Algebraic Solution Research Articles

Algebraic entropy for systems of quad equations

In this work I discuss briefly the calculation of the algebraic entropy for systems of quad equations. In particular, I observe that since systems of multilinear equations can have algebraic solution, in some cases one might need to restrict the direction of evolution only to the pair of vertices yielding a birational evolution. Some examples from the exiting literature are presented and discussed within this framework.

Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation

It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$_7$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation.

Fault-prognosability, [formula omitted]-step prognosis and [formula omitted]-step predictive diagnosis in partially observed petri nets by means of algebraic techniques

In this paper, we explore some issues relevant to fault monitoring in discrete event systems modeled by partially observed LPNs with, possibly indistinguishable observable events and acyclic unobservable subnet. Firstly, we address the (offline) fault-prognosability analysis problem. Subsequently, we tackle the two online problems of K-step fault prognosis and K-step predictive diagnosis. We propose algebraic formulations and solutions to these problems. Namely, a necessary and sufficient condition for fault-prognosability is established based on solving an integer optimization problem. The proposed approach is applicable for bounded Petri nets. As for the K-step prognosis and K-step predictive diagnosis, algebraic approaches based on state estimation on a sliding horizon are elaborated to produce relevant verdicts. The established results for K-step prognosis and K-step predictive diagnosis are applicable for both bounded and unbounded Petri nets.

The Perturbed Chen–Lee–Liu Equation: Diverse Optical Soliton Solutions and Other Wave Solutions

The objective of the present study is to extract the optical soliton solutions (OSSs) of the perturbed Chen–Lee–Liu equation by exerting three techniques, which are the extended Wang’s direct mapping method, tanh/coth function method and the Subequation method. Various OSSs, including the dark, bright, and singular soliton solutions, as well as other wave solutions, such as the singular periodic and algebraic solitary wave solutions, are successfully found. By assigning the appropriate parameters, the dynamic characteristics of the attained solutions are depicted graphically via the Mathematica software. The outcomes confirm that the methods are highly effective for constructing the OSSs and can also be adopted to investigate the other PDEs in optics.

A Feature-Space Theory of the Production Effect in Recognition.

Mathematical models explaining production effects assume that production leads to the encoding of additional features, such as phonological ones. This improves memory with a combination of encoding strength and feature distinctiveness, implementing aspects of propositional theories. However, it is not clear why production differs from other manipulations such as study time and spaced repetition, which are also thought to influence strength. Here we extend attentional subsetting theory and propose an explanation based on the dimensionality of feature spaces. Specifically, we suggest phonological features are drawn from a compact feature space. Deeper features are sparsely subselected from a larger subspace. Algebraic and numerical solutions shed light on several findings, including the dependency of production effects on how other list items are encoded (differing from other strength factors) and the production advantage even for homophones. This places production within a continuum of strength-like manipulations that differ in terms of the feature subspaces they operate upon and leads to novel predictions based on direct manipulations of feature-space properties.

Algebraic Solution of the Quantum Harmonic Oscillator

The Wave-particle duality and the Schrodinger cat paradox pose important notions on quantum mechanical systems, in which a quantum system can exist in two physically opposite or distinct states. This motivates scientists from the days of Louis de Broglie to confidently look at matter as having a dual nature, which is a physical particle that can also be visualized as a wave entity. This paper seeks to provide an elegant method of solving the time-independent Schrodinger Equation (TISE) for a quantum simple harmonic oscillator (QSHO). A fully algebraic method is presented, the paper explains in detail how simple algebra, coupled with basic quantum mechanical operator commutation algebra can be employed to solve the TISE without resorting to complicated mathematical methods of solving second-order differential equations. The construction of the Hamiltonian is explained and an analogy between the classical simple harmonic oscillator (CSHO) and the QSHO is drawn to develop the solution method. The action of creation and annihilation operators on wave functions is exploited to generate states with higher and lower energies respectively and to generate a Hamiltonian that can act on a system and generate required quantum states. The aim is to generate the ground state of the QSHO and its associated energy value. From this state, other higher energy states can be generated by the action of the creation operator. The use of algebraic methods has been seen to be versatile as it does not explicitly rely on any particular basis or coordinate system. The method allows operators to be manipulated in different bases for example the momentum operator can be given as P ̂ x=-iℏ (∂/∂x) in a spatial coordinate basis or as a pure number p ̂ =p in the momentum basis.

Algebraic Solution of Gaunt Coefficients via the Angular Momentum Ladder Operators

In this study, Gaunt coefficients, frequently encountered in quantum mechanical calculations of atomic and molecular structures, have been algebraically derived. Firstly, the Gaunt coefficient, equal to the integral over the solid angle of the product of three spherical harmonics, is written in terms of angular momentum ladder operators. Subsequently, raising or lowering operators are applied to spherical harmonics, and the obtained integrals are solved using the recurrence and orthogonality relations of spherical harmonics. As a result, algebraic expressions for Gaunt coefficients are obtained in terms of quantum numbers.

Existence of the solutions to the Brocard–Ramanujan problem for norm forms

The Brocard–Ramanujan problem, which is an unsolved problem in number theory, is to find integer solutions ( x , ℓ ) (x,\ell ) of x 2 − 1 = ℓ ! x^2-1=\ell ! . Many analogs of this problem are currently being considered. As one example, it is known that there are at most only finitely many algebraic integer solutions ( x , ℓ ) (x, \ell ) , up to a unit factor, to the equations N K ( x ) = ℓ ! N_K(x) = \ell ! , where N K N_K are the norms of number fields K / Q K/\mathbf Q . In this paper, we construct infinitely many number fields K K such that N K ( x ) = ℓ ! N_K(x) = \ell ! has at least 22 22 solutions for positive integers ℓ \ell .

An improved model for predicting froth recovery

In froth flotation, froth recovery is a key factor as it quantifies the effect that the froth has on the overall cell recovery. Froth recovery is affected by a number of variables, including froth stability, particle size and density, and the operating conditions of the flotation cell. Despite its importance, modelling froth recovery is challenging due to the large number of competing effects involved. In this paper, we present a model for froth recovery that takes into account the effects of detachment and subsequent transport of particles. The model is phenomenological and is derived from a set of coupled partial differential equations used to describe both liquid and particle motion within a froth. An analytical algebraic solution is obtained for the froth recovery by making a number of assumptions, which are discussed and justified in the paper. This model is related to a previous model produced by the authors, in which froth recovery was described in terms of a constant particle detachment fraction. Conversely, this enhanced model includes an improved description for particle detachment behaviour during coalescence events, allowing for a variable particle detachment fraction. The key parameters in the model are two dimensionless groups. The first group combines the particle settling velocity, the air addition rate and froth stability parameters such as air recovery and the change in bubble size over the froth, while the second key dimensionless group is the ratio of the surface loading on the incoming bubbles to the maximum surface loading. This model is compared to experimental results and is shown to accurately describe the evolution of the froth recovery behaviour down a bank of cells in response to the decreasing froth stability.

Optimization of Weight Matrices for the Linear Quadratic Regulator Problem Using Algebraic Closed-Form Solutions

This work proposes an analytical gradient-based optimization approach to determine the optimal weight matrices that make the state and control input at the final time close to zero for the linear quadratic regulator problem. Most existing methodologies focused on regulating the diagonal elements using only bio-inspired approaches or analytical approaches. The method proposed, contrarily, optimizes both diagonal and off-diagonal matrix elements based on the gradient. Moreover, by introducing a new variable composed of the steady-state and time-varying terms for the Riccati matrix and using the coordinate transformation for the state, one develops algebraic equationsbased closed-form solutions to generate the required states and numerical partial derivatives for an optimization strategy that does not require the computationally intensive numerical integration process. The authors test the algorithm with one- and two-degrees-of-freedom linear plant models, and it yields the weight matrices that successfully satisfy the pre-defined requirement, which is the norm of the augmented states less than 10−5. The results suggest the broad applicability of the proposed algorithm in science and engineering.

Distributed Algorithms for Boolean Equations Over Networks

In this paper, we study systems of Boolean equations over a network, where each node in the network possesses only one Boolean equation from the system. The Boolean equation assigned at any particular node is a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">private</i> equation known to this node only, and the aim of the paper is to develop distributed algorithms that allow all the nodes to obtain solutions to the network Boolean equations without exchanging their local Boolean equations. First, we observe that the Boolean equations can be locally lifted to a system of linear algebraic equations under a basis of Boolean vectors, which is distributedly solvable using existing distributed linear equation algorithms as a subroutine. Next, we construct a distributed Boolean equation solver by the nodes solving the lifted linear network equation for a number of randomly selected initial values, and then converting the algebraic solutions into solutions to the original Boolean equations by a novel Boolean vector search algorithm. We prove that for solvable Boolean equations, when the initial values of the nodes for the distributed linear equation solving step are i.i.d selected according to a uniform distribution in a high-dimensional cube, such an algorithm returns the exact solution set of the Boolean equations at each node with high probability. We also present an algorithm for distributed verification of the satisfiability of Boolean equations when the solvability is not known beforehand, and prove its correctness. Finally, we show that by utilizing linear equation solvers with differential privacy to replace the in-network computing routines, the distributed Boolean equation algorithms can be made differentially private.

On Abel’s Problem and Gauss Congruences

Abstract A classical problem due to Abel is to determine if a differential equation $y^{\prime}=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta $ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta $, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta $ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty \}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y^{\prime}=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta (x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y^{\prime}=\eta y$ with an algebraic solution when $x\eta (x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks.

On the existence and generation of non- trivial BIBDs

A balanced incomplete block design (BIBD) is an efficient tool to infer parameters of a linear model. With a BIBD treatments are compared in blocks of size A variety of algebraic methods were developed to determine BIBDs depending on the parameter pair (v,k). Despite the progress in the field, there are many pairs (v,k) without an algebraic solution. For such pairs it is solely known, that a trivial BIBD exists, which is commonly large. In this study we present new results on the existence and generation of BIBDs which are smaller than the trivial ones. Such so-called non- trivial BIBDs exist if a trivial BIBDs is not elementary. With a recursive method of BIBD- generation it is shown that for and there exist non- trivial BIBDs. The same is shown for and Numerical programs were written for the generation of new non- trivial BIBDs, which delivered smaller BIBDs than the recursive method for Commonly used R-programs like “crossdes” and “ibd” failed to determine such BIBDs.

Joint Moving Target and Antenna Localization for Distributed MIMO Radar With a Calibration Object

This paper investigates the problem of simultaneously locating the moving target and refining erroneous antenna parameters with a calibration object in distributed multiple-input multiple-out (MIMO) radars. Albeit traditional methods achieve moving target localization with antenna parameter errors, they still have some disadvantages, such as the high computational complexity of joint localization or the high noise sensitivity of estimating the target parameters. To overcome the drawbacks of existing algorithms, we first formulate a new localization model that joint target and antenna localization with a calibration object and then present an algebraic solution. Unlike previous work, the proposed estimator has additional unknowns of antenna parameter errors and presents new matrix equations by incorporating the calibration measurements. The proposed method is shown theoretically to attain the Cramer–Rao lower bound (CRLB) accuracy both in theory and simulations under small measurement noise and antenna error levels. Furthermore, simulation results show that the proposed method significantly improves the localization performance compared with existing methods.

Algebraic Solution of Tropical Best Approximation Problems

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

Top-down metacomputing with algebraic dimensionality raising for automating theory-building to enable directly computable multiphysics models

The need to address the fundamental issues associated with the computational model generation process in the context of multiphysics theories that observe conservation laws and relevant assumptions motivates the present work. Accordingly, the goal of this paper is to describe, discuss, and demonstrate the benefits and implications of using metacomputing (i.e., the utilization of computational technologies for computing or generating programs that, in turn, compute the desired results) for theory or model generation, further assisted by metacomputing for raising the semantic and syntactic dimensionality of physics models for continua. The issues associated with the model generation process are identified at the outset. Then, a top-down approach that contains three distinct metacomputing activities is described. First, a meta-theoretical description of generating continuum multiphysics models defined in terms of quantities instantiated from the field of real numbers is described, along with its computational implementation. Then, a computational framework for raising the semantic dimensionality from the reals to the hypercomplex numbers is presented for both 2D and 3D continuum multiphysics problems. The semantic dimensionality is expressed in terms of the order of the algebra used to define the tensorial quantities of interest and their associated operations, as they participate in representing physics models expressed initially as systems of partial differential equations. The utilization of hypercomplex algebras, such as those for complex numbers and quaternions, is demonstrated for representative applications of models. Subsequently, syntactic dimensionality is defined in terms of the basis of the space utilized to write equational systems. Finally, a prototype of a computational system for converting equational systems to directed graphs termed “Algebraic Solution Graphs” is described. Examples of how to utilize these technologies are also presented to convey the efficacy and efficiency of the proposed approach.

Metacomputing for Directly Computable Multiphysics Models

Abstract When it comes to multiphysics modeling and simulation, the ever-improving advances of computational technologies have forced the user to manage higher resource complexity while at the same time they are motivating the modeling of more complex systems than before. Consequently, the time for the user’s iterations within the context space characterizing all choices required for a successful computation far exceeds the time required for the runtime software execution to produce acceptable results. This paper presents metacomputing as an approach to address this issue, starting with describing this high-dimensional context space. Then it highlights the abstract process of multiphysics model generation/solution and proposes performing top-down and bottom-up metacomputing. In the top-down approach, metacomputing is used for automating the process of generating theories, raising the semantic dimensionality of these theories in higher dimensional algebraic systems that enable simplification of the equational representation, and raising the syntactic dimensionality of equational representation from 1D equational forms to 2D and 3D algebraic solution graphs that reduce solving to path-following. In the bottom-up approach, already existing legacy codes evolving over multiple decades are encapsulated at the bottom layer of a multilayer semantic framework that utilizes category theory based operations on specifications to enable the user to spend time only for defining the physics of the relevant problem and not have to deal with the rest of the details involved in deploying and executing the solution of the problem at hand. Consequently, these two metacomputing approaches enable the automated generation, composition, deployment, and execution of directly computable multiphysics models.

An effects of mass transpiration and inclined MHD on nanoboundary layer of an ostwald-de waele fluid due to a shrinking boundary

This paper examines the power-law nanofluid flows due to a shrinking sheet with mass transpiration under inclined MHD. The governing partial differential equations are similarly transformed into nonlinear ordinary differential equations and solved analytically. The mathematical model through analytical solutions graphically shows that the flow's dynamics depending on the Chandraselhar number, mass transpiration, nanofluid and inclined angle parameters. The main findings and methods of the present work includes the closed-form exact solutions are examined for some special cases and one of them is an algebraic solution. Specific and quantitative analytical solutions provide important understanding to the boundary layer flow for power-law nanofluid. The present study reveals that the magnetic field significantly increases the magnitude of the skin friction and mass transpiration. Cu-H2O nanofluid is used to examine the entaire calculation. Nanofluids are suspensions of nanoparticles in fluids that show significant enhancement of their properties at modest nanoparticle concentration.

Algebraic solution for aplanatic cemented doublets with aspherical surfaces of second degree.

An algebraic method of an initial design of the aplanatic cemented doublet with a unit focal length is described. Based on the third-order aberration theory, algebraic equations are derived for calculating the design parameters of an aplanatic thin cemented doublet with spherical surfaces and aspherical surfaces of the second degree. These equations make it easy to determine whether there exists a real solution of the doublet design for required parameters. The results of the calculations of selected cemented doublet solutions are presented in the examples. The obtained parameters can be used as the starting solution for the final optimization of doublet designs.

Underwater Wireless Sensor Network-Based Localization Method under Mixed Line-of-Sight/Non-Line-of-Sight Conditions

Source localization in underwater sensor networks (UWSNs) presents complex challenges due to sensor nodes drift caused by ocean currents, non-line-of-sight (NLOS) propagation resulting from underwater multipath effects, and environmental noise. This paper proposes a practical and innovative algebraic solution based on the time difference of arrival (TDOA) for source localization in shallow seas. The proposed solution effectively addresses the issues arising from sensor position errors and multipath effects by incorporating the sea-surface reflection non-line-of-sight (SNLOS) link and optimizing the algorithm, thereby significantly improving positioning accuracy. The core concept of the method involves utilizing the weighted least squares algorithm to obtain an initial estimate of the source position, followed by direct estimation of the bias and subsequent refinement of the solution. In contrast to traditional closed-form solutions, this method avoids the introduction of intermediate parameters and directly handles the estimated bias from the previous step. Even when only considering the line-of-sight (LOS) link, the proposed solution achieves precise localization with a minimal number of sensors. Theoretical analysis demonstrates that the solution can achieve the Cramér–Rao lower bound (CRLB) accuracy under low noise conditions, and simulation results validate the superior performance of the proposed solution.