Derivative of the Analytic Kernel

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Explicit Formula ◽  
General Formula ◽  
Kernel Function ◽  
Schwartz Function ◽  
Explicit Result ◽  
Analytic Kernel ◽  
Holomorphic Projection ◽  
Period Integral ◽  
Local Term

This chapter computes the derivative of the analytic kernel. It first decomposes the kernel function into a sum of infinitely many local terms indexed by places v of Fnonsplit in E. Each local term is a period integral of some kernel function. The chapter then considers the v-part for non-archimedean v. An explicit formula is given in the unramified case, and an approximation is presented in the ramified case assuming the Schwartz function is degenerate. An explicit result of the v-part for archimedean v is also introduced. The chapter proceeds by reviewing a general formula of holomorphic projection, and estimates the growth of the kernel function in order to apply the formula. It also computes the holomorphic projection of the analytic kernel function and concludes with a discussion of the holomorphic kernel function.

Assumptions on the Schwartz Function

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Local Field ◽  
Kernel Functions ◽  
Quadratic Space ◽  
Schwartz Function ◽  
Schwartz Functions ◽  
First Case ◽  
Analytic Kernel

This chapter introduces two classes of degenerate Schwartz functions which significantly simplify the computations and arguments of both the analytic kernel and the geometric kernel functions. It first restates the kernel identity in terms of un-normalized kernel functions before stating the assumptions on the Schwartz function and claiming that these assumptions can be “added” to the kernel identity without losing the generality. It then considers some simple properties of the assumptions and proceeds by discussing the two classes of degenerate Schwartz functions. In the first case, a non-archimedean local field and a non-degenerate quadratic space are described. In the second case, since all the data are unramified, the lemma can be verified by explicit computations.

Mordell-Weil Groups and Generating Series

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Kernel Function ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
Generating Series ◽  
Analytic Kernel

This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before introducing a theorem, which is an identity between the analytic kernel and the geometric kernel. It then defines the generating series and uses it to describe the geometric kernel. It also presents a theorem, which is an identity formulated in terms of projectors, and reviews some basic notations and results on Shimura curves. Other topics covered include the Eichler–Shimura theory for abelian varieties parametrized by Shimura curves, normalization of the geometric kernel, and the analytic kernel function. The chapter concludes with an analysis of the kernel identity implied in the first theorem.

Local Heights of CM Points

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Kernel Function ◽  
Quadratic Extension ◽  
Multiplicity Function ◽  
Schwartz Function ◽  
Shimura Curve ◽  
Fourth Case ◽  
Cm Points ◽  
Ordinary Case ◽  
Archimedean Place

This chapter computes the local heights and compares them with the derivatives computed before. It checks the theorem place by place and takes into account all the assumptions on the Schwartz function. According to the reduction of the Shimura curve, the situation is divided to the following four cases: archimedean case, supersingular case, superspecial case, and ordinary case. The treatments in different cases are similar in spirit, except that the fourth case is slightly different. The supersingular case is divided into two subcases: unramified case and ramified case. The chapter also describes local heights of CM points at any archimedean place v. The discussion covers the multiplicity function, the kernel function, unramified quadratic extension, ramified quadratic extension, ordinary components, supersingular components, and superspecial components.

scholarly journals A Formula for Popp’s Volume in Sub-Riemannian Geometry

2013 ◽  
Vol 1 ◽  
pp. 42-57 ◽  
Author(s):  
Davide Barilari ◽  
Luca Rizzi
Keyword(s):  
Riemannian Manifold ◽  
Explicit Formula ◽  
General Formula ◽  
Riemannian Geometry ◽  
Isometry Group ◽  
Natural Generalization ◽  
Riemannian Structure ◽  
The One

Abstract For an equiregular sub-Riemannian manifold M, Popp’s volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp’s volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub- Laplacian, namely the one associated with Popp’s volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp’s volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp’s volume is essentially the unique volume with such a property.

X-ray Structure Refinement and Vibrational Spectroscopy of Ca8Gd2 (PO4)6 O2

2013 ◽  
Vol 12 (10) ◽  
pp. 719-726
Author(s):  
R. Ayadi ◽  
Mohamed Boujelbene ◽  
T. Mhiri
Keyword(s):  
Solid State ◽  
General Formula ◽  
Vibrational Spectroscopy ◽  
Powder Diffraction ◽  
Infrared Spectra ◽  
Solid State Reaction ◽  
Structure Refinement ◽  
Unit Cell ◽  
Cell Group ◽  
X Ray

The present paper is interested in the study of compounds from the apatite family with the general formula Ca10 (PO4)6A2. It particularly brings to light the exploitation of the distinctive stereochemistries of two Ca positions in apatite. In fact, Gd-Bearing oxyapatiteCa8 Gd2 (PO4)6O2 has been synthesized by solid state reaction and characterized by X-ray powder diffraction. The site occupancies of substituents is0.3333 in Gd and 0.3333 for Ca in the Ca(1) position and 0. 5 for Gd in the Ca (2) position.  Besides, the observed frequencies in the Raman and infrared spectra were explained and discussed on the basis of unit-cell group analyses.

Algorithm of target tracking based on Mean-Shift with adaptive bandwidth of kernel function

2009 ◽  
Vol 29 (6) ◽  
pp. 1680-1682
Author(s):  
Chang-tao CHEN ◽  
Qin ZHU ◽  
Sheng-yi ZHOU ◽  
Jia-ming ZHANG
Keyword(s):  
Target Tracking ◽  
Kernel Function ◽  
Mean Shift ◽  
Adaptive Bandwidth
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