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Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)?k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)?C(q)k1/q for ?(t)?q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ?(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a2+b2+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a2+b2=p. 相似文献
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Tomśš Cipra 《Applications of Mathematics》2006,51(6):597-604
Various types of exponential smoothing for data observed at irregular time intervals are surveyed. Double exponential smoothing
and some modifications of Holt’s method for this type of data are suggested. A real data example compares double exponential
smoothing and Wright’s modification of Holt’s method for data observed at irregular time intervals.
This research has been supported by the Ministry of Education, Youth and Sports of the Czech Republic, Research Project MSM
0021620839, and by the Grant Agency of Charles University, Grant No. 342/2005. 相似文献
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Tomáš Cipra 《Annals of Operations Research》1991,30(1):95-105
Three possible approaches to stochastic programming problems defined in time (so that they contain random processes) are described in this paper: (1) an application of the extremal theory of random processes; (2) an exponential penalty model approach related to scenario analysis; (3) a modification of the entropic penalty approach. Explicit results are derived for some special cases. 相似文献
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S. Khandelwal T. Siegmund R.J. Cipra J.S. Bolton 《International Journal of Solids and Structures》2012,49(18):2394-2403
Topologically interlocked materials (TIMs) are a class of materials made by a structured assembly of an array of identically shaped and sized unit elements that are held in a confining framework. The assembly can resist transverse forces in the absence of adhesives between the unit elements. The present study focuses on topologically interlocked materials with cellular unit elements. The resulting materials achieve their properties by a combination of deformation of the individual unit elements and their contact interaction. Drop tower tests were conducted to characterize the mechanical behavior of the cellular TIMs made of an intrinsically brittle base material. The TIMs were found to exhibit perfect softening, independent of the relative density of the cellular units. The analysis of the experiments revealed a positive correlation between strength and toughness in contrast to more conventional materials. An analytical model for the prediction of the observed material behavior is developed. Model predictions are in agreement with experimental data. The implications of the present findings for the design of these novel materials are discussed. 相似文献
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