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.
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...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.展开更多
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 ...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.展开更多
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 equivalen...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展开更多
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 mo...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}.展开更多
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 Revie...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.展开更多
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 max...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.展开更多
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.
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 expr...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.展开更多
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 establi...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.展开更多
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 giv...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.展开更多
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 ...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.展开更多
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 expectatio...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.展开更多
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 al...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.展开更多
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 differe...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.展开更多
Let Q(x) be a nonnegative definite, symmetric matrix such that √Q(X) is Lipschitz con- tinuous. Given a real-valued function b(x) and a weak solution u(x) of div(QVu) = b, we find sufficient conditions in o...Let Q(x) be a nonnegative definite, symmetric matrix such that √Q(X) is Lipschitz con- tinuous. Given a real-valued function b(x) and a weak solution u(x) of div(QVu) = b, we find sufficient conditions in order that √Qu has some first order smoothness. Specifically, if is a bounded open set in Rn, we study when the components of vVu belong to the first order Sobolev space W1'2(Ω) defined by Sawyer and Wheeden. Alternately we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of ^-^-1 X^Xiu + b = O. We do not assume that {Xi}is a HSrmander collection of vector fields in ~. The results signal ones for more general equations.展开更多
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 d...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.展开更多
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 integra...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.展开更多
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 m...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.展开更多
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 ...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.展开更多
文摘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.
基金supported by National Natural Science Foundation of China (Grant No. 12171223)the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021A1515010396)。
文摘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.
文摘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.
文摘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
基金supported by National Natural Science Foundation of China(Grant No.11571328).
文摘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}.
基金This research was supported by National Natural Science Foundation of China(Nos.11001006 and 91130019/A011702)by the Fundamental Research Funds for the Central Universities(No.YWF-13-A01)+1 种基金d by the Fund of State Key Laboratory of Software Development Environment(No.SKLSDE-2013ZX-13)The author is grateful to the two anonymous referees whose comments improved this paper.The author also thanks Prof.Yun-Bin Zhao,Prof.Henry Wolkowicz and his doctoral student Minghua Lin for valuable comments.
文摘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.
基金This work of Jiayan Zhu is partially supported by seeding project funding(2019ZZX026)scientific research project funding of talent recruitment,and start up funding for scientific research of Hubei University of Chinese MedicineThis work of Zhengbang Li is partially supported by self-determined research funds of Central China Normal University from colleges'basic research of MOE(CCNU18QN031).
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
基金The Starting Research Projects for Young Teachers of Southwest Jiaotong University (No.2007Q090)
文摘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.
基金supported by the Natural Science Foundation of China under Grant Nos.11271014 and 11971045。
文摘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.
基金Supported in part by the National Natural Science Foundation of China (Grant Nos 60573028, 60803156)the Open Research Fund of the National Mobile Communications Research Laboratory of Southeast University (Grant No W200805)in part by Singapore Ministry of Education Academic Research Fund (Grant No T206B2204)
文摘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.
基金supported by the Danish Council for Technology and Innovation
文摘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.
文摘Let Q(x) be a nonnegative definite, symmetric matrix such that √Q(X) is Lipschitz con- tinuous. Given a real-valued function b(x) and a weak solution u(x) of div(QVu) = b, we find sufficient conditions in order that √Qu has some first order smoothness. Specifically, if is a bounded open set in Rn, we study when the components of vVu belong to the first order Sobolev space W1'2(Ω) defined by Sawyer and Wheeden. Alternately we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of ^-^-1 X^Xiu + b = O. We do not assume that {Xi}is a HSrmander collection of vector fields in ~. The results signal ones for more general equations.
基金supported by the National Natural Science Foundation of China(Grant No.11971476).
文摘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.
基金This research is partially supported by the National Science Foundation of China(Grant No.11271079,10671095)RG 11/2015-2016R from the Education University of Hong Kong。
文摘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.
基金supported by the Fundamental Research Funds for the Central Universities(No.DL13BB14)
文摘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.
基金the Major Research plan of the National Natural Science Foundation of China(90604023);the Natural Science Foundation of Beijing(4072020);the National Research Foundation for the Doctoral Program of Higher Education of China(20040013007); the National High Technology Research and Development Program of China(2006AA01Z419).
文摘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.