Comparison and hybridization of crossover operators for the nurse scheduling problem

In this paper, we present a hybrid genetic algorithm for the well-known nurse scheduling problem (NSP). The NSP involves the construction of roster schedules for nursing staff in order to maximize the quality of the roster schedule subject to various hard constraints. In the literature, several genetic algorithms have been proposed to solve the NSP under various assumptions. The contribution of this paper is twofold. First, we extensively compare the various crossover operators and test them on a standard dataset in a solitary approach. Second, we propose several options to hybridize the various crossover operators.     

Uniqueness of the Unconditional Basis of ?1 (?p) and ?p (?1), 0 < p < 1

We prove that the quasi-Banach spaces ₁ (_p) and _p (₁), 0 < p < 1 have a unique unconditional basis up to permutation     

Counting derangements, involutions and unimodal elements in the wreath product Cr ? Sn

We count derangements, involutions and unimodal elements in the wreath product C _r ≀ S _n by the numbers of excedances, fixed points and 2-cycles. Properties of the generating functions, including combinatorial formulas, recurrence relations and real-rootedness are studied. The results obtained specialize to those on the symmetric group S _n and on the hyperoctahedral group B _n when r = 1, 2, respectively.     

New computer-based search strategies for extreme functions of the Gomory–Johnson infinite group problem

We describe new computer-based search strategies for extreme functions for the Gomory–Johnson infinite group problem. They lead to the discovery of new extreme functions, whose existence settles several open questions.     

Ueber das volle Formensystem der tern?ren biquadratischen Formf=x13x2+x23x3+x33x1

Ohne ZusammenfassungHieizu cme Tale     

The Lindel?f number of Cp(X)×Cp(X) for strongly zero-dimensional X

We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C _p (X, M) is a continuous image of a closed subspace of C _p (X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindel?f number of C _p (X)×C _p (X) coincides with the Lindel?f number of C _p (X). We also prove that l(C _p (X ⁿ )^κ) ≤ l(C _p (X)^κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.     

Asymptotics of the Best Polynomial Approximation of?|x|p and?of?the Best Laurent Polynomial Approximation of sgn(x) on Two Symmetric Intervals

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of |x| ^p on two symmetric intervals. Note that, in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of sgn(x) on [−1,−a]∪[a,1] by Laurent polynomials.      

New series representations for Jacobi?s triple product identity and more via the q-Markov method

We derive new series representations for Jacobi?s triple product identity, the q-binomial theorem, q-analogs of the exponential function, and more with several special cases using the q-Markov-WZ method.     

Necessary and sufficient condition for the generalized solution of a mixed problem for the wave equation to belong to the class L p for p ≥ 1

In terms of requirements imposed on the boundary function, we obtain a necessary and sufficient condition for the generalized solution of a mixed problem for the wave equation with zero initial conditions and with boundary conditions of the first kind to belong to the class L _p .     

Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the p−1 problem

We utilize the method of Bellman functions to derive new $L^p$-estimates of Littlewood–Paley type involving $p?1$. Among the applications to singular integrals we improve the $2(p?1)$ bounds for the Ahlfors–Beurling operator on $L^p(\mathbb{C})$ when $p\to \infty$. In addition, dimensionless estimates of Riesz transforms in the classical as well as in the Ornstein–Uhlenbeck setting are attained. To cite this article: O. Dragi?evi?, A. Volberg, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Weyl’s theorem for algebraically wF(p, r, q) operators with p, r?>?0 and q?≥?1

If T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1 acting in an infinite-dimensional separable Hilbert space, then we prove that Weyl’s theorem holds for f(T) for any f ∈ Hol(σ(T)), where Hol(σ(T)) is the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is a wF(p, r, q) operator with p, r > 0 and q ≥ 1, then the a-Weyl’s theorem holds for f(T). In addition, if T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1, then we establish the spectral mapping theorems for the Weyl spectrum and for the essential approximate point spectrum of T for any f ∈ Hol(σ(T)), respectively. Finally, we examine the stability of Weyl’s theorem and the a-Weyl’s theorem under commutative perturbations by finite-rank operators.     

Pairwise ?-independence of the sets TiA for a mixing transformation T

If an ergodic automorphism T of a probability space is not partially rigid, then for any numbers a ∈ (0, 1) and ɛ > 0 there exists a set A such that all sets T ⁱ A, i > 0, are pairwise ɛ-independent. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 43, No. 2, pp. 88–91, 2009 Original Russian Text Copyright ? by V. V. Ryzhikov This research was carried out under the Program for Support of Leading Scientific Schools in the Russian Federation (grant no. 6849.2006.1).     

Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

Given k identical salesmen, where k ? 2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k = 2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k ? 2, closing a question that has remained open for a decade. Along with this, we have further developed a (1 + ?)-approximation algorithm for any ? > 0.     

Topological hochschild homology of extensions of ?/p? by polynomials, and of ?[x]/(xn) and ?[x]/(xn?1)

Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x ⁿ) and [x]/(x ⁿ–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra.     

Dealing with the multiplicity of solutions of the ℓ1 and ℓ∞ regression models

It is well known that the least absolute value (?_∞) and the least sum of absolute deviations (?₁) algorithms produce estimators that are not necessarily unique. In this paper it is shown how the set of all solutions of the ?₁ and ?_∞ regression problems for moderately large sample sizes can be obtained. In addition, if the multiplicity of solutions wants to be avoided, two new methods giving the same optimal ?₁ and ?_∞ values, but supplying unique solutions, are proposed. The idea consists of using two steps: in the first step the optimal values of the ?₁ and ?_∞ errors are calculated, and in the second step, in case of non-uniqueness of solutions, one of the multiple solutions is selected according to a different criterion. For the ?_∞ the procedure is used sequentially but removing, in each iteration, the data points with maximum absolute residual and adding the corresponding constraints for keeping these residuals, and this process is repeated until no change in the solution is obtained. In this way not only the maximum absolute residual values are minimized in the modified method, but also the maximum absolute residual values of the remaining points sequentially, until no further improvement is possible. In the ?₁ case a least squares criterion is used but restricted to the ?₁ residual condition. Thus, in the modified ?₁ method not only the ?₁ residual is minimized, but also the sum of squared residuals subject to the ?₁ residual. The methods are illustrated by their application to some well known examples and their performances are tested by some simulations, which show that the lack of uniqueness problem cannot be corrected for some experimental designs by increasing the sample size.     

Necessary and sufficient conditions for a generalized solution to the initial-boundary value problem for the wave equation to belong to W p 1 with p ≥ 1

We establish necessary and sufficient conditions on the boundary function under which a generalized solution to the initial-boundary value problem for the wave equation with boundary conditions of the first kind belongs to W _p ¹ .     

Homogenization of the elliptic Dirichlet problem: Error estimates in the (L2 → H1)-norm

Let      

Surfaces with pg?=?q?=?1, K2?=?8 and nonbirational bicanonical map

We prove that if the bicanonical map of a minimal surface of general type S with p _g  = q = 1 and K_S²=8{K_{S}^2=8} is nonbirational, then it is a double cover onto a rational surface. An application of this theorem is the complete classification of minimal surfaces of general type with p_g=q=1, K_S²=8{p_{g}=q=1, K_{S}^2=8} and nonbirational bicanonical map.