An L-function free proof of Hua's Theorem on sums of five prime squares

Abstract We provide a new proof of Hua's result that every sufficiently large integer N ≡ 5 (mod 24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of L-functions. Here, we present a proof based on the transference principle first introduced in[5]. Using a sieve theoretic approach similar to ([10]), we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudo-randomness of the majorant of the characteristic function of the W-tricked primes which requires a precise evaluation of the occurring Gaussian sums and Jacobi symbols.

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Spotting Neural Spike Patterns Using an Adversary Background Model

Neural Computation ◽

10.1162/089976601317098493 ◽

2001 ◽

Vol 13 (12) ◽

pp. 2681-2708 ◽

Cited By ~ 1

Author(s):

Itay Gat ◽

Naftali Tishby

Keyword(s):

Technical Difficulty ◽

Background Model ◽

Theoretic Approach ◽

Information Theoretic ◽

Pattern Recognition Problem ◽

Recording Conditions ◽

Specific Meaning ◽

Information Theoretic Approach ◽

Behaving Monkeys ◽

Spike Patterns

The detection of a specific stochastic pattern embedded in an unknown background noise is a difficult pattern recognition problem, encountered in many applications such as word spotting in speech. A similar problem emerges when trying to detect a multineural spike pattern in a single electrical recording, embedded in the complex cortical activity of a behaving animal. Solving this problem is crucial for the identification of neuronal code words with specific meaning. The technical difficulty of this detection is due to the lack of a good statistical model for the background activity, which rapidly changes with the recording conditions and activity of the animal. This work introduces the use of an adversary background model. This model assumes that the background “knows” the pattern sought, up to a first-order statistics, and this “knowledge” creates a background composed of all the permutations of our pattern. We show that this background model is tightly connected to the type-based information-theoretic approach. Furthermore, we show that computing the likelihood ratio is actually decomposing the log-likelihood distribution according to types of the empirical counts. We demonstrate the application of this method for detection of the reward patterns in the basal ganglia of behaving monkeys, yielding some unexpected biological results.

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Ultrastructural autoradiography of L6 skeletal-muscle cells exposed to a tritiated macrolide antibiotic (LY237216) in vitro

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100123465 ◽

1992 ◽

Vol 50 (1) ◽

pp. 612-613

Author(s):

S.L. White ◽

C.B. Jensen ◽

D.D. Giera ◽

D.A. Laska ◽

M.N. Novilla ◽

...

Keyword(s):

Skeletal Muscle ◽

Quantitative Analysis ◽

Macrolide Antibiotic ◽

Circle Method ◽

Apical Surface ◽

Polycarbonate Membrane ◽

L6 Cells ◽

Low Viscosity ◽

Erythromycin A

In vitro exposure to LY237216 (9-Deoxo-11-deoxy-9,11-{imino[2-(2-methoxyethoxy)ethylidene]-oxy}-(9S)-erythromycin), a macrolide antibiotic, was found to induce cytoplasmic vacuolation in L6 skeletal muscle myoblast cultures (White, S.L., unpubl). The present study was done to determine, by autoradiographic quantitative analysis, the subcellular distribution of 3H-LY237216 in L6 cells.L6 cells (ATCC, CRL 1458) were cultured to confluency on polycarbonate membrane filters (Millipore Corp., Bedford, MA) in M-199 medium (GIBCO® Labs) with 10% fetal bovine serum. The cells were exposed from the apical surface for 1-hour to unlabelled-compound (0 μCi/ml) or 50 (μCi/ml of 3H-LY237216 at a compound concentration of 0.25 mg/ml. Following a rapid rinse in compound-free growth medium, the cells were slam-frozen against a liquid nitrogen cooled, polished copper block in a CF-100 cryofixation unit (LifeCell Corp., The Woodlands, TX). Specimens were dried in the MDD-C Molecular Distillation Drier (LifeCell Corp.), vapor osmicated and embedded in Spurrs low viscosity resin. Ultrathin sections collected on formvar coated stainless steel grids were counter-stained, then individually mounted on corks. A monolayer of Ilford L4 nuclear emulsion (Polysciences, Inc., Warrington, PA) was placed on the sections, utilizing a modified “loop method”. The emulsions were exposed for 7-weeks in a light-tight box at 4°C. Autoradiographs were developed in Microdol-X developer and examined on a Philips EM410LS transmission electron microscope. Quantitative analysis of compound localization employed the point and circle approach of Williams; incorporating the probability circle method of Salpeter and McHenry.

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