scholarly journals Chebyshev Distance

2016 ◽  
Vol 24 (2) ◽  
pp. 121-141 ◽  
Author(s):  
Roland Coghetto
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Definition Of ◽  
Topological Sense

Summary In [21], Marco Riccardi formalized that ℝN-basis n is a basis (in the algebraic sense defined in [26]) of ${\cal E}_T^n $ and in [20] he has formalized that ${\cal E}_T^n $ is second-countable, we build (in the topological sense defined in [23]) a denumerable base of ${\cal E}_T^n $ . Then we introduce the n-dimensional intervals (interval in n-dimensional Euclidean space, pavé (borné) de ℝn [16], semi-intervalle (borné) de ℝn [22]). We conclude with the definition of Chebyshev distance [11].

scholarly journals On the convergence of a Newton-like method inℝnand the use of Berinde's exit criterion

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
Rabindranath Sen ◽  
Sulekha Mukherjee ◽  
Rajesh Patra
Keyword(s):  
Euclidean Space ◽  
Newton’S Method ◽  
Newton's Method ◽  
Scalar Equation ◽  
Dimensional Euclidean Space ◽  
Definition Of

Berinde has shown that Newton's method for a scalar equationf(x)=0converges under some conditions involving onlyfandf′and notf″when a generalized stopping inequality is valid. Later Sen et al. have extended Berinde's theorem to the case where the condition thatf′(x)≠0need not necessarily be true. In this paper we have extended Berinde's theorem to the class ofn-dimensional equations,F(x)=0, whereF:ℝn→ℝn,ℝndenotes then-dimensional Euclidean space. We have also assumed thatF′(x)has an inverse not necessarily at every point in the domain of definition ofF.

scholarly journals The Observable Representation

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 310
Author(s):  
L. Schulman
Keyword(s):  
Phase Transition ◽  
Euclidean Space ◽  
Complex Systems ◽  
Random Walks ◽  
Stochastic Dynamics ◽  
Dimensional Euclidean Space ◽  
Significant Feature ◽  
Original Space ◽  
Definition Of ◽  
Low Dimensional

The observable representation (OR) is an embedding of the space on which a stochastic dynamics is taking place into a low dimensional Euclidean space. The most significant feature of the OR is that it respects the dynamics. Examples are given in several areas: the definition of a phase transition (including metastable phases), random walks in which the OR recovers the original space, complex systems, systems in which the number of extrema exceed convenient viewing capacity, and systems in which successful features are displayed, but without the support of known theorems.

scholarly journals A GEOMETRICAL INTERPRETATION OF RENORMALIZATION GROUP FLOW

1994 ◽  
Vol 09 (08) ◽  
pp. 1261-1286 ◽  
Author(s):  
BRIAN P. DOLAN
Keyword(s):  
Renormalization Group ◽  
Euclidean Space ◽  
Point Of View ◽  
Dimensional Euclidean Space ◽  
Lie Derivative ◽  
Coupling Constant ◽  
Four Dimensions ◽  
Anomalous Dimensions ◽  
Definition Of ◽  
Group Flow

The renormalization group (RG) equation in D-dimensional Euclidean space, RD, is analyzed from a geometrical point of view. A general form of the RG equation is derived which is applicable to composite operators as well as tensor operators (on RD) which may depend on the Euclidean metric. It is argued that physical N-point amplitudes should be interpreted as rank N covariant tensors on the space of couplings, [Formula: see text], and that the RG equation can be viewed as an equation for Lie transport on [Formula: see text] with respect to the vector field generated by the β functions of the theory. In one sense it is nothing more than the definition of a Lie derivative. The source of the anomalous dimensions can be interpreted as being due to the change of the basis vectors on [Formula: see text] under Lie transport. The RG equation acts as a bridge between Euclidean space and coupling constant space in that the effect on amplitudes of a diffeomorphism of RD (that of dilations) is completely equivalent to a diffeomorphism of [Formula: see text] generated by the β functions of the theory. A form of the RG equation for operators is also given. These ideas are developed in detail for the example of massive λφ4 theory in four dimensions.

The Spiric Sections of Perseus and the Uniform Parameterizations of the Cassinian Ovals

2020 ◽  
Vol 58 ◽  
pp. 81-97
Author(s):  
Ivaïlo M. Mladenov ◽  
Keyword(s):  
Euclidean Space ◽  
Surface Area ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Geometrical Parameters ◽  
Explicit Formulas ◽  
Ancient Time ◽  
Definition Of

Here we derive explicit formulas that parameterize the Cassinian ovals based on their recognition as the so called spiric sections of the standard tori in the three-dimensional Euclidean space which was suggested in the ancient time by Perseus. These formulas derived originally in terms of the toric parameters are expressed through the usual geometrical parameters that enter in the present day definition of the Cassinian curves. All three types of morphologically different curves are illustrated graphically using the corresponding sets of parameters and respective formulas. The geometry of the ovals can be studied in full details and this is done here to some extent. As examples explicit formulas for the embraced volume and the surface area of the dumbbell like surface generated by the oval are presented. Last, but not least, new alternative explicit parameterizations of the Cassinian ovals are derived in polar, and even in non-canonical Monge forms.

Full Development of Tarski's Geometry of Solids

2008 ◽  
Vol 14 (4) ◽  
pp. 481-540 ◽  
Author(s):  
Rafaŀ Gruszczyński ◽  
Andrzej Pietruszczak
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Final Part ◽  
Dimensional Euclidean Space ◽  
Short Paper ◽  
Logical Status ◽  
Full Development ◽  
Definition Of ◽  
The Universe ◽  
Universe Of Discourse

AbstractIn this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper [20, 21]. We show that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. We show that once having adopted such a solution Tarski's Postulate 4 can be omitted, together with its versions 4′ and 4″. We also prove that the equivalence of postulates 4, 4′ and 4″ is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms.We build a model (in three-dimensional Euclidean space) of the theory of so called T*-structures and present the proof of the fact that this is the only (up to isomorphism) model of this theory.Moreover, we propose different categorical axiomatizations of the geometry of solids. In the final part of the paper we answer the question concerning the logical status (within the theory of T*-structures) of the definition of the concentricity relation given by Tarski.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

scholarly journals Geometric Algebra Framework Applied to Symmetrical Balanced Three-Phase Systems for Sinusoidal and Non-Sinusoidal Voltage Supply

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1259
Author(s):  
Francisco G. Montoya ◽  
Raúl Baños ◽  
Alfredo Alcayde ◽  
Francisco Manuel Arrabal-Campos ◽  
Javier Roldán Roldán Pérez
Keyword(s):  
Geometric Algebra ◽  
Dimensional Euclidean Space ◽  
Active Components ◽  
Electrical Circuits ◽  
Current Harmonics ◽  
Operation Conditions ◽  
Multiple Phase ◽  
Three Phase ◽  
Phase Systems ◽  
Definition Of

This paper presents a new framework based on geometric algebra (GA) to solve and analyse three-phase balanced electrical circuits under sinusoidal and non-sinusoidal conditions. The proposed approach is an exploratory application of the geometric algebra power theory (GAPoT) to multiple-phase systems. A definition of geometric apparent power for three-phase systems, that complies with the energy conservation principle, is also introduced. Power calculations are performed in a multi-dimensional Euclidean space where cross effects between voltage and current harmonics are taken into consideration. By using the proposed framework, the current can be easily geometrically decomposed into active- and non-active components for current compensation purposes. The paper includes detailed examples in which electrical circuits are solved and the results are analysed. This work is a first step towards a more advanced polyphase proposal that can be applied to systems under real operation conditions, where unbalance and asymmetry is considered.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

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