The special values of the standard 𝐿-functions for 𝐺𝑆𝑝_{2𝑛}×𝐺𝐿₁

In this paper the problem of constructing space–time from string theory is addressed in the context of D-brane physics. It is suggested that the knowledge of discrete configurations of D-branes is sufficient to reconstruct the motivic building blocks of certain Calabi–Yau varieties. The collections of D-branes involved have algebraic base points, leading to the notion of K-arithmetic D-crystals for algebraic number fields K. This idea can be tested for D0-branes in the framework of toroidal compactifications via the conjectures of Birch and Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these conjectures can be interpreted as formulae that relate the canonical Néron–Tate height of the base points of the D-crystals to special values of the motivic L-function at the central point. In simple cases the knowledge of the D-crystals of Heegner type suffices to uniquely determine the geometry.

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Stark–Heegner points and special values of L-series

L-functions and Galois Representations ◽

10.1017/cbo9780511721267.002 ◽

2010 ◽

pp. 1-23 ◽

Cited By ~ 1

Author(s):

Massimo Bertolini ◽

Henri Darmon ◽

Samit Dasgupta

Keyword(s):

Heegner Points ◽

Special Values

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Diffusion of dust particles emitted from a fixed source

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v4i4.5058 ◽

2015 ◽

Vol 4 (4) ◽

pp. 454

Author(s):

Khaled Al-mashrafi

Keyword(s):

Source Function ◽

Strong Dependence ◽

Vertical Direction ◽

Dust Particles ◽

Special Values ◽

Diffusion Parameters ◽

Special Cases ◽

Small Times ◽

The Mathematical Model ◽

General Time

<p>In this paper, we investigate the mathematical model for the diffusion of dust particles emitted from a fixed source. Mathematically, the time-dependent diffusion equation in the presence of a point source whose strength is dependent on time is solved. The solution in closed form for a source of general time dependence is obtained. A number of special cases, in which the source function of time is explicitly given and special values of the diffusion parameters are taken are examined in detail. The numerical calculations show the strong dependence of the concentration of dust on the speed of the wind, the source, and its position in the vertical direction. It is also found that the diffusion parameters play an important role in the spread of the dust particles in the atmosphere. When diffusion is present only in the vertical direction, it is found that for small times the dust spreads with a front that travels with the speed of the wind.</p>

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Gamma Factors, Root Numbers, and Distinction

Canadian Journal of Mathematics ◽

10.4153/cjm-2017-011-6 ◽

2018 ◽

Vol 70 (3) ◽

pp. 683-701 ◽

Cited By ~ 1

Author(s):

Nadir Matringe ◽

Omer Offen

Keyword(s):

Linear Group ◽

General Linear Group ◽

Quadratic Extension ◽

Root Number ◽

Converse Problem ◽

Root Numbers ◽

Special Values ◽

Distinguished Representations ◽

Local Invariants ◽

Gamma Factor

AbstractWe study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of p-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.

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On the theory of finite deformations of elastic crystals

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1942.0041 ◽

1942 ◽

Vol 180 (982) ◽

pp. 285-304 ◽

Cited By ~ 4

Keyword(s):

Order Approximation ◽

Finite Deformations ◽

Second Order ◽

Order Effects ◽

Zero Temperature ◽

Cubic Crystals ◽

Special Values ◽

Lennard Jones ◽

Second Order Effects ◽

Elastic Crystals

The general theory of finite deformation of cubic crystals at zero temperature is developed to a second-order approximation, and the cases of (1) a uniform hydrostatic pressure, (2) a tension in the direction of one of the axes, (3) a shear along the (0, 1, 0) planes, and (4) a shear along the (0, 1, 1) planes of the lattice, are worked out in detail. A number of ‘second-order effects’ (deviations from Hooke’s law) are predicted which in case (1) have been observed and measured by Bridgman, and in the remaining cases certainly can be detected and measured by suitable experimental arrangements. Assuming the particular force law between the particles of the lattice which was first introduced by Mie and Grüneisen, and later used in the investigations of Lennard-Jones and of Born and his collaborators, and using some of the results of the latter authors, the constants governing the above-mentioned second-order effects are expressed in terms of the constants governing the force law, and calculated numerically for a number of special values of these constants. Thus by comparing the calculated values of these constants with the results of measurements at low temperature the unknown force law could probably be determined.

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