Generalized hyper-heuristics for solving 2D Regular and Irregular Packing Problems

The idea behind hyper-heuristics is to discover some combination of straightforward heuristics to solve a wide range of problems. To be worthwhile, such a combination should outperform the single heuristics. This article presents a GA-based method that produces general hyper-heuristics that solve two-dimensional regular (rectangular) and irregular (convex polygonal) bin-packing problems. A hyper-heuristic is used to define a high-level heuristic that controls low-level heuristics. The hyper-heuristic should decide when and where to apply each single low-level heuristic, depending on the given problem state. In this investigation two kinds of heuristics were considered: for selecting the figures (pieces) and objects (bins), and for placing the figures into the objects. Some of the heuristics were taken from the literature, others were adapted, and some other variations developed by us. We chose the most representative heuristics of their type, considering their individual performance in various studies and also in an initial experimentation on a collection of benchmark problems. The GA included in the proposed model uses a variable-length representation, which evolves combinations of condition-action rules producing hyper-heuristics after going through a learning process which includes training and testing phases. Such hyper-heuristics, when tested with a large set of benchmark problems, produce outstanding results for most of the cases. The testbed is composed of problems used in other similar studies in the literature. Some additional instances for the testbed were randomly generated.     

Weak Type Inequalities for Moduli of Smoothness: The Case of Limit Value Parameters

In this paper we obtain Ul’yanov type inequalities for fractional moduli of smoothness/K-functionals for the limit value parameters: p=1 or q=∞. Needed versions of Nikol’skii type inequalities for trigonometric polynomials are given. We show that these estimates are sharp. Corresponding embedding theorems for the Lipschitz spaces are investigated.     

Regular Balanced Cayley Maps for Cyclic, Dihedral and Generalized Quaternion Groups

A Cayley map is a Cayley graph embedded in an orientable surface such that. the local rotations at every vertex are identical. In this paper, balanced regular Cayley maps for cyclic groups, dihedral groups, and generalized quaternion groups are classified.     

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle,andthat if this system has a separatrix cycle passing through the weak saddle,then the stability of theseparatrix cycle is contrary to that of the singular point surrounded by it.     

Weak and strong convergence theorems for asymptotically nonexpansive mappings

The purpose of this paper is to prove weak and strong convergences of a modified implicit iteration process to common fixed points for a finite family of asymptotically nonexpansive mappings. The results presented in this paper improve and extend some well known results.     

Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem

Let (X _i ) be a stationary and ergodic Markov chain with kernel Q and f an L ² function on its state space. If Q is a normal operator and f=(I?Q)^1/2 g (which is equivalent to the convergence of \(\sum_{n=1}^{\infty}\frac{\sum_{k=0}^{n-1}Q^{k}f}{n^{3/2}}\) in L ²), we have the central limit theorem [cf. (Derriennic and Lin in C.R. Acad. Sci. Paris, Sér. I 323:1053–1057, 1996; Gordin and Lif?ic in Third Vilnius conference on probability and statistics, vol. 1, pp. 147–148, 1981)]. Without assuming normality of Q, the CLT is implied by the convergence of \(\sum_{n=1}^{\infty}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}\), in particular by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=o(\sqrt{n}/\log^{q}n)\), q>1 by Maxwell and Woodroofe (Ann. Probab. 28:713–724, 2000) and Wu and Woodroofe (Ann. Probab. 32:1674–1690, 2004), respectively. We show that if Q is not normal and f∈(I?Q)^1/2 L ², or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to \(\sum_{n=1}^{\infty}c_{n}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}<\infty\) for some sequence c _n ↘0, or by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=O(\sqrt{n}/\log n)\), the CLT need not hold.     

Generalized Qualification and Qualification Levels for Spectral Regularization Methods

The concept of qualification for spectral regularization methods (SRM) for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error (Engl et al. in Regularization of inverse problems. Mathematics and its applications, vol. 375, Kluwer Academic, Dordrecht, 1996; Mathé in SIAM J. Numer. Anal. 42(3):968–973, 2004; Mathé and Pereverzev in Inverse Probl. 19(3):789–803, 2003; Vainikko in USSR Comput. Math. Math. Phys. 22(3): 1–19, 1982). In this article, the definition of qualification is extended and three different levels are introduced: weak, strong and optimal. It is shown that the weak qualification extends the definition introduced by Mathé and Pereverzev (Inverse Probl. 19(3):789–803, 2003), mainly in the sense that the functions associated with orders of convergence and source sets need not be the same. It is shown that certain methods possessing infinite classical qualification (e.g. truncated singular value decomposition (TSVD), Landweber’s method and Showalter’s method) also have generalized qualification leading to an optimal order of convergence of the regularization error. Sufficient conditions for a SRM to have weak qualification are provided and necessary and sufficient conditions for a given order of convergence to be strong or optimal qualification are found. Examples of all three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced in Mathé and Pereverzev (Inverse Probl. 19(3):789–803, 2003) are shown. In particular, SRMs having extended qualification in each one of the three levels and having zero or infinite classical qualification are presented. Finally, several implications of this theory in the context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown. This work was supported by DARPA/SPO, NASA LaRC and the National Institute of Aerospace under Grant VT-03-1, 2535, by AFOSR Grants F49620-03-1-0243 and FA9550-07-1-0273, by Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET, and by Universidad Nacional del Litoral, U.N.L., Argentina, through Project CAI+D 2006, P.E. 236.     

On Weak Regular ^＊-semigroups

In this paper, a special kind of partial algebras called projective partial groupoids is defined.It is proved that the inverse image of all projections of a fundamental weak regular ^*-semigroup under the homomorphism induced by the maximum idempotent-separating congruence of a weak regular ^*-semigroup has a projective partial groupoid structure. Moreover, a weak regular ^*-product which connects a fundamental weak regular ^*-semigroup with corresponding projective partial groupoid is defined and characterized. It is finally proved that every weak regular ^*-product is in fact a weak regular ^*-semigroup and any weak regular ^*-semigroup is constructed in this way.     

A Weak Property β Equals Reflexivity for a Banach Space

We introduce a weak property () in Aerms of the De Blasi measure of weak noncompactness and prove that this property characterizes the reflexivity.     

On the Fourier-Sine and Kontorovich–Lebedev Generalized Convolution Transforms and Their Applications

Ukrainian Mathematical Journal - We study generalized convolutions for the Fourier sine and Kontorovich–Lebedev transforms $$ left(hunderset{F_s,K}{ast }fright)(x) $$ in a two-parameter...     

Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems

We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme.     

Generalized Gibbs phenomenon for Fourier partial sums and de la Vallée-Poussin sums

It is well-known that the Fourier partial sums of a function exhibit the Gibbs phenomenon at a jump discontinuity. We study the same question for de la Vallée-Poussin sums. Here we find a new Gibbs function and a new Gibbs constant. When the function is continuous, a behavior similar to the Gibbs phenomenon also occurs at a kink. We call it the “generalized Gibbs phenomenon”. Let $F_{n}(x):=\frac{k_{n}(g,x)-g(x)}{k_{n}(g,x_{0})-g(x_{0})}$ , where x ₀ is a kink and where k _n (g,x) represents Fourier partial sums and de la Vallée-Poussin sums. We show that F _n (x) exhibits the “generalized Gibbs phenomenon”. New universal Gibbs functions for both sums are derived.     

Central Limit Theorems for a Super-Diffusion over a Stochastic Flow

Central limit theorems of the occupation time of a superprocess over a stochastic flow are proved. For the critical and higher dimensions d≥4, the limits are Gaussian variables. For d=3, the limit is conditional Gaussian. When the stochastic flow disappears, the results degenerate to those for the ordinary super-Brownian motion.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

Continuous Singularity of the Weak Limit of Convolution Powers of a Discrete Probability Measure on 2 × 2 Stochastic Matrices

Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim _n ⁿ , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .     

Characterization of Polynomial Systems Satisfying Generalized Convolution Differential-Difference Equations

I. Introduction. The present paper has been motivated by the desire to find all polynomial solutions of the convolution type differential -difference equation (1.1) D_xg_n(x)=sum from i=1 to n-1 (g_i(x)g_(n-i)(x),n≥2,) where g_1(x) is assumed to be a constant. This problem arose in work by one of the authors (Kerr) with a differential equation arising in a coal research project     

On the Weak Kowalevski–Painlevé Property for Hyperelliptically Separable Systems

We consider so called hyperelliptically separable systems (h.s.s.) arising in various physical problems, whose generic invariant manifolds can be completed either to hyperelliptic Jacobians or to their nonlinear subvarieties (strata) or their finite coverings. In the case of strata the algebraic geometrical structure of such systems has much in common with that of algebraic completely integrable systems (a.c.i.s.). Using this property we study formal singular solutions of a.c.i.s. and h.s.s., which may contain fractional powers of time. We give estimates for the number and leading behavior of their principal and lower balances both for a generic and for the so called physical direction of the flow. This can be regarded as an useful extension of the Kowalevski–Painlevé integrability test. We also prove that when the system is h.s. but not a.c.i., its generic solutions are single-valued on an infinitely sheeted ramified covering of the complex time plane. Some model examples are considered, such as the hierarchy of integrable generalizations of the Henon–Heiles and the Neumann systems.     

A Certain Regular Property of the Method I Construction and Packing Measure

Let τ be a premeasure on a complete separable metric space and let τ＊ be the Method I measure constructed from τ. We give conditions on T such that τ＊ has a regularity as follows： Every τ＊-measurable set has measure equivalent to the supremum of premeasures of its compact subsets. Then we prove that the packing measure has this regularity if and only if the corresponding packing premeasure is locally finite.     

The α-comparison Property and Finite Nuclear Dimension of Generalized Inductive Limits for C~*-algebras

The paper is devoted to the study of generalized inductive limit of C~*-algebras with coherent maps being completely positive contractions of order zero: the nuclear dimension of generalized inductive limit of C~*-algebras with finite nuclear dimension is finite; the generalized inductive limits of C~*-algebras with the α-comparison property also have the α-comparison property.