AN INVARIANT FOR HYPERGRAPHS

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Balancing exploration and exploitation with adaptive variation for evolutionary multi-objective optimization

Although recent studies have shown that evolutionary algorithms are effective tools for solving multi-objective optimization problems, their performances are often bottlenecked by the suitability of the evolutionary operators with respect to the optimization problem at hand and their corresponding parametric settings. To adapt the search dynamic of evolutionary operation in multi-objective optimization, this paper proposes an adaptive variation operator that exploits the chromosomal structure of binary representation and synergizes the function of crossover and mutation. The overall search ability is deterministically tuned online to maintain a balance between extensive exploration and local fine-tuning at different stages of the evolutionary search. Also, the coordination between the two variation operators is achieved by means of an adaptive control that ensures an efficient exchange of information between the different chromosomal sub-structures throughout the evolutionary search. Extensive comparative studies with several representative variation operators are performed on different benchmark problems and significant algorithmic performance improvements in terms of proximity, uniformity and diversity are obtained with the incorporation of the proposed adaptive variation operator into the evolutionary multi-objective optimization process.     

Regularized additive operator-difference schemes

The construction of additive operator-difference (splitting) schemes for the approximate solution Cauchy problem for the first-order evolutionary equation is considered. Unconditionally stable additive schemes are constructed on the basis of the Samarskii regularization principle for operator-difference schemes. In the case of arbitrary multicomponent splitting, these schemes belong to the class of additive full approximation schemes. Regularized additive operator-difference schemes for evolutionary problems are constructed without the assumption that the regularizing operator and the operator of the problem are commutable. Regularized additive schemes with double multiplicative perturbation of the additive terms of the problem’s operator are proposed. The possibility of using factorized multicomponent splitting schemes, which can be used for the approximate solution of steadystate problems (finite difference relaxation schemes) are discussed. Some possibilities of extending the proposed regularized additive schemes to other problems are considered.     

Possibly non-unital operator system structures on a possibly non-unital function system

In this paper, we first give the definition of possibly non-unital function system, which is a characterization of the self-adjoint subspace of the space of continuous functions on a compact Hausdorff space with the induced order and norm structure. Similar to operator system case, we define the unitalization of a possibly non-unital function system. Then we construct two possibly non-unital operator system structures on a given possibly non-unital function system, which are the analogues of minimal and maximal operator spaces on a normed space. These two structures have many interesting relations with the minimal and maximal operator system structures on a given function system.     

Jacobi structures of evolutionary partial differential equations

In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under the action of reciprocal transformations that only change the spatial variable. The main technical tool is in a suitable generalization of the classical Schouten–Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermanifolds. These constructions are used to compute the Lichnerowicz–Jacobi cohomologies and to prove a Darboux theorem for Jacobi structures with hydrodynamic leading terms. We also introduce the notion of bi-Jacobi structures, and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure.     

Lyapunov Stability of a Class of Operator Integro-differential Equations with Applications to Viscoelasticity

The Lyapunov stability is analysed for a class of integro-differential equations with unbounded operator coefficients. These equations arise in the study of non-conservative stability problems for viscoelastic thin-walled elements of structures. Some sufficient stability conditions are derived by using the direct Lyapunov method. These conditions are formulated for arbitrary kernels of the Volterra integral operator in terms of norms of the operator coefficients. Employing these conditions the supersonic flutter of a viscoelastic panel is studied and explicit expressions for the critical gas velocity are derived. Dependence of the critical flow velocity on the material characteristics and compressive load is analysed numerically.     

基于Hadamard算子的二维离散量子行走的概率测度估计

二维离散时间量子行走是直线上的量子行走的推广.通过演化算子的作用,行走者能够按照一定规律进行移动.在本文中,我们将Hadamard算子作为控制行走者方向的硬币算子,通过与控制行走者位置的条件转移算子结合,构成完整的演化算子.通过傅里叶变换,将行走者所处的时域空间转换成频域空间后,用傅里叶积分的平稳相位法得到了行走者在t步后处于位置(x,y)的振幅以及此时的概率估计.     

SM-stability of operator-difference schemes

The spectral mimetic (SM) properties of operator-difference schemes for solving the Cauchy problem for first-order evolutionary equations concern the time evolution of individual harmonics of the solution. Keeping track of the spectral characteristics makes it possible to select more appropriate approximations with respect to time. Among two-level implicit schemes of improved accuracy based on Padé approximations, SM-stability holds for schemes based on polynomial approximations if the operator in an evolutionary equation is self-adjoint and for symmetric schemes if the operator is skew-symmetric. In this paper, additive schemes (also called splitting schemes) are constructed for evolutionary equations with general operators. These schemes are based on the extraction of the self-adjoint and skew-symmetric components of the corresponding operator.     

基于指数分布的进化策略

典型的进化策略受自然进化过程的启发而成为求解全局优化问题的重要方法。传统的ES变异算子作为一个主要的进化技术是建立在正态分布的随机变量基础上的，本文提出了基于指数分布的进化策略由于采用了新的变异算子有效地减少了产生探试解的成本，从而优于传统的进化策略。     

On Axioms Contituting the Foundation of Hypergraph Theory

Hypergraphs are systems of finite sets, being the most general structures in discrete mathematics and powerful tools in dealing with discrete systems. In general, a branch of mathematics is built on some axioms. Informational scientists introduced the acyclic axiom for hypergraphs. In this paper, we first list several results concerning acyclic hypergraphs, in order to show that Acyclic-Axioms constitute the foundation of acyclic hypergraph theory. Then we give the basic theorem which shows that the Cycle-Axiom covers the Acyclic-Axioms and constitutes the foundation of hypergraph theory.     

EMSO-Definability of Some Languages with Event Structures

Languages of words, trees, and pictures with related event structures are considered. An event structure is defined for a word (tree, picture) from the event structures of individual letters and homogeneous relations between the events of letters on neighboring positions. The main problem considered is the problem of EMSO-definability of two language classes: the class of languages with cyclic event structures and the class of languages with acyclic event structures. Bibliography: 5 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 304, 2003, pp. 128–140.     

Random fixed point theorems for multivalued acyclic random maps

Some random fixed point theorems for acyclic valued random operators are proved. The class of acyclic – valued random operators includes convex – valued and star – shaped – valued random operators. This leads to the discovery of some new results     

Nonunique Solvability of Certain Differential Equations and Their Connection with Geometric Approximation Theory

In this paper, we study the structural and approximative properties of sets admitting an upper semicontinuous acyclic selection from an almost-best approximation operator. We study the questions of nonunique solvability of a nonlinear inhomogeneous Dirichlet problem on the basis of these properties.     

Towards the evolution of social structure

To what extent can social structure result from evolutionary processes as popposed to being deliberately organised? To begin to answer this questions five different but releated social simulations are reviewed, and a map of which mechanisms might results in what structures under what conditions being started. These show that different structures can be brought about by evolutionary processes based on the abilities and propensities of the individuals. The article ends with some challenges—to construct a credible simulations of more sophisticated structures: social group selection and self-organised value chains.     

Grey prediction evolution algorithm for global optimization

This article uses the grey prediction theory to structure a new metaheuristic: grey prediction evolution algorithm based on the even grey model. The proposed algorithm considers the population series of evolutionary algorithms as a time series, and uses the even grey model as a reproduction operator to forecast the next population (without employing any mutation and crossover operators). It is theoretically proven that the reproduction operator based on the even grey model is adaptive. Additionally, the algorithmic search mechanism and its differences with other evolutionary algorithms are analyzed. The performance of the proposed algorithm is validated on CEC2005 benchmark functions and a test suite composed of six engineering constrained design problems. The comparison experiments show the effectiveness and superiority of the proposed algorithm.The proposed algorithm can be regarded as the first case of structuring metaheuristics by using the prediction theory. The novel algorithm is anticipated to influence two future works. The first is to propose more metaheuristics inspired by prediction theories (including some statistical algorithms). Another is that the theoretical results of these prediction systems can be used for this novel type of metaheuristics.     

On the structure of F-indistinguishability operators

Some properties concerning the structure of F-indistinguishability operators are analyzed. It is shown that any of such operators is generated by a family of fuzzy subsets. This result, since it gives the way to construct F-indistinguishabilities, facilitates new applications of fuzzy relations. The links between F-indistinguishability operators and a kind of generalized metrics in the unit interval - which are also explored - are used to define the canonical generators of an F-indistinguishability operator, that is, the fuzzy partition associated with the operator.     

The Notion of Vertex Operator Coalgebra and a Geometric Interpretation

The notion of vertex operator coalgebra is presented and motivated via the geometry of conformal field theory. Specifically, we describe the category of geometric vertex operator coalgebras, whose objects have comultiplicative structures meromorphically induced by conformal equivalence classes of worldsheets. We then show this category is isomorphic to the category of vertex operator coalgebras, which is defined in the language of formal algebra. The latter has several characteristics which give it the flavor of a coalgebra with respect to the structure of a vertex operator algebra and several characteristics that distinguish it from a standard dual—both of them will be highlighted.     

Some Geometric Properties of Analytic Functions Involving a New Fractional Operator

In this paper, we introduce and study a new fractional operator and its implications in terms of the Ruscheweyh derivative operator, the Sălăgean operator and a certain fractional differintegral operator. Some geometric properties of the analytic functions involving this operator are derived. We also consider some applications and derive certain corollaries of our main results. Some useful consequences and relationship of certain results with known results are also pointed out.

     

Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails

We consider a continuous-time symmetric supercritical branching random walk on a multidimensional lattice with a finite set of the particle generation centres, i.e. branching sources. The main object of study is the evolutionary operator for the mean number of particles both at an arbitrary point and on the entire lattice. The existence of positive eigenvalues in the spectrum of an evolutionary operator results in an exponential growth of the number of particles in branching random walks, called supercritical in the such case. For supercritical branching random walks, it is shown that the amount of positive eigenvalues of the evolutionary operator, counting their multiplicity, does not exceed the amount of branching sources on the lattice, while the maximal of these eigenvalues is always simple. We demonstrate that the appearance of multiple lower eigenvalues in the spectrum of the evolutionary operator can be caused by a kind of ‘symmetry’ in the spatial configuration of branching sources. The presented results are based on Green’s function representation of transition probabilities of an underlying random walk and cover not only the case of the finite variance of jumps but also a less studied case of infinite variance of jumps.     

Efficient tree traversal to reduce code growth in tree-based genetic programming

Genetic programming is an evolutionary optimization method following the principle of program induction. Genetic programming often uses variable-length tree structures for representing candidate solutions. A serious problem with variable-length representations is code growth: during evolution these tree structures tend to grow in size without a corresponding increase in fitness. Many anti-bloat methods focus solely on size reduction and forget about fitness improvement, which is rather strange when using an “optimization” method. This paper reports the application of a semantically driven local search operator to control code growth and improve best fitness. Five examples, two theoretical benchmark applications and three real-life test problems are used to illustrate the obtained size reduction and fitness improvement. Performance of the local search operator is also compared with various other anti-bloat methods such as size and depth delimiters, an expression simplifier, linear and adaptive parsimony pressure, automatically defined functions and Tarpeian bloat control.