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.

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.

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.

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.

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.

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.

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.

Manin’s conjecture for a class of singular cubic hypersurfaces

Letℓ≥2 be a fixed positive integer and Q(y)be a positive definite quadratic form inℓvariables with integral coefficients.The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u^(3)=Q(y)z.We can get a power-saving result for a class of special quadratic forms and improve on some previous work.

On the constructions of constant-composition codes from perfect nonlinear functions

A new construction of constant-composition codes based on all known perfect nonlinear functions from Fqm to itself is presented, which provides a kind of unified constructions of constant-composition codes based on all known perfect nonlinear functions from F qm to itself.It is proved that the new constant-composition codes are optimal with respect to the Luo-Fu-Vinck-Chen bound, when m is an odd positive integer greater than 1.Finally, we point out that two constructions of constant-composition codes, proposed by Ding Cunsheng et al.in 2005, are equivalent to two special types of the new constant-composition codes.

Krein-space approach to robust fltering for uncertain systems with time-varying bias

In this paper, a novel Krein space approach to robust estimation for uncertain systems with accumulated bias is proposed. The bias is impacted by system uncertainties and exists in both state transition and observer matrices. Initial conditions and cross-correlated uncertainty inputs are described by the sum quadratic constraint （SQC）. Without modifying the SQC, the minimal state of the SQC is obtained through Krein space method. The inertia condition for a minimum of a deterministic quadratic form is derived when the coefficient of observer uncertainty input is non-unit matrix. Recursions of Krein space state filtering and bias filtering are developed respectively. Since the cross correlation between uncertainties is considered, a cross correlation gain is introduced into the posteriori estimator. Finally, a numerical example illustrates the performance of the proposed filter.

The Cartesian analytical solutions for the Ndimensional compressible Navier-Stokes equations with density-dependent viscosity

In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.

New families of p-ary sequences with low correlation and large linear span

This article presents a new family of p-ary sequences. The proposed sequences are proved to have not only low correlation property, but also large linear span and large family size. Furthermore, it shows that the new family of sequences contains Tang＇s construction as a subset if m-sequences are excluded from both constructions.

Optimal AK composite estimators in current population survey

The Current Population Survey(CPS)is a monthly household sample survey with a sample consisting of eight rotation groups.Sampled individuals of a rotation group are interviewed four consecutive months and another four consecutive months after resting eight consecutive months.A composite-type estimator is adopted in the CPS for the estimation of the monthly population total,which combines sample information from the current month’s survey and previous months using the fact that 75%households have data for two consecutive months.There are two values,A and K,in the composite estimator to decide how to combine the available information,and thus this estimator is called the AK composite estimator.In this paper,we use a formula of the meansquared error of AK composite estimator and propose an easy-to-use method of choosing A and K based on data,which evolves the estimation of some population quantities using the method of moments and replication.Some numerical studies are conducted to illustrate the effectiveness of the proposed method.

Realization of Numbers As the Degrees of Maps between Manifolds

We determine the set of degrees between some classes of oriented closed （n - 1）-connected 2n-manifolds by using the arithmetic theory of quadratic forms.

Subfield Codes of Linear Codes from Perfect Nonlinear Functions and Their Duals

Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)_(x∈F_(p)^(m)),T r(a)):a,b∈F_(p)^(m),c∈F_(p)}for f(x)=x^(2) and f(x)=x p k+1,respectively,where Tr(⋅)is the trace function from F_(p)^(m) to F_(p),and k is a nonnegative integer.In this paper,we further investigate the subfield code C f for f(x)being a known perfect nonlinear function over F_(p)^(m) and generalize some results in[17,35].The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields.In addition,the parameters of the duals of these codes are also determined.Several examples show that some of our codes and their duals have the best known parameters according to the code tables in[16].The duals of some proposed codes are optimal according to the Sphere Packing bound if p≥5.