveness of the localization approach is validated by a numerical simulation experiment.

A problem considered by Friedlander & Iwaniec and the discrete Hardy-Littlewood method

AbstractFollowing Friedlander & Iwaniec [FRIEDLANDER, J. B.—IWANIEC, H.:Summation formulae for coefficients of L-functions,Canad. J. Math.57(2005), 494—505], the objective of this note are the coefficientsanof the Dirichlet series forL(s,χ1)L(s,χ2)L(s,χ3) whereχ1,χ2,χ3are primitive Dirichlet characters with modulesD1,D2,D3. For∑n≤xan$\sum\limits_{n\le x}a_n$, withxlarge, sharp asymptotics are established which are uniform inD1,D2,D3. To this end, the modern method for the estimation of exponential sums, due to [HUXLEY, M. N.:Area, Lattice Points, and Exponential Sums,LMS Monographs, New Ser.13, University Press, Oxford, 1996] and others, is applied with gain.

A note on character sums in finite fields

We prove a character sum estimate in Fq[t] and answer a question of Shparlinski.

On certain character sums

Let q ≥ 2 be an integer, χ be a non-principal character mod q, A = A(q) ≤ q, B = B(q) ≤ q, and H = H(q) ≤ q. In this paper, we shall use the estimates for Kloosterman sums and the properties of trigonometric sums to give some sharp estimates for character sums of the formwhere ħ(A, B, H) = {a ∈ ℤ| (a, q) = 1, 1 ≤ a ≤ A, 1 ≤ b ≤ B, ab ≡ 1 (mod q), |a − b| ≤ H}.

On the Isotropic Constant of Random Polytopes with Vertices on an $$\ell _p$$ ℓ p -Sphere

The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the \(\ell _p\)-unit sphere of \({{\mathbb {R}}}^n\) for some \(1\le p < \infty \) is considered. We prove that these random polytopes have uniformly absolutely bounded isotropic constants with overwhelming probability. This generalizes the result for the Euclidean sphere \((p=2)\) obtained by Alonso-Gutierrez. The proof requires several different tools including a probabilistic representation of the cone measure due to Schechtman and Zinn and moment estimates for sums of independent random variables with log-concave tails originating in a paper of Gluskin and Kwapien.

On the distribution of αpγ + β modulo one

Let ‖⋅‖ denote the minimum distance to an integer. For 0<γ<1, θ>0 and (α,β)∈R∖{0}×R we study when‖αpγ+β‖<p−θ, holds for infinitely many primes p of a special type. In particular, we consider when this inequality holds for primes p such that p+2 has few prime factors counted with multiplicity. This is done using an exponential sum estimate of the author and the linear sieve of Iwaniec with bilinear error term. This is related to recent work of Tolev, Todorova, Matomäki and Cai.