Quantum Computingvia Entanglement in Geometric Algebra Approach

The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers would only provide dramatic speedups for a few specific problems, for example, factoring integers and breaking cryptographic codes in the conventional quantum computing approach. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In the conventional approach, it is implemented through the tensor product of qubits. In the suggested geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on a three-dimensional sphere, which is very different from the usual Hilbert space scheme.

Singularity Analysis of a 3-RPS Parallel Manipulator Using Geometric Algebra

Singular configurations must be avoided in path planning and control of a parallel manipulator. However, most studies rarely focus on an overall singularity loci distribution of lower-mobility parallel mechanisms. Geometric algebra is employed in analysis of singularity of a 3-RPS parallel manipulator. Twist and wrench in screw theory are represented in geometric algebra. Linear dependency of twists and wrenches are described by outer product in geometric algebra. Reciprocity between twists and constraint wrenches are reflected by duality. To compute the positions of the three spherical joints of the 3-RPS parallel manipulator, Tilt-and-Torsion angles are used to describe the orientation of the moving platform. The outer product of twists and constraint wrenches is used as an index for closeness to singularity（ICS） of the 3-RPS parallel manipulator. An overall and thorough perspective of the singularity loci distribution of the 3-RPS parallel manipulator is disclosed, which is helpful to design, trajectory planning and control of this kind of parallel manipulator.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry)

This article refers to the “Mathematics of Harmony” by Alexey Stakhov in 2009, a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries–New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry (λ > 0 is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scientific ideas-the “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—the “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

Research Progress of the Algebraic and Geometric Signal Processing

The investigation of novel signal processing tools is one of the hottest research topics in modern signal processing community. Among them, the algebraic and geometric signal processing methods are the most powerful tools for the representation of the classical signal processing method. In this paper, we provide an overview of recent contributions to the algebraic and geometric signal processing. Specifically, the paper focuses on the mathematical structures behind the signal processing by emphasizing the algebraic and geometric structure of signal processing. The two major topics are discussed. First, the classical signal processing concepts are related to the algebraic structures, and the recent results associated with the algebraic signal processing theory are introduced. Second, the recent progress of the geometric signal and information processing representations associated with the geometric structure are discussed. From these discussions, it is concluded that the research on the algebraic and geometric structure of signal processing can help the researchers to understand the signal processing tools deeply, and also help us to find novel signal processing methods in signal processing community. Its practical applications are expected to grow significantly in years to come, given that the algebraic and geometric structure of signal processing offer many advantages over the traditional signal processing.

A Novel Signal Processing Coprocessor for <i>n-</i>Dimensional Geometric Algebra Applications

This paper provides an implementation of a novel signal processing co-processor using a Geometric Algebra technique tailored for fast and complex geometric calculations in multiple dimensions. This is the first hardware implementation of Geometric Algebra to specifically address the issue of scalability to multiple (1 - 8) dimensions. This paper presents a detailed description of the implementation, with a particular focus on the techniques of optimization used to improve performance. Results are presented which demonstrate at least 3x performance improvements compared to previously published work.

Scattering of Geometric Algebra Wave Functions and Collapse in Measurements

The research considers wavelike objects that are elements of even subalgebra of geometric algebra in three dimensions. The used formalism particularly eliminates long existing confusion about the reasons behind the appearance of the imaginary unit in quantum mechanics and introduces clear definition of wave functions. When a wave function acts through the Hopf fibration on a localized geometric algebra element, that is executing a measurement, the result can be named as “collapse” of the wave function.

ON JUSTESEN'S ALGEBRAIC GEOMETRY CODES

An isomorphism preserving Hamming distance between two algebraic geometry(AG)codes is presented to obtain the main parameters of Justesen’s algebraic geometry(JAG)codes.To deduce a simple approach to the decoding algorithm,a code word in a“small”JAG codeis used to correspond to error-locator polynomial.By this means,a simple decoding procedureand its ability of error correcting are explored obviously.The lower and upper bounds of thedimension of AG codes are also obtained.

Fractal Geometry:Axioms,Fractal Derivative and Its Geometrical Meaning

Physics success is largely determined by using mathematics.Physics often themselves create the necessary mathematical apparatus.This article shows how you can construct a fractal calculus-mathematics of fractal geometry.In modem scientific literature often write from a firm that"there is no strict definition of fractals",to the more moderate that"objects in a certain sense,fractal and similar."We show that fractal geometry is a strict mathematical theory,defined by their axioms.This methodology allows the geometry of axiomatised naturally define fractal integrals and differentials.Consistent application on your input below the axiom gives the opportunity to develop effective methods of measurement of fractal dimension,geometri-cal interpretation of fractal derivative gain and open dual symmetry.

Solving Algebraic Problems with Geometry Diagrams Using Syntax-Semantics Diagram Understanding

Solving Algebraic Problems with Geometry Diagrams(APGDs)poses a significant challenge in artificial intelligence due to the complex and diverse geometric relations among geometric objects.Problems typically involve both textual descriptions and geometry diagrams,requiring a joint understanding of these modalities.Although considerable progress has been made in solving math word problems,research on solving APGDs still cannot discover implicit geometry knowledge for solving APGDs,which limits their ability to effectively solve problems.In this study,a systematic and modular three-phase scheme is proposed to design an algorithm for solving APGDs that involve textual and diagrammatic information.The three-phase scheme begins with the application of the statetransformer paradigm,modeling the problem-solving process and effectively representing the intermediate states and transformations during the process.Next,a generalized APGD-solving approach is introduced to effectively extract geometric knowledge from the problem’s textual descriptions and diagrams.Finally,a specific algorithm is designed focusing on diagram understanding,which utilizes the vectorized syntax-semantics model to extract basic geometric relations from the diagram.A method for generating derived relations,which are essential for solving APGDs,is also introduced.Experiments on real-world datasets,including geometry calculation problems and shaded area problems,demonstrate that the proposed diagram understanding method significantly improves problem-solving accuracy compared to methods relying solely on simple diagram parsing.

Valid Geometric Solutions for Indentations with Algebraic Calculations

This paper analyzes the force vs depth loading curves of conical, pyramidal, wedged and for spherical indentations on a strict mathematical basis by explicit use of the indenter geometries rather than on still world-wide used iterated “contact depths” with elastic theory and violation of the energy law. The now correctly analyzed loading curves provide as yet undetectable phase-transition. For the spherical indentations, this includes an obvious correction for the varying depth/radius ratio, which had previously been disregarded. Only algebraic formulas are now used for the calculation of material’s properties without data-fittings, or simplifications, or false simulations. Penetration resistance differences of materials’ polymorphs provide precise intersection values as kink unsteadiness by equalization of linear regression lines from mathematically linearized loading curves. These intersections indicate phase transition onset values for depth and force. The precise and correct determination of phase-transition onsets allows for energy and phase-transition energy calculations. The unprecedented algebraic equations are most simply and mathematically reproducibly deduced. There are no restrictions for elastic and/or plastic behavior and no use of different formulas for different force ranges. The novel indentation formulas reveal unprecedented access to the onset, energy and transition energy of phase-transitions. This is now also achieved for spherical indentations. Their formula as deduced for plotting is reformulated for integrations. The distinction of applied work (Wapplied) and indentation work (Windent) allows now for comparing spherical with pyramidal indentation phase-transitions. Only low energy phase-transitions from pyramidal indentation may be missed in spherical indentations. The rather low penetration depths of sphere calottes calculate very close for cap and flat area values. This allows for the calculation of the indentation phase-transition onset pressure and thus the successful comparison with hydrostatic anvil pressurizing results. This is very helpful for their interpretations, as low energy phase-transitions are often missed under the anvil, and it further strengthens the unparalleled ease of the indentation techniques. Exemplification is reported for pyramidal, spherical, and hydrostatic anvil stressing by the numerical analysis of published germanium data. The previous widely accepted historical indentation theories and standards are challenged. Falsely simulated and even published so-called “experimental” indentation data from the literature can most easily be checked. They are mathematically unsound and their correction is urgently necessary for scientific reasons and daily safety with stressed materials. The motivation for this paper is the challenge of worldwide incorrect ISO 14577 standards for false and incomplete characterization of materials. The minimization of catastrophic failures e.g. in aviation requires the strengthening and the advancements of the mathematical truth by using our closed formulas that are based on undeniable geometric and algebraic calculation rules.

ON THE DECODING OF ALGEBRAIC GEOMETRIC CODES BASED ON FIA

Suppose C is an irreducible algebraic curve of genus g, C*(D,G) is an algebraic geometric code with designed minimum distance d* = deg(G)-2g + 2. In this paper, a decoding algorithm based on Fundamental Iterative Algorithm(FIA) is presented, also its reasonableness is proved. In fact, our decoding algorithm is a modification of the algorithm proposed by G. L. Fend and T. R. N. Rao(1993) and can correct any received words with errors not more than (d*-1)/2, whereas the complexity is only about one half as much as Feng and Rao’s. The procedure can be implemented easily by hardware or software.

TOPOLOGICAL AND GEOMETRIC PROPERTY OF MATRIX ALGEBRA

Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group and also a smooth hypersurface in B（R^n） with the dimension n × n.

Investigation of pore geometry influence on fluid flow in heterogeneous porous media:A pore-scale study

Brazilian pre-salt reservoirs are renowned for their intricate pore networks and vuggy nature,posing significant challenges in modeling and simulating fluid flow within these carbonate reservoirs.Despite possessing excellent petrophysical properties,such as high porosity and permeability,these reservoirs typically exhibit a notably low recovery factor,sometimes falling below 10%.Previous research has indicated that various enhanced oil recovery(EOR)methods,such as water alternating gas(WAG),can substantially augment the recovery factor in pre-salt reservoirs,resulting in improvements of up to 20%.Nevertheless,the fluid flow mechanism within Brazilian carbonate reservoirs,characterized by complex pore geometry,remains unclear.Our study examines the behavior of fluid flow in a similar heterogeneous porous material,utilizing a plug sample obtained from a vugular segment of a Brazilian stromatolite outcrop,known to share analogies with certain pre-salt reservoirs.We conducted single-phase and multi-phase core flooding experiments,complemented by medical-CT scanning,to generate flow streamlines and evaluate the efficiency of water flooding.Subsequently,micro-CT scanning of the core sample was performed,and two cross-sections from horizontal and vertical plates were constructed.These cross-sections were then employed as geometries in a numerical simulator,enabling us to investigate the impact of pore geometry on fluid flow.Analysis of the pore-scale modeling and experimental data unveiled that the presence of dead-end pores and vugs results in a significant portion of the fluid remaining stagnant within these regions.Consequently,the injected fluid exhibits channeling-like behavior,leading to rapid breakthrough and low areal swept efficiency.Additionally,the numerical simulation results demonstrated that,irrespective of the size of the dead-end regions,the pressure variation within the dead-end vugs and pores is negligible.Despite the stromatolite's favorable petrophysical properties,including relatively high porosity and permeability,as well as the presence of interconnected large vugs,the recovery factor during water flooding remained low due to early breakthrough.These findings align with field data obtained from pre-salt reservoirs,providing an explanation for the observed low recovery factor during water flooding in such reservoirs.

Polarizations as States and Their Evolution in Geometric Algebra Terms with Variable Complex Plane

Recently suggested scheme?[1] of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors.

Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry

The space of internal geometry of a model of a real crystal is supposed to be finite, closed, and with a constant Gaussian curvature equal to unity, permitting the realization of lattice systems in accordance with Fedorov groups of transformations. For visualizing computations, the interpretation of geometrical objects on a Clifford surface (SK) in Riemannian geometry with the help of a 2D torus in a Euclidean space is used. The F-algorithm ensures a computation of 2D sections of models of point systems arranged perpendicularly to the symmetry axes l3, l4, and l6. The results of modeling can be used for calculations of geometrical sizes of crystal structures, nanostructures, parameters of the cluster organization of oxides, as well as for the development of practical applications connected with improving the structural characteristics of crystalline materials.

Strapdown Navigation Using Geometric Algebra: Screw Blade Algorithm

The rigid body motion can be represented by a motor in geometric algebra, and the motor can be rewritten as a trinometric function of the screw blade. In this paper, a screw blade strapdown inertial navigation system (SDINS) algorithm is developed. The trigonometric function form of the motor is derived and utilized to deduce the Bortz equation of the screw blade. The screw blade SDINS algorithm is proposed by using the procedure similar to that of the conventional rotation vector attitude updating algorithm. The superiority of the screw blade algorithm over the conventional ones in precision is analyzed. Simulation results reveal that the screw blade algorithm is more suitable for the high-pre- cision SDINS than the conventional ones.

THE BSE PROPERTY FOR SOME VECTOR-VALUED BANACH FUNCTION ALGEBRAS

In this paper,X is a locally compact Hausdorff space and A is a Banach algebra.First,we study some basic features of C0(X,A)related to BSE concept,which are gotten from A.In particular,we prove that if C0(X,A)has the BSE property then A has so.We also establish the converse of this result,whenever X is discrete and A has the BSE-norm property.Furthermore,we prove the same result for the BSE property of type I.Finally,we prove that C0(X,A)has the BSE-norm property if and only if A has so.

Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L∞(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L2(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .

A Generalized Array Factor for Time-Modulated Hexagonal Based Antenna Array Geometry With Novel Trapezoidal Switching

The concept of the time-modulated array has been emerging as an alternative to the complex phase shifters,which lowers the cost of the array feeding network due to the utilization of radio frequency(RF)switches.The various forms of hexagonal antenna array geometries can be used for applications like surveillance tracking in phased array radar and wireless communication systems.This work proposes the generalized array factor(AF)for the hexagonal antenna array geometry based on time modulation.The time modulation in generalized hexagonal geometry can maintain the fixed static amplitude excitation,giving more flexibility over time.Furthermore,a novel trapezoidal switching function is also proposed and applied to the generalized array factor to enable future researchers to use this array factor in the field of advancement to observe how switching schemes like trapezoidal and rectangular affect the array pattern's side lobe level(SLL).The generalized equation can be utilized for the analysis and synthesis of radiation characteristics of the time-modulated hexagonal array(TMHA),time-modulated concentric hexagonal array(TMCHA),time-modulated hexagonal cylindrical array(TMHCA),and time-modulated hexagonal concentric cylindrical array(TMHCCA).The numerical result illustrates the generation of AF of time-modulated hexagonal structures and also shows that the trapezoidal switching sequence outperforms the rectangular switch using the cat swarm optimization(CSO)approach.