Separable free quadratic algebras over quadratic integers

The aim of the paper is to determine all free separable quadratic algebras over the rings of integers of quadratic fields in terms of the properties of the fundamental unit in the real case. The paper corrects some earlier published results on the subject.     

On the quadratic character of quadratic units

Let be a prime. Let a,b∈Z with p?a(a²+b²). In the paper we mainly determine by assuming p=c²+d² or p=Ax²+2Bxy+Cy² with AC−B²=a²+b². As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1).     

Convex hull of two quadratic or a conic quadratic and a quadratic inequality

Mathematical Programming - 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...     

Solving quadratic equations using reduced unimodular quadratic forms

Let be an symmetric matrix with integral entries and with , but not necesarily positive definite. We describe a generalized LLL algorithm to reduce this quadratic form. This algorithm either reduces the quadratic form or stops with some isotropic vector. It is proved to run in polynomial time. We also describe an algorithm for the minimization of a ternary quadratic form: when a quadratic equation is solvable over , a solution can be deduced from another quadratic equation of determinant . The combination of these algorithms allows us to solve efficiently any general ternary quadratic equation over , and this gives a polynomial time algorithm (as soon as the factorization of the determinant of is known).

     

Approximating quadratic programming with bound and quadratic constraints

Received May 20, 1997 / Revised version received March 9, 1998 Published online October 9, 1998     

Bound constrained quadratic programming via piecewise quadratic functions

1 , the smallest eigenvalue of a symmetric, positive definite matrix, and is solved by Newton iteration with line search. The paper describes the algorithm and its implementation including estimation of λ₁, how to get a good starting point for the iteration, and up- and downdating of Cholesky factorization. Results of extensive testing and comparison with other methods for constrained QP are given. Received May 1, 1997 / Revised version received March 17, 1998 Published online November 24, 1998     

Pairs of quadratic forms over a quadratic field extension

Let F be a field of characteristic distinct from 2, $L=F(\sqrt{d})$ a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, $M_f$, $M_g$ their matrices. We say that the pair $(f,g)$ is a k-pair if there exist $S\in G{L}_n(L)$ such that all the entries of the $k\times k$ upper-left corner of the matrices $S{M}_f{S}^t$ and $S{M}_g{S}^t$ are in F. We give certain criteria to determine whether a given pair $(f,g)$ is a k-pair. We consider the transfer ${\mathrm{cor}}_{L(t)/F(t)}$ determined by the $F(t)$-linear map $s:L(t)\to F(t)$ with $s(1)=0$, $s(\sqrt{d})=1$, and prove that if $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a $[\frac{k+1}{2}]$-pair. If, additionally, the form $f+tg$ does not have a totally isotropic subspace of dimension $p+1$ over $L(t)$, we show that $(f,g)$ is a $(k?2p)$-pair. In particular, if the form $f+tg$ is anisotropic, and $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a k-pair.     

A robust algorithm for quadratic optimization under quadratic constraints

Most existing methods of quadratically constrained quadratic optimization actually solve a refined linear or convex relaxation of the original problem. It turned out, however, that such an approach may sometimes provide an infeasible solution which cannot be accepted as an approximate optimal solution in any reasonable sense. To overcome these limitations a new approach is proposed that guarantees a more appropriate approximate optimal solution which is also stable under small perturbations of the constraints.     

On runs of consecutive quadratic residues and quadratic nonresidues

Based on results of Weil and of Burgess, we have obtained a boundK(l) such that all primesp K(l) have a sequence of at leastl consecutive quadratic residues and a sequence of at leastl consecutive nonresidues in the interval [1,p – 1]. The bound forl=9 being 414463, we have computed, for primes less than 420000, the lengths of the longest sequences of consecutive residues and of nonresidues. We present these data and make some observations concerning them. One of the observations is that there is an observed difference in the length of the maximal sequence between primes congruent to 1 (mod 4) and primes congruent to 3 (mod 4).     

Nonnegative quadratic estimation and quadratic sufficiency in general linear models

Notions of linear sufficiency and quadratic sufficiency are of interest to some authors. In this paper, the problem of nonnegative quadratic estimation for β^′Hβ+hσ² is discussed in a general linear model and its transformed model. The notion of quadratic sufficiency is considered in the sense of generality, and the corresponding necessary and sufficient conditions for the transformation to be quadratically sufficient are investigated. As a direct consequence, the result on (ordinary) quadratic sufficiency is obtained. In addition, we pose a practical problem and extend a special situation to the multivariate case. Moreover, a simulated example is conducted, and applications to a model with compound symmetric covariance matrix are given. Finally, we derive a remark which indicates that our main results could be extended further to the quasi-normal case.     

Hidden conic quadratic representation of some nonconvex quadratic optimization problems

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.     

Rank restrictions on the quadratic form in indefinite quadratic programming

Summary A quadratic programming problem, whereq(x) =a ^T x +x ^T Qx is an indefinite objective function, can be solved withSwarup's approach of optimizing (c ^T x + )(d ^T x + ) only if the rank ofQ is two; ifQ is definite, the rank ofQ must be one.

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Zusammenfassung Ein quadratisches Optimierungsproblem, in demq(x) =a ^T x +x ^T Qx eine indefinite Zielfunktion ist, kann mitSwarups Optimierungsansatz (c ^T x + )(d ^T x + ) nur gelöst werden, wenn der Rang vonQ gleich zwei ist; wennQ definit ist, muß der Rang vonQ gleich eins sein.

     

On the complexity of quadratic programming with two quadratic constraints

Mathematical Programming - The complexity of quadratic programming problems with two quadratic constraints is an open problem. In this paper we show that when one constraint is a ball constraint...     

Global optimization of a quadratic functional with quadratic equality constraints

In this paper, we investigate a constrained optimization problem with a quadratic cost functional and two quadratic equality constraints. While it is obvious that, for a nonempty constraint set, there exists a global minimum cost, a method to determine if a given local solution yields the global minimum cost has not been established. We develop a necessary and sufficient condition that will guarantee that solutions of the optimization problem yield the global minimum cost. This constrained optimization problem occurs naturally in the computation of the phase margin for multivariable control systems. Our results guarantee that numerical routines can be developed that will converge to the global solution for the phase margin.     

Free quadratic bialgebra

In this paper, we obtain the following main theorem for a free quadratic bialgebraJ:

1. Forp≠0,J is a pointed cosemisimple coalgebra. Forp=0,J is a hyperalgebra.
2. Forp≠0 andq≠0,J has antipodeS iffp·q+2=0 andS(x)=x. Forp=0 orq=0,J has antipode andS(x)=×.
3. All leftJ ^*-modules are rational.

Also, we give some applications in homological theory and algebraicK-theory.     

Free quadratic harness

Free quadratic harness is a Markov process from the class of quadratic harnesses, i.e. processes with linear regressions and quadratic conditional variances. The process has recently been constructed for a restricted range of parameters in Bryc et al. (2010) [7] using Askey-Wilson polynomials. Here we provide a self-contained construction of the free quadratic harness for all values of the parameters.