QUATERNIONIC SLICE REGULAR FUNCTIONS AND QUATERNIONIC LAPLACE TRANSFORMS

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.

The Similarity Transformation of Regularizing Functionals and Simularity Property of Their Fourier Transformations

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 matrix learning with a general regularized risk functional criterion

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.

THE PRODUCT OPERATOR BETWEEN BLOCH-TYPE SPACES OF SLICE REGULAR FUNCTIONS

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.

Application of the Regularization Method to the Numberical Inversion of a Class of Generalized Radon Transform

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.

α-times Integrated Regularized Cosine Functions and Second Order Abstract Cauchy Problems

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.

Monotonicity of the tail dependence for multivariate t-copula

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.

Boundary value problems for two types of degenerate elliptic systems in R^4

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.

Erratum to “The Riemann Hypothesis-Millennium Prize Problem” [Advances in Pure Mathematics 6 (2016) 915-920]

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.

Limit theorems for supremum of Gaussian processes over a random interval

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 → ∞.

ON POSITIVE SOLUTIONS OF ONE CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

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.

The Riemann Hypothesis-Millennium Prize Problem

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.

The Fixed Point Theorem and the Iterative Approximation of Modified Cauchy Integral Operator of Regular Functions

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^-.

Regioregular and Nondestructive Graphene Functionalization for High-Performance Electrochromic and Supercapacitive Devices

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.

REGULARIZED SEMIGROUPS GENERATED BY PSEUDODIFFERENTIAL OPERATORS ON C~α

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.

Limsup Results and LIL for Partial Sum Processes of a Gaussian Random Field

Let {ξ j ; j ∈ ℤ+ d be a centered stationary Gaussian random field, where ℤ+ d is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝd, having nonnegative integer coordinates. For each j = (j 1 , ..., jd) in ℤ+ d , we denote |j| = j 1 ... j d and for m, n ∈ ℤ+ d , define S(m, n] = Σ m【j≤n ζ j , σ2(|n−m|) = ES 2 (m, n], S n = S(0, n] and S 0 = 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.

Path properties of l_p-valued Gaussian random fields

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.

On the Seneta Sequences

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.