Representation of an Integer by a Quadratic Form through the Cornacchia Algorithm

Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2+y2=n). Starting from a quadratic form with two variables f(x,y)=ax2+bxy+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn/| D |) (possibly α itself) and the equation f(x,y)=n. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form n=ax2+bxy+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.

MAXIMUM TEST FOR A SEQUENCE OF QUADRATIC FORM STATISTICS ABOUT SCORE TEST IN LOGISTIC REGRESSION MODEL

This article proposes the maximum test for a sequence of quadratic form statistics about score test in logistic regression model which can be applied to genetic and medicine fields.Theoretical properties about the maximum test are derived.Extensive simulation studies are conducted to testify powers robustness of the maximum test compared to other two existed test.We also apply the maximum test to a real dataset about multiple gene variables association analysis.

CROSS-CORRELATIONS BETWEEN QUADRATIC FORM SEQUENCES AND GEOMETRIC SEQUENCES OVER FINITE FIELDS

For all types of quadratic forms,the cross-correlations between geometric sequences and the newly defined quadratic form sequences are determined to extend the results presented by Klapper in 1993 and 1997.The technique for computing cross-correlations is based on counting the number of solutions for a system of equations that consists of a quadratic form and a linear function.

Representations by Certain Sextenary Quadratic Forms Whose Coefficients Are 1, 2, 3 and 6

Here, we determine formulae, for the numbers of representations of a positive integer by certain sextenary quadratic forms whose coefficients are 1, 2, 3 and 6.

Generalized Theta-Functions with Characteristics and Cusp Form Corresponding to Quadratic Forms in Ten Variables

The modular properties of generalized theta-functions with characteristics are used to build cusp form corresponding to quadratic forms in ten variables.

On n-universal quadratic forms over dyadic local fields

Let n≥2 be an integer. We give necessary and sufficient conditions for an integral quadratic form over dyadic local fields to be n-universal by using invariants from Beli's theory of bases of norm generators.Also, we provide a minimal set for testing n-universal quadratic forms over dyadic local fields, as an analogue of Bhargava and Hanke's 290-theorem(or Conway and Schneeberger's 15-theorem) on universal quadratic forms with integer coefficients.

On Quasi-Reduced Quadratic Forms

With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.

On Binary Quadratic Forms Modulo n

Given a binary quadratic polynomial f(x_(1),x_(2))=αx_(1)^(2)+βx_(1)x_(2)+γx_(2)^(2)∈Z[x_(1),x_(2)],for every c∈Z and n≥2,we study the number of solutions NJ(f;c,n)of the congruence equation f(x_(1),x_(2))≡c mod n in(Z/nZ)^(2) such that xi∈(Z/nZ)^(×)for i∈J⊆{1,2}.

A Note on Legendre–Fenchel Conjugate of the Product of Two Positive-Definite Quadratic Forms

The Legendre–Fenchel conjugate of the product of two positive-definite quadratic forms was posted as an open question in the field of nonlinear analysis and optimization by Hiriart-Urruty[‘Question 11’in SIAM Review 49,255–273,2007].Under a convexity assumption on the function,it was answered by Zhao[SIAM J.Matrix Analysis&Applications 31(4),1792–1811,2010].In this note,we answer the open question without making the convexity assumption.

Representation of Numbers by Binary Quadratic Forms

Letf(x,y)=ax2+bxy+cy2,g(x,y)=Ax2+Bxy+Cy2,be two binary quadratic forms with real coefficients.A real number m is said to be represented by fif f(x,y)=m has a(rational)integer solution(x,y).We say f and g are equivalent if there exists aninteger matrlx(r s t u)with determinant±1 such that f(x′,y′)=g(x,y),where

MULTI-VALUED CROSSCORRELATION FUNCTION FOR m-SEQUENCES

Based on the theory of exponential sums an d quadratic forms over finite field, the crosscorrelation function values betwee n two maximal linear recursive sequences are determined under some conditions.

ON THE ABSOLUTE STABILITY OF A CLASS OF INDIRECT CONTROL SYSTEMS

The absolute stability of a class of indirect control systems was studied by applying the theory of Hermitian quadratic form and Jordan normal form. The algebraic formal criteria for the absolute stability are established, and these results are new and useful.

A class of exact solutions for N-dimensional incompressible magnetohydrodynamic equations

In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic （MHD） equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations axe directly constructed.

The 0 ,1 - Distribution of the First CoordinateSequences Derived from ML- Sequences over Z/(2~e)

The 0,1 distribution of the first coordinate sequences derived from the maximal lenght sequences (ML sequences) over Z/(2~e) is exactly given.

New Sarwate Bounds on Aperiodic Correlation of Sequences over pth Roots of Unity

The minimum aperiodic crosscorrelation of binary sequences of size M and length n over the alphabet E={1, -1} has been obtained by Levenshtein for M≥4 and n≥2 These bounds improve a long standing bound given by Welch. In this paper, the Sarwate bounds for codes over the p th roots of unity with the same parameters M and n are discussed, that is,the lower bounds and trade off are established for the maximum magnitude of the aperiodic crosscorrelation function and the maximum magnitude of the out of phase aperiodic autocorrelation function for the sets of periodic sequences with the same parameters M and n by using the modified Levenshtein method. The results show that new bounds are tighter than Sarwate bounds and Levenshtein bounds.

Cascaded Construction of Semi-Bent and Bent Functions

Based on the theory of quadratic forms over finite fields, a new construction of semi-bent and bent functions is presented. The proposed construction has a cascaded characteristic. Some previously known constructions of semi-bent and bent functions are special cases of the new construction.

A Nonparametric Model Checking Test for Functional Linear Composite Quantile Regression Models

This paper is focused on the goodness-of-fit test of the functional linear composite quantile regression model.A nonparametric test is proposed by using the orthogonality of the residual and its conditional expectation under the null model.The proposed test statistic has an asymptotic standard normal distribution under the null hypothesis,and tends to infinity in probability under the alternative hypothesis,which implies the consistency of the test.Furthermore,it is proved that the test statistic converges to a normal distribution with nonzero mean under a local alternative hypothesis.Extensive simulations are reported,and the results show that the proposed test has proper sizes and is sensitive to the considered model discrepancies.The proposed methods are also applied to two real datasets.

On formalism and stability of switched systems

In this paper, we formulate a uniform mathematical framework for studying switched systems with piecewise linear partitioned state space and state dependent switching. Based on known results from the theory of differential inclusions, we devise a Lyapunov stability theorem suitable for this class of switched systems. With this, we prove a Lyapunov stability theorem for piecewise linear switched systems by means of a concrete class of Lyapunov functions. Contrary to existing results on the subject, the stability theorems in this paper include Filippov （or relaxed） solutions and allow infinite switching in finite time. Finally, we show that for a class of piecewise linear switched systems, the inertia of the system is not sufficient to determine its stability. A number of examples are provided to illustrate the concepts discussed in this paper.

Elliptic curves and positive definite ternary forms

For two given ternary quadratic forms f( x, y, z) and g( x, y, z), let r( f, n) and r( g,n) be the numbers of representations of n represented by f( x, y, z) and g( x, y, z) respectively. In this paper we study the following problem: when will we have r( f, n) = r( g, n) or r( f, n) ≠ r( g, n).Our method is to use elliptic curves and the corresponding new forms.

A Refinement of the Weighted Hilbert Inequality

In this paper it is shown that a refinement on the weighted Hilbert inequalityfor double series can be established by introducing a proper non-zero real number R_ω.The expression of R_ω is given by means of the positive definiteness of a Gram matrix.