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.     

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.      

The Gradient Flow of <Emphasis Type="Italic">∫</Emphasis><Subscript><Emphasis Type="Italic">M</Emphasis></Subscript>|Rm|<Superscript>2</Superscript>

We study the gradient flow of the Riemannian functional ℱ(g):=∫ _M |Rm|². This flow corresponds to a fourth-order degenerate parabolic equation for a Riemannian metric. We prove that the degeneracies may be accounted for entirely by diffeomorphism flow, and hence we show short-time existence using the DeTurck method. We prove L ² derivative estimates of Bernstein-Bando-Shi type and use these to give a basic obstruction to long time existence and prove a compactness theorem.      

Eigenvalue spectra and geometric transformation of polyhedra

Starting from one fullerene, the three geometric transformations, cap, face dual and edge dual produce series of carbon clusters and deltahedra. The geometric relation between these polyhedra implies that their topological matrices and eigenvalue spectra must be relative. We have developed a matrix algebra method to research some polyhedra with high symmetry and one kind, two or three kinds of equivalent vertices such as C₆₀(I_h), resolve their exact eigenvalues, and proved this point.     

Fuzzy profit measures for a fuzzy economic order quantity model

Changing economic conditions make the selling price and demand quantity more and more uncertain in the market. The conventional inventory models determine the selling price and order quantity for a retailer’s maximal profit with exactly known parameters. This paper develops a solution method to derive the fuzzy profit of the inventory model when the demand quantity and unit cost are fuzzy numbers. Since the parameters contained in the inventory model are fuzzy, the profit value calculated from the model should be fuzzy as well. Based on the extension principle, the fuzzy inventory problem is transformed into a pair of two-level mathematical programs to derive the upper bound and lower bound of the fuzzy profit at possibility level α. According to the duality theorem of geometric programming, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs to solve. By enumerating different α values, the upper bound and lower bound of the fuzzy profit are collected to approximate the membership function. Since the profit of the inventory problem is expressed by the membership function rather than by a crisp value, more information is provided for making decisions.     

More on pooling spaces

A pooling space is a ranked poset P such that the subposet w⁺ induced by the elements above w is atomic for each element w of P. Pooling spaces were introduced in [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Math. 282 (2004) 163-169] for the purpose of giving a uniform way to construct pooling designs, which have applications to the screening of DNA sequences. Many examples of pooling spaces were given in that paper. In this paper, we clarify a few things about the definition of pooling spaces. Then we find that a geometric lattice, a well-studied structure in literature, is also a pooling space. This provides us many classes of pooling designs, some old and some new. We study the pooling designs constructed from affine geometries. We find that some of them meet the optimal bounds related to a conjecture of Erdös, Frankl and Füredi.