Padé Approximations of and preservation of quadratic Lyapunov functions

We investigate whether or not quadratic Lyapunov functions are preserved under Padé approximations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetries of quadratic form classes and of quadratic surd continued fractions. Part I: A Poincaré tiling of the de Sitter world

The problem of classifying the indefinite binary quadratic forms with integer coefficients is solved by introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Under the action of the special linear group acting on the integer plane lattice, each class of indefinite forms has a well-defined finite number of representatives inside each such domain. In the second part, we will show how to obtain the symmetry type of a class and also the number of its points in all domains from a single representative of that class.     

Computation of the smallest positive eigenvalue of a quadratic λ-matrix

Summary The computation of the smallest positive eigenvalue ^* of a quadratic -matrix is used to determine the stability of structures. In addition to existence results we derive two efficient and reliable methods to calculate ^*. Both methods are based on shift techniques which are discussed thoroughly with respect to convergence.     

Hardy’s theorem for zeta-functions of quadratic forms

LetQ(u ₁,…,u ₁) =Σd _ij u _i u _j (i,j = 1 tol) be a positive definite quadratic form inl(≥3) variables with integer coefficientsd _ij (=d _ji ). Puts=σ+it and for σ>(l/2) write $$Z_Q (s) = \Sigma '(Q(u_1 ,...,u_l ))^{ - s} ,$$ where the accent indicates that the sum is over alll-tuples of integer (u ₁,…,u _l ) with the exception of (0,…, 0). It is well-known that this series converges for σ>(l/2) and that (s-(l/2))Z _Q (s) can be continued to an entire function ofs. Let σ be any constant with 0<σ<1/100. Then it is proved thatZ _Q (s)has ?_δTlogT zeros in the rectangle(|σ-1/2|≤δ, T≤t≤2T).     

Γ-Convergence of some super quadratic functionals with singular weights

We study the Γ-convergence of the following functional (p > 2)

On the use of simplex methods in constructing quadratic models

In this paper,we investigate the quadratic approximation methods.After studying the basic idea of simplex methods,we construct several new search directions by combining the local information progressively obtained during the iterates of the algorithm to form new subspaces.And the quadratic model is solved in the new subspaces.The motivation is to use the information disclosed by the former steps to construct more promising directions.For most tested problems,the number of function evaluations have been reduced obviously through our algorithms.     

Visualization of the $$\varepsilon $$-subdifferential of piecewise linear–quadratic functions

Computing explicitly the \(\varepsilon \)-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class of (convex univariate) piecewise linear–quadratic functions for which existing numerical libraries allow practical computations. We visualize the results in a primal, dual, and subdifferential views through several numerical examples. We also provide a visualization of the Brøndsted–Rockafellar theorem.     

Direct-dual Fejér methods for problems of quadratic programming

The paper deals with the S-technology, which reduces convex problems of quadratic programming to the solution of systems of several linear, and one convex, inequalities. A certain variant of the Fejér method is applied to these systems. In particular, the problem of the constructive separability of convex polyhedral sets by a layer of maximal thickness is solved. This algorithm plays an important role in problems of discriminant analysis.     

Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion

Dzhaparidze and Spreij (Stoch Process Appl, 54:165–174, 1994) showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same approximation is valid for a special class of continuous stochastic processes. This class contains both semimartingales and non-semimartingales. The motivation comes partially from the recent work by Bender et al. (Finance Stoch, 12:441–468, 2008), where it is shown that the quadratic variation of the log-returns determines the hedging strategy.     

Some generalization of Cauchy’s and the quadratic functional equations

We find the solutions ${f,g,h \colon S \to H}$ of each of the functional equations $$\sum\limits_{\lambda \in \Lambda} f(x+\lambda y)=|\Lambda| g(x)+h(y),\quad x,y\in S,$$ where (S,?+) is an abelian semigroup, Λ is a finite subgroup of the automorphism group of S,?(H,?+) is an abelian group.     

Solvable quadratic Lie algebras

A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.     

The effect of torsion on the distribution of Ш among quadratic twists of an elliptic curve

Let E be an elliptic curve of rank zero defined over Q and ? an odd prime number. For E of prime conductor N, in Quattrini (2006) [Qua06], we remarked that when ?||ETor(Q)|, there is a congruence modulo ? among a modular form of weight 3/2 corresponding to E and an Eisenstein series. In this work we relate this congruence in weight 3/2, to a well-known one occurring in weight 2, which arises when E(Q) has an ? torsion point. For N prime, we prove that this last congruence can be lifted to one involving eigenvectors of Brandt matrices Bp(N) in the quaternion algebra ramified at N and infinity. From this follows the congruence in weight 3/2. For N square free we conjecture on the coefficients of a weight 3/2 Cohen-Eisenstein series of level N, which we expect to be congruent to the weight 3/2 modular form corresponding to E.     

Polynomial behavior of special values of partial zeta functions of real quadratic fields at s = 0

We compute the special values of partial zeta functions at s = 0 for family of real quadratic fields K _n and ray class ideals ${\mathfrak{b}_n}$ such that ${\mathfrak{b}_n^{-1} = [1, \delta(n)]}$ where the continued fraction expansion of δ(n) ? 1 is purely periodic and terms are polynomials in n of degree bounded by d. With additional assumptions, we prove that the special values of the partial zeta functions at s = 0 are given by a quasi-polynomial of degree less than or equal to d as a function of n. We apply this to conclude that the special values of the Hecke’s L-functions at s = 0 for the family ${(K_n, \mathfrak{b}_n, \chi_n:= \chi \circ N_{K_n/\mathbb{Q}})}$ for any Dirichlet character χ behave like quasi-polynomial as well. We compute explicitly the coefficients of the quasi-polynomials. Two examples satisfying the condition are presented, and for these two families, the special values of the partial zeta functions at s = 0 are given.     

Global existence of small solutions to quadratic nonlinear schrödinger equations

Abstract. We study the global existence of small solutions to some quadratic nonlinear Schrödinger equations in three or four space dimensions.     

Effect of Cabri-assisted instruction on secondary school students’ misconceptions about graphs of quadratic functions

Pre-existing misconceptions are serious impediments to learning in mathematics. Means for detecting and correcting them have received much attention in the literature of educational research. Dynamic geometry software has been tried at different grade levels. This quasi-experimental study investigates the effect of Cabri-assisted instruction on tenth graders’ misconceptions about graphs of quadratic functions. The results indicate that Cabri-assisted instruction did not have a statistically significant effect on their misconceptions, but the misconception scores of the experimental group were better than those of the control group. Also, there was a significant difference between the achievement scores of the two groups. Scores of the experimental group on a quadratic functions test were significantly higher than those of the control group. In conclusion, some implications and limitations of the study are discussed.     

How regular can the boundary of a quadratic Siegel disk be?

In the family of quadratic polynomials with an irrationally indifferent fixed point, we show the existence of Siegel disks with a fine control on the degree of regularity of the linearizing map on their boundary. A general theorem is stated and proved. As a particular case, we show that in the quadratic family, there are Siegel disks whose boundaries are but not Jordan curves.

     

On triviality of the Kashiwara–Vergne problem for quadratic Lie algebras

We show that the Kashiwara–Vergne (KV) problem for quadratic Lie algebras (that is, Lie algebras admitting an invariant scalar product) reduces to the problem of representing the Campbell–Hausdorff series in the form $\ln (e^x{e}^y)=x+y+[x,a(x,y)]+[y,b(x,y)]$, where $a(x,y)$ and $b(x,y)$ are Lie series in x and y. This observation explains the existence of explicit rational solutions of the quadratic KV problem, whereas constructing an explicit rational solution of the full KV problem would probably require the knowledge of a rational Drinfeld associator. It also gives, in the case of quadratic Lie algebras, a direct proof of the Duflo theorem (implied by the KV problem). To cite this article: A. Alekseev, C. Torossian, C. R. Acad. Sci. Paris, Ser. I 347 (2009).