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Acta Mathematicae Applicatae Sinica(English Series) 2021 Vol.37

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Some Explorations on Two Conjectures About Rademacher Sequences
Ze-chun HU, Guo-lie LAN, Wei SUN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 1-16.   DOI: 10.1007/s10255-021-0993-0
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In this paper, we explore two conjectures about Rademacher sequences. Let (εi) be a Rademacher sequence, i.e., a sequence of independent {-1, 1}-valued symmetric random variables. Set Sn=a1ε1+…+anεn for a=(a1, …, an) ∈ Rn. The first conjecture says that P (|Sn|≤||a||) ≥ 1/2 for all a ∈ Rn and n ∈ N. The second conjecture says that P (|Sn|≥||a||) ≥ 7/32 for all a ∈ Rn and n ∈ N. Regarding the first conjecture, we present several new equivalent formulations. These include a topological view, a combinatorial version and a strengthened version of the conjecture. Regarding the second conjecture, we prove that it holds true when n ≤ 7.
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Estimating Differences in Restricted Mean Lifetime Using Additive Hazards Models under Dependent Censoring
Qun LI, Bao-xue ZHANG, Liu-quan SUN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 17-34.   DOI: 10.1007/s10255-021-0987-y
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In epidemiological and clinical studies, the restricted mean lifetime is often of direct interest quantity. The differences of this quantity can be used as a basis of comparing several treatment groups with respect to their survival times. When the factor of interest is not randomized and lifetimes are subject to both dependent and independent censoring, the imbalances in confounding factors need to be accounted. We use the mixture of additive hazards model and inverse probability of censoring weighting method to estimate the differences of restricted mean lifetime. The average causal effect is then obtained by averaging the differences in fitted values based on the additive hazards models. The asymptotic properties of the proposed method are also derived and simulation studies are conducted to demonstrate their finite-sample performance. An application to the primary biliary cirrhosis (PBC) data is illustrated.
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Number of Induced Matchings of Graphs
Yan CHEN, Yan LIU
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 35-47.   DOI: 10.1007/s10255-021-0996-x
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A matching M of a graph G is an induced matching if no two edges in M are joined by an edge of G. Let iz(G) denote the total number of induced matchings of G, named iz-index. It is well known that the Hosoya index of a graph is the total number of matchings and the Hosoya index of a path can be calculated by the Fibonacci sequence. In this paper, we investigate the iz-index of graphs by using the Fibonacci-Narayana sequence and characterize some types of graphs with minimum and maximum iz-index, respectively.
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Uniform Lipschitz Estimates of Homogenization of Elliptic Systems in Divergence Form with Dini Conditions
Rong DONG, Dong-sheng LI, Hai-liang ZHANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 48-68.   DOI: 10.1007/s10255-021-1001-4
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The paper is devoted to the homogenization of elliptic systems in divergence form. We obtain uniform interior as well as boundary Lipschitz estimates in a bounded C1,γ domain when the coefficients are Dini continuous, inhomogeneous terms are divergence of Dini continuous functions and the boundary functions have Dini continuous derivatives. The results extend Avellaneda and Lin's work[Comm. Pure Appl. Math., 40:803-847 (1987)], where Hölder continuity is the main assumption on smoothness of the data.
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p-arrangeable Graphs are Folkman Linear
Xun CHEN, Qi-zhong LIN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 69-74.   DOI: 10.1007/s10255-021-1000-5
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For graphs F and G, let F → (G, G) denote that any red/blue edge coloring of F contains a monochromatic G. Define Folkman number f(G; t) to be the smallest order of a graph F such that F → (G, G) and ω(F) ≤ t. It is shown that f(G; t) ≤ cn for p-arrangeable graphs with n vertices, where p ≥ 1, c=c(p) and t=t(p) are positive constants.
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A Necessary and Sufficient Condition for the Solvability of the Nonlinear Schrödinger Equation on a Finite Interval
Ruo-meng LI, Xian-guo GENG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 75-100.   DOI: 10.1007/s10255-021-0994-z
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The admissibility of the initial-boundary data, which characterizes the existence of solution for the initial-boundary value problem, is important. Based on the Fokas method and the inverse scattering transformation, an approach is developed to solve the initial-boundary value problem of the nonlinear Schrödinger equation on a finite interval. A necessary and sufficient condition for the admissibility of the initial-boundary data is given, and the reconstruction of the potential is obtained.
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Maximum Likelihood Estimator of the Location Parameter under Moving Extremes Ranked Set Sampling Design
Wang-xue CHEN, Chun-xian LONG, Rui YANG, Dong-sen YAO
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 101-108.   DOI: 10.1007/s10255-021-0998-8
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Cost effective sampling design is a problem of major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming. In the current paper, a modification of ranked set sampling (RSS) called moving extremes RSS (MERSS) is considered for the estimation of the location parameter for location family. A maximum likelihood estimator (MLE) of the location parameter for this family is studied and its properties are obtained. We prove that the MLE is an equivariant estimator under location transformation. In order to give more insight into the performance of MERSS with respect to (w.r.t.) simple random sampling (SRS), the asymptotic efficiency of the MLE using MERSS w.r.t. that using SRS is computed for some usual location distributions. The relative results show that the MLE using MERSS can be real competitors to the MLE using SRS.
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Modelling the Effects of Pest Control with Development of Pesticide Resistance
Bing LIU, Bao-lin KANG, Feng-mei TAO, Gang HU
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 109-125.   DOI: 10.1007/s10255-021-0988-x
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In this paper, we assume that the pest population is divided into susceptible pests and infected pests, and only susceptible pests do harm to crops. Considering the two methods of spraying pesticides and releasing infected pests and natural enemies to control susceptible pests (the former is applied more frequently), and assuming that only susceptible pests develop resistance to pesticides, a pest control model with resistance development is established. By using the basic theory of impulsive differential systems and analytical methods, the sufficient condition for the global attractiveness of the susceptible pest eradication periodic solution is given. Combined with numerical simulations, the effects of spraying frequency of pesticides on critical threshold conditions for eradicating susceptible pests are discussed. The results confirm that it is not that the more frequently the pesticides are sprayed, the better the result of the pest control is. Two control strategies for eradicating susceptible pests are proposed:switching pesticides and releasing natural enemies elastically. Finally, the parameters in the critical threshold are analyzed from the following two aspects:(1) The key factors affecting pest control are determined by parameter sensitivity analyses. The results indicate that the correlation of the critical threshold concerning the killing efficiency rate and the decay rate of pesticides to susceptible pests varies due to the resistance development of susceptible pests. (2) Three-dimensional graphs and contours of susceptible pest eradication critical threshold with two parameters are simulated, and the effects of the main parameters on the critical threshold are analyzed.
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Convergence of Stochastic Gradient Descent in Deep Neural Network
Bai-cun ZHOU, Cong-ying HAN, Tian-de GUO
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 126-136.   DOI: 10.1007/s10255-021-0991-2
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Stochastic gradient descent (SGD) is one of the most common optimization algorithms used in pattern recognition and machine learning. This algorithm and its variants are the preferred algorithm while optimizing parameters of deep neural network for their advantages of low storage space requirement and fast computation speed. Previous studies on convergence of these algorithms were based on some traditional assumptions in optimization problems. However, the deep neural network has its unique properties. Some assumptions are inappropriate in the actual optimization process of this kind of model. In this paper, we modify the assumptions to make them more consistent with the actual optimization process of deep neural network. Based on new assumptions, we studied the convergence and convergence rate of SGD and its two common variant algorithms. In addition, we carried out numerical experiments with LeNet-5, a common network framework, on the data set MNIST to verify the rationality of our assumptions.
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Reflection and Refraction of Waves Across an Interface of Two-phase Flow
Kai HU, Hui KAN, Chun-lei TANG, Xiao-zhou YANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 137-147.   DOI: 10.1007/s10255-021-0992-1
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We investigate a hyperbolic system of one-dimensional isothermal fluid with liquid-vapor phase transition. The refraction-reflection phenomena are intensively analyzed when elementary waves travel across the two-phase interface. We apply the characteristic method and hodograph transform of Riemann to reduce the nonlinear PDEs to a concise form. Specially for the case of incident rarefaction wave, reduced linear equations are convenient to solve by Laplace transform. Then an integral formula in wave interaction region is derived in this paper, instead of the hypergeometric functions solutions for non-isothermal polytropic gases. It is also observed that when incident waves travel from the vapor phase to the liquid phase, the refracted waves must be accelerated and move forward.
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An Explicit Lower Bound for Blow Up Time in a Class of Nonlinear Wave Equations with Nonlinear Damping and Source Terms
Xiao-ming PENG, Ya-dong SHANG, Xue-qin WANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 148-154.   DOI: 10.1007/s10255-021-0995-y
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This paper deals with an initial boundary value problem for a class of nonlinear wave equation with nonlinear damping and source terms whose solution may blow up in finite time. An explicit lower bound for blow up time is determined by means of a differential inequality argument if blow up occurs.
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Multiple Solutions for the Klein-Gordon-Maxwell System with Steep Potential Well
Xiao-qi LIU, Chun-lei TANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 155-165.   DOI: 10.1007/s10255-021-0986-z
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In this paper, we concern the Klein-Gordon-Maxwell system with steep potential well
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Without global and local compactness, we can tell the difference of multiple solutions from their norms in Lp(R3).
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A New Nonmonotone Trust Region Barzilai-Borwein Method for Unconstrained Optimization Problems
Xing LI, Wen-li DONG, Zheng PENG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 166-175.   DOI: 10.1007/s10255-021-0997-9
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In this paper, we propose a new nonmonotone trust region Barzilai-Borwein (BB for short) method for solving unconstrained optimization problems. The proposed method is given by a novel combination of a modified Metropolis criterion, BB-stepsize and trust region method. The new method uses the reciprocal of BB-stepsize to approximate the Hessian matrix of the objective function in the trust region subproblems, and accepts some bad solutions according to the modified Metropolis criterion based on simulated annealing idea. Under some suitable assumptions, the global convergence of the new method is established. Some preliminary numerical results indicate that, the new method is more efficient compared with the existing trust region BB method.
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A New Lower Bound on the Potential-Ramsey Number of Two Graphs
Jin-zhi DU, Jian-hua YIN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 176-182.   DOI: 10.1007/s10255-021-0999-7
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A nonincreasing sequence π=(d1, …, dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π. Given a graph H, a graphic sequence π is potentially H-graphic if π has a realization containing H as a subgraph. For graphs G1 and G2, the potential-Ramsey number rpot(G1, G2) is the smallest integer k such that for every k-term graphic sequence π, either π is potentially G1-graphic or the complementary sequence π=(k-1-dk, …, k-1-d1) is potentially G2-graphic. For 0 ≤ k ≤ ⎣ t/2 」, denote Kt-k to be the graph obtained from Kt by deleting k independent edges. If k=0, Busch et al. (Graphs Combin., 30(2014)847-859) present a lower bound on rpot(G, Kt) by using the 1-dependence number of G. In this paper, we utilize i-dependence number of G for i ≥ 1 to give a new lower bound on rpot(G, Kt-k) for any k with 0 ≤ k ≤ ⎣ t/2 」. Moreover, we also determine the exact values of rpot(Kn, Kt-k) for 1 ≤ k ≤ 2.
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Markov Jump Processes in Estimating Sharing of Identity by Descent
Xian CHEN, Wei GUO, Xu-min NI
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 183-191.   DOI: 10.1007/s10255-021-0989-9
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Identity by descent (IBD) sharing is a very important genomic feature in population genetics which can be used to reconstruct recent demographic history. In this paper we provide a framework to estimate IBD sharing for a demographic model called two-population model with migration. We adopt the structured coalescent theory and use a continuous-time Markov jump process {X(t), t ≥ 0} to describe the genealogical process in such model. Then we apply Kolmogorov backward equation to calculate the distribution of coalescence time and develop a formula for estimating the IBD sharing. The simulation studies show that our method to estimate IBD sharing for this demographic model is robust and accurate.
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On d-row (Column) Antimagic Matrices and Subset Partitions
Zhi-he LIANG, Shi-xin LIANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (1): 192-200.   DOI: 10.1007/s10255-021-0990-3
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An m×k matrix is said to be a d-row (column) antimagic matrix if its row-sums (column-sums) form an arithmetic progression with a difference d. The goal of this paper is to obtain the existence theorems and construction methods of some d-row (column) antimagic matrices. Using these results we give the necessary and sufficient condition for the existence of an (m, d)-partition of[1, mk].
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Some Kinds of Bargaining Equilibria of Multi-objective Games
Chun WANG, Hui YANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 201-213.   DOI: 10.1007/s10255-021-1003-2
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To solve the choice of multi-objective game’s equilibria, we construct general bargaining games called self-bargaining games, and define their individual welfare functions with three appropriate axioms. According to the individual welfare functions, we transform the multi-objective game into a single-objective game and define its bargaining equilibrium, which is a Nash equilibrium of the single-objective game. And then, based on certain continuity and concavity of the multi-objective game’s payoff function, we proof the bargaining equilibrium still exists and is also a weakly Pareto-Nash equilibrium. Moreover, we analyze several special bargaining equilibria, and compare them in a few examples.
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Sphericity and Identity Test for High-dimensional Covariance Matrix Using Random Matrix Theory
Shou-cheng YUAN, Jie ZHOU, Jian-xin PAN, Jie-qiong SHEN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 214-231.   DOI: 10.1007/s10255-021-1004-1
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This paper addresses the issue of testing sphericity and identity of high-dimensional population covariance matrix when the data dimension exceeds the sample size. The central limit theorem of the first four moments of eigenvalues of sample covariance matrix is derived using random matrix theory for generally distributed populations. Further, some desirable asymptotic properties of the proposed test statistics are provided under the null hypothesis as data dimension and sample size both tend to infinity. Simulations show that the proposed tests have a greater power than existing methods for the spiked covariance model.
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Degree Conditions for k-Hamiltonian [a, b]-factors
Jie WU, Si-zhong ZHOU
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 232-239.   DOI: 10.1007/s10255-021-1005-0
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Let a, b, k be nonnegative integers with 2 ≤ a < b. A graph G is called a k-Hamiltonian graph if G - U contains a Hamiltonian cycle for any subset UV (G) with |U| = k. An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. If G - U admits a Hamiltonian [a, b]-factor for any subset UV (G) with |U| = k, then we say that G has a k-Hamiltonian [a, b]-factor. Suppose that G is a k-Hamiltonian graph of order n with n ≥ ((a+b-4)(2a+b+k-6))/(b-2) + k and δ(G) ≥ a + k. In this paper, it is proved that G admits a k-Hamiltonian [a, b]-factor if max{dG(x),dG(y)} ≥ ((a-2)n+(b-2)k)/(a+b-4) + 2 for each pair of nonadjacent vertices x and y in G.
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A Construction of Variable Strength Covering Arrays
Ling JIANG, Ce SHI
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 240-250.   DOI: 10.1007/s10255-021-1006-z
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Covering arrays (CA) of strength t, mixed level or fixed level, have been applied to software testing to aim for a minimum coverage of all t-way interactions among components. The size of CA increases with the increase of strength interaction t, which increase the cost of software testing. However, it is quite often that some certain components have strong interactions, while others may have fewer or none. Hence, a better way to test software system is to identify the subsets of components which are involved in stronger interactions and apply high strength interaction testing only on these subsets. For this, in 2003, the notion of variable strength covering arrays was proposed by Cohen et al. to satisfy the need to vary the size of t in an individual test suite. In this paper, an effective deterministic construction of variable strength covering arrays is presented. Based on the construction, some series of variable strength covering arrays are then obtained, which are all optimal in the sense of their sizes. In the procedure, two classes of new difference matrices of strength 3 are also mentioned.
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Optimality Conditions for Minimax Optimization Problems with an Infinite Number of Constraints and Related Applications
Li-nan ZHONG, Yuan-feng JIN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 251-263.   DOI: 10.1007/s10255-021-1019-7
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This paper is concerned with the study of optimality conditions for minimax optimization problems with an infinite number of constraints, denoted by (MMOP). More precisely, we first establish necessary conditions for optimal solutions to the problem (MMOP) by means of employing some advanced tools of variational analysis and generalized differentiation. Then, sufficient conditions for the existence of such solutions to the problem (MMOP) are investigated with the help of generalized convexity functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, some of the obtained results are applied to formulating optimality conditions for weakly efficient solutions to a related multiobjective optimization problem with an infinite number of constraints, and a necessary optimality condition for a quasi ε-solution to problem (MMOP).
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Quantile Regression for Thinning-based INAR(1) Models of Time Series of Counts
Dan-shu SHENG, De-hui WANG, Kai YANG, Zi-ang WU
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 264-277.   DOI: 10.1007/s10255-021-1014-z
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In this paper, we develop the quantile regression (QR) estimation for the first-order integer-valued autoregressive (INAR(1)) models by defining the smoothing INAR(1) process. Jittering method is used to derive the QR estimators for the autoregressive coefficient and the quantile of innovations. The consistency and asymptotic normality of the proposed estimators are established. The performances of the proposed estimation procedures are evaluated by Monte Carlo simulations. The results show that the proposed procedures perform well for simulations and a real data application.
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Conflict-free Connection Number and Independence Number of a Graph
Jing WANG, Meng JI
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 278-286.   DOI: 10.1007/s10255-021-1013-0
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An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the minimum number of colors that are required in order to make G conflict-free connected. In this paper, we investigate the relation between the conflict-free connection number and the independence number of a graph. We firstly show that cfc(G) ≤ α(G) for any connected graph G, and give an example to show that the bound is sharp. With this result, we prove that if T is a tree with ∆(T) ≥ (α(T)+2)/2, then cfc(T) = ∆(T).
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Ambarzumyan Theorems for Dirac Operators
Chuan-fu YANG, Feng WANG, Zhen-you HUANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 287-298.   DOI: 10.1007/s10255-021-1007-y
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We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable selfadjoint matrix potential. The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators, which are subject to separation boundary conditions or periodic (semi-periodic) boundary conditions, and some analogs of Ambarzumyan’s theorem are obtained. The proof is based on the existence and extremal properties of the smallest eigenvalue of corresponding vectorial Sturm-Liouville operators, which are the second power of Dirac operators.
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The Consistency for the Estimators of Semiparametric Regression Model with Dependent Samples
Yi WU, Xue-jun WANG, Ling CHEN, Kun JIANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 299-318.   DOI: 10.1007/s10255-021-1008-x
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For the semiparametric regression model: Y(j)(xin, tin) = tinβ+g(xin)+e(j)(xin), 1 ≤ jk, 1 ≤ in, where tin ∈ R and xin ∈ Rp are known to be nonrandom, g is an unknown continuous function on a compact set A in Rp, ej(xin) are m-extended negatively dependent random errors with mean zero, Y(j)(xin, tin) represent the j-th response variables which are observable at points xin, tin. In this paper, we study the strong consistency, complete consistency and r-th (r > 1) mean consistency for the estimators βk,n and gk,n of β and g, respectively. The results obtained in this paper markedly improve and extend the corresponding ones for independent random variables, negatively associated random variables and other mixing random variables. Moreover, we carry out a numerical simulation for our main results.
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Infinite Horizon Forward-Backward Doubly Stochastic Differential Equations and Related SPDEs
Qing-feng ZHU, Liang-quan ZHANG, Yu-feng SHI
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 319-336.   DOI: 10.1007/s10255-021-1009-9
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A type of infinite horizon forward-backward doubly stochastic differential equations is studied. Under some monotonicity assumptions, the existence and uniqueness results for measurable solutions are established by means of homotopy method. A probabilistic interpretation for solutions to a class of stochastic partial differential equations combined with algebra equations is given. A significant feature of this result is that the forward component of the FBDSDEs is coupled with the backward variable.
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Continuous Forcing Spectra of Even Polygonal Chains
He-ping ZHANG, Xiao-yan JIANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 337-347.   DOI: 10.1007/s10255-021-1010-3
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Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The cardinality of a forcing set of M with the smallest size is called the forcing number of M, denoted by f(G, M). The forcing spectrum of G is defined as: Spec(G) = {f(G, M)|M is a perfect matching of G}. In this paper, by applying the Z-transformation graph (resonance graph) we show that for any polyomino with perfect matchings and any even polygonal chain, their forcing spectra are integral intervals. Further we obtain some sharp bounds on maximum and minimum forcing numbers of hexagonal chains with given number of kinks. Forcing spectra of two extremal chains are determined.
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Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Shear-thinning Flow
Ya-zhou CHEN, Hai-liang LI, Xiao-ding SHI
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 348-363.   DOI: 10.1007/s10255-021-1011-2
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This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shear-thinning flow in a bounded domain Ω ⊂ Rn, n ≥ 2. It is proved that there exists a suitable weak solution of the shear-thinning fluid in the n-D smooth bounded domain (for n ≥ 2). For 3D model, it is proved that the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.
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A Direct Method of Moving Planes to Fractional Power SubLaplace Equations on the Heisenberg Group
Xin-jing WANG, Peng-cheng NIU
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 364-379.   DOI: 10.1007/s10255-021-1016-x
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We give the direct method of moving planes for solutions to the conformally invariant fractional power subLaplace equation on the Heisenberg group. The method is based on four maximum principles derived here. Then symmetry and nonexistence of positive cylindrical solutions are proved.
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Lp Decay Rate for a Nonlinear Convection Diffusion Reaction Equation in Rn
Guo-wei LIU, Hong-mei XU, Yuan-mei XIA
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 380-392.   DOI: 10.1007/s10255-021-1018-8
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This paper studies the asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in Rn. Firstly, the global existence and uniqueness of classical solutions for small initial data are established. Then, we obtain the Lp, 2 ≤ p ≤ +∞ decay rate of solutions. The approach is based on detailed analysis of the Green function of the linearized equation with the technique of long wave-short wave decomposition and the Fourier analysis.
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Exact Confidence Limits for the Parameter of an Exponential Distribution in the Accelerated Life Tests under Type-I Censoring
De-qiang ZHENG, Xiang-zhong FANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 393-408.   DOI: 10.1007/s10255-021-1021-0
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Life data frequently arise in many reliability studies, such as accelerated life tests studies. This paper considers the part of life data where failure and censoring observations may exist. To develop statistical methods and theory for the analysis of these data, a new approach was proposed to obtain the exact lower and upper confidence limits for the mean life of the exponential distribution with Type-I censoring data. It is assumed that the acceleration factor is a random variable, and that the distribution of the acceleration factor is known from some empirical information or the meta analysis. A method for constructing the lower and upper confidence limits for the parameter based on an ordering relation among the sample space was proposed. Simulation studies and analyses of two examples suggest that the proposed method performed well.
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The Number of Circles of a Maximum State of a Plane Graph with Applications
Xian-an JIN, Jun GE, Xiao-Sheng CHENG, Yu-qing LIN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 409-420.   DOI: 10.1007/s10255-021-1020-1
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Motivated by the connection with the genus of the corresponding link and its application on DNA polyhedral links, in this paper, we introduce a parameter smax(G), which is the maximum number of circles of states of the link diagram D(G) corresponding to a plane (positive) graph G. We show that smax(G) does not depend on the embedding of G and if G is a 4-edge-connected plane graph then smax(G) is equal to the number of faces of G, which cover the results of S. Y. Liu and H. P. Zhang as special cases.
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A Worst-Case Risk Measure by G-VaR
Zi-ting PEI, Xi-shun WANG, Yu-hong XU, Xing-ye YUE
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (2): 421-440.   DOI: 10.1007/s10255-021-1002-3
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G-VaR, which is a type of worst-case value-at-risk (VaR), is defined as measuring risk incorporating model uncertainty. Compared with most extant notions of worst-case VaR, G-VaR can be computed using an explicit formula, and can be applied to large portfolios of several hundred dimensions with low computational cost. We also apply G-VaR to robust portfolio optimization, thereby providing a tractable means to facilitate optimal allocations under the condition of market ambiguity.
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The Analysis of Impact of Brexit on the Post-Brexit EU Using Intervented Multivariate Time Series
Yu TIAN, Shao-pei MA, Rong-xiang RUI, Zhen YU, Mao-zai TIAN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (3): 441-458.   DOI: 10.1007/s10255-021-1022-z
Abstract218)      PDF(pc) (663KB)(447)       Save
The UK is the most important partner of the EU in terms of economic and other fields due to the geographical proximity. It was one of the largest economies in the EU and its per capita income is higher than the EU average, so it is a net contributor to the EU. With UKs membership of the EU ended on 31 January 2019, there are concerns that the Brexit may have a significant impact on the EU, resulting in social, economic, political, and institutional changes, etc. in EU. While the impact of Brexit on the UK has always been the subject of considerable scholarly interest in recent years, there is relatively little literature on the impact of Brexit on the EU. This paper focuses on the evaluation of the impact of Brexit on the EU economy and other relevant aspects along three dimensions: GDP, PPP, Quarterly GDP growth. Employing powerful quantitative analysis technology that includes vector autoregression model, multivariate time series model with intervention variables, and autoregression integrated moving average, this paper obtains the important and novel evidence about the potential impact of Brexit on the EU economy, pointing out that Brexit is of far-reaching significance to the EU. This analysis uses several statistical models to screen out several key influencing factors, which can be used to predict the total GDP of EU in the next five years. The results show that EU economy will react negatively to "no-deal" Brexit, and its growth rate of economy will slow down significantly in next 5 years. Finally, we put forward relevant policy suggestions on how to deal with the negative impact of Brexit on EU.
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Asymmetric Information, Heterogeneous Prior Beliefs and Market Regulation
Hong LIU, Ying JIANG, Huai-nian ZHANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (3): 459-476.   DOI: 10.1007/s10255-021-1023-y
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This paper studies the trading behavior of an irrational insider and its influence on the market equilibrium in the presence of market regulation. We find that the market with only one insider with private information is almost close to a strong efficient market, under the condition of market regulation. In the equilibrium, the probability of the insider being caught trading with private information is zero, which shows that the reasonable behavior of the regulator is to essentially give up regulation. But the market efficiency and the irrational trader’s trading intensity all greatly improve because of the existence of the market regulation.
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Existence and Properties of Solutions for a Class of Fractional Differential Equations
Yong-qiang XU, Shu-hong CHEN, Zhong TAN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (3): 477-484.   DOI: 10.1007/s10255-021-1025-9
Abstract169)      PDF(pc) (131KB)(376)       Save
In this paper, we consider the initial value problem of a class of fractional differential equations. Firstly, we obtain the existence and uniqueness of the solutions by using Picard’s method of successive approximation. Then we discuss the dependence of the solutions on the initial value.
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Hosoya and Merrifield-Simmons Indices in Random Polyphenyl Chains
Wei-ling YANG
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (3): 485-494.   DOI: 10.1007/s10255-021-1026-8
Abstract47)      PDF(pc) (195KB)(155)       Save
The Hosoya index of a graph is the total number of matchings in it. And the Merrifield-Simmons index is the total number of independent sets in it. They are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In this paper, we obtain explicit analytical expressions for the expectations of the Hosoya index and the Merrifield-Simmons index of a random polyphenyl chain.
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Least Squares Model Averaging Based on Generalized Cross Validation
Xin-min LI, Guo-hua ZOU, Xin-yu ZHANG, Shang-wei ZHAO
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (3): 495-509.   DOI: 10.1007/s10255-021-1024-x
Abstract173)      PDF(pc) (295KB)(259)       Save
Frequentist model averaging has received much attention from econometricians and statisticians in recent years. A key problem with frequentist model average estimators is the choice of weights. This paper develops a new approach of choosing weights based on an approximation of generalized cross validation. The resultant least squares model average estimators are proved to be asymptotically optimal in the sense of achieving the lowest possible squared errors. Especially, the optimality is built under both discrete and continuous weigh sets. Compared with the existing approach based on Mallows criterion, the conditions required for the asymptotic optimality of the proposed method are more reasonable. Simulation studies and real data application show good performance of the proposed estimators.
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The Minimum Stretch Spanning Tree Problem for Typical Graphs
Lan LIN, Yi-xun LIN
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (3): 510-522.   DOI: 10.1007/s10255-021-1028-6
Abstract42)      PDF(pc) (193KB)(130)       Save
With applications in communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The problem has been proved NP-hard and fixed-parameter polynomial algorithms have been obtained for some special families of graphs. In this paper, we concentrate on the optimality characterizations for typical classes of graphs. We determine the exact formulae for the complete k-partite graphs, split graphs, generalized convex graphs, and several planar grids, including rectangular grids, triangular grids, and triangulated-rectangular grids.
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Stabilization and Synchronization of Discrete-time Fractional Chaotic Systems with Non-identical Dimensions
Samir BENDOUKHA
Acta Mathematicae Applicatae Sinica(English Series)    2021, 37 (3): 523-538.   DOI: 10.1007/s10255-021-1029-5
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This paper investigates the stabilization and synchronization of two fractional chaotic maps proposed recently, namely the 2D fractional Hénon map and the 3D fractional generalized Hénon map. We show that although these maps have non–identical dimensions, their synchronization is still possible. The proposed controllers are evaluated experimentally in the case of non–identical orders or time–varying orders. Numerical methods are used to illustrate the results.
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