Existence, multiplicity and concentration of bound states for a quasilinear elliptic field equation

In this paper we study existence, multiplicity and concentration of solutions for the following nonlinear field equation where the potential V is positive and W is an appropriate singular function. Here is regarded as a small parameter. Under suitable conditions on V and W we find solutions exhibiting a concentration behaviour at an absolute minimum of V as Such solutions are obtained as local minima for the associated functional; the proofs of our results rely on a careful analysis of the behaviour of minimizing sequences and use arguments inspired by the concentration-compactness principle. Received July 21, 1999; Accepted April 9, 2000 / Published online September 14, 2000     

On toric h-vectors of centrally symmetric polytopes

We prove tight lower bounds for the coefficients of the toric h-vector of an arbitrary centrally symmetric polytope generalizing previous results due to R. Stanley and the author using toric varieties. Our proof here is based on the theory of combinatorial intersection cohomology for normal fans of polytopes developed by G. Barthel, J.-P. Brasselet, K. Fieseler and L. Kaup, and independently by P. Bressler and V. Lunts. This theory is also valid for nonrational polytopes when there is no standard correspondence with toric varieties. In this way we can establish our bounds for centrally symmetric polytopes even without requiring them to be rational. Received: 24 March 2004     

Relative symplectic subquadrangle hemisystems of the Hermitian surface

We introduce the notion of relative subquadrangle regular system of a generalized quadrangle. A relative subquadrangle regular system of order m on a generalized quadrangle S of order (s, t) is a set \({\mathcal R}\) of embedded subquadrangles with a prescribed intersection property with respect to a given subquadrangle T such that every point of S T lies on exactly m subquadrangles of \({\mathcal R}\) . If m is one half of the total number of such subquadrangles on a point we call \({\mathcal R}\) a relative subquadrangle hemisystem with respect to T. We construct two infinite families of symplectic relative subquadrangle hemisystems of the Hermitian surface \({{\mathcal H}(3,q^2)}\) , q even.     

A new semifield of order 2

In [G. Lunardon, Semifields and linear sets of PG(1,q^t), Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta (in press)], G. Lunardon has exhibited a construction method yielding a theoretical family of semifields of order q²ⁿ,n>1 and n odd, with left nucleus F_qⁿ, middle and right nuclei both F_q² and center F_q. When n=3 this method gives an alternative construction of a family of semifields described in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti, On a generalization of cyclic semifields, J. Algebraic Combin. 26 (2009), 1-34], which generalizes the family of cyclic semifields obtained by Jha and Johnson in [V. Jha, N.L. Johnson, Translation planes of large dimension admitting non-solvable groups, J. Geom. 45 (1992), 87-104]. For n>3, no example of a semifield belonging to this family is known.In this paper we first prove that, when n>3, any semifield belonging to the family introduced in the second work cited above is not isotopic to any semifield of the family constructed in the former. Then we construct, with the aid of a computer, a semifield of order 2¹⁰ belonging to the family introduced by Lunardon, which turns out to be non-isotopic to any other known semifield.     

On the strong convergence of a general-type Krasnosel’skii–Mann’s algorithm depending on the coefficients

Let H be a Hilbert space, \({(W_n)_{n \in \mathbb{N}}}\) a suitable family of mappings, S a nonexpansive mapping and D a strongly monotone operator. We are interested in the strong convergence of the general scheme

$$x_{n + 1} = \gamma x_{n} + (1 - \gamma)W_{n} (\alpha_{n}S_{x_{n}} + (1 - \alpha_{n})(I - \mu_{n}D)x_{n}),\quad \gamma \in [0, 1),$$

in dependence of the coefficients \({(\alpha_{n})_{n \in \mathbb{N}}}\) and \({(\mu_{n})_{n \in \mathbb{N}}}\) .

     

Factor probabilistic distance clustering (FPDC): a new clustering method

Factor clustering methods have been developed in recent years thanks to improvements in computational power. These methods perform a linear transformation of data and a clustering of the transformed data, optimizing a common criterion. Probabilistic distance (PD)-clustering is an iterative, distribution free, probabilistic clustering method. Factor PD-clustering (FPDC) is based on PD-clustering and involves a linear transformation of the original variables into a reduced number of orthogonal ones using a common criterion with PD-clustering. This paper demonstrates that Tucker3 decomposition can be used to accomplish this transformation. Factor PD-clustering alternatingly exploits Tucker3 decomposition and PD-clustering on transformed data until convergence is achieved. This method can significantly improve the PD-clustering algorithm performance; large data sets can thus be partitioned into clusters with increasing stability and robustness of the results. Real and simulated data sets are used to compare FPDC with its main competitors, where it performs equally well when clusters are elliptically shaped but outperforms its competitors with non-Gaussian shaped clusters or noisy data.     

Integer linear programming models for a cement delivery problem

We consider a cement delivery problem with an heterogeneous fleet of vehicles and several depots. The demands of the customers are typically larger than the capacity of the vehicles which means that most customers are visited several times. This is a split delivery vehicle routing problem with additional constraints. We first propose a two phase solution method that assigns deliveries to the vehicles, and then builds vehicle routes. Both subproblems are formulated as integer linear programming problems. We then show how to combine the two phases in a single integer linear program. Experiments on real life instances are performed to compare the performance of the two solution methods.     

Interior-Point Solver for Large-Scale Quadratic Programming Problems with Bound Constraints

In this paper, we present an interior-point algorithm for large and sparse convex quadratic programming problems with bound constraints. The algorithm is based on the potential reduction method and the use of iterative techniques to solve the linear system arising at each iteration. The global convergence properties of the potential reduction method are reassessed in order to take into account the inexact solution of the inner system. We describe the iterative solver, based on the conjugate gradient method with a limited-memory incomplete Cholesky factorization as preconditioner. Furthermore, we discuss some adaptive strategies for the fill-in and accuracy requirements that we use in solving the linear systems in order to avoid unnecessary inner iterations when the iterates are far from the solution. Finally, we present the results of numerical experiments carried out to verify the effectiveness of the proposed strategies. We consider randomly generated sparse problems without a special structure. Also, we compare the proposed algorithm with the MOSEK solver. Research partially supported by the MIUR FIRB Project RBNE01WBBB “Large-Scale Nonlinear Optimization.”