The use in additive number theory of numbers without large prime factors

1993 ◽  
Vol 345 (1676) ◽  
pp. 363-376
Keyword(s):  
Historical Perspective ◽  
Exponential Sums ◽  
Upper Bounds ◽  
Mean Value ◽  
Additive Number Theory ◽  
The Past ◽  
Classical Work ◽  
Prime Factors ◽  
Special Case ◽  
Large Natural Number

In the past few years considerable progress has been made with regard to the known upper bounds for G ( k ) in Waring’s problem, that is, the smallest s such that every sufficiently large natural number is the sum of at most 8 k th powers of natural numbers. This has come about through the development of techniques using properties of numbers having only relatively small prime factors. In this article an account of these developments is given, and they are illustrated initially in a historical perspective through the special case of cubes. In particular the connection with the classical work of Davenport on smaller values of k is demonstrated. It is apparent that the fundamental ideas and the underlying mean value theorems and estimates for exponential sums have numerous applications and a brief account is given of some of them.

scholarly journals A Hybrid Mean Value Involving Dedekind Sums and the General Exponential Sums

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Jianghua Li ◽  
Tingting Wang
Keyword(s):  
Asymptotic Formula ◽  
Upper Bound ◽  
Exponential Sums ◽  
Mean Value ◽  
Gauss Sums ◽  
Dedekind Sums ◽  
Analytic Method ◽  
Hybrid Mean Value ◽  
Classical Work ◽  
Upper Bound Estimate

The main purpose of this paper is using the analytic method, A. Weil’s classical work for the upper bound estimate of the general exponential sums, and the properties of Gauss sums to study the hybrid mean value problem involving Dedekind sums and the general exponential sums and give a sharp asymptotic formula for it.

scholarly journals Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers

2016 ◽  
Vol 152 (6) ◽  
pp. 1121-1158 ◽  
Author(s):  
Adam J. Harper
Keyword(s):  
Riemann Hypothesis ◽  
Exponential Sums ◽  
Upper Bounds ◽  
Weighted Sums ◽  
Number Of Solutions ◽  
Mean Values ◽  
Smooth Numbers ◽  
Prime Factors ◽  
Moment Bounds ◽  
Smooth Integers

We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant y\leqslant x^{1/3}$. Then we prove sharp upper bounds for the $p$th moment of (possibly weighted) sums, for any real $p>2$ and $\log ^{C(p)}x\leqslant y\leqslant x$. Our proof develops an argument of Bourgain, showing that this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of $a+b=c$ in $y$-smooth integers less than $x$ whenever $\log ^{C}x\leqslant y\leqslant x$. Previously this was only known assuming the generalised Riemann hypothesis. Combining them with transference machinery of Green, we prove Roth’s theorem for subsets of the $y$-smooth numbers whenever $\log ^{C}x\leqslant y\leqslant x$. This provides a deterministic set, of size ${\approx}x^{1-c}$, inside which Roth’s theorem holds.

scholarly journals Cardiorespiratory Functioning in Youth with Persistent Post-Concussion Symptoms: A Pilot Study

2021 ◽  
Vol 10 (4) ◽  
pp. 561
Author(s):  
Aliyah Snyder ◽  
Christopher Sheridan ◽  
Alexandra Tanner ◽  
Kevin Bickart ◽  
Molly Sullan ◽  
...  
Keyword(s):  
Processing Speed ◽  
Mean Value ◽  
Exhaled Breath ◽  
Post Injury ◽  
The Past ◽  
Neurobehavioral Outcomes ◽  
End Tidal ◽  
Relationship Of ◽  
The Relationship ◽  
End Tidal Co2

Dysregulation of the autonomic nervous system (ANS) may play an important role in the development and maintenance of persistent post-concussive symptoms (PPCS). Post-injury breathing dysfunction, which is influenced by the ANS, has not been well-studied in youth. This study evaluated cardiorespiratory functioning at baseline in youth patients with PPCS and examined the relationship of cardiorespiratory variables with neurobehavioral outcomes. Participants were between the ages of 13–25 in two groups: (1) Patients with PPCS (concussion within the past 2–16 months; n = 13) and (2) non-injured controls (n = 12). Capnometry was used to obtain end-tidal CO2 (EtCO2), oxygen saturation (SaO2), respiration rate (RR), and pulse rate (PR) at seated rest. PPCS participants exhibited a reduced mean value of EtCO2 in exhaled breath (M = 36.3 mmHg, SD = 2.86 mmHg) and an altered inter-correlation between EtCO2 and RR compared to controls. Neurobehavioral outcomes including depression, severity of self-reported concussion symptoms, cognitive catastrophizing, and psychomotor processing speed were correlated with cardiorespiratory variables when the groups were combined. Overall, results from this study suggest that breathing dynamics may be altered in youth with PPCS and that cardiorespiratory outcomes could be related to a dimension of neurobehavioral outcomes associated with poorer recovery from concussion.

scholarly journals Bounding the Size and Probability of Epidemics on Networks

2008 ◽  
Vol 45 (2) ◽  
pp. 498-512 ◽  
Author(s):  
Joel C. Miller
Keyword(s):  
Infectious Disease ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Marginal Probability ◽  
Transmission Properties ◽  
Disease Spreading ◽  
Special Case

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.

scholarly journals Historical perspective of neurology in Iran

2019 ◽  
Author(s):  
Shri Kant-Mishra ◽  
Hadi Mohammad Khanli ◽  
Golnoush Akhlaghipour ◽  
Ghazaleh Ahmadi Jazi ◽  
Shaweta Khosa1
Keyword(s):  
Historical Perspective ◽  
Medical Center ◽  
Medical Science ◽  
Early Period ◽  
Medieval Period ◽  
Neurologic Diseases ◽  
The Past ◽  
Modern Period ◽  
History Of ◽  
Modern Era

Iran is an ancient country, known as the cradle of civilization. The history of medicine in Iran goes back to the existence of a human in this country, divided into three periods: pre-Islamic, medieval, and modern period. There are records of different neurologic terms from the early period, while Zoroastrian (religious) prescription was mainly used until the foundation of the first medical center (Gondishapur). In the medieval period, with the conquest of Islam, prominent scientists were taught in Baghdad, like Avicenna, who referred to different neurologic diseases including stroke, paralysis, tremor, and meningitis. Several outstanding scientists developed the medical science of neurology in Iran, the work of whom has been used by other countries in the past and present. In the modern era, the Iranian Neurological Association was established with the efforts of Professor Jalal Barimani in 1991.

scholarly journals Sliding-Window Thompson Sampling for Non-Stationary Settings

2020 ◽  
Vol 68 ◽  
pp. 311-364
Author(s):  
Francesco Trovo ◽  
Stefano Paladino ◽  
Marcello Restelli ◽  
Nicola Gatti
Keyword(s):  
Real World ◽  
State Of The Art ◽  
Sliding Window ◽  
Upper Bounds ◽  
Decision Problems ◽  
Sequential Decision ◽  
Thompson Sampling ◽  
The Past ◽  
Real World Applications ◽  
Window Approach

Multi-Armed Bandit (MAB) techniques have been successfully applied to many classes of sequential decision problems in the past decades. However, non-stationary settings -- very common in real-world applications -- received little attention so far, and theoretical guarantees on the regret are known only for some frequentist algorithms. In this paper, we propose an algorithm, namely Sliding-Window Thompson Sampling (SW-TS), for nonstationary stochastic MAB settings. Our algorithm is based on Thompson Sampling and exploits a sliding-window approach to tackle, in a unified fashion, two different forms of non-stationarity studied separately so far: abruptly changing and smoothly changing. In the former, the reward distributions are constant during sequences of rounds, and their change may be arbitrary and happen at unknown rounds, while, in the latter, the reward distributions smoothly evolve over rounds according to unknown dynamics. Under mild assumptions, we provide regret upper bounds on the dynamic pseudo-regret of SW-TS for the abruptly changing environment, for the smoothly changing one, and for the setting in which both the non-stationarity forms are present. Furthermore, we empirically show that SW-TS dramatically outperforms state-of-the-art algorithms even when the forms of non-stationarity are taken separately, as previously studied in the literature.

scholarly journals A note on mean value of mixed exponential sums

2014 ◽  
Vol 144 ◽  
pp. 234-243 ◽  
Author(s):  
Chen Hua ◽  
Ai Xiaochuan ◽  
Cai Guangxing
Keyword(s):  
Exponential Sums ◽  
Mean Value

On Some Issues in Shakedown Analysis

2001 ◽  
Vol 68 (5) ◽  
pp. 799-808 ◽  
Author(s):  
G. Maier
Keyword(s):  
Heterogeneous Media ◽  
Direct Methods ◽  
Cost Effective ◽  
Upper Bounds ◽  
Saturated Porous Media ◽  
Safety Factors ◽  
Shakedown Analysis ◽  
Time Stepping ◽  
Kinematic Theorem ◽  
Special Case

Shakedown analysis, and its more classical special case of limit analysis, basically consists of “direct” (as distinct from time-stepping) methods apt to assess safety factors for variable repeated external actions and procedures which provide upper bounds on history-dependent quantities. The issues reviewed and briefly discussed herein are: some recent engineering-oriented and cost-effective methods resting on Koiter’s kinematic theorem and applied to periodic heterogeneous media; recent extensions (after the earlier ones to dynamics and creep) to another area characterized by time derivatives, namely poroplasticity of fluid-saturated porous media. Links with some classical or more consolidated direct methods are pointed out.

Digital Prepress

2011 ◽  
pp. 24-39
Author(s):  
Mark Snyder
Keyword(s):  
Historical Perspective ◽  
Print Media ◽  
Digital Technologies ◽  
Printing Industry ◽  
New Techniques ◽  
Digital Revolution ◽  
The Past ◽  
Problems And Solutions ◽  
Common Problems ◽  
Work Done

This chapter identifies changes that the printing industry has undergone during the past 25 years as a result of the digital revolution. It also provides a brief historical perspective of the printing industry and how it has evolved. It is undeniable that the computer has had an impact on the development of print media and today it is rare to find any prepress work done without the use of some digital technology. The workflow of a traditional printed piece is described from start to finish and is compared to a more modern digital workflow to familiarize readers with the processes and contrast the old with the new techniques. This chapter will identify common problems that occur in the preparation of print media using digital technologies. In particular, it will explore a variety of problems and solutions related to the use of digital prepress as well as identifying new innovations intended to improve prepress operations in the future.

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