The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space

Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (F _k ) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k ≥ 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k − 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs. Received: December 2005, Accepted: March 2006     

On a single discrete scale for preventive maintenance with two shock processes affecting a complex system

A new approach to optimal maintenance of systems (networks) is suggested. It is applied to systems subject to two external independent shock processes. A system ‘consists’ of two parts, and each shock process affects only its own part. A new notion of bivariate signature is suggested and used for obtaining survival characteristics of a system and further optimization of the preventive maintenance actions. The preventive maintenance optimization is considered in the univariate discrete scale that counts the overall numbers of shocks of both types. An example of a transportation network is considered. Copyright © 2016 John Wiley & Sons, Ltd.     

Computational thinking and constructionism: creating difference equations in spreadsheets

The invention of the computer has led to the establishment of a new research paradigm, computation, which has recently become more and more popular in scientific exploration. However, computation is not well represented in high school and university curricula in science and engineering, although it applies to a wide range of disciplines beyond computer science and software engineering. In light of the increasing need to provide students with computational education, this paper presents a novel way to develop computational thinking among students. The proposed approach is based on the implementation of Papert's theory of constructionism in electronic spreadsheets. In this approach, students build their knowledge while constructing the difference equation that describes a physical (or engineering) phenomenon, based on specific cases investigated in the spreadsheet. The method does not require the students to write code or perform complex calculations in the spreadsheet and makes it possible to teach advanced subjects at a relatively early stage. The method is demonstrated through contents taken from the secondary and tertiary curricula in mechanics and electromagnetism.     

The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Let F _k be a free group of rank k ≥ 2 with a fixed set of free generators. We associate to any homomorphism φ from F _k to a group G with a left-invariant semi-norm a generic stretching factor, λ(φ), which is a noncommutative generalization of the translation number. We concentrate on the situation where φ: F _k → Aut(X) corresponds to a free action of F _k on a simplicial tree X, in particular, where φ corresponds to the action of F _k on its Cayley graph via an automorphism of F _k . In this case we are able to obtain some detailed “arithmetic” information about the possible values of λ = λ(φ). We show that λ ≥ 1 and is a rational number with 2kλ ∈ ℤ[1/(2k − 1)] for every φ ∈ Aut(F _k ). We also prove that the set of all λ(φ), where φ varies over Aut(F _k ), has a gap between 1 and 1+(2k−3)/(2k ²−k), and the value 1 is attained only for “trivial” reasons. Furthermore, there is an algorithm which, when given φ, calculates λ(φ). The second and the third author were supported by the NSF grant DMS#0404991 and the NSA grant DMA#H98230-04-1-0115.     

Fredholmness and invertibility of Toeplitz operators with matrix almost periodic symbols

We consider Toeplitz operators with symbols that are almost periodic matrix functions of several variables. It is shown that under certain conditions on the group generated by the Fourier support of the symbol, a Toeplitz operator is Fredholm if and only if it is invertible.

     

Numerical ranges of cube roots of the identity

The numerical range of a bounded linear operator T on a Hilbert space H is defined to be the subset W(T)={〈Tv,v〉:v∈H,∥v∥=1} of the complex plane. For operators on a finite-dimensional Hilbert space, it is known that if W(T) is a circular disk then the center of the disk must be a multiple eigenvalue of T. In particular, if T has minimal polynomial z³-1, then W(T) cannot be a circular disk. In this paper we show that this is no longer the case when H is infinite dimensional. The collection of 3×3 matrices with three-fold symmetry about the origin are also classified.     

Modal group time spreads in weakly range-dependent deep ocean environments

The temporal spread of modal group arrivals in weakly range-dependent deep ocean environments is considered. It is assumed that the range dependence is sufficiently weak that mode coupling is predominantly local in mode number. The phrase "modal group arrival" is taken here to mean the contribution to a transient wave field corresponding to a fixed mode number. There are three contributions to modal group time spreads which combine approximately in quadrature. These are the reciprocal bandwidth (the minimal pulse width), a deterministic dispersive contribution that is proportional to bandwidth and grows like range r, and a scattering-induced contribution that grows approximately like r(32). The latter two contributions are shown to be proportional to the waveguide invariant beta, a property of the background sound speed profile. The results presented, based mostly on asymptotic theory, are shown to agree well with full-wave numerical wave field simulations and available exact mode theoretical results. Simulations are shown that correspond approximately to conditions during the LOAPEX acoustic propagation experiment.     

Self‐assembly of p‐tert‐butyl thiacalix[4]arenes and metal cations into nanoscale three‐dimensional particles

The shape of supramolecular aggregates based on stereoisomers of p‐tert‐butyl thiacalix[4]arenes functionalized with secondary, tertiary amide and hydrazide groups at the lower rim in cone, partial cone and 1,3‐alternate conformations with several metal cations were investigated by atomic force microscopy. The examined p‐tert‐butyl thiacalix[4]arenes form host–guest complexes; dimers, spherics ellipsoids and elongated nanoscale particles depending on the conformation of macrocycles, the nature of the binding centers and the nature of the metal cation. Only associates formed by p‐tert‐butyl thiacalix[4]arenes with morpholide groups at the lower rim in cone conformation with silver cations exhibited a higher antimicrobial activity. Copyright © 2012 John Wiley & Sons, Ltd.