局部熵高维重分形谱的上界估计

We discuss the problem of higher-dimensional multifractal spectrum of local entropy for arbitrary invariant measures. By utilizing characteristics of a dynamical system, namely, higher-dimensional entropy capacities and higher-dimensional correlation entropies, we obtain three upper estimates on the hlgher-dimensional multifractal spectrum of local entropies. We also study the domain of higher-dimensional multifractai spetrum of entropies.     

Explicit exact solutions for (2 + 1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, we construct explicit exact solutions for the coupled Boiti–Leon–Pempinelli equation (BLP equation) by using a extended tanh method and symbolic computation system Mathematica. By means of the method, many new exact travelling wave solutions for the BLP system are successfully obtained. the extended tanh method can be applied to other higher-dimensional coupled nonlinear evolution equations in mathematical physics.     

Arithmetical properties of multiple Ramanujan sums

In the present paper, we introduce a multiple Ramanujan sum for arithmetic functions, which gives a multivariable extension of the generalized Ramanujan sum studied by D.R. Anderson and T.M. Apostol. We then find fundamental arithmetic properties of the multiple Ramanujan sum and study several types of Dirichlet series involving the multiple Ramanujan sum. As an application, we evaluate higher-dimensional determinants of higher-dimensional matrices, the entries of which are given by values of the multiple Ramanujan sum.     

A generating function of higher-dimensional Apostol-Zagier sums and its reciprocity law

We introduce higher-dimensional Dedekind sums with a complex parameter z, generalizing Zagier's higher-dimensional Dedekind sums. The sums tend to Zagier's higher-dimensional Dedekind sums as z→∞. We show that the sums turn out to be generating functions of higher-dimensional Apostol-Zagier sums which are defined to be hybrids of Apostol's sums and Zagier's sums. We prove reciprocity law for the sums. The new reciprocity law includes reciprocity formulas for both Apostol and Zagier's sums as its special case. Furthermore, as its application we obtain relations between special values of Hurwitz zeta function and Bernoulli numbers, as well as new trigonometric identities.     

Engineering magnetic polariton system with distributed coefficients: Applications to soliton management

In the wake of the recent design of a powerful method for generating higher-dimensional evolution systems with distributed coefficients Kuetche (2014) [15] illustrated on the dynamics of the current-fed membrane of zero Young’s modulus, we construct the general Lax-representation of a new higher-dimensional coupled evolution equations with varying coefficients. Discussing the physical meanings of these equations, we show that the coupled system above describes the propagation of magnetic polaritons within saturated ferrites, resulting structurally from the fast-near adiabatic magnetization dynamics combined to the Maxwell’s equations. Accordingly, we address some practical issues of the nonautonomous soliton managements underlying in the fast remagnetization process of data inputs within magnetic memory devices.     

Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combination of the theory introduced in Global smooth solution curves using rigorous branch following (van den Berg et al., Math. Comput. 79(271):1565–1584, 2010) and in Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs (Gameiro and Lessard, J. Diff. Equ. 249(9):2237–2268, 2010). Using this method, one can obtain proofs of existence of global smooth solution curves of equilibria for large (continuous) parameter ranges and about local uniqueness of the solutions on the curve. As an application, we compute several smooth branches of equilibria for the three-dimensional Cahn–Hilliard equation.     

An Algorithm for Portfolio Optimization with Variable Transaction Costs,Part 1: Theory

A portfolio optimization problem consists of maximizing an expected utility function of n assets. At the end of a typical time period, the portfolio will be modified by buying and selling assets in response to changing conditions. Associated with this buying and selling are variable transaction costs that depend on the size of the transaction. A straightforward way of incorporating these costs can be interpreted as the reduction of portfolios’ expected returns by transaction costs if the utility function is the mean-variance or the power utility function. This results in a substantially higher-dimensional problem than the original n-dimensional one, namely (2K+1)n-dimensional optimization problem with (4K+1)n additional constraints, where 2K is the number of different transaction costs functions. The higher-dimensional problem is computationally expensive to solve. This two-part paper presents a method for solving the (2K+1)n-dimensional problem by solving a sequence of n-dimensional optimization problems, which account for the transaction costs implicitly rather than explicitly. The key idea of the new method in Part 1 is to formulate the optimality conditions for the higher-dimensional problem and enforce them by solving a sequence of lower-dimensional problems under the nondegeneracy assumption. In Part 2, we propose a degeneracy resolving rule, address the efficiency of the new method and present the computational results comparing our method with the interior-point optimizer of Mosek. This research was supported by the National Science and Engineering Research Council of Canada and the Austrian National Bank. The authors acknowledge the valuable assistance of Rob Grauer and Associate Editor Franco Giannessi for thoughtful comments and suggestions.     

On higher-dimensional contrast structure of singularly perturbed Dirichlet problem

In this paper,we address the existence and asymptotic analysis of higher-dimensional contrast structure of singularly perturbed Dirichlet problem.Based on the existence,an asymptotical analysis of a steplike contrast structure (i.e.,an internal transition layer solution) is studied by the boundary function method via a proposed smooth connection.In the framework of this paper,we propose a first integral condition,under which the existence of a heteroclinic orbit connecting two equilibrium points is ensured in a higher-dimensional fast phase space.Then,the step-like contrast structure is constructed,and the internal transition time is determined.Meanwhile,the uniformly valid asymptotical expansion of such an available step-like contrast structure is obtained.Finally,an example is presented to illustrate the result.     

Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we construct new explicit exact solutions for the coupled the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation (KD equation) by using a improved mapping approach and variable separation method. By means of the method, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) for the KD system are successfully obtained. The improved mapping approach and variable separation method can be applied to other higher-dimensional coupled nonlinear evolution equations.     

Hyperdeterminants associated with multiple even functions

In this paper, we introduce a new method to give the formulae for the hyperdeterminants of higher-dimensional matrices associated with multiple even function (modr) on the gcd-closed sets. Our result generalizes the result of Yamasaki obtained in 2010 and also extends the results of Haukkanen and Hong obtained in 1992 and 2002, respectively.     

Dimensional reduction of gravity and relation between static states,cosmologies, and waves

We introduce generalized dimensional reductions of an integrable (1+1)-dimensional dilaton gravity coupled to matter down to one-dimensional static states (black holes in particular), cosmological models, and waves. An unusual feature of these reductions is that the wave solutions depend on two variables: space and time. They are obtained here both by reducing the moduli space (available because of complete integrability) and by a generalized separation of variables (also applicable to nonintegrable models and to higher-dimensional theories). Among these new wavelike solutions, we find a class of solutions for which the matter fields are finite everywhere in space-time, including infinity. These considerations clearly demonstrate that a deep connection exists between static states, cosmologies, and waves. We argue that it should also exist in realistic higher-dimensional theories. Among other things, we also briefly outline the relations existing between the low-dimensional models that we discuss here and the realistic higher-dimensional ones. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 422–452, December, 2007.     

Higher-dimensional Contou-Carrère symbol and continuous automorphisms

We prove that the higher-dimensional Contou-Carrère symbol is invariant under the continuous automorphisms of algebras of iterated Laurent series over a ring. Applying this property, we obtain a new explicit formula for the higher-dimensional Contou-Carrère symbol. Unlike previously known formulas, this formula holds over an arbitrary ring, not necessarily a Q-algebra, and its derivation does not employ algebraic K-theory.     

运用漂移布朗族的高维狄利克莱问题的数值解

本文对高维狄利克莱问题的数值解提出了一种新的有效的求解方法。这种方法运用了解的随机表达式、球面击中时和位置的分布以及漂移布朗族的强马氏性。     

A new chaos optimization algorithm based on symmetrization and levelling approaches for global optimization

Chaos optimization algorithm is a recently developed method for global optimization based on chaos theory. It has many good features such as easy implementation, short execution time and robust mechanisms for escaping from local minima compared with existing stochastic searching algorithms. In the present paper, we propose a new chaos optimization algorithm (COA) approach called SLC (symmetric levelled chaos) based on new strategies including symmetrization and levelling: the proposed SLC method is, to our knowledge, the first chaos approach that can efficiently and successfully operates in higher-dimensional spaces. The proposed method is tested on a number of benchmark functions, and its performance comparisons are provided against previous COAs. The experiment results show that the proposed method has a marked improvement in performance over the classical COA approaches. Moreover, among all COA approaches, SLC is the only one to work efficiently in higher-dimensional spaces.     

Visibility of the higher-dimensional central projection into the projective sphere

We can describe higher-dimensional classical spaces by analytical projective geometry, if we embed the d-dimensional real space onto a d + 1-dimensional real projective metric vector space. This method allows an approach to Euclidean, hyperbolic, spherical and other geometries uniformly [8]. To visualize d-dimensional solids, it is customary to make axonometric projection of them. In our opinion the central projection gives more information about these objects, and it contains the axonometric projection as well, if the central figure is an ideal point or an s-dimensional subspace at infinity. We suggest a general method which can project solids into any picture plane (space) from any central figure, complementary to the projection plane (space). Opposite to most of the other algorithms in the literature, our algorithm projects higher-dimensional solids directly into the two-dimensional picture plane (especially into the computer screen), it does not use the three-dimensional space for intermediate step. Our algorithm provides a general, so-called lexicographic visibility criterion in Definition and Theorem 3.4, so it determines an extended visibility of the d-dimensional solids by describing the edge framework of the two-dimensional surface in front of us. In addition we can move the central figure and the image plane of the projection, so we can simulate the moving position of the observer at fixed objects on the computer screen (see first our figures in reverse order). Supported by DAAD 2008 Multimedia Technology for Mathematics and Computer Science Education.     

高维Dirichlet问题数值解的概率方法

本文用概率方法求得高维Dirichlet内问题和外问题在一般区域上的数值解\bd 高维漂移布朗族对停时具有强马氏性, 它在球面上的击中时和位置分布已知, 再利用Dirichlet问题解的随机表达式, 我们可以获得高维Dirichlet问题的数值解     

Image Projection and Representation on S<Superscript>n</Superscript>

In signal processing, communications, and other branches of information technologies, it is often desirable to map the higher-dimensional signals on Sⁿ. In this article we introduce a novel method of representing signals on Sⁿ. This approach is based on geometric function theory, in particular on the theory of quasiregular mappings. The importance of sampling is underlined, and new geometric sampling theorems for general manifolds are presented.     

A new construction of caps

In this paper we present a general method to construct caps in higher-dimensional projective spaces. As an application, for q≥8 even we obtain caps in PG(5,q) larger than the caps known so far, and a new class of caps of size (q+1)(q²+3) for q≥7 odd.     

高维Dirchlet问题数值解的概率方法

本文用概率方法求得高维Dirichlet内问题和外问题在一般区域上的数值解.高维漂移布朗族对停时具有强马氏性,它在球面上的击中时和位置分布已知,再利用Dirichlet问题解的随机表达式,我们可以获得高维Dirichlet问题的数值解.     

MATCHING PURSUITS AMONG SHIFTED CAUCHY KERNELS IN HIGHER-DIMENSIONAL SPACES

Appealing to the Clifford analysis and matching pursuits, we study the adaptive decompositions of functions of several variables of finite energy under the dictionaries consisting of shifted Cauchy kernels. This is a realization of matching pursuits among shifted Cauchy kernels in higher-dimensional spaces. It offers a method to process signals in arbitrary dimensions.