Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygon...Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygonal numbers of consecutive orders in matrix form. We also try to find the solution of a Diophantine equation in terms of polygonal numbers.展开更多
In this paper,the problems of forward reachable set estimation and safety verification of uncertain nonlinear systems with polynomial dynamics are addressed.First,an iterative sums of squares(SOS)programming approach ...In this paper,the problems of forward reachable set estimation and safety verification of uncertain nonlinear systems with polynomial dynamics are addressed.First,an iterative sums of squares(SOS)programming approach is developed for reachable set estimation.It characterizes the over-approximations of the forward reachable sets by sub-level sets of time-varying Lyapunovlike functions that satisfy an invariance condition,and formulates the problem of searching for the Lyapunov-like functions as a bilinear SOS program,which can be solved via an iterative algorithm.To make the over-approximation tight,the proposed approach seeks to minimize the volume of the overapproximation set with a desired shape.Then,the reachable set estimation approach is extended for safety verification,via explicitly encoding the safety constraint such that the Lyapunov-like functions guarantee both reaching and avoidance.The efficiency of the presented method is illustrated by some numerical examples.展开更多
The four-parameter lag-lead compensator design has received much attention in the last two decades. However, most approaches have been either trial-and-error or only for special cases. This paper presents a non-trial-...The four-parameter lag-lead compensator design has received much attention in the last two decades. However, most approaches have been either trial-and-error or only for special cases. This paper presents a non-trial-and-error design method for four-parameter lag-lead compensators. Here, the compensator design problem is formulated into a polynomial function optimization problem and solved by using the recently developed sum-of-squares (SOS) techniques. This result not only provides a useful design method but also shows the power of the SOS techniques.展开更多
We characterize all functions f : N → C such that f(m^2 + n^2) = f(m)^2 + f(n)^2 for all m, n ∈ N. It turns out that all such functions can be grouped into three families, namely f ≡ 0, f(n) = ±n (...We characterize all functions f : N → C such that f(m^2 + n^2) = f(m)^2 + f(n)^2 for all m, n ∈ N. It turns out that all such functions can be grouped into three families, namely f ≡ 0, f(n) = ±n (subject to some restrictions on when the choice of the sign is possible) and f(n) =±l/2(again subject to some restrictions on when the choice of the sign is possible).展开更多
In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
Minimization of the weighted nonlinear sum of squares of differences may be converted to the minimization of sum of squares. The Gauss-Newton method is recalled and the length of the step of the steepest descent metho...Minimization of the weighted nonlinear sum of squares of differences may be converted to the minimization of sum of squares. The Gauss-Newton method is recalled and the length of the step of the steepest descent method is determined by substituting the steepest descent direction in the Gauss-Newton formula. The existence of minimum is shown.展开更多
The quartic minimization over the sphere is an NP-hard problem in the general case.There exist various methods for computing an approximate solution for any given instance.In practice,it is quite often that a global o...The quartic minimization over the sphere is an NP-hard problem in the general case.There exist various methods for computing an approximate solution for any given instance.In practice,it is quite often that a global optimal solution was found but without a certification.We will present in this article two classes of methods which are able to certify the global optimality,i.e.,algebraic methods and semidefinite program(SDP)relaxation methods.Several advances on these topics are summarized,accompanied with some emerged new results.We want to emphasize that for mediumor large-scaled instances,the problem is still a challenging one,due to an apparent limitation on the current force for solving SDP problems and the intrinsic one on the approximation techniques for the problem.展开更多
This paper presents an Expanding Annular Domain(EAD)algorithm combined with Sum of Squares(SOS)programming to estimate and maximize the domain of attraction(DA)of power systems.The proposed algorithm can systematicall...This paper presents an Expanding Annular Domain(EAD)algorithm combined with Sum of Squares(SOS)programming to estimate and maximize the domain of attraction(DA)of power systems.The proposed algorithm can systematically construct polynomial Lyapunov functions for power systems with transfer conductance and reliably determine a less conservative approximated DA,which are quite difficult to achieve with traditional methods.With linear SOS programming,we begin from an initial estimated DA,then enlarge it by iteratively determining a series of so-called annular domains of attraction,each of which is characterized by level sets of two successively obtained Lyapunov functions.Moreover,the EAD algorithm is theoretically analyzed in detail and its validity and convergence are shown under certain conditions.In the end,our method is tested on two classical power system cases and is demonstrated to be superior to existing methods in terms of computational speed and conservativeness of results.展开更多
Suppose f,g1,…,gm are multivariate polynomials in x∈R^(n)and their degrees are at most 2d.Consider the problem:Minimize f(x)subject to g1(x)≥0,…,gm(x)≥0.Let f_(min)(resp.,f_(max))be the minimum(resp.,maximum)of f...Suppose f,g1,…,gm are multivariate polynomials in x∈R^(n)and their degrees are at most 2d.Consider the problem:Minimize f(x)subject to g1(x)≥0,…,gm(x)≥0.Let f_(min)(resp.,f_(max))be the minimum(resp.,maximum)of f on the feasible set S,and f_(sos)be the lower bound of_(fmin)given by Lasserre’s relaxation of order d.This paper studies its approximation bound.Under a suitable condition on g1,…,gm,we prove that(f_(max)−f_(sos))≤Q(f_(max)−f_(min))with Q a constant depending only on g1,…,gm but not on f.In particular,if S is the unit ball,Q=O(n^(d));if S is the hypercube,Q=O(n^(2d));if S is the boolean set,Q=O(n^(d)).展开更多
In this article, we introduce a robust sparse test statistic which is based on the maximum type statistic. Both the limiting null distribution of the test statistic and the power of the test are analysed. It is shown ...In this article, we introduce a robust sparse test statistic which is based on the maximum type statistic. Both the limiting null distribution of the test statistic and the power of the test are analysed. It is shown that the test is particularly powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature. The numerical results show that the test proposed significantly outperforms those tests in a range of settings, especially for sparse alternatives.展开更多
文摘Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygonal numbers of consecutive orders in matrix form. We also try to find the solution of a Diophantine equation in terms of polygonal numbers.
基金supported in part by the National Natural Science Foundation of China under Grant Nos.12171159 and 61772203in part by the Zhejiang Provincial Natural Science Foundation of China under Grant No.LY20F020020。
文摘In this paper,the problems of forward reachable set estimation and safety verification of uncertain nonlinear systems with polynomial dynamics are addressed.First,an iterative sums of squares(SOS)programming approach is developed for reachable set estimation.It characterizes the over-approximations of the forward reachable sets by sub-level sets of time-varying Lyapunovlike functions that satisfy an invariance condition,and formulates the problem of searching for the Lyapunov-like functions as a bilinear SOS program,which can be solved via an iterative algorithm.To make the over-approximation tight,the proposed approach seeks to minimize the volume of the overapproximation set with a desired shape.Then,the reachable set estimation approach is extended for safety verification,via explicitly encoding the safety constraint such that the Lyapunov-like functions guarantee both reaching and avoidance.The efficiency of the presented method is illustrated by some numerical examples.
基金Supported in part by the National High-Tech Research and Development (863) Program of China (Nos.2007AA11Z215 and 2007AA11Z222)the National Key Technology Research and Development Program (No.2006CBJ18B02)
文摘The four-parameter lag-lead compensator design has received much attention in the last two decades. However, most approaches have been either trial-and-error or only for special cases. This paper presents a non-trial-and-error design method for four-parameter lag-lead compensators. Here, the compensator design problem is formulated into a polynomial function optimization problem and solved by using the recently developed sum-of-squares (SOS) techniques. This result not only provides a useful design method but also shows the power of the SOS techniques.
基金Supported by the Ministry of Science and Technological Development of Serbia(Grant No.174006)
文摘We characterize all functions f : N → C such that f(m^2 + n^2) = f(m)^2 + f(n)^2 for all m, n ∈ N. It turns out that all such functions can be grouped into three families, namely f ≡ 0, f(n) = ±n (subject to some restrictions on when the choice of the sign is possible) and f(n) =±l/2(again subject to some restrictions on when the choice of the sign is possible).
基金Supported by the National Natural Science Foundation of China(No.11271300)
文摘In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
文摘Minimization of the weighted nonlinear sum of squares of differences may be converted to the minimization of sum of squares. The Gauss-Newton method is recalled and the length of the step of the steepest descent method is determined by substituting the steepest descent direction in the Gauss-Newton formula. The existence of minimum is shown.
基金This work is partially supported by the National Natural Science Foundation of China(No.11771328)Young Elite Scientists Sponsorship Program by Tianjin,and the Natural Science Foundation of Zhejiang Province,China(No.LD19A010002).
文摘The quartic minimization over the sphere is an NP-hard problem in the general case.There exist various methods for computing an approximate solution for any given instance.In practice,it is quite often that a global optimal solution was found but without a certification.We will present in this article two classes of methods which are able to certify the global optimality,i.e.,algebraic methods and semidefinite program(SDP)relaxation methods.Several advances on these topics are summarized,accompanied with some emerged new results.We want to emphasize that for mediumor large-scaled instances,the problem is still a challenging one,due to an apparent limitation on the current force for solving SDP problems and the intrinsic one on the approximation techniques for the problem.
基金supported in part by the State Key Program of National Natural Science Foundation of China under Grant No.U1866210Young Elite Scientists Sponsorship Program by CSEE under Grant No.CSEE-YESS-2018007.
文摘This paper presents an Expanding Annular Domain(EAD)algorithm combined with Sum of Squares(SOS)programming to estimate and maximize the domain of attraction(DA)of power systems.The proposed algorithm can systematically construct polynomial Lyapunov functions for power systems with transfer conductance and reliably determine a less conservative approximated DA,which are quite difficult to achieve with traditional methods.With linear SOS programming,we begin from an initial estimated DA,then enlarge it by iteratively determining a series of so-called annular domains of attraction,each of which is characterized by level sets of two successively obtained Lyapunov functions.Moreover,the EAD algorithm is theoretically analyzed in detail and its validity and convergence are shown under certain conditions.In the end,our method is tested on two classical power system cases and is demonstrated to be superior to existing methods in terms of computational speed and conservativeness of results.
基金The research was partially supported by the National Science Foundation grant DMS-0844775.
文摘Suppose f,g1,…,gm are multivariate polynomials in x∈R^(n)and their degrees are at most 2d.Consider the problem:Minimize f(x)subject to g1(x)≥0,…,gm(x)≥0.Let f_(min)(resp.,f_(max))be the minimum(resp.,maximum)of f on the feasible set S,and f_(sos)be the lower bound of_(fmin)given by Lasserre’s relaxation of order d.This paper studies its approximation bound.Under a suitable condition on g1,…,gm,we prove that(f_(max)−f_(sos))≤Q(f_(max)−f_(min))with Q a constant depending only on g1,…,gm but not on f.In particular,if S is the unit ball,Q=O(n^(d));if S is the hypercube,Q=O(n^(2d));if S is the boolean set,Q=O(n^(d)).
基金supported by the National Natural Science Foundation of China(Grant No.11571052)Social Science Research Foundation of Hu’nan Provincial Department(Grant No.15YBA066)Outstanding Youth Foundation of Hu’nan Provincial Department of Education(Grant No.17B047)
文摘In this article, we introduce a robust sparse test statistic which is based on the maximum type statistic. Both the limiting null distribution of the test statistic and the power of the test are analysed. It is shown that the test is particularly powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature. The numerical results show that the test proposed significantly outperforms those tests in a range of settings, especially for sparse alternatives.