scholarly journals Tropical geometry over higher dimensional local fields

2015 ◽  
Vol 2015 (698) ◽  
Author(s):  
Soumya D. Banerjee
Keyword(s):  
Local Field ◽  
Fundamental Theorem ◽  
Tropical Geometry ◽  
Local Fields ◽  
Higher Dimensional

AbstractWe introduce the tropicalization of closed sub-schemes of a torus defined over a higher dimensional local field. We study the basic invariants of such tropicalizations. This is a generalization of the fundamental theorem of tropical geometry to higher dimensional local fields.

scholarly journals Fundamental theorem of prehomogeneous vector spaces of characteristic p

1997 ◽  
Vol 56 (2) ◽  
pp. 331-341
Author(s):  
Tatsuo Kimura ◽  
Takeyoshi Kogiso ◽  
Makiko Fujinaga
Keyword(s):  
Local Field ◽  
Fundamental Theorem ◽  
Functional Equations ◽  
Local Fields ◽  
Vector Spaces ◽  
Characteristic P ◽  
Prehomogeneous Vector Spaces

For a local field of characteristic 0, the functional equations of zeta distributions of prehomogeneous vector spaces have been obtained by M. Sato, Shintani, Igusa, F. Sato and Gyoja. In this paper, we shall consider the case of local fields of characteristic p > 0.

scholarly journals Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos A. M. André ◽  
João Dias
Keyword(s):  
Local Field ◽  
Unit Group ◽  
Basic Algebra ◽  
Local Fields ◽  
Dimensional Representation ◽  
Smooth Representation ◽  
One Dimensional ◽  
Finite Dimensional

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

The Shafarevich Basis in Higher-Dimensional Local Fields

2014 ◽  
Vol 202 (3) ◽  
pp. 410-421 ◽  
Author(s):  
E. V. Ikonnikova ◽  
E. V. Shaverdova
Keyword(s):  
Local Fields ◽  
Higher Dimensional

On the Integral Extensions of Quadratic Forms Over Local Fields

1970 ◽  
Vol 22 (2) ◽  
pp. 297-307 ◽  
Author(s):  
Melvin Band
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Ring Of Integers ◽  
Finite Dimensional ◽  
Integral Analogue ◽  
Necessary And Sufficient ◽  
Special Case

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

Local Field Effects Near Surfaces of Small Systems

1994 ◽  
Vol 366 ◽  
Author(s):  
David Beaglehole
Keyword(s):  
Dielectric Constant ◽  
External Field ◽  
Electric Field ◽  
Local Field ◽  
Computer Calculation ◽  
Local Fields ◽  
Small Systems ◽  
Shape Dependence ◽  
Self Consistent ◽  
Reflection Of Light

ABSTRACTThe interaction of light with a system of molecules depends upon the polarisation induced by an external electric field, which depends not only upon the external field but also upon the local fields due to neighboring polarised molecules. These local fields result in the traditional Clausius-Mossotti (CM) dielectric constant for a molecule deeply imbedded in a medium. Near the surface the local fields are altered, and the dielectric constant becomes anisotropic and dependent upon depth into the medium. The local fields are shape dependent in small systems and differ substantially from the CM value.A self-consistent computer calculation of the local fields has been implemented, and these effects will be shown using molecule positions and polarisabilities typical of liquids and crystals. The shape dependence of small systems, the reflection of light from liquids with fluctuating surfaces, and the effect of supporting substrates will be described.

scholarly journals Dualwavelet frames in Sobolev spaces on local fields of positive characteristic

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 2091-2099
Author(s):  
Ishtaq Ahmad ◽  
Neyaz Sheikh
Keyword(s):  
Sobolev Spaces ◽  
Local Field ◽  
Positive Characteristic ◽  
Local Fields ◽  
Wavelet Frames ◽  
The Past ◽  
Dual Wavelet Frames ◽  
Dual Wavelet ◽  
Field Of Positive Characteristic

Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. In this article, we obtain the characterization of nonhomogeneous wavelet frames and nonhomogeneous dual wavelet frames in a Sobolev spaces on a local field of positive characteristic by means of a pair of equations.

scholarly journals Turning a hot spot into a cold spot: polarization-controlled Fano-shaped local-field responses probed by a quantum dot

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Juan Xia ◽  
Jianwei Tang ◽  
Fanglin Bao ◽  
Yongcheng Sun ◽  
Maodong Fang ◽  
...  
Keyword(s):  
Quantum Dot ◽  
Local Field ◽  
Quantum Control ◽  
Temporal Dynamics ◽  
Hot Spot ◽  
Local Fields ◽  
Cold Spot ◽  
Spectral Dispersion ◽  
Field Response ◽  
Single Quantum Dot

Abstract Optical nanoantennas can convert propagating light to local fields. The local-field responses can be engineered to exhibit nontrivial features in spatial, spectral and temporal domains, where local-field interferences play a key role. Here, we design nearly fully controllable local-field interferences in the nanogap of a nanoantenna, and experimentally demonstrate that in the nanogap, the spectral dispersion of the local-field response can exhibit tuneable Fano lineshapes with nearly vanishing Fano dips. A single quantum dot is precisely positioned in the nanogap to probe the spectral dispersions of the local-field responses. By controlling the excitation polarization, the asymmetry parameter q of the probed Fano lineshapes can be tuned from negative to positive values, and correspondingly, the Fano dips can be tuned across a broad spectral range. Notably, at the Fano dips, the local-field intensity is strongly suppressed by up to ~50-fold, implying that the hot spot in the nanogap can be turned into a cold spot. The results may inspire diverse designs of local-field responses with novel spatial distributions, spectral dispersions and temporal dynamics, and expand the available toolbox for nanoscopy, spectroscopy, nano-optical quantum control and nanolithography.

scholarly journals Brownian motion and finite approximations of quantum systems over local fields

2017 ◽  
Vol 29 (05) ◽  
pp. 1750016 ◽  
Author(s):  
Erik Makino Bakken ◽  
Trond Digernes ◽  
David Weisbart
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Local Field ◽  
Schrödinger Operators ◽  
Quantum Systems ◽  
Local Fields ◽  
Schrodinger Operators ◽  
Finite Systems ◽  
Finite Approximations ◽  
Finite Approximability

We give a stochastic proof of the finite approximability of a class of Schrödinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman–Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.

WEIERSTRASS-TYPE FUNCTIONS IN p-ADIC LOCAL FIELDS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050043
Author(s):  
YIN LI
Keyword(s):  
Dynamical System ◽  
Differentiable Function ◽  
Local Field ◽  
Discrete Dynamical System ◽  
Linear Connection ◽  
Local Fields ◽  
Type Function ◽  
Nowhere Differentiable Function

The Weierstrass nowhere differentiable function has been studied often as example of functions whose graphs are fractals in [Formula: see text]. This paper investigates the Weierstrass-type function in the [Formula: see text]-adic local field [Formula: see text] whose graph is a repelling set of a discrete dynamical system, and proves that there exists a linear connection between the orders of the [Formula: see text]-adic calculus and the dimensions of the corresponding graphs.

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