The functions studied in the paper are the quaternion-valued functions of a quaternionic variable.It is shown that the left slice regular functions and right slice regular functions are related by a particular involut...The functions studied in the paper are the quaternion-valued functions of a quaternionic variable.It is shown that the left slice regular functions and right slice regular functions are related by a particular involution,and that the intrinsic slice regular functions play a central role in the theory of slice regular functions.The relation between left slice regular functions,right slice regular functions and intrinsic slice regular functions is revealed.As an application,the classical Laplace transform is generalized naturally to quaternions in two different ways,which transform a quaternion-valued function of a real variable to a left or right slice regular function.The usual properties of the classical Laplace transforms are generalized to quaternionic Laplace transforms.展开更多
There is little work concerning the properties of quaternionic operators acting on slice regular function spaces defined on quaternions.In this paper,we present an equivalent characterization for the boundedness of th...There is little work concerning the properties of quaternionic operators acting on slice regular function spaces defined on quaternions.In this paper,we present an equivalent characterization for the boundedness of the product operator C_(φ)D^(m) acting on Bloch-type spaces of slice regular functions.After that,an equivalent estimation for its essential norm is established,which can imply several existing results on holomorphic spaces.展开更多
In the first part of this paper, we discuss the Holder continuity of the cauchy integral operator for regular functions and the relation between ‖T{f}‖α and ‖f‖α. In the second part of this paper, we introduce t...In the first part of this paper, we discuss the Holder continuity of the cauchy integral operator for regular functions and the relation between ‖T{f}‖α and ‖f‖α. In the second part of this paper, we introduce the modified cauchy integral operator T^- for regular functions. Firstly, we prove that the operator T^- has a unique fixed point by the Banach＇s Contraction Mapping Principle. Secondly, we give the Mann iterative sequence, and then we show the iterative sequence strongly converges to the fixed point of the operator T^-.展开更多
In this paper, α-times integrated C-regularized cosine functions and mild α-times integrated C-existence families of second order are introduced. Equivalences are proved among α-times integrated C-regularized cosin...In this paper, α-times integrated C-regularized cosine functions and mild α-times integrated C-existence families of second order are introduced. Equivalences are proved among α-times integrated C-regularized cosine function for a linear operator A, C-wellposed of (α+1)-times abstract Cauchy problem and mild a -times integrated C-existence family of second order for A when the commutable condition is satisfied. In addition, if A = C-1AC, they are also equivalent to A generating the α -times integrated C-regularized cosine function. The characterization of an exponentially bounded mild α -times integrated C-existence family of second order is given out in terms of a Laplace transform.展开更多
Let and denote respectively the functionswhere λ≥1, The author discusses the similarity transformation of the regularizing functionals of these functions and the similar property of their Fourier transformation.
Kernel-based methods work by embedding the data into a feature space and then searching linear hypothesis among the embedding data points. The performance is mostly affected by which kernel is used. A promising way is...Kernel-based methods work by embedding the data into a feature space and then searching linear hypothesis among the embedding data points. The performance is mostly affected by which kernel is used. A promising way is to learn the kernel from the data automatically. A general regularized risk functional （RRF） criterion for kernel matrix learning is proposed. Compared with the RRF criterion, general RRF criterion takes into account the geometric distributions of the embedding data points. It is proven that the distance between different geometric distdbutions can be estimated by their centroid distance in the reproducing kernel Hilbert space. Using this criterion for kernel matrix learning leads to a convex quadratically constrained quadratic programming （QCQP） problem. For several commonly used loss functions, their mathematical formulations are given. Experiment results on a collection of benchmark data sets demonstrate the effectiveness of the proposed method.展开更多
In the research of bistatic tomography imaging of translating object, we get a class of generalized Radon transformation. In this paper, first we prove the existence and uniguenness of its solution in theory and poin...In the research of bistatic tomography imaging of translating object, we get a class of generalized Radon transformation. In this paper, first we prove the existence and uniguenness of its solution in theory and point out this problem is ill-posed with an especial example.Secondly by means of multiplicative interpolation functions to approximate models, we constracted regularizing functional. Finally we simplify calculation by Fourier transformation,get regularizing solutions that converge to accurate solution.展开更多
This paper considers the upper orthant and extremal tail dependence indices for multivariate t-copula. Where, the multivariate t-copula is defined under a correlation structure. The explicit representations of the tai...This paper considers the upper orthant and extremal tail dependence indices for multivariate t-copula. Where, the multivariate t-copula is defined under a correlation structure. The explicit representations of the tail dependence parameters are deduced since the copula of continuous variables is invariant under strictly increasing transformation about the random variables, which are more simple than those obtained in previous research. Then, the local monotonicity of these indices about the correlation coefficient is discussed, and it is concluded that the upper extremal dependence index increases with the correlation coefficient, but the monotonicity of the upper orthant tail dependence index is complex. Some simulations are performed by the Monte Carlo method to verify the obtained results, which are found to be satisfactory. Meanwhile, it is concluded that the obtained conclusions can be extended to any distribution family in which the generating random variable has a regularly varying distribution.展开更多
Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Cliffor...Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system＇s solution, the boundary value problem as stated above is trans- formed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R4 are derived.展开更多
The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mist...The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.展开更多
Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the...Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) > x) is considered, as x → ∞.展开更多
In this paper, we establish a 1-1 correspondence between positive solutionsof one class of nonlinear differential equations and a class of harmonic functions.These results give an explicit description of E.B.Dynkin...In this paper, we establish a 1-1 correspondence between positive solutionsof one class of nonlinear differential equations and a class of harmonic functions.These results give an explicit description of E.B.Dynkin's class Ha of positive harmonic functions.展开更多
This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice o...This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem-Millennium Prize Problem.展开更多
Let Op(f) be a pseudodifferential operator with symbol f∈ S m ρ,0 having constant coefficients. We prove that Op(f) generates a regularized semigroup or cosine function on C α (R n) (0<...Let Op(f) be a pseudodifferential operator with symbol f∈ S m ρ,0 having constant coefficients. We prove that Op(f) generates a regularized semigroup or cosine function on C α (R n) (0<α<1) when the symbol f(ξ) and its derivatives satisfy certain growth conditions.展开更多
To overcome two obstacles in graphene functionalization—the random distribution of functional groups and substantial lattice defects,in this contribution,we rationally bypass the universal yet destructive graphene ox...To overcome two obstacles in graphene functionalization—the random distribution of functional groups and substantial lattice defects,in this contribution,we rationally bypass the universal yet destructive graphene oxide(GO)-derived methodologies and adopt reductive covalent functionalization of natural graphite.In this strategy,ultrahigh density graphene sheets with evenly distributed negative charges were intermediately yielded by potassium reduction to graphite.Subsequently,they were regioregularly and efficiently brominated by the benchmark electrophile of molecular bromine.The combined characterizations determined the graphene bromide derivative to have a molecular formula of C_(24)-Br,corresponding well with the brominated entity of the inseparable C_(24)-K^(+)graphite intercalation compounds.Due to the regular distribution of Br groups and intact hexagonal lattice of the graphene matrix,the C_(24)-Br_GBr delivers exceptional electrical properties although theπ-conjugation is partially blocked by C(sp^(3))-Br sites and greatly outperforms its counterparts derived from GO bromination.In both model applications of quickly switchable electrochromic devices and all-solid-state supercapacitors,C_(24)-Br_GBr exhibits impressive performance,which highlights the great significance and prospect of regioregular and lattice-nondestructive graphene functionalization.展开更多
Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP>...Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in ℤ<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n ∈ ℤ<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = Σ<SUB> m【j≤n </SUB>ζ<SUB> j </SUB>, σ<SUP>2</SUP>(|n−m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t 】 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 【 α 【 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.展开更多
General limit theorems are established for l^p-valued Gaussian random fields indexed by a multidimensional parameter,which contain both almost sure moduli of continuity and limits of large increments for the l^p-value...General limit theorems are established for l^p-valued Gaussian random fields indexed by a multidimensional parameter,which contain both almost sure moduli of continuity and limits of large increments for the l^p-valued Gaussian random fields under(?)explicit conditions.展开更多
In this paper, we prove some properties of the Seneta sequences and functions, and in particular we prove a representation theorem in the Karamata sense for the sequences from the Seneta class SOc.
基金supported by NSFC(12071422)Zhejiang Province Science Foundation of China(LY14A010018)。
文摘The functions studied in the paper are the quaternion-valued functions of a quaternionic variable.It is shown that the left slice regular functions and right slice regular functions are related by a particular involution,and that the intrinsic slice regular functions play a central role in the theory of slice regular functions.The relation between left slice regular functions,right slice regular functions and intrinsic slice regular functions is revealed.As an application,the classical Laplace transform is generalized naturally to quaternions in two different ways,which transform a quaternion-valued function of a real variable to a left or right slice regular function.The usual properties of the classical Laplace transforms are generalized to quaternionic Laplace transforms.
基金supported by the National Natural Science Foundation of China(11701422).
文摘There is little work concerning the properties of quaternionic operators acting on slice regular function spaces defined on quaternions.In this paper,we present an equivalent characterization for the boundedness of the product operator C_(φ)D^(m) acting on Bloch-type spaces of slice regular functions.After that,an equivalent estimation for its essential norm is established,which can imply several existing results on holomorphic spaces.
基金the National Natural Science Foundation of China （No. 10771049 10771050）+1 种基金 the Natural Science Foundation of Hebei Province （No. A2007000225） and the Foundation of Hebei Normal University （No. L2007Q05） the 11th Five-Year Plan Educational and Scientific Issues of Hebei Province （No. O8020147）.
文摘In the first part of this paper, we discuss the Holder continuity of the cauchy integral operator for regular functions and the relation between ‖T{f}‖α and ‖f‖α. In the second part of this paper, we introduce the modified cauchy integral operator T^- for regular functions. Firstly, we prove that the operator T^- has a unique fixed point by the Banach＇s Contraction Mapping Principle. Secondly, we give the Mann iterative sequence, and then we show the iterative sequence strongly converges to the fixed point of the operator T^-.
基金This project is supported by the Natural Science Foundation of China and Science Development Foundation of the Colleges and University of Shanghai.
文摘In this paper, α-times integrated C-regularized cosine functions and mild α-times integrated C-existence families of second order are introduced. Equivalences are proved among α-times integrated C-regularized cosine function for a linear operator A, C-wellposed of (α+1)-times abstract Cauchy problem and mild a -times integrated C-existence family of second order for A when the commutable condition is satisfied. In addition, if A = C-1AC, they are also equivalent to A generating the α -times integrated C-regularized cosine function. The characterization of an exponentially bounded mild α -times integrated C-existence family of second order is given out in terms of a Laplace transform.
文摘Let and denote respectively the functionswhere λ≥1, The author discusses the similarity transformation of the regularizing functionals of these functions and the similar property of their Fourier transformation.
基金supported by the National Natural Science Fundation of China (60736021)the Joint Funds of NSFC-Guangdong Province(U0735003)
文摘Kernel-based methods work by embedding the data into a feature space and then searching linear hypothesis among the embedding data points. The performance is mostly affected by which kernel is used. A promising way is to learn the kernel from the data automatically. A general regularized risk functional （RRF） criterion for kernel matrix learning is proposed. Compared with the RRF criterion, general RRF criterion takes into account the geometric distributions of the embedding data points. It is proven that the distance between different geometric distdbutions can be estimated by their centroid distance in the reproducing kernel Hilbert space. Using this criterion for kernel matrix learning leads to a convex quadratically constrained quadratic programming （QCQP） problem. For several commonly used loss functions, their mathematical formulations are given. Experiment results on a collection of benchmark data sets demonstrate the effectiveness of the proposed method.
文摘In the research of bistatic tomography imaging of translating object, we get a class of generalized Radon transformation. In this paper, first we prove the existence and uniguenness of its solution in theory and point out this problem is ill-posed with an especial example.Secondly by means of multiplicative interpolation functions to approximate models, we constracted regularizing functional. Finally we simplify calculation by Fourier transformation,get regularizing solutions that converge to accurate solution.
基金The National Natural Science Foundation of China(No.11001052,11171065)the National Science Foundation of Jiangsu Province(No.BK2011058)the Science Foundation of Nanjing University of Posts and Telecommunications(No.JG00710JX57)
文摘This paper considers the upper orthant and extremal tail dependence indices for multivariate t-copula. Where, the multivariate t-copula is defined under a correlation structure. The explicit representations of the tail dependence parameters are deduced since the copula of continuous variables is invariant under strictly increasing transformation about the random variables, which are more simple than those obtained in previous research. Then, the local monotonicity of these indices about the correlation coefficient is discussed, and it is concluded that the upper extremal dependence index increases with the correlation coefficient, but the monotonicity of the upper orthant tail dependence index is complex. Some simulations are performed by the Monte Carlo method to verify the obtained results, which are found to be satisfactory. Meanwhile, it is concluded that the obtained conclusions can be extended to any distribution family in which the generating random variable has a regularly varying distribution.
基金Supported by the National Science Foundation of China(11401162,11571089,11401159,11301136)the Natural Science Foundation of Hebei Province(A2015205012,A2016205218,A2014205069,A2014208158)Hebei Normal University Dr.Fund(L2015B03)
文摘Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system＇s solution, the boundary value problem as stated above is trans- formed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R4 are derived.
文摘The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.
基金Supported by the Scientific Research Fund of Sichuan Provincial Education Department(12ZB082)the Scientific research cultivation project of Sichuan University of Science&Engineering(2013PY07)+1 种基金the Scientific Research Fund of Shanghai University of Finance and Economics(2017110080)the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing(2018QZJ01)
文摘Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) > x) is considered, as x → ∞.
文摘In this paper, we establish a 1-1 correspondence between positive solutionsof one class of nonlinear differential equations and a class of harmonic functions.These results give an explicit description of E.B.Dynkin's class Ha of positive harmonic functions.
文摘This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem-Millennium Prize Problem.
文摘Let Op(f) be a pseudodifferential operator with symbol f∈ S m ρ,0 having constant coefficients. We prove that Op(f) generates a regularized semigroup or cosine function on C α (R n) (0<α<1) when the symbol f(ξ) and its derivatives satisfy certain growth conditions.
基金The authors gratefully acknowledge financial support from the National Natural Science Foundation of China(no.21504080)the Natural Science Foundation of Zhejiang Province(no.LQ15B040003)the Fundamental Research Funds of Zhejiang Sci-Tech University(no.2020Q040).
文摘To overcome two obstacles in graphene functionalization—the random distribution of functional groups and substantial lattice defects,in this contribution,we rationally bypass the universal yet destructive graphene oxide(GO)-derived methodologies and adopt reductive covalent functionalization of natural graphite.In this strategy,ultrahigh density graphene sheets with evenly distributed negative charges were intermediately yielded by potassium reduction to graphite.Subsequently,they were regioregularly and efficiently brominated by the benchmark electrophile of molecular bromine.The combined characterizations determined the graphene bromide derivative to have a molecular formula of C_(24)-Br,corresponding well with the brominated entity of the inseparable C_(24)-K^(+)graphite intercalation compounds.Due to the regular distribution of Br groups and intact hexagonal lattice of the graphene matrix,the C_(24)-Br_GBr delivers exceptional electrical properties although theπ-conjugation is partially blocked by C(sp^(3))-Br sites and greatly outperforms its counterparts derived from GO bromination.In both model applications of quickly switchable electrochromic devices and all-solid-state supercapacitors,C_(24)-Br_GBr exhibits impressive performance,which highlights the great significance and prospect of regioregular and lattice-nondestructive graphene functionalization.
基金NSERC Canada grants of Miklos Csorgo and Barbara Szyszkowicz at Carleton University,Ottawa,and by KRF-2003-C00098NSERC Canada grants at Carleton University,Ottawa
文摘Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in ℤ<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n ∈ ℤ<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = Σ<SUB> m【j≤n </SUB>ζ<SUB> j </SUB>, σ<SUP>2</SUP>(|n−m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t 】 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 【 α 【 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.
基金This work was supported by NSERC Canada grants at Carleton University and by KOSEF-R01-2005-000-10696-0
文摘General limit theorems are established for l^p-valued Gaussian random fields indexed by a multidimensional parameter,which contain both almost sure moduli of continuity and limits of large increments for the l^p-valued Gaussian random fields under(?)explicit conditions.
基金Supported by the Grant No.144031 by Ministary of Science of Republic of Serbia
文摘In this paper, we prove some properties of the Seneta sequences and functions, and in particular we prove a representation theorem in the Karamata sense for the sequences from the Seneta class SOc.
