In this appendix we complete the results of Chapter 10 concerning the decomposition of tensor space under the action of the orthogonal group or the symplectic group. Following ideas of Weyl [7], we decompose the full tensor space into spaces of partially harmonic tensors. This is the full tensor generalization of the decompositions of the algebras of symmetric tensors and skew-symmetric tensors...

Let k be a number field. Let Ω denote the set of places of k and for v ∈ Ω, let kv denote the completion of k at v. A classical theorem of Hasse-Minkowski states that a quadratic form q over k represents zero non-trivially provided it represents zero non-trivially over kv for all v ∈ Ω; in particular, two quadratic forms over k are isomorphic if they are isomorphic over kv for all v ∈ Ω. Anothe...

We consider random walks where each path is equipped with a random weight which is stationary and independent in space and time. We show that under some assumptions the arising probability distributions are in a sense uniformly absolutely continuous with respect to the usual probability distribution for symmetric random walks. We consider random walks on the d-dimensional lattice Z with each pa...

Let ω(n) be the number of distinct prime factors of n. For any positive integer k let n = nk be the smallest positive integer such that ω(n + 1), . . . , ω(n + k) are mutually distinct. In this paper, we give upper and lower bounds for nk. We study the same quantity when ω(n) is replaced by Ω(n), the total number of prime factors of n counted with repetitions. Let ω(n) and Ω(n) denote respectiv...

Let Ω be a domain in a smooth complete Finsler manifold, and let G be the largest open subset of Ω such that for every x in G there is a unique closest point from ∂Ω to x (measured in the Finsler metric). We prove that the distance function from ∂Ω is in C k,α loc (G ∪ ∂Ω), k ≥ 2 and 0 < α ≤ 1, if ∂Ω is in C k,α .

We consider the following singularly perturbed Neumann problem ε∆u− u+ u = 0 in Ω, u > 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where p is subcritical and Ω is a smooth and bounded domain in R with its unit outward normal ν. Lin-Ni-Wei [18] proved that there exists ε0 such that for 0 < ε < ε0 and for each integer k bounded by 1 ≤ k ≤ δ(Ω, n, p) (ε| log ε|)n (0.1) where δ(Ω, n, p) is a constant depending only o...

Consider a bounded strongly pseudo-convex domain Ω with smooth boundary in Cn. Let T be the Toeplitz algebra on Bergman space La2(Ω). That is, is C⁎-algebra generated by operators {Tf:f∈L∞(Ω)}. Extending work [27], [28] special case of unit ball, we show that any such Ω, and {Tf:f∈VObdd}+K are essential commutants each other, where K collection compact On general considered this paper, proofs r...

For a finite rank one simple Lie-type group acting 2-transitively on set Ω and k∈ℕ we derive formulae for the number of G-orbits all k-subsets Ω.

We present in this chapter an overview of the Mumford and Shah model for image segmentation. We discuss its various formulations, some of its properties, the mathematical framework, and several approximations. We also present numerical algorithms and segmentation results using the Ambrosio–Tortorelli phase-field approximations on one hand, and using the level set formulations on the other hand....