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Agraïments: The second author is supported by AGAUR grant PIV-DGR-2010 and by FCT through CAMGDS, Lisbon. We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse. More precisely, a quadratic system having an invariant ellipse can be written into the form ˙x = x2+y2−1+y(ax+by+c), ˙y = −x(ax+by+c), and the ellipse becomes x2+y2 = 1. We prove that (i) this quadratic system is analytic integrable if and only if a = 0; (ii) if x2 + y2 = 1 is a periodic orbit, then this quadratic system is Liouvillian integrable if and only if x2 + y2 = 1 is not a limit cycle; and (iii) if x2 + y2 = 1 is not a periodic orbit, then this quadratic system is Liouvilian integrable if and only if a = 0.

We construct an infinite family of pairs of imaginary quadratic fields Q ( D ) and Q ( D + 1 ) with D ∈ Z whose class numbers are both divisible by 3.

A closed-form representation of the distribution function of the ratio of two linear combinations of Chi-squared variables is derived. The ratio is of the following form R = (X + aY)/(bY + Z), wher...

In this paper, a generalization of the quadratic manifold approach for the reduction of geometrically nonlinear structural dynamics problems is presented. This generalization is obtained by a linearization of the static force with respect to the generalized coordinates, resulting in a shift of the quadratic behavior from the force to the manifold. In this framework, static derivatives emerge as natural extensions to the modal derivatives for displacement fields other than the vibration modes, such as the Krylov subspace vectors. In the nonlinear projection framework employed here, the dynamic problem is projected onto the tangent space of the quadratic manifold, allowing for a much lower number of generalized coordinates compared to linear basis methods. The potential of the quadratic manifold approach is investigated in a numerical study, where several variations of the approach are compared on different examples, giving a clear indication of where the proposed approach is applicable.

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...

Ellipsoid bounds for strictly convex quadratic integer programs have been proposed in the literature. The idea is to underestimate the strictly convex quadratic objective function q of the problem by another convex quadratic function with the same continuous minimizer as q and for which an integer minimizer can be easily computed. We initially propose in this paper a different way of constructing the quadratic underestimator for the same problem and then extend the idea to other quadratic integer problems, where the objective function is convex (not strictly convex), and where the objective function is nonconvex and box constraints are introduced. The quality of the bounds proposed is evaluated experimentally and compared to the related existing methodologies.

Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power q, we present a class of linear codes over finite fields $$F_q$$ with quadratic forms via a general construction and then determine the explicit complete weight enumerators of these linear codes. Our construction covers some related ones via quadratic form functions and the linear codes may have applications in cryptography and secret sharing schemes.

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

In this paper, general quadratic forms of nonstationary, α-mixing time series are considered. Under mixing and moment assumptions, asymptotically normality of these forms are derived. These results...

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.

Quadratic residue is an important concept in number theory because it has both theoretical and practical application in mathematics and other areas such as computer science and communication. We also have a same concept of quadratic residue in general finite fields. Finding a square root of a quadratic residue in finite fields is an essential problem in computational algebra. In this paper we present an algorithm of computing square root of quadratic residue in finite fields using a primitive element.

Given an isotropic quadratic form over a number field which assumes a value $t$, we investigate the distribution of points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of quadratic forms, we produce bounds on height of points outside of some algebraic sets in a quadratic space at which the form assumes the value $t$. Our bounds on height are explicit in terms of the heights of the form, the space, the algebraic set and the value $t$., 11 pages; to appear in Journal of Number Theory

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)

Let φ1 and φ2 be quadratic forms over a field K of characteristic different from 2. In this paper, we complete the proof of the following result : if dim(φ1)=dim(φ2)=7, then φ1 is isotropic...

This paper is a sequel to [6] , in which we introduced quadratic forms on a module over a supertropical semiring R and analyzed the set of bilinear companions of a single quadratic form V → R in case the module V is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering , which is a partial order iff the semiring is “upper bound.” Any polynomial map q (or quadratic form) then induces a pre-order, which can be studied in terms of “ q -minimal elements,” which are elements a which cannot be written in the form b + c where b a but q ( b ) = q ( a ) . We determine the q -minimal elements by examining their support. But the class of all polynomial maps (in up to rank ( V ) variables) is itself a module over R , so the basic properties of the minimal ordering are applied to this R -module, or its submodule Quad ( V ) consisting of quadratic forms on V . This is a significant initial step in the classification of quadratic forms over semirings arising in tropical mathematics. Quad ( V ) is the sum of two disjoint submodules QL ( V ) and Rig ( V ) , consisting of the quasilinear and the rigid quadratic forms on V respectively (cf. [6] ). Both QL ( V ) and Rig ( V ) are free with explicitly known bases, but Quad ( V ) itself is almost never free.

The properties of surjective real quadratic maps are investigated. Sufficient conditions for the property of surjectivity to be stable under various perturbations are obtained. Examples of surjective quadratic maps whose surjectivity breaks down after an arbitrarily small perturbation are constructed. Sufficient conditions for quadratic maps to have nontrivial zeros are obtained. For a smooth even map in a neighbourhood of the origin an inverse function theorem in terms of the degree of the corresponding quadratic map is obtained. A canonical form of surjective quadratic maps from to is constructed. Bibliography: 27 titles.

This paper proposes quadratic log-aesthetic curves that are curves whose logarithmic curvature graphs are quadratic. In previous work, generalized log-aesthetic curves are derived by shifting either the curvature or the radius of curvature of log-aesthetic curves. Quadratic log-aesthetic curves are another generalization of log-aesthetic curves by making logarithmic curvature graphs quadratic. We derive the equations of quadratic log-aesthetic curves and clarify their characteristics. For drawing quadratic log-aesthetic curves, we need to compute the inverses of the error and imaginary error functions. We present a method for computing these inverses and confirm that the curves can be generated in real time.

An explicit formula is given for the representation number of each of the 75 reduced, positive-definite, integral, primitive, quaternary quadratic forms [Formula: see text], which belong to a genus with discriminant [Formula: see text] containing one and only one form class.

We give an interpretation of the reduction algorithm of Gauss in terms of carks, which are certain types of infinite ribbon graphs (or infinite dessins). We then describe an alternative reduction which is slightly faster than Gauss'. We also solve the minimal value problem and describe an algorithmic solution to the representation problem of indefinite binary 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.

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.

We determine the indexes of all orthogonal Grassmannians of a generic 16-dimensional quadratic form in $$I^3_q$$ . This is applied to show that the 3-Pfister number of the form is $$\ge $$ 4. Other consequences are: a new and characteristic-free proof of a recent result by Chernousov–Merkurjev on proper subforms in $$I^2_q$$ (originally available in characteristic 0) as well as a new and characteristic-free proof of an old result by Hoffmann–Tignol and Izhboldin–Karpenko on 14-dimensional quadratic forms in $$I^3_q$$ (originally available in characteristic $$\ne $$ 2). We also suggest an extension of the method, based on investigation of the topological filtration on the Grothendieck ring of a maximal orthogonal Grassmanian, which applies to quadratic forms of dimension higher than 16.

In this paper we prove a conjecture of David Masser on small height integral equivalence between integral quadratic forms. Using our resolution of Masser’s conjecture we show that integral orthogonal groups are generated by small elements which is essentially an effective version of Siegel’s theorem on the finite generation of these groups. We also obtain new estimates on reduction theory and representation theory of integral quadratic forms. Our line of attack is to make and exploit the connections between certain problems about quadratic forms and group actions, whence we may study the problem in the well-developed theory of homogeneous dynamics, arithmetic groups, and the spectral theory of automorphic forms.

We study simple quadratic approximations for general Hamilton-Jacobi-Bellman equations. The theoretical error bounds are shown to be composed of the time discretization errors and quadratic approximation ones at each time step. Several numerical experiments show that the quadratic approximations work for the equations with the boundary and Hamiltonian well approximated by the quadratic functions.

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.

In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Both oscillators are modeled as ordinary second-order differential equations with strong quadratic nonlinearities: one with positive quadratic term and the second with a quadratic term which changes the sign. Solutions for both equations are obtained in the form of Jacobi elliptic functions. For the asymmetric oscillator, conditions for the periodic motion are determined, while for the symmetric oscillator a new approximate solution procedure based on averaging is developed. Obtained results are tested on an optomechanical system where the motion of a cantilever in the intracavity field is oscillatory. Two types of quadratic nonlinearities in the system are investigated: symmetric and asymmetric. The advantage and disadvantage of both models is discussed. The analytical procedure suggested in the paper is applied. The obtained solution agrees well with a numerical one.

Let k be a field of characteristic distinct from 2, f 1 , f 2 , f 3 quadratic forms in 3 variables over k. We prove that these forms have a common zero over k if and only if the quadratic form u 1 f 1 + u 2 f 2 + u 3 f 3 is isotropic over the field k ( u 1 , u 2 , u 3 ) and the cubic form det ( t 1 f 1 + t 2 f 2 + t 3 f 3 ) in variables t 1 , t 2 , t 3 is isotropic over k. We also consider a similar question for n quadratic forms in 3 variables where n ≥ 4 .

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.

We show the solution for the quadratic wave equation associated to the harmonic oscillator exists over a longer time interval than the one given by local existence theory. The key point is that the nonlinearity is quadratic so that we can estimate the small divisor and perform a normal form process.

In the framework of integral quadratic constraints a more general stability theorem is derived by relaxing the assumptions on the uncertainty. In contrast to existing results, input-output relations are only required to hold on a subspace of the full signal space. This result is employed to incorporate uncertainties exhibiting sampling behavior into the framework of integral quadratic constraints. The arising infinite dimensional stability test is exactly reduced to a finite dimensional linear matrix inequality. Advantages of this approach are illustrated for the specific case of an interconnection including a pulse-width modulator.

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.

The Quadratic Finite Element Model Updating Problem (QFEMUP) concerns with updating a symmetric second-order finite element model so that it remains symmetric and the updated model reproduces a given set of desired eigenvalues and eigenvectors by replacing the corresponding ones from the original model. Taking advantage of the special structure of the constraint set, it is first shown that the QFEMUP can be formulated as a suitable constrained nonlinear programming problem. Using this formulation, a method based on successive optimizations is then proposed and analyzed. To avoid that spurious modes (eigenvectors) appear in the frequency range of interest (eigenvalues) after the model has been updated, additional constraints based on a quadratic Rayleigh quotient are dynamically included in the constraint set. A distinct practical feature of the proposed method is that it can be implemented by computing only a few eigenvalues and eigenvectors of the associated quadratic matrix pencil.

A theorem of Albert-Draxl states that if a tensor product of two quaternion division algebras $Q_1$, $Q_2$ over a field $F$ is not a division algebra, then there exists a separable quadratic extension of $F$ that embeds as a subfield in $Q_1$ and in $Q_2$. We establish a modified version of this result where the tensor product of quaternion algebras is replaced by the corestriction of a single quaternion algebra over a separable field extension. As a tool in the proof, we show that if the transfer of a nonsingular quadratic form $\varphi$ over a quadratic extension is isotropic for a linear functional $s$ such that $s(1)=0$, then $\varphi$ contains a nondegenerate subform defined over the base field.

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.

We prove a general theorem that evaluates the Legendre symbol [Formula: see text] under certain conditions on the integers A, B, m and the prime p. The evaluation is in terms of parameters appearing in a binary quadratic form representing p. The theorem has applications to quartic residuacity.

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 our attempt is to investigate the class number problem of imaginary quadratic fields. We establish that the class number of imaginary quadratic field ℚ(−q), for certain positive integer q, is a multiple of 3. We also show that there are infinitely many imaginary quadratic fields whose class numbers are multiples of 3.

International audience; We propose a solution approach for the general problem (QP) of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints. The resolution is based on the reformulation of the original problem (QP) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based on quadratic convex relaxation. We concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible. Furthermore, we extend our method to the case of mixed-integer quadratic problems with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. Finally, we illustrate the different results of the article by small examples and we present some computational experiments on pure-integer and mixed-integer instances of (QP). Most of the considered instances with up to 53 variables can be solved by our approach combined with the use of Cplex.

In this paper, we show that perturbing a simple 3-d quadratic system with a center-type singular point can yield at least 10 small-amplitude limit cycles around a singular point. This result improves the 7 limit cycles obtained recently in a simple 3-d quadratic system around a Hopf singular point. Compared with Bautin’s result for quadratic planar vector fields, which can only have 3 small-amplitude limit cycles around an elementary center or focus, this result of 10 limit cycles is surprisingly high. The theory and methodology developed in this paper can be used to consider bifurcation of limit cycles in higher-dimensional systems.

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.

Completing the study initiated by Mounaix and Collet [J. Stat. Phys. {\bf 143}, 139-147 (2011)], we investigate the realizations of a Gaussian random field in the limit where a given (general) quadratic form of the field is large. Concentration in $L^2$ and in probability is proved under mild conditions and the resulting quasi-deterministic behavior of the field is given. Applications to a large {\it local} quadratic form are considered in two specific cases. In particular, the quasi-deterministic structure of a Gaussian random flow with a large local helicity at some given point is determined explicitly., REVTeX file, 27 pages, submitted to J. Stat. Phys; acknowledgements specified

The mathematical approximation of scanned data by continuous functions like quadratic forms is a common task for monitoring the deformations of artificial and natural objects in geodesy. We model the quadratic form by using a high power structured errors-in-variables homogeneous equation. In terms of Euler-Lagrange theorem, a total least squares algorithm is designed for iteratively adjusting the quadratic form model. This algorithm is proven as a universal formula for the quadratic form determination in 2D and 3D space, in contrast to the existing methods. Finally, we show the applicability of the algorithm in a deformation monitoring.

Let G be a simple linear algebraic group over a field F, and V an absolutely irreducible representation of G. We show that under some mild hypotheses there exists an invariant homogeneous polynomial f for the action of G on V defined over F, such that twisted forms of f up to a scalar multiple classify twisted forms of G for which the representation V is defined over F. This result extends the classical case of a quadratic form q and its orthogonal group O ( q ) .