Chinese Quarterly Journal of Mathematics

On the Distribution of Values of Euler's Function over Integers in Arithmetic Progressions

FENG Bin

2017, 32(2): 111-117. doi:10.13371/j.cnki.chin.q.j.m.2017.02.001

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Let φ（n） denote the Euler-totient function, we study the distribution of solutions of φ（n） ≤ x in arithmetic progressions, where n ≡ l（mod q） and an asymptotic formula was obtained by Perron formula.

Mathematical Model for Diabetes Mellitus with Impulsive Injections of Glucose-insulin

WANG Xia, ZHANG Ying, SONG Xin-yu

2017, 32(2): 118-133. doi:10.13371/j.cnki.chin.q.j.m.2017.02.002

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Impulsive injections of glucose and insulin analogues are very important strategies for the control of diabetes mellitus. We mainly imitate diabetes patients take insulin before eating, and eating approximately as a pulse blood glucose injection, as a result, a new mathematical model with impulsive injections of both glucose and insulin at different fixed times is formulated in this paper. Using the discrete dynamical system determined by the stroboscopic map, we show that the existence and uniqueness of a positive globally asymptotically stable periodic solution for type I diabetes. By impulsive comparison theorem, we obtain the glucose concentration level of the system is uniformly bounded above and below for type Ⅱ diabetes. Numerical analysis verifies our theoretical results.

A New Variational Formulation for a Kind of Reaction-diffusion Problem in Broken Sobolev Space

GE Zhi-hao, CAO Ji-wei

2017, 32(2): 134-141. doi:10.13371/j.cnki.chin.q.j.m.2017.02.003

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In this paper, a new variational formulation for a reaction-diffusion problem in broken Sobolev space is proposed. And the new formulation in the broken Sobolev space will be proved that it is well-posed and equivalent to the standard Galerkin variational formulation. The method will be helpful to easily solve the original partial differential equation numerically. And the method is novel and interesting, which can be used to deal with some complicated problem, such as the low regularity problem, the differential-integral problem and so on.

Majorization Results and Coefficients Bounds for a Class of Univalent Functions Associated with Operator

XIONG Liang-peng

2017, 32(2): 142-151. doi:10.13371/j.cnki.chin.q.j.m.2017.02.004

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In this paper we introduce a new general subclass n,G∑ a,λ（A, B, α） of univalent functions related the known integral operator and differential operator. Some majorization results for n,G∑ a,λ（A, B, 1） as well as the other functions are given. Furthermore, we find the coefficients bounds on |a2| and |a3| for functions in*n,G∑ a,λ（A₁, B₁, A₂, B₂, α₁, α₂）, which is the bi-univalent functions defined by n,G ∑a,λ（A, B, α） and subordination. By giving specific values of the parameters of our main results, several（known or new） consequences of main results are also discussed.

The Convergence of a Moving Average Process of AANA Random Variables

TAN Jia-xin, HUANG Qian, HU Qi, YANG Wen-zhi

2017, 32(2): 152-160. doi:10.13371/j.cnki.chin.q.j.m.2017.02.005

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Based on the asymptotically almost negatively associated（AANA） random variables, we investigate the complete moment convergence for a moving average process under the moment condition E[Y log（1 + Y）] < ∞. As an application, Marcinkiewicz-Zygmundtype strong law of large numbers for this moving average process is presented in this paper.

Complete Convergence for Weighted Sums of \tilde{φ} -mixing Sequence

LIU Ting-ting, WANG Xue-jun, WANG Xing-hui

2017, 32(2): 161-171. doi:10.13371/j.cnki.chin.q.j.m.2017.02.006

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In this paper, the complete convergence and strong law of large numbers for weighted sums of φ-mixing sequence with different distribution are investigated under some weaker moment conditions. Our results extend ones of independent sequence with identical distribution to the case of φ-mixing sequence with different distribution.

Multi-symplectic Geometry and Preissmann Scheme for GSDBM Equation

WANG Jun-jie, LI Sheng-ping

2017, 32(2): 172-180. doi:10.13371/j.cnki.chin.q.j.m.2017.02.007

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The multi-symplectic geometry for the GSDBM equation is presented in this paper. The multi-symplectic formulations for the GSDBM equation are presented and the local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. The multi-symplectic discretization of each formulation is exemplified by the multisymplectic Preissmann scheme. The numerical experiments are given, and the results verify the efficiency of the Preissmann scheme.

A Sort of Gray-scale Morphological Dilations and Erosions

GUO Yan-yan, GUO Qi

2017, 32(2): 181-186. doi:10.13371/j.cnki.chin.q.j.m.2017.02.008

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We introduce first a sort of gray-scale morphological dilations and erosions, which might have some further applications in image analysis. Then we show that the dilation and the erosion defined here form adjunctive pairs. The duality between the dilation and the erosion and some other properties, such as the commuting property with translation and homothety, of these operators are discussed as well.

An Improvement of Existence Theorems of Additive Selection Maps for Sub-additive Set-valued Maps

PIAO Yong-jie

2017, 32(2): 187-193. doi:10.13371/j.cnki.chin.q.j.m.2017.02.009

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In this paper, we will construct a new class of subadditive set-valued maps and use Cantor theorem to prove that the set-valued map has an unique additive selection map when the set-valued map satisfies some certain conditions, and then compare the obtained result with the well-known results.

Stable Determination of Convex Bodies from Intersection Bodies

MA Tong-yi

2017, 32(2): 194-207. doi:10.13371/j.cnki.chin.q.j.m.2017.02.010

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In this paper, by means of the dual Brunn-Minkowski theories and methods as well as the integral transform, we have established a stability version of nonsymmetric convex bodies from intersection bodies. In addition, a stability results of symmetric convex bodies from Lp-counterparts is established.

On Conformable Delta Fractional Derivative on Time Scales

YOU Xue-xiao, ZHAO Da-fang, CHENG Jian

2017, 32(2): 208-215. doi:10.13371/j.cnki.chin.q.j.m.2017.02.011

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In this paper, we introduce and investigate the concept of conformable delta fractional derivative on time scales. By using the theory of time scales, we obtain some basic properties of the conformable delta fractional derivative. Our results extend and improve both the results in [9] and the usual delta derivative.

On Gorenstein Resolution Dimensions of Complexes

CUI Jun-feng

2017, 32(2): 216-220. doi:10.13371/j.cnki.chin.q.j.m.2017.02.012

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Let W be a self-orthogonal class of R-modules. We prove that W-Gorenstein resolution dimension of a complex X is equivalent to the supremum of W-Gorenstein resolution dimension of modules X_ifor all i ∈ Z.

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