Class Of Quadratic Forms Research Articles

Schwartz operators

In this paper, we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them, in particular, with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Fréchet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show, in particular, that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally, we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators.

Impurity of the corner angles in certain special families of simplices

Focusing on the fact that the sum of the angles of any Euclidean triangle is constant and equals π for all triangles, Hajja and Martini raised, in [Math Intell 35(3):16–28, 2013, Problem 9], the question whether an analogous statement holds for higher dimensional d-simplices. An interesting answer was given by Hajja and Hammoudeh in (Beit Algebra Geom (to appear), 2014), where they proved that for the measure arising from what is known as the polar sine, the sum of measures of the corner angles of an orthocentric tetrahedron is constant and equals π. A crucial ingredient in that treatment is the fact that orthocentric d-simplices are pure, in the sense that the planar subangles of every corner angle are all acute, all obtuse, or all right. In this article, it is shown that this property is not shared by any of the three other special families of d-simplices that appear in the literature, namely, the families of circumscriptible, isodynamic, and isogonic (or rather tetra-isogonic) d-simplices, thus answering Problem 3 of (Hajja and Martini in Math Intell 35(3):16–28, 2013). Specifically, it is proved that there are d-simplices in each of these families in which one of the corner angles has an acute, an obtuse, and a right planar subangle. The tools used are expected to be useful in various other contexts. These tools include formulas for the volumes of d-simplices in these families in terms of the parameters in their standard parameterizations, simple characterizations of the Cayley–Menger determinants of such d-simplices, embeddability of a given d-simplex belonging to any of these families in a (d + 1)-simplex in the same family, formulas for some special determinants, and a nice property of a certain class of quadratic forms.

Hilbertʼs Tenth Problem for rational function fields over p-adic fields

Let K be a p-adic field (a finite extension of some Qp) and let K(t) be the field of rational functions over K. We define a kind of quadratic reciprocity symbol for polynomials over K and apply it to prove isotropy for a certain class of quadratic forms over K(t). Using this result, we give an existential definition for the predicate “vt(x)⩾0” in K(t). This implies undecidability of diophantine equations over K(t).

On average consensus in directed networks of agents with switching topology and time delay

In this article, the average consensus problem in directed networks of agents with both switching topology and coupling delay is investigated. First, based on a specific orthogonal transformation of the Laplacian matrix, an important proposition for verifying the positive definiteness of a class of quadratic forms is provided, which is of independent interest in matrix theory. And the relation between weakly connected and strongly connected digraphs is also investigated. Then, it is proved that all the agents reach the average consensus asymptotically for appropriate time delay if the communication topology keeps weakly connected and balanced. Finally, a numerical example is given to demonstrate the effectiveness and advantage of the new result.

Quadratic Form made a Perfect Power by a New Composition Theorem on Arbitrary Quadratic Forms

This paper deals with the diophantine equation Q(x_1,\,x_2,\,\ldots,\,x_m)= y^n , where m and n are arbitrary positive integers and Q(x_1,\,x_2,\,\ldots,\,x_m) is an arbitrary quadratic form in the m variables x_1,\,x_2,\,\ldots,\,x_m . While solutions of special cases of this equation have been published earlier, the general equation of this type has not been solved till now. To solve this equation, we first show that, given an arbitrary quadratic form Q(x_1,\,x_2,\,\ldots,\,x_m) in m variables, there exists a composition formula Q(u_i)\,Q^2(v_i)=Q(w_i) where u_i and v_i ( i= 1,\,2,\,\ldots,\,m ) are arbitrary variables and the w_i ( i=1,\,2,\,\ldots,\,m ) are cubic forms in the variables u_i and v_i ( i=1,\,2,\,\ldots,\,m ). This is a new composition formula, different from the standard composition formulae of the type Q(u_i)Q(v_i)=Q(w_i) which are known for certain classes of quadratic forms. As the equation Q(x_i)=y^n is not always solvable, we prove a theorem giving a necessary and sufficient condition for its solvability. We use the aforementioned composition formula to obtain parametric solutions of the equation Q(x_i)=y^n , and also give some numerical examples.

Characterization of robust stability of a class of interconnected systems

We consider robust stability analysis of a class of large scale interconnected systems. The individual subsystems may be different but they are assumed to share a property that characterizes a set of interconnection matrices which lead to stable overall systems. The main contribution of the paper is to show that, for the case where the network interconnection matrix is normal, (robust) stability verification can be simplified to a low complexity problem of checking whether the frequency response of the individual subsystems and the eigenvalues of the interconnection matrix can be mutually separated using a class of quadratic forms. Most interestingly, it is shown that this criterion is also necessary, in the sense that if the criterion is violated, an interconnection matrix of the same eigenvalue distribution can be found to make the overall system unstable.

Testing for Covariate Effects in the Fully Nonparametric Analysis of Covariance Model

Traditional inference questions in the analysis of covariance mainly focus on comparing different factor levels by adjusting for the continuous covariates, which are believed to also exert a significant effect on the outcome variable. Testing hypotheses about the covariate effects, although of substantial interest in many applications, has received relatively limited study in the semiparametric/nonparametric setting. In the context of the fully nonparametric analysis of covariance model of Akritas et al., we propose methods to test for covariate main effects and covariate–factor interaction effects. The idea underlying the proposed procedures is that covariates can be thought of as factors with many levels. The test statistics are closely related to some recent developments in the asymptotic theory for analysis of variance when the number of factor levels is large. The limiting normal distributions are established under the null hypotheses and local alternatives by asymptotically approximating a new class of quadratic forms. The test statistics bear similar forms to the classical F-test statistics and thus are convenient for computation. We demonstrate the methods and their properties on simulated and real data.

Four Dimensional Quadratic Forms Over F(X) Where and a Failure of the Strong Hasse Principle#

We consider four dimensional quadratic forms over F(X) where F is a field of characteristic not two and . Let q be a regular four dimensional quadratic form over F(X) such that its signatures with respect to all orderings on F(X) lie in either {0, 2} or {0, − 2}. (E.g., a four dimensional torsion form over F(X) or the form ⟨ 1, 1, 1, − α⟩ where α ∈ F(X) is totally positive.) We determine conditions equivalent to the isotropy of q over F(X). Using these conditions we show that a class of quadratic forms over k(X, Y) (where k is hereditarily quadratically closed) is anisotropic. In a separate argument we show that this class of forms is locally isotropic everywhere. Thus the strong Hasse principle does not hold for quadratic forms over such k(X, Y). These counter-examples include those of Kim and Roush [Kim, K. H., Roush, W. (1991). Quadratic forms over C[t 1, t 2]. J. Algebra 140:65–82] and of Jaworski [Jaworski, P. (2001). On the strong Hasse principle for fields of quotients of power series rings in two variables. Math. Z. 236:531–566].

On the distribition of residue classes of quadratic forms and integer-detecting sequences in number fields

The extensive study of residue classes to given moduli which are represented by a quadratic form leads to an application to sequences in algebraic number fields. So-called integer-detecting sequences were introduced over the rationals in earlier work. We discuss the respective concept for sequences in number fields. 1. Introduction and definitions In this paper, we study quadratic forms x 2 + Dy 2 mod s for given moduli s and try to decide which residue classes are represented by them. Quadratic forms of this type may be considered as norms in quadratic number fields. These norms and, more generally, norms in arbitrary number fields occur naturally when studying certain sequences, so-called integer-detecting sequences. We first shall explain this notion for rational numbers. Integer-detecting sequences were introduced in [12] (see also [3]), and a bunch of necessary or sufficient criteria was derived. For the convenience of the reader let us review the definition: A sequence (mn) n 0 of positive integers with mn 0) if any rational number r ,s atisfy� � .

Construction of optimal Lyapunov function for systems with structured uncertainties

The robust stability problem for nominally linear system with nonlinear, time-varying structured perturbations p/sub j/, j=1,...,q, is considered. The system is of the form x/spl dot/=A/sub N/x+/sup q//spl Sigma//sub j=1/p/sub j/A/sub j/x. When Lyapunov direct method is utilized to solve the problem, usually some quadratic form is chosen as a Lyapunov function. The note presents a procedure of selection of optimal Lyapunov function from the class of quadratic forms. Under some weak conditions the procedure is effective in solving the robust stability problem. The procedure is simple, requires only such numerical routines as inverting positive definite symmetric matrices and determining the eigenvalues and eigenvectors of symmetric matrices. It is expected that when optimal Lyapunov function is used for a robust linear feedback controller design, the resulting controller will have the smallest in some sense gains. The examples demonstrate the effectiveness of the method. The method is effective for large scale systems, it worked well for 24 dimensional system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Levels of positive definite ternary quadratic forms

The level N and squarefree character q of a positive definite ternary quadratic form are defined so that its associated modular form has level N and character χ q {\chi _q} . We define à collection of correspondences between classes of quadratic forms having the same level and different discriminants. This makes practical a method for finding representatives of all classes of ternary forms having a given level. We also give a formula for the number of genera of ternary forms with a given level and character.

Levels of Positive Definite Ternary Quadratic Forms

The level N and squarefree character q of a positive definite ternary quadratic form are defined so that its associated modular form has level N and character ${\chi _q}$. We define à collection of correspondences between classes of quadratic forms having the same level and different discriminants. This makes practical a method for finding representatives of all classes of ternary forms having a given level. We also give a formula for the number of genera of ternary forms with a given level and character.

Bounding Approximations to Some Quadratic Limit States

In second‐order reliability problems the limit state surface is (locally) replaced by a quadratic surface. This leads to a study of quadratic forms in the basic uncertainty variables. The present paper exploits a simple, closed‐form expression for the distribution function of a special class of quadratic forms in zero‐mean Gaussian variables to obtain bounds on the distribution function of general quadratic forms in such variables. It is shown that these bounds can be used to obtain useful estimates of the failure probability for such models.

Lévy’s functional analysis in terms of an infinite dimensional Brownian motion III

The purpose of this paper is to define minimality of surfaces in an infinite dimensional space E by probabilistic methods with the description of the relation between minimal surfaces and harmonic functions on the space E, and to analyze purely analytic properties of a certain class of quadratic forms on the space E.

Conditional Positivity of Quadratic Forms in Hilbert Space

If X and Y are real Hilbert spaces, $A:X \to Y$ is a bounded linear operator, and $\Gamma \subseteq Y$ is a closed convex cone, an immediate sufficient condition for a quadratic form Q on X to be positive subject to the constraint $Ax \in \Gamma $, is that Q be decomposable as a sum $Q(x) = C(Ax) + S(x)$, where C is a quadratic form on Y which is positive on $\Gamma $, and S is positive definite on X. The necessity of such a decomposition is not obvious, but is established here for a class of quadratic forms which commonly occur in variational problems—the Legendre forms. The proof furnishes formulas for C and S which are explicit apart from the occurrence of an unknown scalar. The usefulness of the result is illustrated by the determination of the focal (conjugate) time of a linear-quadratic control problem with inequality constraints on the final state.

Ein Kriterium f�r die Richtigkeit der Milnor-Vermutung

Let F denote a field of characteristic ≠2. The Witt ring WF (i.e. the ring of similarity classes of quadratic forms with coefficients in F) will be characterized by taking the similarity classes of certain Pfister forms [2] as generators and using suitable relations among them (theorem 1). In order to characterize GF, the graded ring associated to WF, in terms of generators and relations Milnor [4] constructed a homomorphism from his ring k*F onto GF and conjectured this homomorphism to be an isomorphism. This conjecture turns out to be equivalent to a problem on ideals in a commutative ring (theorem 2).

On testing a set of correlation coefficients for equality: Some asymptotic results

This note considers tests of the hypothesis H that the off-diagonal elements of a correlation matrix are equal. A test statistic T3 proposed by Aitkin, Nelson & Reinfurt (1968) is shown to be asymptotically equivalent, under H, to a member of a certain class of quadratic forms in the sample correlations. The asymptotic null distribution of each member of this class is obtained, the distribution of T3 and of statistics proposed by other authors following as a special case. The asymptotic probability of type-one error for the critical region suggested by Aitkin et al. depends upon an unspecified null parameter in such a way as to make questionable the applicability of the test.