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.     

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.     

Subset currents on free groups

We introduce and study the space ${{\mathcal{S}{\rm Curr} (F_N)}}$ of subset currents on the free group F _N , and, more generally, on a word-hyperbolic group. A subset current on F _N is a positive F _N -invariant locally finite Borel measure on the space ${{\mathfrak{C}_N}}$ of all closed subsets of ?F _N consisting of at least two points. The well-studied space Curr(F _N ) of geodesics currents–positive F _N -invariant locally finite Borel measures defined on pairs of different boundary points–is contained in the space of subset currents as a closed ${{\mathbb{R}}}$ -linear Out(F _N )-invariant subspace. Much of the theory of Curr(F _N ) naturally extends to the ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ context, but new dynamical, geometric and algebraic features also arise there. While geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in F _N . If a free basis A is fixed in F _N , subset currents may be viewed as F _N -invariant measures on a “branching” analog of the geodesic flow space for F _N , whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of F _N with respect to A. Similarly to the case of geodesics currents, there is a continuous Out(F _N )-invariant “co-volume form” between the Outer space cv _N and the space ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ of subset currents. Given a tree ${{T \in {\rm cv}_N}}$ and the “counting current” ${{\eta_H \in \mathcal{S}\;{\rm Curr} (F_N)}}$ corresponding to a finitely generated nontrivial subgroup H ≤  F _N , the value ${{\langle T, \eta_H \rangle}}$ of this intersection form turns out to be equal to the co-volume of H, that is the volume of the metric graph T _H /H, where ${{T_H \subseteq T}}$ is the unique minimal H-invariant subtree of T. However, unlike in the case of geodesic currents, the co-volume form ${{{\rm cv}_N \times \mathcal{S}\;{\rm Curr}(F_N)\; \to [0,\infty)}}$ does not extend to a continuous map ${{\overline{{\rm cv}}_N \times \mathcal{S}\; {\rm Curr} (F_N) \to [0,\infty)}}$ .     

On Numerical Ranges of Rank-Two Operators

Possible shapes of numerical ranges of rank-two operators are studied. In particular it is proved that for 4-by-4 unitarily irreducible matrices with an eigenvalue of geometric multiplicity two, the numerical ranges have at most one flat portion on the boundary and there are no multiply generated round boundary points.     

Tensor Rank: Matching Polynomials and Schur Rings

We study the polynomial equations vanishing on tensors of a given rank. By means of polarization we reduce them to elements  $A$ of the group algebra ${\mathbb {Q}}[S_n\times S_n]$ and describe explicit linear equations on the coefficients of  $A$ to vanish on tensors of a given rank. Further, we reduce the study to the Schur ring over the group $S_n\times S_n$ that arises from the diagonal conjugacy action of  $S_n$ . More closely, we consider elements of ${\mathbb {Q}}[S_n\times S_n]$ vanishing on tensors of rank $n-1$ and describe them in terms of triples of Young diagrams, their irreducible characters, and nonvanishing of their Kronecker coefficients. Also, we construct a family of elements in ${\mathbb {Q}}[S_n\times S_n]$ vanishing on tensors of rank $n-1$ and illustrate our approach by a sharp lower bound on the border rank of an explicitly produced tensor. Finally, we apply this construction to prove a lower bound $5n^2/4$ on the border rank of the matrix multiplication tensor (being, of course, weaker than the best known one $(2-\epsilon )\cdot n^2$ , due to Landsberg, Ottaviani).     

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.     

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     

Performance of a silicon monochromator under high heat load

The performance of a cryogenically cooled double‐crystal silicon monochromator was studied under high‐heat‐load conditions with total absorbed powers and power densities ranging from 8 to 780 W and from 8 to 240 W mm^?2, respectively. When the temperature of the first crystal is maintained close to the temperature of zero thermal expansion of silicon, the monochromator shows nearly ideal performance with a thermal slope error of 0.6 µrad. By tuning the size of the first slit, the regime of the ideal performance can be maintained over a wide range of heat loads, i.e. from power densities of 110 W mm^?2 (at total absorbed power of 510 W) to 240 W mm^?2 (at total absorbed power of 240 W).