System Of Homogeneous Linear Equations Research Articles

Nontrivial solutions for homogeneous linear equations over some non-quotient hyperfields

We introduce a class of hyperfields which includes several constructions of non-quotient hyperfields. We then use it to partially answer a question posed by M. Baker and T. Zhang: Does a system of homogeneous linear equations with more unknowns than equations always have a nonzero solution? We also consider a class of hyperfields that was claimed in the literature to be non-quotient, and show that this is false.

АЛГОРИТМИ ПОБУДОВИ МІНІМАЛЬНОЇ ПОРОДЖУВАЛЬНОЇ МНОЖИНИ РОЗВ’ЯЗКІВ СИСТЕМ ЛІНІЙНИХ РІВНЯНЬ

We consider optimizing transformations of the algorithm for construction of minimal generating sets of solutions of systems of linear homogeneous equations (SLHE) over the set of natural numbers. The features of such SLHEs are described, optimization transforms are substantiated, and examples of algorithm operation before and after optimization transforms are given. The application of the algorithm is illustrated by examples of the analysis of the properties of Petri nets and the construction of a set of basic solutions in the fields of complex, real, and rational numbers and over finite fields. Keywords: systems of linear equations, algorithms, solutions, optimization, complexity.

Embedding of Unimodular Row Vectors

In this paper, we mainly study the embedding problem of unimodular row vectors, focusing on avoiding the identification of polynomial zeros. We investigate the existence of the minimal syzygy module of the ZLP polynomial matrix and demonstrate that the minimal syzygy module has structural properties that are similar to the fundamental solution system of homogeneous linear equations found in linear algebra. Finally, we provide several embedding methods for unimodular vectors in certain cases.

Linear equations on Drinfeld modules

Let L be a finite extension of the rational function field in one variable over a finite field Fq and E be a Drinfeld module defined over L. Given finitely many elements in E(L), this paper aims to prove that linear relations among these points can be characterized by solutions of an explicitly constructed system of homogeneous linear equations over Fq[t]. As a consequence, we show that there is an explicit upper bound for the size of the generators of linear relations among these points. This result can be regarded as an analogue of a theorem of Masser for finitely many K-rational points on an elliptic curve defined over a number field K.

Дифференцирование линейных алгебр с единицей над полем

Linear algebras over a given field arise when studying various problems of algebra, analysis and geometry. The operation of differentiation, which originated in mathematical analysis, was transferred to the theory of linear algebras over a field, as well as to the theory of rings. The set of all differentiations of a linear algebra themselves form a linear algebra. This algebra is called the algebra of differentiations. At the same time, this algebra admits the structure of a Lie algebra. If the algebra whose differentiations are considered is finite-dimensional, then its Lie algebra of differentiations will also be finite-dimensional. Therefore, the­re is a natural problem of determining the dimension of the Lie algebras of the differentiations of the linear algebra under consideration or to obtain an estimate from above of the dimension of the algebra of differentiations. To solve these problems, a system of linear homogeneous equations is obtained, which is satisfied by the components of arbitrary differen­tia­tion. Evaluation of the rank of this system allows us to obtain an estimate from below of the rank of the matrix under consideration.

Scaffoldings of totally positive matrices and line insertion

Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line insertion. In this work, we revisit this problem. First, we show that every totally positive matrix can be associated with a certain vertex-weighted graph in such a way that the entries of the matrix are equal to sums over certain paths in this graph. We call this graph a scaffolding of the matrix. We then use this to give a complete characterization of all possible line insertions as the strongly positive solutions to a given homogeneous system of linear equations.

Automated Discovery of Geometric Theorems Based on Vector Equations

Automated discovery of geometric theorems has attracted considerable attention from the research community. In this paper, a new method is proposed to discover geometric theorems automatically. This method first generates vector equations based on given geometric relations about a geometric figure and then transforms the vector equations into a system of homogeneous linear equations; after computing the determinants of the coefficient matrices corresponding to the system of equations, the elimination method is applied to obtain a large number of geometric relationships. The test on more than 200 geometric problems shows that the geometric relationships discovered automatically by the proposed method are of obvious geometric meaning.

Internal impedance of a circular cylindrical wire under external, submillimeter wave illumination

AbstractA complete, non‐approximate expression of the internal impedance for an open‐ended circular cylindrical wire being subject to externally incident radiation is derived. A comparison is made with the well‐known formula given by Stratton and later given by Ramo. The modes in which the electromagnetic field oscillates are due to the incoming wave, and the corresponding boundary conditions do not lead to the homogeneous system of linear equations that produce the usual and relatively complex spectrum of damped modes discussed by Stratton. The approximations made to obtain Stratton and Ramo's formula are applicable again to the expression found here and produce the same result. Such approximations are adequate for good conductors at frequencies in the microwave region. However, it is shown that the new formulation, which includes the displacement current and the real part of the permittivity of the wire, is needed for the case of submillimeter radiation. In effect, the values of the impedance computed by using the new expression are considerably different from those resulting from Stratton or Ramo's formula in the Terahertz domain. In particular, these values show now a larger resistive part and a smaller reactive part in this domain.

Rado's theorem for rings and modules

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show that a system of homogeneous linear equations over an infinite integral domain is partition regular if and only if the corresponding matrix satisfies the columns conditions. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.

On the nontrivial solvability of systems of homogeneous linear equations over $\mathbb Z$ in ZFC

Following a recent paper by Herrlich and Tachtsis, we investigate in ZFC the following compactness question: for which unountable cardinals $\kappa$, an arbitrary nonempty system $S$ of homogeneous $\mathbb Z$-linear equations is nontrivially solvable in $\mathbb Z$ provided that each its nonempty subsystem of cardinality $<\kappa$ is nontrivially solvable in $\mathbb Z$?

Is There any Impact of Teaching Vector Spaces From Real Problems? The Case of First Year Engineering Students

In some linear algebra courses at the university level in engineering majors, the vector spaces are presented to students in an abstract way with scarce connections with other subjects and real problems. The goal of this study was to examine the effectiveness, regarding content knowledge and motivation, of a didactic proposal based on a problem based learning and the necessity principle, PBL-NP, modelling real engineering problems through homogeneous systems of linear equations, to introduce the concept of vector space. A quasi-experiment (post-test) was designed with a convenience sample composed of two groups: the experimental group, EG, amounting 33 students who were taught using the PBL-NP, and the control group, CG, composed by 79 students, taught by following an abstract approach. Inferential statistics was used to compare the learning outcomes between groups, by using as contrast variable an external test. The results show that the students in the EG group felt more relaxed and put less effort than CG students, while both groups gather the abstract concepts in a similar extent. Also the percentage who passed the course is higher in the EG compared with CG. Although both groups value positively the subject, a percentage of students in the CG group add some comments referred to the lack of practice related with real problems in the algebra courses taught with the abstract approach.

Trapped flexural waves supported by a pair of identical cylinders in a two-layer fluid

Existence of trapped waves in ocean shows the presence of discrete wave frequencies in the continuous spectrum. In the present work, we compute those trapped mode frequencies due to a pair of identical horizontal circular cylinders submerged in one of the layers of a two-layer fluid with a thin ice-cover at the upper layer and an infinite depth for the lower layer. Theory is developed for multiple cylinders but numerical computation is carried out only for a pair of cylinders. Due to a thin ice-cover replacing the free surface, a fifth-order boundary condition is to be considered in the upper layer that makes the problem complex and challenging but more practical. Considering linear water wave theory, the boundary value problem is developed through modified Helmholtz equation and associated conditions. Subsequently, applying multipole expansion method, an infinite system of homogeneous linear equations with complex coefficients is obtained and solved. By fixing the geometrical parameters and density ratio, the trapped mode frequencies are computed numerically by tracing the zeros of the determinant generated from the truncated system of the above mentioned equations. In the first instance, the cylinders are placed in the lower layer and the variation of trapped modes is examined by varying different parameters such as upper layer depth, submergence depth, ice-cover thickness etc. In the second instance, the same conditions and configurations are considered in the upper layer and existence of trapped modes is looked into in a similar manner. For the considered parameter values, the number of trapped modes enclosed in the continuous spectrum decreases corresponding to an increase in the flexural rigidity of the ice-cover. The trapped mode frequency decreases when either the the upper layer depth or the submergence depth increases. Further, corresponding to a small change in the separation parameter, embedded trapped modes are observed to cease to exist for the free surface and also for a very thin ice-cover. Our findings are supported by graphs depicting various modes. Further, comparison of present result with an established result shows excellent agreement.

A Note on Parallel Distinguishability of two Quantum Operations

In this work, the authors consider a homogeneous system of linear equations of the form $A_\alpha^{\otimes N} \mathbf{x} = 0$ arising from the distinguishability of two quantum operations by $N$ uses in parallel, where the coefficient matrix $A_\alpha$ depends on a real parameter $\alpha$. It was conjectured by Duan et al. that the system has a non-trivial nonnegative solution if and only if $\alpha$ lies in a certain interval $R_N$ depending on $N$. The authors affirm the necessity part of the conjecture and establish the sufficiency of the conjecture for $N\leq 10$ by presenting explicit non-trivial nonnegative solutions for the linear system.

Solution sets of systems of equations over finite lattices and semilattices

Solution sets of systems of homogeneous linear equations over fields are characterized as being subspaces, i.e., sets that are closed under linear combinations. Our goal is to characterize solution sets of systems of equations over arbitrary finite algebras by a similar closure condition. We show that solution sets are always closed under the centralizer of the clone of term operations of the given algebra; moreover, the centralizer is the only clone that could characterize solution sets. If every centralizer-closed set is the set of all solutions of a system of equations over a finite algebra, then we say that the algebra has Property {{,mathrm{(SDC)},}}. Our main result is the description of finite lattices and semilattices with Property {{,mathrm{(SDC)},}}: we prove that a finite lattice has Property {{,mathrm{(SDC)},}} if and only if it is a Boolean lattice, and a finite semilattice has Property {{,mathrm{(SDC)},}} if and only if it is distributive.

Foliation of the Space $$ \mathfrak{s}{\mathfrak{l}}^{\ast}\left(n,\mathbb{R}\right) $$ into Coadjoint Orbits

We suggest a method for constructing parameters on the coadjoint orbits in $$ \mathfrak{s}{\mathfrak{l}}^{\ast}\left(n,\mathbb{R}\right) $$ . The method is based on the fact that parameters are invariant under the action of vector fields normal to the tangent space of the orbit with respect to the Killing form. The construction reduces to solving a homogeneous system of linear equations.

Weighted counting of solutions to sparse systems of equations

AbstractGiven complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ &gt; 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β &gt; 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

Trapped-Energy Thickness-Extensional Mode of a Partially Electroded ZnO Thin-Film Resonator.

Free and forced vibration analysis of ZnO thin-film resonator operating with trapped-energy thickness-extensional (TE) mode is performed based on the dispersion curves for both the unbounded fully electroded and unelectroded films. The thickness solutions to the free vibration problem consist of the mode branches of the dispersion curves for the electroded and the unelectroded regions, respectively. The mode branches in the unelectroded region that carry energy away from the vibration zone are neglected because this effective energy loss corresponds to a complex frequency in the frequency spectrum. Since the thickness solutions in each region have satisfied the differential equations and the boundary conditions on the major surfaces exactly, the substitution of the thickness solutions into the modified Hamilton variational principle derived by Tiersten gives an approximate continuity condition in the form of integral over the thickness of the resonator at the interface between the electroded and unelectroded regions. The stationary condition of the integral continuity condition leads to a system of homogeneous linear equations, which determines the frequency spectrum ranging from the first TE cutoff frequency of the fully electroded film to that of the unelectroded film where the conventional trapped-energy vibration occurs. Forced vibration analysis of ZnO thin film driven into TE mode by applying a voltage to the top and bottom electrodes is also performed, which further verifies the validity of the obtained results from the free vibration analysis.

Colorings beyond Fox: The other linear Alexander quandles

This article is about applications of linear algebra to knot theory. For example, for odd prime p, there is a rule (given in the article) for coloring the arcs of a knot or link diagram from the residues mod p. This is a knot invariant in the sense that if a diagram of the knot under study admits such a coloring, then so does any other diagram of the same knot. This is called p-colorability. It is also associated to systems of linear homogeneous equations over the residues mod p, by regarding the arcs of the diagram as variables and assigning the equation “twice the over-arc minus the sum of the under-arcs equals zero” to each crossing. The knot invariant is here the existence or non-existence of non-trivial solutions of these systems of equations, when working over the integers modulo p (a non-trivial solution is such that not all variables take up the same value). Another knot invariant is the minimum number of distinct colors (values) these non-trivial solutions require, should they exist. This corresponds to finding a basis, supported by a diagram, in which these solutions take up the least number of distinct values. The actual minimum is hard to calculate, in general. For the first few primes, less than 17, it depends only on the prime, p, and not on the specific knots that admit non-trivial solutions, modulo p. For primes larger than 13 this is an open problem. In this article, we begin the exploration of other generalizations of these colorings (which also involve systems of linear homogeneous equations mod p) and we give lower bounds for the number of colors.

A Unique Mathematical Derivation of the Fundamental Laws of Nature Based on a New Algebraic-Axiomatic (Matrix) Approach ‡

In this article, as a new mathematical approach to origin of the laws of nature, using a new basic algebraic axiomatic (matrix) formalism based on the ring theory and Clifford algebras (presented in Section 2), “it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms.” In agreement with the rational Lorentz group, it is also basically assumed that the components of relativistic energy-momentum can only take rational values. In essence, the main scheme of this new mathematical axiomatic approach to the fundamental laws of nature is as follows: First, based on the assumption of the rationality of D-momentum and by linearization (along with a parameterization procedure) of the Lorentz invariant energy-momentum quadratic relation, a unique set of Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford and symmetric algebras) is derived. Then by an initial quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a set of two classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is derived uniquely as well.

Free Vibration Analysis of Rectangular Cantilever Plates Using the Hyperbolic-Trigonometric Series

The purpose of this paper is to develop a numerical analytical method for an accurate solution to the problem on frequencies and mode shapes of a rectangular cantilever plate. The problem is reduced to an infinite system of linear algebraic equations relative to ratios of trigonometric series, which contains vibration frequency as a parameter. The core of the method is using two hyperbolic-trigonometric series by two coordinates with six undetermined ratios. Functional series are subject to the main differential equation of vibrations and undetermined ratios are obtained from boundary conditions of the problem. Symmetric and asymmetric mode shapes are considered separately. The symmetric solution required the introduction of an additional function to compensate free terms in the decomposition of hyperbolic functions into Fourier series. The infinite system relative to six successions of undetermined ratios was reduced to a homogeneous infinite system relative to one (basic) succession of ratios. The iterative process of its solution at the chosen vibration frequency was presented. A compact resolving system of homogeneous linear equations was obtained, relative to basic ratios of mode shapes of a rectangular cantilever plate. The search for natural frequencies was done with simple exhaustion of a frequency parameter up to the values, at which the basic ratios become invariable, starting with some iteration. The simplicity of the algorithm and the resolving system allows fast obtaining natural frequencies with high accuracy. The calculation accuracy is analyzed. The results in this study are well coincided with the results of the authors, who fulfilled all the problem's conditions most accurately. The obtained results can be used to do highly accurate dynamic calculations in nanoengineering. The calculation accuracy with this algorithm can be enhanced by increasing the number of terms in series, the number of iterations and the size of the mantissa.