We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24. Contents 1. Introduction 2. Lattices, Fourier transforms, and Poisson summation 3. Principal theorems 4. Homogeneous spaces 5. Conditions for a sharp bound 6. Stationary points 7. Numerical results 8. Uniqueness Appendix A. ...
The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schr¨dinger equation o
By Frank Merle and Pierre Raphael
Abstract We consider the critical nonlinear Schr¨dinger equation iut = −∆u−|u| N u o with initial condition u(0, x) = u0 in dimension N = 1. For u0 ∈ H 1 , local existence in the time of solutions on an interval [0, T ) is known, and there exist ﬁnite time blow-up solutions, that is, u0 such that limt↑
We develop an abstract component language and a static type system that can tells us the maximum resources a program may use. We prove that the upper resource bound is sharp and we point out a polynomial algorithm that can infer the sharp bound. Knowing the maximal resources a program may request allows us to adjust resource usage of the program and to prevent it from raising exceptions or behaving unexpectedly on systems that do not have enough resources.
We investigate the controversial issue about the upper bound of interjudge agreement in the use of a low-level grammatical representation. Pessimistic views suggest that several percent of words in running text are undecidable in terms of part-of-speech categories. Our experiments with 55kW data give reason for optimism: linguists with only 30 hours' training apply the EngCG-2 morphological tags with almost 100% interjudge agreement.
Bacterial biofilms are regarded to be the primary aetiological factor in the initiation of
gingival inflammation and subsequent destruction of periodontal tissues (Offenbacher 1996)
and three major specific pathogens have been repeatedly identified as etiologic agents,
namely Aggregatibacter (Actinobacillus) actinomycetemcomitans (Aa), Porphyromonas gingivalis
(Pg) and Tannerella forsythia (Tf) (Socransky et al. 1998).