Siegel’s Lemma Is Sharp

An upper estimate for the norm of some positive integer solution of an arbitrary system of linear homogeneous equations with integer coefficients with nonempty set of positive integer solutions is obtained. This estimate is used to estimate from above the norm of a vector in which the components are powers of binomialswhose product is a difference polynomial of minimal possibleweight. The last result can be applied in the theory of beam steerable antenna arrays

The Gibbs approach to study of the critical state is expanded. It is noted that the well known critical state condition D=0 was obtained by Gibbs as a necessary condition for a non-zero solution of a system of homogeneous linear equations and defines a line of spinoidal solutions. To determine the critical point on this line the explicit form of solutions of a system of homogeneous equations is considered and their relationship to the slope of the phase equilibrium line is established. For the condition D=0 four possible cases of critical system thermodynamic behavior are considered. The relationship between each of these cases and the slope of the phase equilibrium line at the critical point is established. General thermodynamic stability conditions for the critical point are considered as a function of the type of critical behavior at the point. It is shown that one of the consequences of critical point stability is a merger of the binodal line with the spinodal at the critical point.

This paper describes a procedure for determining when a set of rational functions are pseudo-multiplicatively independent, i.e. when no non-trivial power product of the rational functions can be a constant. The method used is to derive a multiplicative basis of factors of the numerators and denominators of the rational functions; the basis is then used to derive a system of homogeneous linear equations which will have a non-trivial solution if and only if the rational functions are pseudo-multiplicatively dependent. Algorithms which find a basis for a set of Gaussian rational functions and the corresponding linear equations are presented in detail and bounds on their theoretical computing times are derived.

COMPUTATION OF CASIMIR INVARIANTS OF LIE ALGEBRAS

In the paper, a method is given for finding all solutions of a system of linear equations with interval coefficients and with additional supposition that these coefficients fulfil a given system of homogeneous linear equations.

Rother and Carlsson introduced the Direct Reference Plane method for obtaining a projective reconstruction from a multi-view system, presuming the availability of a planar reference quadrangle in each view. In this method feature points and camera centres are obtained simultaneously by solving a system of homogeneous linear equations. Each feature-camera visibility pair corresponds to two equations in this system. In this paper we derive a count criterion for characterizing sufficient visibility, yielding a system of linearly independent equations that "determines" feature points and camera centres. Finally we discuss some principles to extend the configuration of feature points and camera centres while maintaining sufficient visibility.

In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each equation contains at most three variables.Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be non-trivial. Here is an informal statement of our result: it is NP-hard to distinguish whether there is a non-trivial assignment that satisfies 1-δ fraction of the equations or every non-trivial assignment fails to satisfy a constant fraction of the equations with a margin of Ω(√δ).We develop linearity and dictatorship testing procedures for functions f: Rn -> R over a Gaussian space, which could be of independent interest.We believe that studying the complexity of linear equations over reals, apart from being a natural pursuit, can lead to progress on the Unique Games Conjecture.

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.

In the chapter we discuss a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of N possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–Cech compactification βN. This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. Using various families of uniformly recurrent words, including Sturmian words, the Thue–Morse word and fixed points of weak mixing substitutions, we generate an assortment of central sets which reflect the rich combinatorial structure of the underlying words. One crucial additive property of central sets is that each central set contains all finite sums of distinct terms for some infinite increasing sequence of natural numbers, i.e. is an IP-set. By a celebrated result of N. Hindman, the collection of all IP-sets is partition regular, i.e., if A is an IP-set then for any finite partition of A, one cell of the partition is an IP-set. We introduce an hierarchy of additive combinatorial properties for subsets of N and study them in terms of partition regularity. The results introduced in the chapter rely on interactions between different areas of mathematics: They include the general theory of combinatorics on words, numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–Cech compactification of N.

Solvability of Rado systems in D-sets

The Lanczos-Fourier series expansion is employed to analyze the guided-mode field in an asymmetrical slab waveguide, the core of which has an anisotropic and inhomogeneous dielectric permittivity. A system of linear homogeneous equations is derived by the collocation technique with consideration of the wave equation and the appropriate boundary conditions at the interfaces between the core and cladding media. The propagation constants are found from a determinant equation that ensures the existence of a nontrivial solution of the system. Numerical results are presented for several cases of dielectric permittivity, including the constant, parabolic, linear, and anisotropic cases. This approach is found to converge reasonably fast, and Richardson's extrapolation technique is applied to accelerate the convergence further. The approach can be easily generalized from the scalar to the vector equation, and, as an example, we consider the guided modes of a circular fiber.

SummaryFor square matrices, if AB = I then BA = I. This is proved from the fact that a homogeneous system of linear equations with more unknowns than equations has a nontrivial solution.

Summary The introduction of vertical and horizontal slownesses as independent variables in the transport equation allows us to obtain an explicit expression for the amplitude variation along a ray in a vertically stratified medium. This expression can be used to introduce the dynamical characteristics in the inverse problem of refraction seismology. For the scalar wave equation the ratio of squared amplitudes is a rational functional of some function of the local slowness, the parametrization of which leads to a system of linear homogeneous equations. For P seismic waves the same functional is obtained for the ratio of squared amplitudes weighted by the density.

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.

The roots of matrix pencils (Ay=λBy): existence, calculations, and relations to game theory