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ISSN 01689673 CN 112041/O1
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On a General Class of Semiparametric Hazards Regression Models for Recurrent Gap Times
Qin JIANG, Hui ZHAO, Hong QIN
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 549563.
DOI:
10.1007/s1025501908319
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In the article, we investigate a general class of semiparametric hazards regression models for recurrent gap times. The general class includes the proportional hazards model, the accelerated failure time model and the accelerated hazards models as special cases. The model is flexible in modelling recurrent gap times since a covariate effect is identified as having two separate components, namely a timescale change on hazard progression and a relative hazards ratio. In order to infer the model parameters, the procedure is proposed based on estimating equations. The asymptotic properties of the proposed estimators are established and the finite sample properties are investigated via simulation studies. In addition, a lack of fit test is presented to assess the adequacy of the model and an application of data from a bladder cancer study is reported for illustration.
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GMM and Misspecification Correction for Misspecified Models with Diverging Number of Parameters
Qi ZHANG, Kaiping WANG, Deli LI, Lu LIN
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 780797.
DOI:
10.1007/s1025501908524
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Misspecified models have attracted much attention in some fields such as statistics and econometrics. When a global misspecification exists, even the model contains a large number of parameters and predictors, the misspecification cannot disappear and sometimes it instead goes further away from the true one. Then the inference and correction for such a model are of very importance. In this paper we use the generalized method of moments (GMM) to infer the misspecified model with diverging numbers of parameters and predictors, and to investigate its asymptotic behaviors, such as local and global consistency, and asymptotic normality. Furthermore, we suggest a semiparametric correction to reduce the global misspefication and, consequently, to improve the estimation and enhance the modeling. The theoretical results and the numerical comparisons show that the corrected estimation and fitting are better than the existing ones.
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A Double Varyingcoefficient Modeling Approach for Analyzing Longitudinal Observations
Qunfang XU, Rui LI
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 671688.
DOI:
10.1007/s1025501908408
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The identification of withinsubject dependence is important for constructing efficient estimation in longitudinal data models. In this paper, we proposed a flexible way to study this dependence by using nonparametric regression models. Specifically, we considered the estimation of varying coefficient longitudinal data model with nonstationary varying coefficient autoregressive error process over observational time quantum. Based on spline approximation and local polynomial techniques, we proposed a twostage nonparametric estimation for unknown functional coefficients and didn't not drop any observations in a hybrid least square loss framework. Moreover, we showed that the estimated coefficient functions are asymptotically normal and derived the asymptotic biases and variances accordingly. Monte Carlo studies and two real applications were conducted for illustrating the performance of our proposed methods.
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Estimating Restricted Common Structural Changes for Panel Data
Liwen ZHANG, Zhongyi ZHU
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 893908.
DOI:
10.1007/s102550190859x
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We consider estimating multiple structural changes occurring at unknown common dates in a panel data regression model with restrictions imposed on the coefficients. We establish the consistency and rate of convergence of the structural change estimates and the asymptotic distribution of the parameter estimates. It is shown that the efficiency of the parameter estimators is increased using the restrictions. An efficient dynamic algorithm is proposed to obtain the break date estimates and the parameter estimates. In addition, we propose a testing procedure for the existence of the change points with the restrictions and derive the asymptotic distribution under the null hypothesis of the test statistics. Simulation studies are presented to investigate the performance of the proposed method in finite samples.
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Confidence Intervals of the Generalized Pareto Distribution Parameters Based on Upper Record Values
Xu ZHAO, Weihu CHENG, Yang ZHANG, Shaojie WEI, Zhenhai YANG
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 909918.
DOI:
10.1007/s1025501908604
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1016
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In this paper, we proposed a new efficient approach to construct confidence intervals for the location and scale parameters of the generalized Pareto distribution (GPD) when the shape parameter is known. The superiority of our method is that the distributions of pivots are exact, but not approximate distributions. The proposed interval estimation provides the shortest interval for the GPD parameter whether or not the confident distribution of the pivot is symmetric. We first estimate the location and scale parameters of the GPD using least squares and then, construct confidence intervals based on the equal probability density principle. The results of various simulation studies illustrate that our interval estimators show the better performance than competing method.
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A Note on Ranking in the PlackettLuce Model for Multiple Comparisons
Jing LUO, Hong QIN
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 885892.
DOI:
10.1007/s102550190857z
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169
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Ranking and rating individuals is a fundamental problem in multiple comparisons. One of the most wellknown approaches is the PlackettLuce model, in which the ordering is decided by the maximum likelihood estimator. However, the maximum likelihood estimate (MLE) does not exist when some individuals are never ranked lower than others or lose all their races. In this note, we proposed a penalized likelihood method to address this problem. As the penalized parameter goes to zero, the penalized MLE converges to the original MLE. Further, there exists a critical point in which the penalized likelihood ranking is independent of the choice of the penalized parameter. Several numerical examples are provided.
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On the Integrability of the Abel and of the Extended Liénard Equations
Man Kwong Mak, Tiberiu HARKO
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 722736.
DOI:
10.1007/s1025501908471
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263
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We present some exact integrability cases of the extended Liénard equation
y
" +
f
(
y
)(
y
')
^{n}
+
k
(
y
)(
y
')
^{m}
+
g
(
y
)
y
' +
h
(
y
)=0, with
n
> 0 and
m
> 0 arbitrary constants, while
f
(
y
),
k
(
y
),
g
(
y
), and
h
(
y
) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given by
dv
/
dy
=
f
(
y
)
v
^{3n}
+
k
(
y
)
v
^{3m}
+
g
(
y
)
v
^{2}
+
h
(
y
)
v
^{3}
, with
v
=1/
y
'. As a first step in our study we obtain three integrability cases of the extended quadraticcubic Liénard equation, corresponding to
n
=2 and
m
=3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with
g
(
y
) ≡ 0 and
h
(
y
) ≡ 0, and arbitrary
n
and
m
, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.
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Feature Screening for Ultrahighdimensional Censored Data with Varying Coefficient Singleindex Model
Yi LIU
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 845861.
DOI:
10.1007/s1025501908613
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163
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In this paper, we study the sure independence screening of ultrahighdimensional censored data with varying coefficient singleindex model. This general model framework covers a large number of commonly used survival models. The property that the proposed method is not derived for a specific model is appealing in ultrahigh dimensional regressions, as it is difficult to specify a correct model for ultrahigh dimensional predictors. Once the assuming data generating process does not meet the actual one, the screening method based on the model will be problematic. We establish the sure screening property and consistency in ranking property of the proposed method. Simulations are conducted to study the finite sample performances, and the results demonstrate that the proposed method is competitive compared with the existing methods. We also illustrate the results via the analysis of data from The National Alzheimers Coordinating Center (NACC).
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Regression Analysis for the Additive Hazards Model with General Biased Survival Data
Xiaolin CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 545556.
DOI:
10.1007/s1025502009499
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149
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In survival analysis, data are frequently collected by some complex sampling schemes, e.g., length biased sampling, casecohort sampling and so on. In this paper, we consider the additive hazards model for the general biased survival data. A simple and unified estimating equation method is developed to estimate the regression parameters and baseline hazard function. The asymptotic properties of the resulting estimators are also derived. Furthermore, to check the adequacy of the fitted model with general biased survival data, we present a test statistic based on the cumulative sum of the martingaletype residuals. Simulation studies are conducted to evaluate the performance of proposed methods, and applications to the shrub and Welsh Nickel Refiners datasets are given to illustrate the methodology.
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L
^{p}
Solutions of BSDEs with a New Kind of NonLipschitz Coefficients
Shengjun FAN, Long JIANG
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 695707.
DOI:
10.1007/s1025501908462
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216
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In this paper, we are interested in solving multidimensional backward stochastic differential equations (BSDEs) with a new kind of nonLipschitz coefficients. We establish an existence and uniqueness result of the
L
^{p}
(
p
> 1) solutions, which includes some known results as its particular cases.
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Concentrated Trading and the Survival of Overconfident Traders
Huainian ZHANG, Deqing ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 753760.
DOI:
10.1007/s102550190849z
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180
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We build a strategic trading model where the overconfident trader earns more than the rational trader and the mechanism for this result differs from that of Kyle and Wang (1997). In this paper, discretionary and nondiscretionary liquidity traders coexist. The overconfident insider is less worried by the market maker and thus induces a lower liquidity cost. In this way, he attracts all the trading from the discretionary liquidity traders, which enables the survival of the overconfident trader.
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Asymptotic Stability of Monotone Decreasing Kink Profile Solitary Wave Solutions for Generalized KdVBurgers Equation
Weiguo ZHANG, Wenxia LI, Shenger DENG, Xiang LI
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 475490.
DOI:
10.1007/s1025501908257
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In this paper, we focus on studying the asymptotic stability of the monotone decreasing kink profile solitary wave solutions for the generalized KdVBurgers equation. We obtain the estimate of the firstorder and secondorder derivatives for monotone decreasing kink profile solitary wave solutions, and overcome the difficulties caused by highorder nonlinear terms in the generalized KdVBurgers equation in the estimate by using
L
^{2}
energy estimating method and Young inequality. We prove that the monotone decreasing kink profile solitary wave solutions are asymptotically stable in
H
^{1}
. Moreover, we obtain the decay rate of the perturbation
ψ
in the sense of
L
^{2}
and
L
^{∞}
norm, respectively, which are (1 +
t
)
^{1/2}
and (1 +
t
)
^{1/4}
by using GargliadoNirenberg inequality.
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On a Spanning
K
tree Containing Specified Vertices in a Graph
Feifei SONG, Zhiquan HU
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 919923.
DOI:
10.1007/s1025501908640
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164
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A
k
tree is a tree with maximum degree at most
k
. In this paper, we give a sharp degree sum condition for a graph to have a spanning
k
tree in which specified vertices have degree less than
t
, where 1 ≤
t
≤
k
. We denote by
σ
_{k}
(
G
) the minimum value of the degree sum of
k
independent vertices in a graph
G
. Let
k
≥ 2,
s
≥ 0 and 1 ≤
t
≤
k
be integers, and suppose
G
is an (
s
+ 1)connected graph with
σ
_{k}
(
G
) ≥ 
G
+ (
kt
)
s
1. Then for any
s
specified vertices,
G
contains a spanning
k
tree in which every specified vertex has degree at most
t
. This improves a result obtained by Matsuda and Matsumura.
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The Efficient Computation of Aircraft Range Problem
Fang YU, Jinchuan CUI
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 862872.
DOI:
10.1007/s102550190858y
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193
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The goal of efficient computation is to determine reasonable computing cost in polynomial time by using data structure of instance, and analyze the computing cost of satisfactory solution which can meet user's requirements. When faced with NPhard problem, we usually assess computational performance in the worst case. Polynomial algorithm cannot handle with NPhard problem, so we research on NPhard problem from efficient computation point of view. The work is intended to fill the blank of computational complexity theory. We focus on the cluster structure of instance data of aircraft range problem. By studying the partition and complexity measurement of cluster, we establish a connection between the aircraft range problem and Nvehicle exploration problem, and construct the efficient computation mechanism for aircraft range problem. The last examples show that the effect is significant when we use efficient computation mechanism on aircraft range problem. Decision makers can calculate the computing cost before actually computing.
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Jensen's Inequality Under Nonlinear Expectation Generated by BSDE with Jumps
Na ZHANG, Guangyan JIA
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 873884.
DOI:
10.1007/s1025501908621
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144
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In this paper, we study Jensen's inequality under
f
expectation, which is a nonlinear expectation generated by backward stochastic differential equations (BSDEs) with jumps. We connect
f
convex functions with the viscosity solutions of a kind of integral partial differential equations (IPDEs) with nonlocal terms. And find that under Lipschitz condition, the
f
convex function is still convex in the usual sense, i.e., the jumps shrink the range of ‘convex’ functions.
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Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Shearthinning Flow
Yazhou CHEN, Hailiang LI, Xiaoding SHI
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(2): 348363.
DOI:
10.1007/s1025502110112
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129
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This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shearthinning flow in a bounded domain Ω ⊂ R
^{n}
,
n
≥ 2. It is proved that there exists a suitable weak solution of the shearthinning 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 New Nonmonotone Trust Region BarzilaiBorwein Method for Unconstrained Optimization Problems
Xing LI, Wenli DONG, Zheng PENG
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 166175.
DOI:
10.1007/s1025502109979
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147
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In this paper, we propose a new nonmonotone trust region BarzilaiBorwein (BB for short) method for solving unconstrained optimization problems. The proposed method is given by a novel combination of a modified Metropolis criterion, BBstepsize and trust region method. The new method uses the reciprocal of BBstepsize 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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Nonparametric Mestimation for Functional Stationary Ergodic Data
Xianzhu XIONG, Zhengyan LIN
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 491512.
DOI:
10.1007/s1025501908266
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This paper considers a nonparametric Mestimator of a regression function for functional stationary ergodic data. More precisely, in the ergodic data setting, we consider the regression of a real random variable
Y
over an explanatory random variable
X
taking values in some semimetric abstract space. Under some mild conditions, the weak consistency and the asymptotic normality of the Mestimator are established. Furthermore, a simulated example is provided to examine the finite sample performance of the Mestimator.
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New Lower Bounds to Wraparound
L
^{2}
discrepancy for Utype Designs with Threelevel
Zhenghong WANG, Hong QIN
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 513520.
DOI:
10.1007/s1025501908284
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The objective of this paper is to study the issue of uniformity on threelevel Utype designs in terms of the wraparound
L
^{2}
discrepancy. Based on the known formula, we present a new lower bound of wraparound
L
^{2}
discrepancy for threelevel Utype designs and compare it with those existing ones through figures, numerical simulation and illustrative examples.
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Linear Arboricity of NICPlanar Graphs
Bei NIU, Xin ZHANG
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 924934.
DOI:
10.1007/s102550190865z
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182
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A graph is NICplanar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. It is proved that every NICplanar graph with minimum degree at least 2 (resp.3) contains either an edge with degree sum at most 23 (resp. 17) or a 2alternating cycle (resp. 3alternating quadrilateral). By applying those structural theorems, we confirm the Linear Arboricity Conjecture for NICplanar graphs with maximum degree at least 14 and determine the linear arboricity of NICplanar graphs with maximum degree at least 21.
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Global Regularity of Solutions for a Onedimensional Nuclear Fluid with Nonmonotone Pressure
Baowei FENG
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 798811.
DOI:
10.1007/s1025501908533
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173
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We consider a onedimensional continuous thermal model of nuclear matter, which is described by a compressible NavierStokesPoission system with a nonmonotone equation of state owing to the effective Skyrme nuclear interaction between particles. We prove the global existence of solutions in
H
^{4}
space for a free boundary value problem with a possible destabilizing influence of the pressure which is not always positive, provided a sufficient thermal dissipation is present and first obtain the existence of classical solutions.
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An Exact Generalized Test for Homogeneity of Inverse Gaussian Scale Parameters
Xuhua LIU, Xingzhong XU, Weiyan MU
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 761769.
DOI:
10.1007/s1025501908506
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160
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In this paper, we propose a new generalized
p
value for testing homogeneity of scale parameters
λ
_{i}
from
k
independent inverse Gaussian populations. The proposed generalized
p
value is proved to have exact frequentist property, and it is also invariant under the group of scale transformation. Simulation results indicate that the proposed test is better than existing approximate
χ
^{2}
test.
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WolfeType Duality for Mathematical Programs with Equilibrium Constraints
Lei GUO, Guihua LIN, Jing ZHAO
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 532540.
DOI:
10.1007/s1025501908293
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This paper considers the mathematical programs with equilibrium constraints (MPEC). It is wellknown that, due to the existence of equilibrium constraints, the MangasarianFromovitz constraint qualification does not hold at any feasible point of MPEC and hence, in general, the developed numerical algorithms for standard nonlinear programming problems can not be applied to solve MPEC directly. During the past two decades, much research has been done to develop numerical algorithms and study optimality, stability, and sensitivity for MPEC. However, there are very few results on duality for MPEC in the literature. In this paper, we present a Wolfetype duality for MPEC and, under some suitable conditions, we establish various duality theorems such as the weak duality, direct duality, converse duality, and strict converse duality theorems. We further show that a linear MPEC is equivalent to a linear programming problem in some sense.
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Convergence of Stochastic Gradient Descent in Deep Neural Network
Baicun ZHOU, Congying HAN, Tiande GUO
Acta Mathematicae Applicatae Sinica(English Series). 2021, 37(1): 126136.
DOI:
10.1007/s1025502109912
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198
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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 LeNet5, a common network framework, on the data set MNIST to verify the rationality of our assumptions.
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Pullback Attractors for a 3D Nonautonomous NavierStokesVoight Equations
Yuming QIN, Xinguang YANG, Xin LIU
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 737752.
DOI:
10.1007/s1025501908480
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168
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This paper is concerned with the existence of pullback attractors for the three dimensional nonautonomous NavierStokesVoight equations for the processes generated by the weak and strong solutions. The main difficulty is how to establish the pullback asymptotic compactness via energy equation approach under suitable assumption on external force.
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The Bahadur Representation for Sample Quantiles Under Dependent Sequence
Wenzhi YANG, Shuhe HU, Xuejun WANG
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 521531.
DOI:
10.1007/s1025501908275
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On the one hand, we investigate the Bahadur representation for sample quantiles under
φ
mixing sequence with
φ
(
n
)=
O
(
n
^{3}
) and obtain a rate as
O
(
n
^{3/4}
log
n
), a.s. On the other hand, by relaxing the condition of mixing coefficients to Σ
_{n=1}
^{∞}
φ
^{1/2}
(
n
) < ∞, a rate
O
(
n
^{1/2}
(log
n
)
^{1/2}
), a.s., is also obtained.
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Nash Embedding, Shape Operator and NavierStokes Equation on a Riemannian Manifold
Shizan FANG
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 237252.
DOI:
10.1007/s1025502009281
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142
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What is the suitable Laplace operator on vector fields for the NavierStokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de RhamHodge Laplacian as well as EbinMarsden's Laplacian. A probabilistic representation formula for NavierStokes equations on a general compact Riemannian manifold is obtained when de RhamHodge Laplacian is involved.
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An Inexact Affine Scaling LevenbergMarquardt Method Under Local Error Bound Conditions
Zhujun WANG, Li CAI, Yifan SU, Zhen PENG
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 830844.
DOI:
10.1007/s1025501908560
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150
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We propose an inexact affine scaling LevenbergMarquardt method for solving boundconstrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition. The affine scaling LevenbergMarquardt equation is based on a minimization of the squared Euclidean norm of linearized model adding a quadratic affine scaling matrix to find a solution which belongs to the bounded constraints on variable. The global convergence and the superlinear convergence rate are proved. Numerical results show that the new algorithm is efficient.
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Inequality Estimates of an Inhomogeneous Semilinear Biharmonic Equation in Entire Space
Fen YANG
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 812819.
DOI:
10.1007/s1025501908542
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186
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In this paper, we consider the inequality estimates of the positive solutions for the inhomogeneous biharmonic equation
△
^{2}
u
+
u
^{p}
+
f
(
x
)=0 in R
^{n}
, (*)
where △
^{2}
is the biharmonic operator,
^{p}
> 1,
^{n}
≥ 5 and 0 ≢
f
∈
C
(R
^{n}
) is a given nonnegative function. We obtain different inequality estimates of Eq.(*), with which the necessary conditions of existence on
f
and
p
are also established.
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Signchanging Solutions to Nonlinear Elliptic Equations with Hardy Potential and Critical Parameter in R
^{2}
Chaodong XIE, Zhihui CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 689694.
DOI:
10.1007/s1025501908435
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In this paper, a new Hilbert space is defined which is embedded into
W
_{0}
^{1,q}
(Ω) for 1 ≤
q
< 2. By using signchanging critical point theory, we prove the existence of infinitely many signchanging solutions in this new space for nonlinear elliptic partial differential equations with Hardy potential and critical parameter in R
^{2}
.
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Solutions in the Discrete Fractional Forms of Two Differential Equations with Singular Points
Okkes OZTURK
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 638644.
DOI:
10.1007/s1025501908373
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In present note, we apply the Leibniz formula with the nabla operator in discrete fractional calculus (DFC) due to obtain the discrete fractional solutions of a class of associated Bessel equations (ABEs) and a class of associated Legendre equations (ALEs), respectively. Thus, we exhibit a new solution method for such second order linear ordinary differential equations with singular points.
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Blow up of Solutions for a Nonlinear Petrovsky Type Equation with Timedependent Coefficients
Xiaoxiao ZHENG, Yadong SHANG, Xiaoming PENG
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(4): 836846.
DOI:
10.1007/s1025502009846
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In this paper, we study a nonlinear Petrovsky type equation with nonlinear weak damping, a superlinear source and timedependent coefficients
utt
+ △
^{2}
u
+
k
_{1}
(
t
)
u
_{t}

^{m2}
u
_{t}
=
k
_{2}
(
t
)
u

^{p2}
u
,
x
∈ Ω,
t
> 0,
where Ω is a bounded domain in
R
^{n}
. Under certain conditions on
k
_{1}
(
t
),
k
_{2}
(
t
) and the initialboundary data, the upper bound for blowup time of the solution with negative initial energy function is given by means of an auxiliary functional and an energy estimate method if
p
>
m
. Also, a lower bound of blowup time are obtained by using a Sobolevtype inequality and a first order differential inequality technique for
n
= 2, 3 and
n
> 4.
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Regularity of Global Attractor for Atmospheric Circulation Equations with Humidity Effect
Jiaojiao PAN, Qian JIANG, Tingwei RUAN, Hong LUO
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 820829.
DOI:
10.1007/s1025501908551
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146
)
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In this article, regularity of the global attractor for atmospheric circulation equations with humidity effect is considered. It is proved that atmospheric circulation equations with humidity effect possess a global attractor in
H
^{k}
(Ω,
R
^{4}
) for any
k
≥ 0, which attracts any bounded set of
H
^{k}
(Ω,
R
^{4}
) in the
H
^{k}
norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor.
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Derivative Formula and Coupling Property for Linear SDEs Driven by Lévy Processes
Zhao DONG, Yulin SONG, Yingchao XIE
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 708721.
DOI:
10.1007/s1025501908631
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187
)
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In this paper we investigate an integration by parts formula for Lévy processes by using lower bound conditions of the corresponding Lévy measure. As applications, derivative formula and coupling property are derived for transition semigroups of linear SDEs driven by Lévy processes.
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An Efficient Parameterized Logarithmic Kernel Function for Semidefinite Optimization
Louiza DERBAL, Zakia KEBBICHE
Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(3): 753770.
DOI:
10.1007/s102550200955y
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131
)
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In this paper, we present a primaldual interior point algorithm for semidefinite optimization problems based on a new class of kernel functions. These functions constitute a combination of the classic kernel function and a barrier term.
We derive the complexity bounds for large and smallupdate methods respectively. We show that the best result of iteration bounds for large and smallupdate methods can be achieved, namely
O
(
q
√
n
(log √
n
)
^{q+1/q}
log
n/ε
) for largeupdate methods and
O
(
q
^{3/2}
(log √
q
)
^{q+1/q}
√
n
log
n/ε
) for smallupdate methods.
We test the efficiency and the validity of our algorithm by running some computational tests, then we compare our numerical results with results obtained by algorithms based on different kernel functions.
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Infinitely Many Solutions to a Class of
p
Laplace Equations
YiHua DENG
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(4): 770779.
DOI:
10.1007/s1025501908515
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185
)
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In this paper, we study a class of
p
Laplace equations. Using variational methods, we prove that there are two solutions and one of these solutions is nonnegative. Using recurrence method, we prove that there are infinitely many solutions to this class of equations.
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Mallows Model Averaging Estimation for Linear Regression Model with Right Censored Data
Zhongqi LIANG, Xiaolin CHEN, Yanqiu ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2022, 38(1): 523.
DOI:
10.1007/s1025502210530
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(
47
)
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This paper is concerned with an optimal model averaging estimation for linear regression model with right censored data. The weights for model averaging are picked up via minimizing the Mallows criterion. Under some mild conditions, it is shown that the identified weights possess the property of asymptotic optimality, that is, the model averaging estimator corresponding to these weights achieves the lowest squared error asymptotically. Some numerical studies are conducted to evaluate the finitesample performance of our method and make comparisons with its intuitive competitors, while an application to the PBC dataset is provided to serve as an illustration.
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Asymptotic Normality of Nonparametric Estimate for ZeroUtility Premiums
Limin WEN, Xiaohong ZHUANG, Guoping MEI, Yi ZHANG
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 607619.
DOI:
10.1007/s1025501908337
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Zeroutility principle is one of the main premium pricing principles, which has been widely used in insurance practice. In this paper, the nonparametric estimation of zeroutility premium is given. In addition, the consistency and asymptotic normality of the estimation are proved. Some special cases including linear, exponential and quadratic utility are discussed. Finally, the Monte Carlo method is used to show the convergence rate of premium estimation. Furthermore, the histogram and NormalProbabilityPlot are given to investigate the asymptotic normality of the estimators. The results show that our estimations are good enough to use in practice.
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Existence Results of Damped Second Order Impulsive Functional Differential Equations with Infinite Delay
Shengli XIE, Yiming XIE
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 564579.
DOI:
10.1007/s1025501908328
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Using the Mönch fixed point theorem and progressive estimation method, we study the existence, uniqueness and regularity of mild solutions for damped second order impulsive functional differential equations with infinite delay in Banach spaces. The compactness assumption on associated family of operators and the impulsive term, some restrictive conditions on a priori estimation, noncompactness measure estimation and the impulsive term have not been used, our results are different from some known results. Finally, a noncompact semigroup example explains the obtained results.
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Evolutionary
p
(
x
)Laplacian Equation with a Convection Term
Huashui ZHAN
Acta Mathematicae Applicatae Sinica(English Series). 2019, 35(3): 655670.
DOI:
10.1007/s1025501908426
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The paper studies an evolutionary
p
(
x
)Laplacian equation with a convection term
u
_{t}
=div(
ρ
^{α}
▽
u

^{p(x)2}
▽
u
) +Σ
_{i=1}
^{N}
∂b
_{i}
(
u
)/
∂x
_{i}
,
where
ρ
(
x
)=dist(
x
,
∂
Ω), ess inf
p
(
x
)=
p
^{}
> 2. To assure the wellposedness of the solutions, the paper shows only a part of the boundary, Σ
_{p}
⊂
∂
Ω), on which we can impose the boundary value. Σ
_{p}
is determined by the convection term, in particular, when 1 <
α
<
p
^{}
2/2, Σ
_{p}
={
x
∈
∂
Ω):
b
_{i}
'(0)
n
_{i}
(
x
) < 0}. So, there is an essential difference between the equation and the usual evolutionary pLaplacian equation. At last, the existence and the stability of weak solutions are proved under the additional conditions
α
<
p
^{}
2/2 and Σ
_{p}
=
∂
Ω).
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