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Two formulas for the central inverse moments of a quadratic form in normal variables and of the ratio of such forms are established. They relate the quadratic form determined by a positive definite matrix to that defined by the inverse matrix.

We establish several convexity results which are concerned with nonconvex quadratic matrix (QM) functions: strong duality of quadratic matrix programming problems, convexity of the image of mappings comprised of several QM functions and existence of a corresponding S-lemma. As a consequence of our results, we prove that a class of quadratic problems involving several functions with similar matrix terms has a zero duality gap. We present applications to robust optimization, to solution of linear systems immune to implementation errors and to the problem of computing the Chebyshev center of an intersection of balls.

Formulas are obtained for the number of representations of positive integers by quadratic forms .

The 3-D quadratic diffeomorphism is defined as a map with a constant Jacobian. A few such examples are well known. In this paper, all possible forms of the 3-D quadratic diffeomorphisms are determined. Some numerical results are also given and discussed.

This note proposes a new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. The unknown parameters are determined by the first four cumulants of the quadratic forms. The proposed method is compared with Pearson's three-moment central @g^2 approximation approach, by means of numerical examples. Our method yields a better approximation to the distribution of the non-central quadratic forms than Pearson's method, particularly in the upper tail of the quadratic form, the tail most often needed in practical work.

Some theta function identities are proved and used to give formulae for the number of representations of a positive integer by certain quaternary forms x2 + ey2 + fz2 + gt2 with e, f, g ∈ {1, 2, 4, 8}.

We classify the phase portraits of planar quadratic vector fields with a invariant straight line passing through two finite saddles, when the singularities of the field form a non-convex quadrilateral.

Made available in DSpace on 2022-04-28T18:59:41Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-12-01 In this paper, we classify the global phase portraits of all quadratic planar systems with two parallel invariant straight lines. The main techniques used are Poincaŕe Compactification and Normal Forms Theory combined with the Neumann's Theorem. © 2008 Birkhäuser Verlag Basel/Switzerland. UNESP-IBILCE, Rua C. Colombo, 2265, 15054-000 S. J. Rio Preto UNICAMP-IMECC, Rua S. B. Holanda, 651, 13081-970 Campinas UNESP-IBILCE, Rua C. Colombo, 2265, 15054-000 S. J. Rio Preto

Let R be a regular local ring, K its field of fractions and (V,ϕ) a quadratic space over R. Assume that R contains a field of characteristic zero we show that if (V,ϕ)⊗ R K is isotropic over K, then (V,ϕ) is isotropic over R. This solves the characteristic zero case of a question raised by J.-L. Colliot-Thelene in [3]. The proof is based on a variant of a moving lemma from [7]. A purity theorem for quadratic spaces is proved as well. It generalizes in the charactersitic zero case the main purity result from [9] and it is used to prove the main result in [2].

We prove Witt’s cancelation and extension theorems for Galois Ring valued quadratic forms. The proof is based on the properties of the invariant I , previously defined by the authors, that classifies, together with the type of the corresponding bilinear form (alternating or not), nonsingular Galois Ring valued quadratic forms. Our results extend the Witt’s theorem for mod four valued quadratic forms. On the other hand, the known relation between the invariant I and the Arf invariant of an ordinary quadratic form (if the associated nonsingular bilinear form is alternating) is extended to the nonalternating case by explaining the invariant I in terms of Clifford algebras.

For a given initial state, a constrained infinite horizon linear quadratic optimal control problem can be reduced to a finite dimensional problem [12]. To find a conservative estimate of the size of the reduced problem, the existing algorithms require the on-line solutions of quadratic programs [10] or a linear program [2]. In this paper, we first show based on the Lyapunov theorem that the closed-loop system with a mixed constrained infinite horizon linear quadratic optimal control is exponentially stable on proper sets. Then the exponentially converging envelop of the closed-loop trajectory that can be computed off-line is employed to obtain a finite dimensional quadratic program equivalent to the mixed constrained infinite horizon linear quadratic optimal control problem without any on-line optimization. The example considered in [2] showed that the proposed algorithm identifies less conservative size estimate of the reduced problem with much less computation.

The Fifteen Theorem proved by Conway and Schneeberger is a criterion for quadratic forms over the rational integer ring to be universal. In this article, we give a proof of an analogy of the Fifteen Theorem for quadratic forms over polynomial rings, which is known as the Four Conjecture proposed by Gerstein.

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.

Let V be an indefinite quadratic space over a number field F and U be a nondegenerate subspace of V . Suppose that M is a lattice on V , and that N is a lattice on U which is represented by M locally everywhere. The main result of this paper is a necessary and sufficient condition for which there exists a representation of N by M that approximates a given family of local representations. This is applied to determine when the variety of representations of U by V has strong approximation with respect to a finite set of primes of F that contains all the archimedean primes.

In this paper, we obtain the general solution and the stability of the 2-dimensional vector variable quadratic functional equation f( x + y, z - w) + f(x - y, z + w) = 2f(x, z ) + 2f(y, ). The quadratic form f( x, y) = + without cross product terms is a solution of the above functional equation.

In a paper of Kim, Chan, and Rhagavan, the universal ternary classical quadratic forms over quadratic fields of positive discriminant were discovered. Here a proof of the universality of some of these quadratic forms is given using a technique of Liouville. Another quadratic form over the field of discriminant 8 is shown universal by a different elementary approach.

Let N ≥ 2 be an integer, F a quadratic form in N variables over $\overline{\mathbb{Q}}$, and $Z \subseteq \overline{\mathbb{Q}}^N$ an L-dimensional subspace, 1 ≤ L ≤ N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z,F). This provides an analogue over $\overline{\mathbb{Q}}$ of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over $\overline{\mathbb{Q}}$. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over $\overline{\mathbb{Q}}$. This extends previous results of the author over number fields. All bounds on height are explicit.

In this paper we introduce nearly quadratic modules and establish their basic properties. We also prove a reduction theorem for finite groups generated by nearly quadratic subgroups.

We present a second order cone programming relaxation with O(n2) variables for quadratic assignment problems, which provides a lower bound not less than the well-known quadratic programming bound. It is further strengthened by additional linear inequalities.

Let F(z)=∑n=1∞A(n)qn denote the unique weight 6 normalized cuspidal eigenform on Γ0(4). We prove that A(p)≡0,2,−1(mod 11) when p≠11 is a prime. We then use this congruence to give an application to the number of representations of an integer by quadratic form of level 4.

We study sums of squares, quadratic forms, and related field invariants in a quadratic extension of the rational function field in one variable over a hereditarily pythagorean base field.

For every odd n, on the sphere Sn, ρ(n) − 1 linear orthonormal tangent vector fields, where ρ(n) is the Hurwitz-Radon number, are explicitly constructed. For each 8 × 8 sign matrix, compositions for infinite-dimensional positive definite quadratic forms are explicitly constructed. The infinite-dimensional real normed algebras thus arising are proved to have certain properties of associativity and divisibility type.

We study infinite dimensional quadratic programming problems of an integral type. The decision variable is taken in the L"p space where 1

Let F be an imaginary quadratic number field and K 2 O F the tame kernel of F. In this article, we determine all possible values of r 4(K 2 O F ) for each type of imaginary quadratic number field F. In particular, for each type of imaginary quadratic number field we give the maximum possible value of r 4(K 2 O F ) and show that each integer between the lower and upper bounds occurs as a value of the 4-rank of K 2 O F for infinitely many imaginary quadratic number fields F.

For a general quadratic problem, an analog is formulated as a homogeneous quadratic problem. The estimates ?* constructed based on Shor's dual quadratic estimates for these problems are proved to be equal. It is shown that, for the case of a homogeneous quadratic problem, finding ?* is reduced to an unconstraint minimization problem for a convex function.

We present algorithms for square classes, quadratic forms and Witt classes of quadratic forms over the field of rational functions of one variable over the reals. The algorithms are capable of: finding the unique representative of a square class, deciding if a given function is a square or a sum of squares and deciding if a quadratic form is isotropic or hyperbolic. Moreover we propose a representation for Witt classes of quadratic forms. With this representation one can manipulate Witt classes without operating directly on their coefficients. We present algorithms both for computing this representation and manipulating Witt classes.

Over commutative rings in which 2 is a zero-divisor, to compose a quadratic form with symmetric bilinear forms or with quadratic forms is not quite the same. In this article, the relation between the two classes of compositions is clarified and the results applied to find the ranks of minimal compositions.

Let $Q$ be a nondegenerate indefinite quadratic form on $\mathbb{R}^n$, $n\geq 3$, which is not a scalar multiple of a rational quadratic form, and let $C_Q=\{v\in \mathbb R^n | Q(v)=0\}$. We show that given $v_1\in C_Q$, for almost all $v\in C_Q \setminus \mathbb R v_1$ the following holds: for any $a\in \mathbb R$, any affine plane $P$ parallel to the plane of $v_1$ and $v$, and $\epsilon >0$ there exist primitive integral $n$-tuples $x$ within $\epsilon $ distance of $P$ for which $|Q(x)-a

Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Any finite set $S$ of closed points of $C$ gives rise to an integral domain $\mathcal{O}_S:=\mathbb{F}_q[C-S]$ in $K$. We show that given an $\mathcal{O}_S$-regular quadratic space $(V,q)$ of rank $n \geq 3$, the set of genera in the proper classification of quadratic $\mathcal{O}_S$-spaces isomorphic to $(V,q)$ in the flat or \'etale topology, is in $1:1$ correspondence with ${_2\text{Br}}(\mathcal{O}_S)$, thus there are $2^{|S|-1}$ such. If $(V,q)$ is isotropic, then $\text{Pic}(\mathcal{O}_S)/2$ classifies the forms in the genus of $(V,q)$. For $n \geq 5$ this is true for all genera, hence the full classification is via the abelian group $H^2_{\text{\'et}}(\mathcal{O}_S,\underline{\mu}_2)$., Comment: 19 pages, no figures

We study the quadratic Lagrange spectrum defined by Parkkonen and Paulin by considering the approximation by quadratic numbers whose regular continued fraction expansion is ultimately periodic with the same period as a fixed quadratic number or its Galois conjugate. We improve the upper bound on the approximation constants involved thereby proving a conjecture stated by Bugeaud.

Let q be a positive squarefree integer. A prime p is said to be q-admissible if the equation p = u2+ qv2has rational solutions u, v. Equivalently, p is q-admissible if there is a positive integer k such that [Formula: see text], where [Formula: see text] is the set of norms of algebraic integers in [Formula: see text]. Let k(q) denote the smallest positive integer k such that [Formula: see text] for all q-admissible primes p. It is shown that k(q) has subexponential but suprapolynomial growth in q, as q → ∞.

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over ${\mathbb Z}[\frac{1 + \sqrt{5}}{2}]$. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over ${\mathbb Z}[\frac{1 + \sqrt{5}}{2}]$, and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

Let K be a finite extension of F2. We consider quadratic forms written as the trace of xR(x), where R(x) is a linearized polynomial. We determine the K and R(x) where the form has a radical of codimension 2. This is applied to constructing maximal Artin–Schreier curves.

We introduce an invariant for nonsingular quadratic forms that take values in a Galois Ring of characteristic 4. This notion extends the invariant in Z8 for Z4-valued quadratic forms defined by Brown [E.H. Brown, Generalizations of the Kervaire invariant, Ann. of Math. (2) 95 (2) (1972) 368–383] and studied by Wood [J.A. Wood, Witt's extension theorem for mod four valued quadratic forms, Trans. Amer. Math. Soc. 336 (1) (1993) 445–461]. It is defined in the associated Galois Ring of characteristic 8. Nonsingular quadratic forms are characterized by their invariant and the type of the associated bilinear form (alternating or not).

Let N and M be quadratic ℤ-lattices, and K be a sublattice of N. A representation σ:K→M is said to be extensible to N if there exists a representation ρ:N→M such that ρ|K=σ. We prove in this paper a local–global principle for extensibility of representation, which is a generalization of the main theorems on representations by positive definite ℤ-lattices by Hsia, Kitaoka and Kneser (J. Reine Angew. Math. 301:132–141, 1978) and Jochner and Kitaoka (J. Number Theory 48:88–101, 1994). Applications to almost n-universal lattices and systems of quadratic equations with linear conditions are discussed.

A quadratic form f is said to have the semigroup property if its values at the points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with the semigroup property. If there is an integer bilinear map s such that f(s(x,y)) = f(x)f(y) for all vectors x and y from the integer two-dimensional lattice, then the form f has the semigroup property. We give an explicit integer parameterization of all pairs (f,s) with the property stated above. We do not know any other examples of forms with the semigroup property.

J. S. Hsia has conjectured an arithmetical version of Springer Theorem, which states that no two spinor genera in the same genus of integral quadratic forms become identified over an odd degree extension. In this paper we prove by examples that the corresponding result for quaternionic skew-hermitian forms does not hold in full generality. We prove that it does hold for unimodular skew-hermitian lattices under all extensions and for lattices whose discriminant is relatively prime to 2 under Galois extensions.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

A characterization is given for the integral binary quadratic forms for which the set of represented values is closed under products. It is also proved that for an integral binary quadratic lattice over a Dedekind domain, the product of three values represented by the form is again a value represented by the form. This generalizes the trigroup property observed by V. Arnold in the case of integral binary quadratic forms.

A positive definite integral quadratic form f is called n-regular if f represents every quadratic form of rank n that is represented by the genus of f. In this paper, we show that for any integer n greater than or equal to 27, every n-regular (even) form f is (even) n-universal, that is, f represents all (even, respectively) positive definite integral quadratic forms of rank n. As an application, we show that the minimal rank of n-regular forms has an exponential lower bound for n as it increases.

We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the non-selfadjoint operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper that under the assumption of ellipticity, as soon as the real part of their Weyl symbols is a non-zero non-positive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time., 26 pages

In the paper we find a set of necessary conditions that must be satisfied by a quadratic system in order to have an algebraic limit cycle. We find a countable set of ⩽5 parameter families of quadratic systems such that every quadratic system with an algebraic limit cycle must, after a change of variables, belong to one of those families. We provide a classification of all the quadratic systems which can have an algebraic limit cycle based on geometrical properties of the embedding of the system in the Poincaré compactification of R2. We propose names for all the classes we distinguish and we classify all known examples of quadratic systems with algebraic limit cycle. We also prove the integrability of certain classes of quadratic systems.

In this correspondence, we construct some new quadratic bent functions in polynomial forms by using the theory of quadratic forms over finite fields. The results improve some previous work. Moreover, we solve a problem left by Yu and Gong in 2006.

In this paper, we solve the Hua Loo-Keng problem with prime numbers representable by given primitive positive-definite binary quadratic forms whose discriminants coincide with those of the imaginary quadratic fields in which the quadratic forms decompose into linear factors.

A new quadratic kernel-free non-linear support vector machine (which is called QSVM) is introduced. The SVM optimization problem can be stated as follows: Maximize the geometrical margin subject to all the training data with a functional margin greater than a constant. The functional margin is equal to W T X + b which is the equation of the hyper-plane used for linear separation. The geometrical margin is equal to $$\frac{1}{||W||}$$ . And the constant in this case is equal to one. To separate the data non-linearly, a dual optimization form and the Kernel trick must be used. In this paper, a quadratic decision function that is capable of separating non-linearly the data is used. The geometrical margin is proved to be equal to the inverse of the norm of the gradient of the decision function. The functional margin is the equation of the quadratic function. QSVM is proved to be put in a quadratic optimization setting. This setting does not require the use of a dual form or the use of the Kernel trick. Comparisons between the QSVM and the SVM using the Gaussian and the polynomial kernels on databases from the UCI repository are shown.

Quadratic programming has been widely applied to solving real world problems. The conventional quadratic programming model requires the parameters to be known constants. In the real world, however, the parameters are seldom known exactly and have to be estimated. This paper discusses the interval quadratic programming problems where the cost coefficients, constraint coefficients, and right-hand sides, are represented by interval data. Since the parameters are interval-valued, the objective value is interval-valued as well. A pair of two-level mathematical programs is formulated to calculate the upper bound and lower bound of the objective values of the interval quadratic program. Based on the duality theorem and by applying the variable transformation technique, the pair of two-level mathematical programs is transformed into conventional one-level quadratic program. Solving the pair of quadratic programs produces the interval of the objective values of the problem. An example illustrates the whole idea and sheds some light on interval quadratic programming.

This article deals with minimizing quadratic functions with a special form of quadratic constraints that arise in 3D contact problems of linear elasticity with isotropic friction [Haslinger, J., Kucera, R. and Dostal, Z., 2004, An algorithm for the numerical realization of 3D contact problems with Coulomb friction. Journal of Computational and Applied Mathematics, 164/165, 387-408.]. The proposed algorithm combines the active set method with the conjugate gradient method. Its general scheme is similar to the algorithms of Polyak's type that solve the quadratic programming problems with simple bounds. As the algorithm does not terminate in a finite number of steps, the convergence is proved. The implementation uses an adaptive precision control of the conjugate gradient loops. Numerical experiments demonstrate the computational efficiency of the method.