Randomized k-server algorithms for growth-rate bounded graphs

The k-server problem is a fundamental online problem where k mobile servers should be scheduled to answer a sequence of requests for points in a metric space as to minimize the total movement cost. While the deterministic competitive ratio is at least k, randomized k-server algorithms have the potential of reaching o(k)-competitive ratios. Prior to this work only few specific cases of this problem were solved. For arbitrary metric spaces, this goal may be approached by using probabilistic metric approximation techniques. This paper gives the first results in this direction, obtaining o(k)-competitive ratio for a natural class of metric spaces, including d-dimensional grids, and wide range of k.     

Non-linear ground state representations and sharp Hardy inequalities

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.     

Classification of sets satisfying the geometric characterization

The geometric characterization identifies the sets of nodes such that the Lagrange polynomials are products of factors of first degree. We offer a detailed classification of all known sets satisfying the geometric characterization in the plane. The defect, which takes into account the number of lines containing more nodes than the degree, plays a fundamental role in this classification. Partially supported by the Spanish Research Grant MTM2006-03388, by Gobierno de Aragón and Fondo Social Europeo.     

Time-dependent harmonic oscillator in a magnetic field and an electric field on the non-commutative plane

In the non-commutative space, wave functions and geometric phases are derived for the time-dependent harmonic oscillator in external time-dependent magnetic and electric field. Explicit forms of the coherent states are also given, which are not the minimum uncertainty states for the coordinates and momenta.     

Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

We prove Helly-type theorems for line transversals to disjoint unit balls in ℝ ^d . In particular, we show that a family of n≥2d disjoint unit balls in ℝ ^d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n≥4d−1 disjoint unit balls in ℝ ^d admits a line transversal if any subfamily of size 4d−1 admits a transversal. Andreas Holmsen was supported by the Research Council of Norway, prosjektnummer 166618/V30. Otfried Cheong and Xavier Goaoc acknowledge support from the French-Korean Science and Technology Amicable Relationships program (STAR).     

A four-node discrete singular convolution for geometric transformation and its application to numerical solution of vibration problem of arbitrary straight-sided quadrilateral plates

A four-node discrete singular convolution (DSC) method is developed for free vibration analysis of arbitrary straight-sided quadrilateral plates. The straight-sided quadrilateral domain is mapped into a square domain in the computational space using a four-node element. By using the geometric transformation, the governing equations and boundary conditions of the plate are transformed from the physical domain into a square computational domain. Numerical examples illustrating the accuracy and convergence of the DSC method for skew, trapezoidal, rhombic and arbitrary quadrilateral plates are presented. The results obtained by DSC method were compared with those obtained by the other numerical methods.     

Straight line embeddings of cubic planar graphs with integer edge lengths

We prove that every simple cubic planar graph admits a planar embedding such that each edge is embedded as a straight line segment of integer length. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:270‐274, 2008     

Geometric and Topological Thinking in Organic Chemistry

The beginning student of organic chemistry is often bewildered by what appears to be an enormous maze of random structural variations and reactions that can be mastered only by tedious memorization. To the organic chemist, however, the same subject is often a beautifully ordered discipline of elegant simplicity. An important value of learning organic chemistry is the mastering of “organic thinking,” an approach to intellectual processing whereby the “sameness” of many families of structures and reactions is revealed. This article offers the author's personal views of organic thinking and explores the intellectual and scientific foundations of organic chemistry and of the powerful methods that provide the field with a platform for making rapid conceptual and experimental advances. It is proposed that these methods involve a geometric and topological approach to scientific reasoning within the framework of scientific paradigms that guide experimental design and execution. The basis of this approach is considered in relation to day-to-day thinking, problem solving, and the psychological drive for intellectual closure. The power of the approach is illustrated by the analysis of several photochemical and chemiluminescent reactions.     

Concentrating solutions of some singularly perturbed elliptic equations

We study singularly perturbed elliptic equations arising from models in physics or biology, and investigate the asymptotic behavior of some special solutions. We also discuss some connections with problems arising in differential geometry.