Geometry of Lobachevskii in connection with certain questions of arithmetic

A new geometric method is developed for investigating the Markov problem of the arithmetic minima of indeterminate binary quadratic forms and problems related to them.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 67, pp. 39–85, 1977.     

有限域上二次型序列与几何序列的互相关性

§1Introduction Letpbeaprimenumber,q=pm,andFqdenotethefinitefieldwithqelements.Fo anyn≥1,Trmnm(·)isthetracefunctionfromFqntoFq.LetαbeaprimitiveelementofFq and{α1,α2,...,αn}beabasisofFqnoverFq.Definition1.ForanonlinearfunctionffromFqtoFp,thesequenceS={Si}qn-1i=1withit termdefinedby Si=f(Trmnm(αi))(1iscalledageometricsequence.SuchageometricsequenceShasperioddividingqn-1.Geometricsequences includingm-sequence[1],GMWsequence[2,3],cascadedGMWsequence[4,5]andman others[6],mayhavelar…     

On minima of rational indefinite quadratic forms

Let f(x) be an indefinite quadratic form with real coefficients in n variables with nonzero determinant d. The collection of all values of $\text{v(f) = |d|}{}^{\mathrm{<mtext>?1</mtext><mtext>n</mtext>}}\text{inf}\text{|f(x)|}$, where infimum is taken over x ∈ Zⁿ such that f(x) ≠ 0 (x ≠ 0) is called the spectrum of nonzero minima (spectrum of minima) of such forms. The spectrum is said to be discrete if for every δ > 0, there are only finitely many values of v(f) > δ. It is proved that for rational quadratic forms in n ≥ 3 variables and real quadratic forms in n ≥ 21 variables the spectra of nonzero minima are discrete. Also the spectra of minima of indefinite ternary and quaternary rational quadratic forms are discrete.     

Monomials in quadratic forms

We obtain some constraints on the zero-nonzero pattern of entries in the matrix of a real quadratic form which attains a minimum on a large set of vertices in the multidimensional cube centered at the origin whose edges are parallel to the coordinate axes. In particular, if the graph of the matrix contains an articulation point then the set of the minima of the corresponding quadratic form is not maximal (with respect to set inclusion) among all such sets for various quadratic forms.     

The Probability Content of Cones in Isotropic Random Fields

This paper provides computable representations for the evaluation of the probability content of cones in isotropic random fields. A decomposition of quadratic forms in spherically symmetric random vectors is obtained and a representation of their moments is derived in terms of finite sums. These results are combined to obtain the distribution function of quadratic forms in spherically symmetric or central elliptically contoured random vectors. Some numerical examples involving the sample serial covariance are provided. Ratios of quadratic forms are also discussed.     

The isometry classification of hermitian forms over division algebras

The isometry classification problem occupies a central role in the theory of quadratic and hermitian forms. This article is a survey of results on the problem for quadratic and hermitian forms over a field and also for hermitian and skew-hermitian forms over a noncommutative division algebra with involution. Rather than adopting a very abstract approach, the problems are stated in matrix or linear-algebraic terms. The known solutions depend crucially on the particular field considered, although there are some general results which are mentioned. While many of the results date back a long time, some recent results, especially those on skew-hermitian forms over a quaternion algebra over a number field, are included.     

Zum Transitivitätsverhalten metrischer Kollineationsgruppen

In this paper connections between several geometric questions on the transitivity of metric collineation groups and connections with algebraic questions on the structure of certain kinds of quadratic forms are studied. Geometric problems are translated into algebraic ones so that the far developed algebraic theory of quadratic forms can be applied. There are no restrictions concerning dimension or characteristic.      

几何直观在特征值问题中的应用

借助MATLAB软件将几何直观方法应用于矩阵特征向量的判定、二次曲线的绘制、二次型的分类和微分方程组动力学性质刻画等线性代数特征值问题教学之中,以实例说明几何直观在线性代数课程教学中的应用.     

Cyclic transformations of n-gons and related quadratic forms

This paper discusses circulants that preserve quadratic forms or that are metric dependent. Several related geometric topics are also discussed.     

The slimmest arrangements of hyperplanes

We show how the results of Dowling and Wilson on Whitney numbers in ‘The slimmest geometric lattices’ imply minimum values for the numbers of k-dimensional flats and d-dimensional cells of a projective d-arrangement of hyperplanes and for the number of d-cells missed by an extra hyperplane. Their theorems also characterize the extremal arrangements. We extend their lattice results to doubly indexed Whitney numbers; thence we obtain minima for the number of k-dimensional cells and the number of pairs of flats with x \(\subseteq\) y and dim x=k, dim y=l. The lower bounds are in terms of the rank and number of points of the geometric lattice, or the dimension d and the number of hyperplanes of the arrangement. The minima for k-cells were conjectured by Grünbaum; R. W. Shannon proved the minima for k-dimensional flats and cells and characterized attainment for the latter by a more strictly geometric, non-latticial technique.     

Quadratic forms of a matric-t variate

Summary Crowther [2] studied the distribution of a quadratic form in a matrix normal variate. This, in some sense, is extended by De Waal [4]. They represented the density function of this quadratic form in terms of generalized Hayakawa polynomials. Application of some specific results of these authors facilitates the derivation of distributions of quadratic forms of the matric-t variate. Attention is also given to the distributions of the characteristic roots and the trace of this quadratic matrix. Special cases are considered and some useful integrals are formulated. Financially supported by the CSIR and the University of the Orange Free State     

Global Optimization of a Quadratic Functional with Quadratic Equality Constraints, Part 2

In this paper, we investigate a constrained optimization problem with a quadratic cost functional and two quadratic equality constraints. It is assumed that the cost functional is positive definite and that the constraints are both feasible and regular (but otherwise they are unrestricted quadratic functions). Thus, the existence of a global constrained minimum is assured. We develop a necessary and sufficient condition that completely characterizes the global minimum cost. Such a condition is of essential importance in iterative numerical methods for solving the constrained minimization problem, because it readily distinguishes between local minima and global minima and thus provides a stopping criterion for the computation. The result is similar to one obtained previously by the authors. In the previous result, we gave a characterization of the global minimum of a constrained quadratic minimization problem in which the cost functional was an arbitrary quadratic functional (as opposed to positive-definite here) and the constraints were at least positive-semidefinite quadratic functions (as opposed to essentially unrestricted here).     

Nonnegativity of quadratic forms on intersections of quadrics and quadratic maps

The question of the nonnegativity of quadratic forms on intersections of quadratic cones is considered. An answer is given in terms of Lagrange multipliers of the quadratic forms under consideration.     

Markov and Lagrange spectra (survey of the literature)

A survey is given of investigations on the Markov problem of the arithmetic minima of indeterminate, binary, quadratic forms and on the Lagrange-Hurwitz problem of Diophantine approximations of irrational numbers by rational numbers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 67, pp. 5–38, 1977.     

Isotropic Tori,Complex Germ and Maslov Index,Normal Forms and Quasimodes of Multidimensional Spectral Problems

More than twenty years ago V. P. Maslov posed the question under what conditions it is possible to assign to invariant isotropic lower-dimensional tori of Hamiltonian systems sequences of asymptotic eigenvalues and eigenfunctions (spectral series) of the corresponding quantum mechanical and wave operators. In the present paper this question is answered in terms of the quadratic approximation to the theory of normal forms. We also discuss the quantization conditions for isotropic tori and their relation to topological, geometric, and dynamical characteristics (Maslov indices, rotation (winding) numbers, eigenvalues of dynamical flows, etc.).     

Accelerations for global optimization covering methods using second derivatives

Two improvements for the algorithm of Breiman and Cutler are presented. Better envelopes can be built up using positive quadratic forms. Better utilization of first and second derivative information is attained by combining both global aspects of curvature and local aspects near the global optimum. The basis of the results is the geometric viewpoint developed by the first author and can be applied to a number of covering type methods. Improvements in convergence rates are demonstrated empirically on standard test functions.Partially supported by an University of Canterbury Erskine grant.     

Dynamical stability and optimization of parameters of cylindrical shells under short-time influence of loads that increase in a power-wise way

Some criteria of dynamical nonstability are derived for smooth cylindrical shells under actions of axial contracting loads that increase fast in a power-wise way. In terms of nonlinear programming the optimization problem is formulated that determines parameters of shells of minimal mass under some restrictions with respect to local and global loss of stability and also with respect to strength. With the help of the Kuhn-Tucker theorem some variants of the optimal solution are obtained. As an example, we consider two characteristic forms of load: linearly increasing load, and a load that increases in the quadratic way. We study the influence of basic parameters of these loads on the mass, and geometric dimensions of the optimal shells. Some numerical examples are given.Translated from Dinamicheskie Sistemy, No. 8, pp. 47–53, 1989.