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In this paper, we consider the problem of minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint. A key difficulty with this problem is its nonconvexity. Us...

In this paper, we consider a nonlinear matrix equation . The uniqueness of positive definite solution without any extra condition when s≥t is obtained. A fixed-point iteration with stepsize parameter for finding the positive definite solution is proposed. A condition number and some new perturbation bounds of the unique positive definite solution are derived. Finally, some numerical examples are given to show the efficiency of the proposed iterative method and perturbation bounds.

In this paper, we study the contact geometry of second-order ordinary differential equations that are quadratic in the highest derivative (the so-called quadratic Abel equations). Namely, we realize each quadratic Abel equation as the kernel of some nonlinear differential operator. This operator is defined by a quadratic form on the Cartan distribution in the 1-jet space. This observation makes it possible to establish a one-to-one correspondence between quadratic Abel equations and quadratic forms on Cartan distribution. Using this realization, we construct a contact-invariant {e}-structure associated with a nondegenerate Abel equation (i.e., the basis of vector fields that is invariant under contact transformations). Finally, in terms of this {e}-structure we solve the problem of contact equivalence of nondegenerate Abel equations

We obtain necessary and sufficient conditions for a continuous real-valued positive definite function of m variables (m > 1) with support in a bounded convex centrally symmetric body in the space Rm to have a real-valued even Boas–Kac root.

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.

In this paper, we prove that the 3-rank of the ideal class group of the imaginary quadratic field Q(√>) is at least 3 for every positive integer n.

In this paper we consider an aggregation technique introduced by Yıldıran (J Math Control Inf 26:417–450, 2009) to study the convex hull of regions defined by two quadratic inequalities or by a conic quadratic and a quadratic inequality. Yıldıran (2009) shows how to characterize the convex hull of open sets defined by two strict quadratic inequalities using Linear Matrix Inequalities. We show how this aggregation technique can be easily extended to yield valid conic quadratic inequalities for the convex hull of open sets defined by two strict quadratic inequalities or by a strict conic quadratic and a strict quadratic inequality. We also show that for sets defined by a strict conic quadratic and a strict quadratic inequality, under one additional containment assumption, these valid inequalities characterize the convex hull exactly. We also show that under certain topological assumptions, the results from the open setting can be extended to characterize the closed convex hull of sets defined with non-strict conic and quadratic inequalities., National Science Foundation (U.S.) (Grant CMMI-1030662)

For some diagonal quadratic forms, we improve the upper bound for the dimension of the space of theta-series with respect to the positive definite quadratic forms.

It has been known, in [7] , that one-mode quadratic Weyl operators are well-defined as unitary operators acting on the quadratic Fock space. These operators were defined by their action on the finite particles space. However, their action on the domain of the quadratic exponential vectors is still unknown. In this paper, we provide a new reformulation (w.r.t. [1] ) of the quadratic exponential vectors and we compute the action of the one-mode quadratic Weyl operators on the set of these exponential vectors. Then, we prove the independence of the one-mode quadratic Weyl operators parameterized by the principal domain associated with the quadratic Heisenberg group obtained in [7] . This significant contribution to the program of developing the quadratic white noise calculus constitutes a step toward the C ⁎ -representation of the renormalized square of white noise algebra.

In this paper, we further generalize the technique for constructing the normal (or positive definite) and skew-Hermitian splitting iteration method for solving large sparse nonHermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method converges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

We enumerate all positive definite ternary quadratic forms over number fields with class number at most 2. This is done by constructing all definite quaternion orders of type number at most 2 over number fields. Finally, we list all definite quaternion orders of ideal class number 1 or 2.

We investigate the relationship between the numbers of representations of certain integers by a primitive integral binary quadratic form [Formula: see text] of discriminant [Formula: see text] and the order of the class of [Formula: see text] in the form class group of discriminant [Formula: see text], in the case when this order is even. The explicit form of the solutions obtained is used to give a partial answer to a question regarding which multiples of [Formula: see text] can be parameterized in a particular way.

Based on the geometric significance of positive definite quadratic programming, the objective function of definite quadratic programming problem is converted into the distance form. At the same time, the objective function is standardized and normalized, so, a new algorithm of positive definite quadratic programming solution is given. Under certain conditions, the algorithm only need simple operation and simple decision, it has good applicability. The numerical experiments show that the algorithm is feasible and effective, it has abvious advantages comparing with other algorithms.

In the paper, quadratic mappings acting from one finite-dimensional space to another are studied. Sufficient conditions for the stable surjectivity of a quadratic surjective mapping (i.e., for the condition that every quadratic mapping sufficiently close to a given one is also surjective) are obtained. The existence problem for nontrivial zeros of a surjective quadratic mapping acting from Rn to Rn is studied. For n = 3, the absence of these zeros is proved.

Quadratic functionals defined on the space of functions differentiable on a geometric graph are considered. Analogs of the Lagrange and Dubois–Raymond lemmas are proved. Necessary extremum conditions for these quadratic functionals are obtained. A boundary value problem with conditions posed locally at the vertices of a geometric graph is shown to be selfadjoint if and only if it is generated by a quadratic functional. A subclass of quadratic energy functionals is singled out. The space of solutions of the homogeneous boundary value problem generated by a quadratic energy functional is described, and nondegeneracy criteria for such boundary value problems are derived.

Let A(1) and A(2) be two nonzero commuling} and {a2, fi2}-quadratic matrices, respectively, where alpha(1), beta(1), alpha(2), beta(2) is an element of C with alpha(1) not equal beta(1) and alpha(2) not equal beta(2). The aim of this work is mainly to characterize all situations, where the linear combination a(1)A(1) + a(2)A(2) is a generalized quadratic matrix. The results established here cover many of the results in the literature related to idempotency, in volutivi ty and tripotency of the linear combinations of idempotent and/or in volutive matrices.

A family of maximal length ź4-sequences was proposed by Tang, Udaya and Fan in 2005 by using binary sequences based on quadratic forms over finite fields, and was shown to possess low correlation property. However, its correlation distribution still remains open. In this paper, the maximal length ź4-sequences constructed by Tang, Udaya and Fan are equivalently expressed as another form via ź4-valued quadratic forms and then the correlation distribution is completely determined from the approach of ź4-valued quadratic forms.

The first part of the paper recalls and enlarges some results which appeared in the author's paper published 50 years ago, characterizing the relationship between the diagonal entries of mutually inverse positive definite matrices. In the second part, sign patterns of positive definite matrices which are also inverse positive are studied.

Real homogeneous quadratic mappings from Rn to R2 are examined. It is known that the image of such a mapping is always convex. A proof of the convexity of the image based on the quadratic extremum principle is given. The following fact is noted: If the quadratic mapping Q is surjective and n > 2 + dimkerQ, then there exists a regular zero of Q. A certain criterion of the linear dependence of quadratic forms is also stated.

In this paper, we consider a nonlinear matrix equation, which has the form , where is a positive integer, is an arbitrary complex matrix and is an Hermitian positive definite matrix. A double of elegant estimates of the Hermitian positive definite solution are obtained. Three iterative methods for computing the Hermitian positive definite solution are proposed. Some numerical examples to show the efficiency of the proposed iterative methods are provided.

In this paper we study Jensen’s inequality under quadratic g -expectation, i.e., the expectation generated by backward stochastic differential equations (BSDEs) with generator of quadratic growth in its component z . In particular, we define a new kind of convexity, the C -convexity, via a second order ODE depending on a real constant C , and we study the relationships between quadratic g -convexity and C -convexity.

In this paper the Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions of the matrix equation AXB=D are considered. The necessary and sufficient conditions for the existence of Re-nnd and Re-pd solutions to the equation are provided and the explicit representations of the general Re-nnd and Re-pd solutions are given when it is solvable.

Our previous study proposed a positive quadratic system representation for molecular interaction in a cell, including a signal transduction pathway and a gene regulatory network, and also presented a method for estimating a positive invariant set depending on the initial state. As an extension towards wider applications of this approach, this paper proposes a system representation called here a singularly perturbed positive quadratic system, and shows that every positive rational system, which is used as a mathematical model expressing biological behavior, can be approximately represented by a quasi-steady state system of a singularly perturbed positive quadratic system. In addition, we prove that the singularly perturbed positive quadratic system preserves stability at an equilibrium point of the positive rational system.

Free vibrations of fluid-solids structures are governed by a nonsymmetric eigenvalue problem. This problem can be transformed into a definite quadratic eigenvalue problem. The present paper considers properties of definite problems and variational characterization for definite quadratic eigenvalue problem. We propose improvement of existing methods which determine whether a quadratic pencil Q ( λ ) is definite. The improved method is reflected in a better localization of start parameters μ and ξ where Q ( μ ) > 0 > Q ( ξ ) and μ > ξ .

We investigate the nonlinear matrix equationXr-∑i=1mAi∗X-δiAi=I, whereris a positive integer andδi∈(0,1], for i=1,2,…,m. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique positive definite solution is established. An iterative algorithm is provided to compute the positive definite solutions for the equation and error estimate. Finally, some numerical examples are given to show the effectiveness and convergence of this algorithm.

An integral quadratic \(\mathbb Z\)-lattice is said to be strictly regular if it primitively represents all integers primitively represented by its genus. In this paper, all strictly regular positive definite primitive integral diagonal quadratic \(\mathbb Z\)-lattices are determined.

In this paper we study of densities of the 4-rank of narrow ideal class groups of real quadratic function fields over the rational function field Fq(T ) when q ≡ 3 mod 4.

Text It will be shown that there exist only finitely many isometry classes of primitive integral positive definite quaternary quadratic Z -lattices that are strictly regular, in the sense that they primitively represent all integers primitively represented by their genus. Video For a video summary of this paper, please visit http://youtu.be/V6EzAKAwpfw .

Stability of nonconvex quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces is investigated. We present several stability properties of the global solution map, and the continuity of the optimal value function, assuming that the problem data undergoes small perturbations., accepted for publication in AMV

We propose new algorithms for (i) the local optimization of bound constrained quadratic programs, (ii) the solution of general definite quadratic programs, and (iii) finding either a point satisfying given linear equations and inequalities or a certificate of infeasibility. The algorithms are implemented in Matlab and tested against state-of-the-art quadratic programming software.

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem, and correction term invariants due to Fr\o yshov, and Ozsv\'ath and Szab\'o. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property., Comment: 17 pages, 5 figures; Typos fixed. We added more results including properties on spherical 3-manifolds. The version to appear in Topology and its Applications

In this chapter, we recall some important results regarding quadratic linear spaces. Their basic purpose is to answer the following question.

An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we introduce the conductor of a quadratic polynomial, and give an effective characterization of almost universal ternary quadratic polynomials with odd prime power conductor.

An exact closed-form analytical expression is obtained for a one-dimensional characteristic function of an arbitrary quadratic form formed by statistically dependent Gaussian random variables with nonzero means. Necessary and sufficient conditions under which an arbitrary quadratic form has the generalized χ2 distribution are formulated. Parameters of the distribution of an arbitrary quadratic form are analyzed for particular cases.

Let k be a field of characteristic distinct from 2, V a finite dimensional vector space over k . We call two pairs of quadratic k -forms ( f 1 , g 1 ) , ( f 2 , g 2 ) on V isomorphic if there exists an isomorphism s : V → V such that f 2 = f 1 ∘ s , g 2 = g 1 ∘ s . We prove that if f 1 + t g 1 ≃ f 2 + t g 2 over k ( t ) and either the form f 1 + t g 1 is anisotropic, or det ( f 1 + t g 1 ) is a squarefree polynomial of degree at least dim V − 1 , then the pairs ( f 1 , g 1 ) and ( f 2 , g 2 ) are isomorphic.

Suppose Q is a definite quadratic form on a vector space V over some totally real field K ≠ Q . Then the maximal integral Z K -lattices in ( V , Q ) are locally isometric everywhere and hence form a single genus. We enumerate all orthogonal spaces ( V , Q ) of dimension at least 3, where the corresponding genus of maximal integral lattices consists of a single isometry class. It turns out, there are 471 such genera. Moreover, the dimension of V and the degree of K are bounded by 6 and 5 respectively. This classification also yields all maximal quaternion orders of type number one.

The weak isotropy index (or equivalently, sublevel) of arbitrary quadratic forms is studied. Its relationship to the level of a form is investigated. The problem of determining the set of values of the weak isotropy index of a form as it ranges over field extensions is addressed, with both admissible and inadmissible numbers being determined. An analogous investigation with respect to the level of a form is also undertaken. A treatment of forms for which the above invariants coincide concludes this article, with some recently-raised questions being resolved. European Commission - Seventh Framework Programme (FP7) Irish Research Council Marie Curie Actions

SummaryConsider the problem of determining, without using a computer or calculator, whether a given quadratic form factors into the product of two linear forms. A solution derived by inspection is often highly nontrivial; however, we can take advantage of equivalent conditions. In this article, we prove the equivalence of five such conditions. Furthermore, we discuss vocabulary such as “reducible,” “degenerate,” and “singular” that are used in the literature to describe these conditions, highlighting the inconsistency with which this vocabulary is applied.

The detailed spectral structure of symmetric, algebraic, quadratic eigenvalue problems has been developed recently. In this paper we take advantage of these canonical forms to provide a detailed analysis of inverse problems of the following form: construct the coefficient matrices from the spectral data including the classical eigenvalue/eigenvector data and sign characteristics for the real eigenvalues. An orthogonality condition dependent on these signs plays a vital role in this construction. Special attention is paid to the cases when the leading and trailing coefficients of the quadratic matrix polynomial are prescribed to be positive definite.

We study ADC quadratic forms and Euclidean quadratic forms over the integers, obtaining complete classification results in the positive case.

For a two-dimensional quadratic system, we obtain necessary conditions for the existence of a triangular quadratic system with the same Mironenko reflecting function as the original system. We suggest an algorithm that permits establishing the coincidence of the reflecting functions of a quadratic nonstationary system and some stationary system.

The problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property, even when the quadratic forms are indefinite. The equivalent convex problem is a semidefinite one, and the equivalence is based on the celebrated S-lemma. In this paper, we show that when the quadratic forms are simultaneously diagonalizable (SD), it is possible to derive an equivalent convex problem, which is a conic quadratic (CQ) one, and as such is significantly more tractable than a semidefinite problem. The SD condition holds for free for many problems arising in applications, in particular, when deriving robust counterparts of quadratic, or conic quadratic, constraints affected by implementation error. The proof of the hidden CQ property is constructive and does not rely on the S-lemma. This fact may be significant in discovering hidden convexity in some nonquadratic problems.

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.

Perturbations of the quadratic form minimization problem under quadratic constraints of the type of equalities are considered. The minimum function ω in this problem which, to each perturbation of the original problem, assigns a sharp lower bound in the perturbed problem is studied. Sufficient conditions for the upper and lower semicontinuity of the minimum function ω both at zero and in its neighborhood are obtained. Examples showing the importance of these conditions are given.

Positive definite piecewise linear functions are applied to the study of the properties of nonnegative trigonometric polynomials and to the analysis of some extremal problems on sets of positive definite functions.