Algebra Equations Research Articles

A note on the paper ‘Ultra discrete permanent and the consistency of max plus linear equations’

This work is concerned with the consistency conditions for the equations in max plus algebra. The three classes of the max plus linear equations presented in the paper Shinzawa [Ultra discrete permanent and the consistency of max plus linear equations, Linear Algebra Appl. 506 (2016) 445–477] can be equivalently converted into the corresponding system of inequalities. The equivalence relation between ultra discrete permanent and maximum cycle mean is suggested. Thus, the necessary and sufficient conditions for solvability of the equations are obtained using the maximum cycle mean in this note.

A Method to Translate Order-Sorted Algebras to Many-Sorted Algebras

Order-sorted algebras and many sorted algebras exist in a long history with many different implementations and applications. A lot of language specifications have been defined in order-sorted algebra frameworks such as the language specifications in K (an order-sorted algebra framework). The biggest problem in a lot of the order-sorted algebra frameworks is that even if they might allow developers to write programs and language specifications easily, but they do not have a large set of tools to provide reasoning infrastructures to reason about the specifications built on the frameworks, which are very common in some many-sorted algebra framework such as Isabelle/HOL, Coq and FDR. This fact brings us the necessity to marry the worlds of order-sorted algebras and many sorted algebras. In this paper, we propose an algorithm to translate a strictly sensible order-sorted algebra to a many-sorted one in a restricted domain by requiring the order-sorted algebra to be strictly sensible. The key idea of the translation is to add an equivalence relation called core equality to the translated many-sorted algebras. By defining this relation, we reduce the complexity of translating a strictly sensible order-sorted algebra to a many-sorted one, make the translated many-sorted algebra equations only increasing by a very small amount of new equations, and keep the number of rewrite rules in the algebra in the same amount. We then prove the order-sorted algebra and its translated many-sorted algebra are bisimilar. To the best of our knowledge, our translation and bisimilar proof is the first attempt in translating and relating an order-sorted algebra with a many-sorted one in a way that keeps the size of the translated many-sorted algebra relatively small.

Purely Hom-Lie bialgebras

In this paper, first we show that there is a Hom-Lie algebra structure on the set of $(\sigma,\sigma)$-derivations of a commutative algebra. Then we construct dual representations of a representation of a Hom-Lie algebra. We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some special Hom-Lie algebras using Hom-$\mathcal O$-operators.

An educational tool for introducing power engineering students with modal extraction concepts

AbstractUndergraduate power engineering students find difficult to apply modal analysis to characterize the dynamical behavior of a power system, although they know some basic concepts of modal analysis from courses such as linear algebra and differential equations. This paper presents an educational software tool for the identification of dominant oscillatory modes from time domain responses. It is aimed to help power engineering systems students to understand and to gain experience on the application of modal analysis for assessing the system dynamics from measured signals of a power system, in terms its dynamic properties such as oscillation frequency and its associated damping.

Universal solvability of interval max-plus matrix equations

This paper deals with the solvability of interval matrix equations in max-plus algebra. Max-plus algebra is the algebraic structure in which classical addition and multiplication are replaced by ⊕ and ⊗, where a⊕b=max{a,b} and a⊗b=a+b.The notation A⊗X⊗C=B, where A, B, and C are given interval matrices, represents an interval max-plus matrix equation. We define three types of solvability of interval max-plus matrix equations, namely the strong universal , universal , and weak universal solvability. We derive the necessary and sufficient conditions which can be verified in polynomial times.

A UNIFIED ANALYSIS OF LINEAR QUATERNION DYNAMIC EQUATIONS ON TIME SCALES

Over the last years, considerable attention has been paid to the role of the quaternion differential equations (QDEs) which extend the ordinary differential equations. The theory of QDEs was recently well established and it has wide applications in physics and life science. This paper establishes a systematic frame work for the theory of linear quaternion dynamic equations on time scales (QDETS), which can be applied to wave phenomena modeling, fluid dynamics and filter design. The algebraic structure of the solutions to the QDETS is actually a left- or right- module, not a linear vector space. On the non-commutativity of the quaternion algebra, many concepts and properties of the classical dynamic equations on time scales (DETS) can not be applied. They should be redefined accordingly. Using $q$-determinant, a novel definition of Wronskian is introduced under the framework of quaternions which is different from the standard one in DETS. Liouville's formula for QDETS is also analyzed. Upon these, the solutions to the linear QDETS are established. The Putzer's algorithms to evaluate the fundamental solution matrix for homogeneous QDETS are presented. Furthermore, the variation of constants formula to solve the nonhomogeneous QDETs is given. Some concrete examples are provided to illustrate the feasibility of the proposed algorithms.

Fracture analysis on an arc-shaped interfacial crack between a superconducting cylinder and its functionally graded coating with transport currents

In this study, an arc-shaped crack between a homogeneous superconducting cylinder and its functionally graded coating is investigated subjected to transport currents. The extended generalized Irie-Yamafuji critical state model in the coating is put forward. The distributions of both current and flux densities in the superconducting composite are presented for the processes of both transport current ascent and descent. The stress intensity factors and energy release rates at crack tips are further given in concise forms and numerically calculated by solving a system of linear algebra equations. It is found that for the present crack model, crack easily propagates and grows with transport current ascent, and that in the process of transport current descent, crack generally does not propagate. According to the maximum energy release rate criterion, the bigger the crack opening angle is, the easier the crack propagates. Also, either increasing the shear modulus graded index of the superconducting coating or decreasing the introduced critical state model parameters can impede crack propagation and growth. The present study should be helpful to design and application of superconducting composites.

Group classification of charged particle motion in stationary electromagnetic fields

In this paper, we classify in terms of Lie point symmetries the three-dimensional nonrelativistic motion of charged particles in arbitrary time-independent electromagnetic fields. The classification is made on the ground of equivalence transformations, and when the system is nonlinear and particularly for inhomogeneous and curved magnetic fields, it is also complete. Using the homogeneous Maxwell’s equations as auxiliary conditions for consistency, in which case the system amounts to a Lagrangian of three degrees of freedom with velocity-dependent potentials, the equivalence group stays the same. Therefore, instead of the actual fields, the potentials are equally employed and their gauge invariance results in an infinite-dimensional equivalence algebra, which nevertheless projects to finite-dimensional symmetry algebras. Subsequently, optimal systems of equivalence subalgebras are obtained that lead to one-, two-, and three-parameter extended symmetry groups, besides the obvious time translations. Finally, based on symmetries of Noether type, aspects of complete integrability are discussed, as well.

Solving the linear inviscid shallow water equations in one dimension, with variable depth, using a recursion formula

When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations.

Linear Representation of Vector Group, Quick Solutions to Maximal Independent Group and Linear Equations in Linear Algebra

Through improving the process of "the linear representation of vector and vector group, and seeking solution to the maximal independent subset of vector group and linear equations" in linear algebra, this article simplifies the conventional two-step solution into one-step. The traditional two steps are: first, the matrix consisting of column vectors or the augmented matrix of linear equations is transformed into a ladder by the means of elementary row transformation; then this ladder matrix is transformed into a simplified row ladder matrix by doing one more elementary row transformation. However, with the improved method, we can work out solutions in one single step of elementary row transformation by transforming the transpose matrix of column vectors or the transpose augmented matrix of linear equations into a ladder matrix. The new method, thus, not only reduces half of the workload but also the difficulty of problem solving.

The equations defining blowup algebras of height three Gorenstein ideals

We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An important step of the proof is the calculation of the degree of the variety parametrized by the forms generating the grade three Gorenstein ideal.

Dual pairs of quantum moment maps and doubles of Hopf algebras

Given a dual pair of topological Hopf algebras A,A⁎, under mild conditions there exists a natural associative algebra homomorphism D(A)→H(A) between the corresponding Drinfeld double D(A) and Heisenberg double H(A). We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck–Springer resolution arises in this context.

Generalized Yangians and their Poisson counterparts

By a generalized Yangian we mean a Yangian-like algebra of one of two classes. One of these classes consists of the so-called braided Yangians, introduced in our previous paper. The braided Yangians are in a sense similar to the reflection equation algebra. The generalized Yangians of second class, called the Yangians of RTT type, are defined by the same formulae as the usual Yangians are but with other quantum $R$-matrices. If such an $R$-matrix is the simplest trigonometrical $R$-matrix, the corresponding Yangian of RTT type is the so-called q-Yangian. We claim that each generalized Yangian is a deformation of the commutative algebra ${\rm Sym}(gl(m)[t^{-1}])$ provided that the corresponding $R$-matrix is a deformation of the flip. Also, we exhibit the corresponding Poisson brackets.

Equations defining the multi-Rees algebras of powers of an ideal

In this paper we describe the defining equations of the multi-Rees algebra R[{Initi}1≤i≤r] over a Noetherian ring R when I is an ideal of linear type. This generalizes a result of Ribbe and recent work of Lin–Polini and Sosa.

Nonlinear interaction between underwater explosion bubble and structure based on fully coupled model

The interaction between an underwater explosion bubble and an elastic-plastic structure is a complex transient process, accompanying violent bubble collapsing, jet impact, penetration through the bubble, and large structural deformation. In the present study, the bubble dynamics are modeled using the boundary element method and the nonlinear transient structural response is modeled using the explicit finite element method. A new fully coupled 3D model is established through coupling the equations for the state variables of the fluid and structure and solving them as a set of coupled linear algebra equations. Based on the acceleration potential theory, the mutual dependence between the hydrodynamic load and the structural motion is decoupled. The pressure distribution in the flow field is calculated with the Bernoulli equation, where the partial derivative of the velocity potential in time is calculated using the boundary integral method to avoid numerical instabilities. To validate the present fully coupled model, the experiments of small-scale underwater explosion near a stiffened plate are carried out. High-speed imaging is used to capture the bubble behaviors and strain gauges are used to measure the strain response. The numerical results correspond well with the experimental data, in terms of bubble shapes and structural strain response. By both the loosely coupled model and the fully coupled model, the interaction between a bubble and a hollow spherical shell is studied. The bubble patterns vary with different parameters. When the fully coupled model and the loosely coupled model are advanced with the same time step, the error caused by the loosely coupled model becomes larger with the coupling effect becoming stronger. The fully coupled model is more stable than the loosely coupled model. Besides, the influences of the internal fluid on the dynamic response of the spherical shell are studied. At last, the case that the bubble interacts with an air-backed stiffened plate is simulated. The associated interesting physical phenomenon is obtained and expounded.

Chebyshev polynomials approach for numerically solving system of two-dimensional fractional PDEs and convergence analysis

In this paper, an efficient numerical technique based on the Chebsyhev orthogonal polynomials is established to obtain the approximate solutions of system of two-dimensional fractional-order PDEs with initial conditions. We construct the corresponding differential operational matrix of fractional-order, and then transform the problem into a system of linear algebra equations. Compared with other analytical or semi-analytical methods, ours can achieve better convergence accuracy only small terms are expanded. Moreover the proposed algorithm is simple in theoretical derivation and numerical simulation. In our study, the convergence analysis of the system is emphatically investigated than other numerical approaches. Lastly, three numerical examples are applied to test the algorithm and that the obtained numerical results show that our approach is effective and robust.

Rise in level of trust and trustworthiness with trust building measures

PurposeThe aim of this paper is to develop a model for presenting level of trust and analyzing the contribution of various trust-building measures undertaken by an organization.Design/methodology/approachThe conceptual framework for the model is based on previous research and the concept of trust and its implications in business environment. This model includes various stages of trust measured against time and trust building measures (TBM). This trust model relates the trustee’s position with the trustor at any point in time and describes its impact on trustee’s position in terms of trustworthiness (sum of “trust deficit” and “trust gain”). 10;Vectors and linear Algebra equations are used to construct the model supplemented with an example from real-life business environment for better understanding of the model. 10.FindingsA trust framework, elaborating level of trust between two parties is explained with the help of a mathematical model. The model includes various stages of trust measured against time and TBM.Research limitations/implicationsIn the practical application of the model, the authors adopted an existing scale to measure trust levels, which can have its limitations and shortcomings. It is however suggested to choose as specific scale for the industry as possible.Practical implicationsThe model can be applied in any situation, person or environment specially to determine the current situation of organizational trust in business which can be helpful in making decisions.Originality/valueThe concept of making trust a part of strategy and a tool for decision-making is novel and applicable in all sectors and situations. By providing a real-time view of the level of trust and impact of TBM will help predict future levels of trust and make it an essential part of decision-making process.

Nonlinear internal wave at the interface of two-layer liquid due to a moving hydrofoil

This paper is concerned with the internal wave at the interface of two layers of liquids due to a hydrofoil in the lower layer liquid. The two-layer fluid is assumed moving parallel to the interface at different velocities. The stratified flow is modeled based on the incompressible potential flow theory, with the nonlinear boundary conditions at the interface. Boundary integral equations are formulated for the fully nonlinear interfacial wave generated by the hydrofoil. The numerical model results in a set of nonlinear algebra equations, which are solved using the quasi-Newton method. We show that the quasi-Newton method is more efficient than Newton’s method, which is often used for solving these types of equations in the literature. The wave profiles were analyzed in terms of the location and thickness of the hydrofoil, the Froude number, and the ratio of the densities of the two fluids. The computations show that the interfacial wave amplitude showed a trend first of increase and then of decrease with the distance between the hydrofoil and the still interface.

Analysis of the mean squared derivative cost function

In this paper, we investigate the mean squared derivative cost functions that arise in various applications such as in motor control, biometrics and optimal transport theory. We provide qualitative properties, explicit analytical formulas and computational algorithms for the cost functions. We also perform numerical simulations to illustrate the analytical results. In addition, as a by‐product of our analysis, we obtain an explicit formula for the inverse of a Wronskian matrix that is of independent interest in linear algebra and differential equations theory. Copyright © 2017 John Wiley & Sons, Ltd.

Cambrian Hopf algebras

Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the classical conditions for binary search trees. Similar to binary trees for the Tamari lattice, Cambrian trees provide convenient combinatorial models for N. Reading's Cambrian lattices of type A. Based on a natural surjection from signed permutations to Cambrian trees, we define the Cambrian Hopf algebra extending J.-L. Loday and M. Ronco's algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. We then construct the Baxter–Cambrian algebra which extends S. Law and N. Reading's Baxter Hopf algebra on rectangulations and S. Giraudo's equivalent Hopf algebra on twin binary trees.