Bound for Minkowski metric or quadratic metric applied to VQ codeword search

1996 ◽  
Vol 143 (1) ◽  
pp. 67 ◽  
Author(s):  
J.S. Pan ◽  
F.R. McInnes ◽  
M.A. Jack
Keyword(s):  
Minkowski Metric

scholarly journals Spectral properties of the equation (∇ + ieA) × u = ±mu

2001 ◽  
Vol 131 (5) ◽  
pp. 1065-1089
Author(s):  
Daniel M. Elton
Keyword(s):  
Spectral Properties ◽  
Time Variable ◽  
Two Dimensional ◽  
Vector Product ◽  
Relativistic Invariance ◽  
Speed Of Light ◽  
Large Parameter ◽  
Real Vector ◽  
Pauli Equation ◽  
Minkowski Metric

We develop a spectral theory for the equation (∇ + ieA) × u = ±mu on Minkowski 3-space (one time variable and two space variables); here, A is a real vector potential and the vector product is defined with respect to the Minkowski metric. This equation was formulated by Elton and Vassiliev, who conjectured that it should have properties similar to those of the two-dimensional Dirac equation. Our equation contains a large parameter c (speed of light), and this motivates the study of the asymptotic behaviour of its spectrum as c → +∞. We show that the essential spectrum of our equation is the same as that of Dirac (theorem 3.1), whereas the discrete spectrum agrees with Dirac to a relative accuracy δλ/mc2 ~ O(c−4) (theorem 3.3). In other words, we show that our equation has the same accuracy as the two-dimensional Pauli equation, its advantage over Pauli being relativistic invariance.

scholarly journals An Analysis of Gravitational waves

2021 ◽  
Author(s):  
Vaibhav Kalvakota
Keyword(s):  
Gravitational Wave ◽  
Gravitational Waves ◽  
Mathematical Structure ◽  
Gauge Condition ◽  
Field Tensor ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Minkowski Metric ◽  
Wave Detector ◽  
The Masses

The September 14, 2015 gravitational wave observations showed the inspiral of two black holes observed from Hanford and Livingston LIGO observatories. This detection was significant for two reasons: firstly, it coupled the result and avoided the possibility of a false alarm by 5σ , meaning that the detected “noise” was indeed from an astronomical source of gravitational waves. We will discuss the primary landscape of gravitational waves, their mathematical structure and how they can be used to predict the masses of the merger system. We will also discuss gravitational wave detector optimisations, and then we will consider the results from the detected merger GW150914. We will consider a straight-forward mathematical approach, and we will primarily be interested in the mathematical modelling of gravitational waves from General Relativity (Section 1). We will first consider a “perturbed” Minkowski metric, and then we will discuss the properties of the perturbation addition tensor. We will then discuss on the gravitational field tensor, and how it arises from the perturbation tensor. We will then talk about the gauge condition, essentially the gauge “freedom” , and then we will talk about the curvature tensor, leading eventually to the effect of gravitational waves on a ring of particles. We will consider the polarisation tensor, which maps the amplitude and polarisation details. The polarisation splits into plus polarised and cross polarised waves, which is technically the effect of a propagating gravitational wave through a ring of particles. We will then talk about the linearized Einstein Field Equations, and how the physical system of merger is encoded into the mathematical structural unity of the metric. We will then talk about the detection of these gravitational waves and how the detector can be optimised, or how the detector can be set so that any “noise” detected can fall in the error margins, and how the detector can prevent the interferometric “photon-noise” from being detected (Section 2.2). Then, we will discuss data results from the source GW150914 detection by LIGO (Section 3).

scholarly journals Lorentz violation with a universal minimum speed as foundation of de Sitter relativity

2018 ◽  
Vol 27 (02) ◽  
pp. 1850011 ◽  
Author(s):  
Cláudio Nassif Cruz ◽  
Rodrigo Francisco dos Santos ◽  
A. C. Amaro de Faria
Keyword(s):  
Cosmological Constant ◽  
Extra Dimension ◽  
Negative Curvature ◽  
Lorentz Violation ◽  
Conformal Factor ◽  
De Sitter ◽  
Minkowski Metric ◽  
Minimum Speed ◽  
New Space ◽  
Cosmological Constants

We aim to investigate the theory of Lorentz violation with an invariant minimum speed called Symmetrical Special Relativity (SSR) from the viewpoint of its metric. Thus, we should explore the nature of SSR-metric in order to understand the origin of the conformal factor that appears in the metric by deforming Minkowski metric by means of an invariant minimum speed that breaks down Lorentz symmetry. So, we are able to realize that there is a similarity between SSR and a new space with variable negative curvature ([Formula: see text]) connected to a set of infinite cosmological constants ([Formula: see text]), working like an extended de Sitter (dS) relativity, so that such extended dS-relativity has curvature and cosmological “constant” varying in time. We obtain a scenario that is more similar to dS-relativity given in the approximation of a slightly negative curvature for representing the current universe having a tiny cosmological constant. Finally, we show that the invariant minimum speed provides the foundation for understanding the kinematics origin of the extra dimension considered in dS-relativity in order to represent the dS-length.

Application of the Cluster Analysis in Computational Paleography

2016 ◽  
pp. 525-543
Author(s):  
Loránd Lehel Tóth ◽  
Raymond Eliza Ivan Pardede ◽  
György András Jeney ◽  
Ferenc Kovács ◽  
Gábor Hosszú
Keyword(s):  
Cluster Analysis ◽  
Unknown Origin ◽  
Canonical Decomposition ◽  
Minkowski Metric ◽  
Clustering Technique ◽  
Future Work

This chapter presents a method to determine the actual version of a script used in constructing of a script relic from unknown origin. The glyphs belong to graphemes as models are realized in the relics as symbols. Some group of glyphs may transform their shape (shapeshifting) through time which produces various versions of scripts that use different glyphs to express the same grapheme. These glyph variants can be identified from extant relics, mainly from historical abecedaries that are used as references. Our algorithm can determine whether or not an abecedary is related to the symbols of a relic from unknown origin by means of the canonical decomposition of the glyphs and symbols. From there an aggregated value called fingerprint is created and it is unique for each relic. The fingerprints then are evaluated by clustering technique using various metrics. As the result of performing comparative evaluations the Minkowski metric provides the most interpretable clustering structure. The results of the evaluations, conclusions, and future work are also presented.

scholarly journals Characterizations of central symmetry for convex bodies in Minkowski spaces

2009 ◽  
Vol 46 (4) ◽  
pp. 493-514
Author(s):  
Gennadiy Averkov ◽  
Endre Makai ◽  
Horst Martini
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Analogous Result ◽  
Area Measure ◽  
Minkowski Metric ◽  
Surface Area Measure ◽  
Shadow Boundary ◽  
Dimensional Minkowski Space ◽  
Centrally Symmetric ◽  
Definition Of

K. Zindler [47] and P. C. Hammer and T. J. Smith [19] showed the following: Let K be a convex body in the Euclidean plane such that any two boundary points p and q of K , that divide the circumference of K into two arcs of equal length, are antipodal. Then K is centrally symmetric. [19] announced the analogous result for any Minkowski plane \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^2$$ \end{document}, with arc length measured in the respective Minkowski metric. This was recently proved by Y. D. Chai — Y. I. Kim [7] and G. Averkov [4]. On the other hand, for Euclidean d -space ℝ d , R. Schneider [38] proved that if K ⊂ ℝ d is a convex body, such that each shadow boundary of K with respect to parallel illumination halves the Euclidean surface area of K (for the definition of “halving” see in the paper), then K is centrally symmetric. (This implies the result from [19] for ℝ 2 .) We give a common generalization of the results of Schneider [38] and Averkov [4]. Namely, let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^d$$ \end{document} be a d -dimensional Minkowski space, and K ⊂ \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^d$$ \end{document} be a convex body. If some Minkowskian surface area (e.g., Busemann’s or Holmes-Thompson’s) of K is halved by each shadow boundary of K with respect to parallel illumination, then K is centrally symmetric. Actually, we use little from the definition of Minkowskian surface area(s). We may measure “surface area” via any even Borel function ϕ: Sd −1 → ℝ, for a convex body K with Euclidean surface area measure dSK ( u ), with ϕ( u ) being dSK ( u )-almost everywhere non-0, by the formula B ↦ ∫ B ϕ( u ) dSK ( u ) (supposing that ϕ is integrable with respect to dSK ( u )), for B ⊂ Sd −1 a Borel set, rather than the Euclidean surface area measure B ↦ ∫ BdSK ( u ). The conclusion remains the same, even if we suppose surface area halving only for parallel illumination from almost all directions. Moreover, replacing the surface are a measure dSK ( u ) by the k -th area measure of K ( k with 1 ≦ k ≦ d − 2 an integer), the analogous result holds. We follow rather closely the proof for ℝ d , which is due to Schneider [38].

scholarly journals COMPLEX BERWALD MANIFOLDS WITH VANISHING HOLOMORPHIC SECTIONAL CURVATURE

2008 ◽  
Vol 50 (2) ◽  
pp. 203-208 ◽  
Author(s):  
RONGMU YAN
Keyword(s):  
Sectional Curvature ◽  
Finsler Metric ◽  
Holomorphic Sectional Curvature ◽  
Minkowski Metric ◽  
Strongly Convex ◽  
Kähler Finsler Metric

AbstractIn this paper, we prove that a strongly convex and Kähler-Finsler metric is a complex Berwald metric with zero holomorphic sectional curvature if and only if it is a complex locally Minkowski metric.

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