Mathematics

Research

18 pages, 872 KiB

Open AccessArticle

Vector Geometric Algebra in Power Systems: An Updated Formulation of Apparent Power under Non-Sinusoidal Conditions

by Francisco G. Montoya, Raúl Baños, Alfredo Alcayde, Francisco Manuel Arrabal-Campos and Javier Roldán-Pérez

Mathematics 2021, 9(11), 1295; https://doi.org/10.3390/math9111295 - 04 Jun 2021

Cited by 12 | Viewed by 3092

Abstract

Traditional electrical power theories and one of their most important concepts—apparent power—are still a source of debate, because they present several flaws that misinterpret the power-transfer and energy-balance phenomena under distorted grid conditions. In recent years, advanced mathematical tools such as geometric algebra [...] Read more.

Traditional electrical power theories and one of their most important concepts—apparent power—are still a source of debate, because they present several flaws that misinterpret the power-transfer and energy-balance phenomena under distorted grid conditions. In recent years, advanced mathematical tools such as geometric algebra (GA) have been introduced to address these issues. However, the application of GA to electrical circuits requires more consensus, improvements and refinement. In this paper, electrical power theories for single-phase systems based on GA were revisited. Several drawbacks and inconsistencies of previous works were identified, and some amendments were introduced. An alternative expression is presented for the electric power in the geometric domain. Its norm is compatible with the traditional apparent power defined as the product of the RMS voltage and current. The use of this expression simplifies calculations such as those required for current decomposition. This proposal is valid even for distorted currents and voltages. Concepts are presented in a simple way so that a strong background on GA is not required. The paper included some examples and experimental results in which measurements from a utility supply were analysed. Full article

(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)

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17 pages, 1001 KiB

Open AccessArticle

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

by Francisco G. Montoya, Raúl Baños, Alfredo Alcayde, Francisco Manuel Arrabal-Campos and Javier Roldán Pérez

Mathematics 2021, 9(11), 1259; https://doi.org/10.3390/math9111259 - 31 May 2021

Cited by 5 | Viewed by 2041

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)

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21 pages, 376 KiB

Open AccessArticle

The μ-Basis of Improper Rational Parametric Surface and Its Application

by Sonia Pérez-Díaz and Li-Yong Shen

Mathematics 2021, 9(6), 640; https://doi.org/10.3390/math9060640 - 17 Mar 2021

Cited by 1 | Viewed by 1222

Abstract

The

μ

-basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of

μ

-bases is still developing, especially of surfaces. We study [...] Read more.

The

μ

-basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of

μ

-bases is still developing, especially of surfaces. We study the

μ

-basis of a rational surface

V

defined parametrically by

P (\bar{t}), \bar{t} = (t_{1}, t_{2})

not being necessarily proper (or invertible). For applications using the

μ

-basis, an inversion formula for a given proper parametrization

P (\bar{t})

is obtained. In addition, the degree of the rational map

ϕ_{P}

associated with any

P (\bar{t})

is computed. If

P (\bar{t})

is improper, we give some partial results in finding a proper reparametrization of

V

. Finally, the implicitization formula is derived from

P

(not being necessarily proper). The discussions only need to compute the greatest common divisors and univariate resultants of polynomials constructed from the

μ

-basis. Examples are given to illustrate the computational processes of the presented results. Full article

(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)

16 pages, 370 KiB

Open AccessArticle

A Topological View of Reed–Solomon Codes

by Alberto Besana and Cristina Martínez

Mathematics 2021, 9(5), 578; https://doi.org/10.3390/math9050578 - 09 Mar 2021

Viewed by 1733

Abstract

We studied a particular class of well known error-correcting codes known as Reed–Solomon codes. We constructed RS codes as algebraic-geometric codes from the normal rational curve. This approach allowed us to study some algebraic representations of RS codes through the study of the [...] Read more.

We studied a particular class of well known error-correcting codes known as Reed–Solomon codes. We constructed RS codes as algebraic-geometric codes from the normal rational curve. This approach allowed us to study some algebraic representations of RS codes through the study of the general linear group

G L (n, q)

. We characterized the coefficients that appear in the decompostion of an irreducible representation of the special linear group in terms of Gromov–Witten invariants of the Hilbert scheme of points in the plane. In addition, we classified all the algebraic codes defined over the normal rational curve, thereby providing an algorithm to compute a set of generators of the ideal associated with any algebraic code constructed on the rational normal curve (NRC) over an extension

F_{q^{n}}

of

F_{q}

. Full article

(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)

15 pages, 335 KiB

Open AccessArticle

Covering Rational Surfaces with Rational Parametrization Images

by Jorge Caravantes, J. Rafael Sendra, David Sevilla and Carlos Villarino

Mathematics 2021, 9(4), 338; https://doi.org/10.3390/math9040338 - 08 Feb 2021

Cited by 2 | Viewed by 1305

Abstract

Let S be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps [...] Read more.

Let S be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps

f, g, h : A^{2} ⇢ S \subset P^{n}

such that the union of the three images covers S. As a consequence, we present a second algorithm that generates two rational maps

f, \tilde{g} : A^{2} ⇢ S

, such that the union of its images covers the affine surface

S \cap A^{n}

. In the affine case, the number of rational maps involved in the cover is in general optimal. Full article

(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)

22 pages, 384 KiB

Open AccessArticle

Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

by Anastasis Kratsios

Mathematics 2021, 9(3), 251; https://doi.org/10.3390/math9030251 - 27 Jan 2021

Viewed by 1132

Abstract

The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector [...] Read more.

The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms

Ω^{n} (X, M)

. Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where

k = C

. Full article

(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)

28 pages, 431 KiB

Open AccessArticle

Computing Birational Polynomial Surface Parametrizations without Base Points

by Sonia Pérez-Díaz and Juan Rafael Sendra

Mathematics 2020, 8(12), 2224; https://doi.org/10.3390/math8122224 - 14 Dec 2020

Cited by 3 | Viewed by 1342

Abstract

In this paper, we present an algorithm for reparametrizing birational surface parametrizations into birational polynomial surface parametrizations without base points, if they exist. For this purpose, we impose a transversality condition to the base points of the input parametrization. Full article

(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)

17 pages, 424 KiB

Open AccessArticle

On the Affine Image of a Rational Surface of Revolution

by Juan G. Alcázar

Mathematics 2020, 8(11), 2061; https://doi.org/10.3390/math8112061 - 19 Nov 2020

Cited by 1 | Viewed by 1347

Abstract

We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines, a concept that appears in the context of [...] Read more.

We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines, a concept that appears in the context of affine differential geometry, created by Blaschke in the first decades of the 20th century, intersect a fixed line. Given a rational surface with this property, which can be algorithmically checked, we provide an algorithmic method to find a surface of revolution, if it exists, whose image under an affine mapping is the given surface; the algorithm also finds the affine transformation mapping one surface onto the other. Finally, we also prove that the only rational affine surfaces of rotation, a generalization of surfaces of revolution that arises in the context of affine differential geometry, and which includes surfaces of revolution as a subtype, affinely transforming into a surface of revolution are the surfaces of revolution, and that in that case the affine mapping must be a similarity. Full article

(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)

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21 pages, 353 KiB

Open AccessArticle

Polarized Rigid Del Pezzo Surfaces in Low Codimension

by Muhammad Imran Qureshi

Mathematics 2020, 8(9), 1567; https://doi.org/10.3390/math8091567 - 11 Sep 2020

Viewed by 1468

Abstract

We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type

\{k_{i} \times \frac{1}{r_{i}} (1, a_{i}) : 3 \leq r_{i} \leq 10, k_{i} \in Z_{\geq 0}\}

as [...] Read more.

We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type

\{k_{i} \times \frac{1}{r_{i}} (1, a_{i}) : 3 \leq r_{i} \leq 10, k_{i} \in Z_{\geq 0}\}

as well-formed and quasismooth varieties embedded in some weighted projective space. In particular, we present a collection of 147 such surfaces such that their image under their anti-canonical embeddings can be described by using one of the following sets of equations: a single equation, two linearly independent equations, five maximal Pfaffians of

5 \times 5

skew symmetric matrix, and nine

2 \times 2

minors of size 3 square matrix. This is a complete classification of such surfaces under certain carefully chosen bounds on the weights of ambient weighted projective spaces and it is largely based on detailed computer-assisted searches by using the computer algebra system MAGMA. Full article

(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)

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