Geometric Calculus Research Articles

An Overview of Generalized Cesàro Sequence Spaces in Geometric Calculus

In this study, we introduce the generalized Cesàro sequence space in geometric calculus and establish a G-modular on this space. The generalized geometric Cesàro sequence space is equipped with the Luxemburg G-norm induced by the G-modular. The connections of the G-modular and Luxemburg G-norm on this space are studied. In addition, we show that the generalized geometric Cesàro sequence space is a G-Banach space under the Luxemburg G-norm and it is G-nonsquare where pn > 1 for all n ∈ ℕ.

Solving Boundary Value Problems by Sinc Method and Geometric Sinc Method

This paper introduces an efficient numerical method for approximating solutions to geometric boundary value problems. We propose the multiplicative sinc–Galerkin method, tailored specifically for solving multiplicative differential equations. The method utilizes the geometric Whittaker cardinal function to approximate functions and their geometric derivatives. By reducing the geometric differential equation to a system of algebraic equations, we achieve computational efficiency. The method not only proves to be computationally efficient but also showcases a valuable symmetric property, aligning with inherent patterns in geometric structures. This symmetry enhances the method’s compatibility with the often-present symmetries in geometric boundary value problems, offering both computational advantages and a deeper understanding of geometric calculus. To demonstrate the reliability and efficiency of the proposed method, we present several examples with both homogeneous and non-homogeneous boundary conditions. These examples serve to validate the method’s performance in practice.

Sturm-Liouville problem in multiplicative fractional calculus

<p>Multiplicative calculus, or geometric calculus, is an alternative to classical calculus that relies on division and multiplication as opposed to addition and subtraction, which are the basic operations of classical calculus. It offers a geometric interpretation that is especially helpful for simulating systems that degrade or expand exponentially. Multiplicative calculus may be extended to fractional orders, much as classical calculus, which enables the analysis of systems having fractional scaling properties. So, in this paper, the well-known Sturm-Liouville problem in fractional calculus is reformulated in multiplicative fractional calculus. The considered problem consists of the Sturm-Liouville operator using multiplicative conformable derivatives on the equation and on boundary conditions. This research aimed to explore some of the problem's spectral aspects, like being self-adjointness of the operator, orthogonality of different eigenfunctions, and reality of all eigenvalues. In this specific situation, Green's function is also recreated.</p>

Vector properties of geometric calculus

After its introduction by Grossman and Katz, the geometry part of geometric calculus has been mostly left unattended. In this paper, we give the elementary theorems to build the geometry on geometric space. We apply the well‐known theorems for inner product in geometric space. The terms geometric determinant and geometric vector product are introduced, and their properties are studied. We give the definitions of the circle, the plane, and the sphere and visualize them along with a line in geometric space. Finally, we discuss the Gram–Schmidt process in geometric space.

Computational Physics of <i>Mathematica</i> and Geometric Calculus

The latter half of the twentieth century yielded two tools of unprecedented power, both of which took decades to mature to their current states. The purpose of this research is to apply these to a theory of gravity and develop the consequences of the model based on these tools. This paper presents such results without mathematical details, which are presented elsewhere. The tools are: Geometric Calculus, developed by David Hestenes, circa 1965 and Mathematica, released in 1988 by Steven Wolfram. Both tools have steep learning curves, requiring several years to acquire expertise in their use. This paper explains in what sense they are optimal.

The Fundamental Lepage Form in Two Independent Variables: A Generalization Using Order-Reducibility

A second-order generalization of the fundamental Lepage form of geometric calculus of variations over fibered manifolds with 2-dimensional base is described by means of insisting on (i) an equivalence relation “Lepage differential 2-form is closed if and only if the associated Lagrangian is trivial” and (ii) the principal component of Lepage form, extending the well-known Poincaré–Cartan form, preserving order prescribed by a given Lagrangian. This approach completes several attempts of finding a Lepage equivalent of a second-order Lagrangian possessing condition (i), which is well-known for first-order Lagrangians in field theory due to Krupka and Betounes.

New Applications of Clifford’s Geometric Algebra

The new applications of Clifford’s geometric algebra surveyed in this paper include kinematics and robotics, computer graphics and animation, neural networks and pattern recognition, signal and image processing, applications of versors and orthogonal transformations, spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces.

Gauge Formulation of Heaviside’s Equations

A primordial field Self-interaction Principle, analyzed in Hestenes’ Geometric Calculus, leads to Heaviside’s equations of the gravitomagnetic field. When derived from Einstein’s nonlinear field equations Heaviside’s “linearized” equations are known as the “weak field approximation”. When derived from the primordial field equation, there is no mention of field strength; the assumption that the primordial field was predominant at the big bang rather suggests that ultra-strong fields are governed by the equations. This aspect has physical significance, so we explore the assumption by formulating the gauge field version of Heaviside’s theory. We compare with recent linearized gravity formulations and discuss the significance of differences.

Geometric-algebra affine projection adaptive filter

Geometric algebra (GA) is an efficient tool to deal with hypercomplex processes due to its special data structure. In this article, we introduce the affine projection algorithm (APA) in the GA domain to provide fast convergence against hypercomplex colored signals. Following the principle of minimal disturbance and the orthogonal affine subspace theory, we formulate the criterion of designing the GA-APA as a constrained optimization problem, which can be solved by the method of Lagrange Multipliers. Then, the differentiation of the cost function is calculated using geometric calculus (the extension of GA to include differentiation) to get the update formula of the GA-APA. The stability of the algorithm is analyzed based on the mean-square deviation. To avoid ill-posed problems, the regularized GA-APA is also given in the following. The simulation results show that the proposed adaptive filters, in comparison with existing methods, achieve a better convergence performance under the condition of colored input signals.

MELITOPOL SCHOOL OF APPLIED GEOMETRY. HISTORY AND WORK EXPERIENCE

The work examines the history of formation, development and the main scientific works of the Melitopol School of Applied Geometry. The materials give a general characteristic of the main scientific directions of the school: variable discrete geometric modeling (VDGM) and point calculus Balyub-Naidish (BN calculus). The materials cover the list and peculiarities of scientific and organizational measures carried out by scientists of the school. The scientific school of applied geometry was founded by Meer Moiseevich Yuditsky in the 60s of the last century. The active personal, scientific and organizational activities of M. M. Yuditsky, the participation of scientists of the department in conferences and seminars of various levels, the defense of dissertations that took place in subsequent years gave the basis for the formation of a regional scientific center for applied geometry on the basis of the department. In 1977, Vladimir Mikhailovich Naydysh became the head of the school, which was called the Melitopol Scientific School of Applied Geometry. One of the significant results of the work of the school and its head personally is the holding in Melitopol of the annual International Scientific and Practical Conference "Modern Problems of Geometric Modeling" (held since 1994). The main scientific direction of the school is the development of the theory and applied aspects of discrete applied geometry. Since 2007, Andrei Vladimirovich Naydysh, Doctor of Technical Sciences, took over the leadership of the scientific school. Today, the basic institution of higher education for the school is the Bogdan Khmelnitsky Melitopol State Pedagogical University. Materials The modern state of the school, the level of its scientific developments, the personal achievements of the school's specialists give reason to consider the Melitopol School of Applied Geometry as having all the features of a scientific school by type - a school as a research team.

A Self-Stabilized Field Theory of Neutrinos

In “A Self-linking Field Formalism” I establish a self-dual field structure with higher order self-induced symmetries that reinforce the first-order dynamics. The structure was derived from Gauss-linking integrals in R3 based on the Biot-Savart law and Ampere’s law applied to Heaviside’s equations, derived in strength-independent fashion in “Primordial Principle of Self-Interaction”. The derivation involves Geometric Calculus, topology, and field equations. My goal in this paper is to derive the simplest solution of a self-stabilized solitonic structure and discuss this model of a neutrino.

Spacetime Geometry with Geometric Calculus

Geometric Calculus is developed for curved-space treatments of General Relativity and comparison is made with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein’s Principle of Equivalence is generalized to a gauge principle that provides the foundation for a new formulation of General Relativity as a Gauge Theory of Gravity on a curved spacetime manifold. Geometric Calculus provides mathematical tools that streamline the formulation and simplify calculations. The formalism automatically includes spinors so the Dirac equation is incorporated in a geometrically natural way.

Berezin integral as a limit of Riemann sum

Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann numbers are endowed with a second associative product coming from a Clifford algebra structure, we show how Berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus. We then show how the concepts of spinors and superspace transform into this framework.

Compressible Navier-Stokes Equations in Cylindrical Passages and General Dynamics of Surfaces—(I)-Flow Structures and (II)-Analyzing Biomembranes under Static and Dynamic Conditions

A new approach to solve the compressible Navier-Stokes equations in cylindrical co-ordinates using Geometric Algebra is proposed. This work was recently initiated by corresponding author of this current work, and in contrast due to a now complete geometrical analysis, particularly, two dimensionless parameters are now introduced whose correct definition depends on the scaling invariance of the N-S equations and the one parameter δ defines an equation in density which can be solved for in the tube, and a geometric Variational Calculus approach showing that the total energy of an existing wave vortex in the tube is made up of kinetic energy by vortex movement and internal energy produced by the friction against the wall of the tube. Density of a flowing gas or vapour varies along the length of the tube due to frictional losses along the tube implying that there is a pressure loss and a corresponding density decrease. After reducing the N-S equations to a single PDE, it is here proven that a Hunter-Saxton wave vortex exists along the wall of the tube due to a vorticity argument. The reduced problem shows finite-time blowup as the two parameters δ and α approach zero. A rearranged form for density is valid for δ approaching infinity for the case of incompressible flow proving positive for the existence of smooth solutions to the cylindrical Navier-Stokes equations. Finally we propose a CMS (Calculus of Moving Surfaces)–invariant variational calculus to analyze general dynamic surfaces of Riemannian 2-Manifolds in R 3 . Establishing fluid structures in general compressible flows and analyzing membranes in such flows for example flows with dynamic membranes immersed in fluid (vapour or gas) with vorticity as, for example, in the lungs there can prove to be a strong connection between fluid and solid mechanics.

Arithmetic summable sequence space over non-Newtonian field

In this article, we introduce the sequence spaces AS(G) and AC(G) of arithmetic summable and arithmetic convergent sequences, respectively, suggested by the geometric sum as k ranges over the divisors of m. We further obtain an analogous of Mobius inversion formula in the sense of geometric calculus and give interesting results in the geometric field.

How to choose the return model for market risk? Getting towards a right magnitude of stressed VaR

Value at Risk (VaR) and stressed value at Risk (SVaR) or expected shortfall are important risk measures widely used in the financial services industry for risk management and market risk capital computation. Fundamental to any (S)VaR model is the choice of the return type model for each risk factor. Because the resulting SVaR numbers are highly sensitive to the chosen return type model it is important to make a prudent choice on the return type modelling. We propose to estimate the return type model from historic data without making an a priori model assumption on the return model. We explain the fundamentals of return type modelling and how it impacts the magnitude of SVaR. We further show how to obtain a global return type model from a set of similar return type models by using geometric calculus. Numerical simulations and illustrations are provided. In this paper, we consider interest rate data, but the proposed methodology is general and can be applied to any other asset class such as inflation, credit spread, equity or fx.

Improvement and validation of Zamak die casting moulds

The automotive industry, like many other industries, uses a wide range of parts produced by the die-cast process. Parts like engine blocks, wheel spacers, alternator housings and command cable terminals, are made by die casting with different kinds of materials like aluminium and zinc alloys. Despite being a reliable process both in terms of quantity and quality, it is very important to keep the process parameters controlled, in order to achieve a minimum percentage of defective parts, which may be caused by several factors such as, porosities, segregations, incomplete fill, soldering, cracks, etc. The main goal of the die casting industry is to achieve the zero per cent defects target, a goal that goes along with the automotive industry and its quality system, and to accomplish this objective the stakeholders need to invest in research and development. In the casting industry, for instance, it is very important to have a complete knowledge of the entire process developed inside the casting machine, from the melting pot to the die, in order to obtain data so one can improve the filling parameters, machine parts, and moulds. The focus of the presented study is the improvement of the methodologies used to design moulds for control cable terminals in Zamak alloys. The work starts by characterizing the flow happening inside the mould at the moment of cavity fill by analysing computer fluid dynamics simulations (CFD). The study proceeds by quantifying the porosities detected on cut terminal surfaces, and the ultimate goal is achieved with the modification of molten metal flow systems, like channels and sprues, and the introduction of venting systems, with a resource to mathematical and geometrical calculus developed in MATLAB® specifically for that purpose. The paper ends with the validation of the improvements, by comparing the initial results with the ones obtained through an improved mould, building bases for novel design concepts of moulds for this kind of parts, as well as new studies trying to improve the results now achieved.

A finite element model for magnetohydrodynamic squeeze-film flows

A computational model is developed to analyze magnetohydrodynamic (MHD) squeeze-film flows featuring an electrically conducting fluid subjected to imposed magnetic and electric fields. The model is based on the so-called MHD Reynolds equation for squeeze-films—an extension of the classical hydrodynamic Reynolds equation. A complete derivation of the MHD Reynolds equation is performed by applying thin-film and quasi-steady assumptions to the Maxwell/Navier-Stokes system coupled by the Lorentz force. The resulting equation is a two-dimensional and variable-coefficient Poisson equation for pressure, which reduces to the purely hydrodynamic form in the limit of vanishing Hartmann number. A geometric calculus formulation facilitates the reduction of the mathematical system into two dimensions, which is a challenge in standard vector calculus due to the cross product. The model permits realistic geometrical representations of the constraining squeeze-surfaces, and we demonstrate the use of a multi-variate Weierstrass-Mandelbrot fractal to numerically generate scale-invariant surface roughness profiles. Ultimately, the governing equation is solved with the Galerkin finite element method. Several numerical examples are conducted to highlight some of the model’s capabilities. MHD forces—as well as the roughness, geometry, and topology of the squeeze surfaces—are shown to significantly influence flow characteristics.

Matrices in computer graphics

In this article, the usage of matrices and matrix operations in computer graphics is shown. A brief overview of geometric transformations in computer graphics is given. The "Matrix - Computer Graphics" application software is created for the representation and easier understanding of relations between geometric transformations and matrix calculus.

How to Choose the Return Model for Market Risk? Getting Towards a Right Magnitude of Stressed VAR

Value at Risk (VaR) and stressed value at Risk (SVaR) or expected shortfall are important risk measures widely used in the financial services industry for risk management and market risk capital computation. Fundamental to any (S)VaR model is the choice of the return type model for each risk factor. Because the resulting SVaR numbers are highly sensitive to the chosen return type model it is important to make a prudent choice on the return type modelling. We propose to reconstruct the return type model from historic data without making an a priori model assumption on the return model. We explain the fundamentals of return type modelling and how it impacts magnitude of SVaR. We further show how to obtain a global return type model from a set of similar return type models by using geometric calculus. Numerical simulations and illustrations are provided. In this paper the focus is on interest rates, but the proposed methodology is general and can be applied to any other asset class such as inflation, credit spread, equity or fx.