Stationary points of O’Hara’s knot energies

In this article we study the regularity of stationary points of the knot energies E ^(α) introduced by O’Hara (Topology 30(2):241–247, 1991; Topol Appl 48(2):147–161, 1992; Topol Appl 56(1):45–61, 1994) in the range ${\alpha\in(2,3)}$ . In a first step we prove that E ^(α) is C ¹ on the set of all regular embedded curves belonging to ${{H^{(\alpha+1)/2,2}(\mathbb {R}{/}\mathbb {Z}, \mathbb {R}^n)}}$ and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of E ^(α) plus a positive multiple of the length. We show that stationary points of finite energy are of class C ^∞—so especially all local minimizers of E ^(α) among curves with fixed length are smooth.     

Constant mean curvature surfaces in warped product manifolds

We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, our results apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds.     

Root Systems In Finite Symplectic Vector Spaces

We study realizations of root systems in possibly degenerate symplectic vector spaces over finite fields, up to symplectic isomorphisms. The main result of this article is the classification of such realizations for the field 𝔽₂. Thereby, each root system requires a specific degree of degeneracy of the symplectic vector space. Our main motivation for this article is that for each such realization of a root system we can construct a Nichols algebra over a nonabelian group.     

An independent set approach for the communication network of the GPS III system

In this article, we describe an algorithm to find the optimal communication network for the new GPS III satellite system. Finding a possible network will be translated to a maximum clique problem and an efficient algorithm for finding all maximum cliques under these special circumstances is described.     

Test-Based Variable Selection via Cross-Validation

Abstract

Test-based variable selection algorithms in regression often are based on sequential comparison of test statistics to cutoff values. A predetermined a level typically is used to determine the cutoffs based on an assumed probability distribution for the test statistic. For example, backward elimination or forward stepwise involve comparisons of test statistics to prespecified t or F cutoffs in Gaussian linear regression, while a likelihood ratio. Wald, or score statistic, is typically used with standard normal or chi square cutoffs in nonlinear settings. Although such algorithms enjoy widespread use, their statistical properties are not well understood, either theoretically or empirically. Two inherent problems with these methods are that (1) as in classical hypothesis testing, the value of α is arbitrary, while (2) unlike hypothesis testing, there is no simple analog of type I error rate corresponding to application of the entire algorithm to a data set. In this article we propose a new method, backward elimination via cross-validation (BECV), for test-based variable selection in regression. It is implemented by first finding the empirical p value α*, which minimizes a cross-validation estimate of squared prediction error, then selecting the model by running backward elimination on the entire data set using α* as the nominal p value for each test. We present results of an extensive computer simulation to evaluate BECV and compare its performance to standard backward elimination and forward stepwise selection.     

The bit‐economy: An artificial model of open‐ended technology discovery

We describe and demonstrate an artificial model of technology discovery called the Bit‐Economy. The model is built from a minimal set of fundamental hallmarks of technology and develops under an open‐ended evolutionary operator which rewards new technology which is able to coordinate both spatially and temporally with the existing technology set. The Bit‐Economy, is able to replicate several features of real technology development including nonmonotonic growth, bunching of creation and destruction events, qualitative topologies of patent networks, and efficiency and waste‐management gains. In contrast to related works, we do not apply an exogenous fitness landscape and so are able to study the process of technology discovery as a self‐guided search toward more complex outcomes. © 2013 Wiley Periodicals, Inc. Complexity 18: 57–67, 2013     

An Infinite-Dimensional Semismooth Newton Method for Elasto-Plastic Contact Problems

A Fenchel dualization scheme for the one-step time-discretized elasto-plastic contact problem with kinematic or isotropic hardening is considered. The associated path is induced by a combined Moreau-Yosida / Tichonov regularization of the dual problem. The sequence of solutions to the regularized problems is shown to converge strongly to the solution of the original problem. This property relies on the density of the intersection of certain convex sets. The corresponding conditions are worked out and customary regularization approaches are shown to be valid in this context. It is also argued that without higher regularity assumptions on the data the resulting problems possess Newton differentiable optimality systems in infinite dimensions [2]. Consequently, each regularized subsystem can be solved mesh-independently at a local superlinear rate of convergence [6]. Numerically the problems are solved using conforming finite elements. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Embedded minimal tori in S 3 and the Lawson conjecture

We show that any embedded minimal torus in S ³ is congruent to the Clifford torus. This answers a question posed by H. B. Lawson, Jr., in 1970.     

Nearest neighbour diagnostic statistics on the accuracy of APT solute cluster characterisation

Diagnostic statistics and information theory techniques have been developed to investigate the accuracy to which solute clusters characterised in atom probe tomography (APT) data can reflect the true nature of the physical clusters in the original specimen. Simulated atom-probe datasets representing a range of atomic solute clustering within a pseudo-binary alloy upon an fcc aluminium lattice were generated for the study. The effectiveness of partitioning the APT-like simulated data based upon a binary classification defined by a distance threshold d _max upon the kth nearest neighbour distance distribution was investigated. Information theory was also used to optimise the selection of the threshold d _max. Analysis of variation was performed upon a factorial design of data simulations with low and high levels of: solute concentration; short-range order; and background to the mass-to-charge-state-ratio spectrum. This meta-analysis showed that the background levels have a significant compromising effect upon the binary classification in low solute systems with relatively low or random levels of clustering. Although the random clustering of higher solute concentrations is better analysed, significantly non-random clustering in both low and high solute concentrations is analysed well despite the presence of high levels of background. A meta-analysis of the binary classification upon a simulated dispersion of coherent precipitates within a similar matrix was also undertaken. Optimal k and d _max parameters are likely a dependent upon the physical dimensions of precipitate size as well as the precipitate/matrix solute concentrations.