Free Subgroups of Groups with Nontrivial Floyd Boundary

Abstract

We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the corresponding well-known results for hyperbolic groups and groups with infinitely many ends. It also shows that no finitely generated amenable group admits a nontrivial boundary of this type. This improves on a theorem by Floyd (Floyd, W. J. (1980). Group completions and limit sets of Kleinian groups. Invent. Math. 57: 205–218) as well as giving an elementary proof of a conjecture stated in that same paper. We also show that if the Floyd boundary of a finitely generated group is nontrivial, then it is a boundary in the sense of Furstenberg and the group acts on it as a convergence group.     

Univariate Geometric Stable Laws

The paper summarizes recent advances in the theory of geometric stable (GS) distributions. The results presented include parametrizations, characterizations, mixture representations, properties, asymptotic and convergent series expansions of densities and distribution functions, moments and tail behavior, simulation, and estimation.     

FRACTOP: A Geometric Partitioning Metaheuristic for Global Optimization

We propose a new metaheuristic, FRACTOP, for global optimization. FRACTOP is based on the geometric partitioning of the feasible region so that search metaheuristics such as Simulated Annealing (SA), or Genetic Algorithms (GA) which are activated in smaller subregions, have increased reliability in locating the global optimum. FRACTOP is able to incorporate any search heuristic devised for global optimization. The main contribution of FRACTOP is that it provides an intelligent guidance (through fuzzy measures) in locating the subregion containing the global optimum solution for the search heuristics imbedded in it. By executing the search in nonoverlapping subregions, FRACTOP eliminates the repetitive visits of the search heuristics to the same local area and furthermore, it becomes amenable for parallel processing. As FRACTOP conducts the search deeper into smaller subregions, many unpromising subregions are discarded from the feasible region. Thus, the initial feasible region gains a fractal structure with many space gaps which economizes on computation time. Computational experiments with FRACTOP indicate that the metaheuristic improves significantly the results obtained by random search (RS), SA and GA.     

Non-interactive geometric probing: Reconstructing non-convex polygons

We address the problem of characterizing polygonal shapes that can be reconstructed from a class of scanners that have asymmetric resolution. We approach this problem using the methodology of non-interactive probing.

Laser raster scanners provide very high precision along the direction of a scan, but it is not practical to place scans very close to each other. A system capable of generating an omni-directional scan pattern can make a series of directional measurements sufficient to permit the reconstruction of a scanned polygon based on the position of edge crossings and the path of the scanning beam between edge crossings. We provide a procedure to reconstruct a polygon from such a data set, as well as a characterization of the shapes that can be reconstructed given a particular scan density. Our system applies to both concave and convex polygons, as well as to polygons containing holes.     

On the geometric autocorrelation function of polycrystalline materials

Herein we derive an expression for direct determination of the geometric autocorrelation function W of a polycrystalline material from images of its grain boundary network (e.g., those delivered by orientation imaging microscopy). We also obtain an identity that relates the mean linear intercept function to a directional derivative of the geometric autocorrelation function. These formulae were applied to examine whether a widely-used formula for W, particularly in theoretical studies of attenuation of elastic waves in polycrystalline media, would be valid for the grain boundary structure of a commercial aluminum alloy. The conclusion was negative.     

A generalized geometric-programming solution to “An economic production quantity model with flexibility and reliability considerations”

The classical economic production quantity (EPQ) model assumes that items are produced by a perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be perfected cost-free, the set-up cost can be reduced by investment in flexibility improvement. In this paper, we propose an EPQ model with a flexible and imperfect production process. We formulate this inventory decision problem using geometric programming (GP), establish more general results using the arithmetic-geometric mean inequality, and solve the problem to obtain a closed-form optimal solution. Following the theoretical treatment, we provide a numerical example to demonstrate that GP has potential as a valuable analytical tool for studying a certain class of inventory control problems. Finally we discuss some aspects of sensitivity analysis of the optimal solution based on the GP approach.     

Geometric phases and coherent states

A cyclic evolution of a pure quantum state is characterized by a closed curve γ in the projective Hilbert space , equipped with the Fubini-Study geometry. It is known that the geometric phase for this evolution is given by the integral of the symplectic form of the Fubini-Study geometry over an arbitrary surface spanning γ. This result extends to an infinite-dimensional Hilbert space for a bosonic quantum field. We prove that is bounded above by the infimum area over all surfaces spanning γ, and that the bound is attained if γ can be spanned by a holomorphic curve. Using an earlier result concerning the intrinsic Euclidean geometry of the coherent state submanifold , we derive an expression for the geometric phase for a cyclic evolution amongst coherent states. We indicate how the intensity of a classical configuration can be inferred from the winding number of the exponential geometric phase about the origin in the complex plane. In the case of photon states we present group theoretic and 2-component spinor representations of . We derive an expression for in the case of a sequence of measurements such that the resulting states are coherent at each step, in terms of a sequence of projection operators. The situation in relation to some earlier experiments of Pancharatnam and Tomita–Chiao is explained.     

A brief study on the mixture rules of the acoustic non-linearity parameter B/A in binary biological mixtures

Using the improved thermodynamic method, the acoustic non-linearity parameter B/A of binary biological mixtures — such as pig blood, sheep blood and bovine blood — is measured in order to test the mixture rules proposed by Apfel and Sehgal. The results show that the B/A values of the above-mentioned blood decrease with an increasing volume fraction of water. The B/A value varies linearly with the volume fraction of protein. It is also shown that, treated as a binary mixture, the blood obeys the trend that the B/A value varies with the volume of water. This is in agreement with the rules proposed by Apfel and Sehgal. However, the measured B/A value is not exactly equal to that of the mixture rules.     

High Order Optimized Geometric Integrators for Linear Differential Equations

In this paper new integration algorithms based on the Magnus expansion for linear differential equations up to eighth order are obtained. These methods are optimal with respect to the number of commutators required. Starting from Magnus series, integration schemes based on the Cayley transform an the Fer factorization are also built in terms of univariate integrals. The structure of the exact solution is retained while the computational cost is reduced compared to similar methods. Their relative performance is tested on some illustrative examples.     

保险联结风险投资模型的赤字问题

保险公司被允许将部分资金投入风险市场,这样保险公司经营的风险来自于未来实际发生索赔的不确定性和投资收益的不确定性。研究了由经典的Cramer-Lundberg模型与按照几何布朗运动股票价格变动的一个风险模型,获得了三种资产分配情况下股票价格波动对赤字发生概率下界的影响。