In this paper A, B, a, b, c, d, e, f , g, h denote sets. One can prove the following three propositions: (1) idA↾B = idA ∩ [:B, B :]. (2) id{a,b,c,d} = {〈a, a〉, 〈b, b〉, 〈c, c〉, 〈d, d〉}. (3) [: {a, b, c, d}, {e, f, g, h} :] = {〈a, e〉, 〈a, f〉, 〈b, e〉, 〈b, f〉, 〈a, g〉, 〈a, h〉, 〈b, g〉, 〈b, h〉} ∪ {〈c, e〉, 〈c, f〉, 〈d, e〉, 〈d, f〉, 〈c, g〉, 〈c, h〉, 〈d, g〉, 〈d, h〉}. Let X, Y be trivial sets. Observe that ...

A (d, d + 1)-graph is a graph whose vertices all have degrees in the set {d, d + 1}. Such a graph is semiregular. An (r, r + 1)-factorization of a graph G is a decomposition of G into (r, r + 1)-factors. For d-regular simple graphs G we say for which x and r G must have an (r, r + 1)-factorization with exactly x (r, r + 1)-factors. We give similar results for (d, d + 1)-simple graphs and for (d...

The articles [25], [20], [8], [9], [1], [23], [18], [24], [19], [26], [22], [6], [3], [7], [14], [4], [17], [15], [16], [2], [11], [12], [13], [21], and [5] provide the terminology and notation for this paper. The following two propositions are true: (1) For every set x and for every non empty set D holds x∩ ⋃ D = ⋃ {x∩d : d ranges over elements of D}. (2) Let R be a non empty reflexive transit...

For more than fifteen years before the International Technology Recommendation Panel (ITRP) decision in August, 2004, there were intensive R&D activities and broad international collaboration among the groups at SLAC, KEK, FNAL, LLNL and other labs for the room temperature X-Band accelerator structures. The goal was to provide an optimized design of the main linac structure for the NLC (Next Li...

We give a counterexample to a conjecture by Wild about binary matroids. We connect two equivalent lines of research in matroid theory: a simple type of basis-exchange property and restrictions on the cardinalities of intersections of circuits and cocircuits. Finally, we characterize direct sums of series-parallel networks by a simple basis-exchange property. In [9], Wild proposed a characteriza...

Abstract We discuss Meyers-Serrin’s type results for smooth approximations of functions $$b=b(t,x): \mathbb {R}\times {R}^n\rightarrow {R}^m$$ b = ( t , x ) : R <m...

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Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain D ⊆ Rd with oblique reflection at ∂D if D 6≡ Rd. For each x ∈ D, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as 2 → 0 of the expected hitting time of the 2-ball centered at x and of the principal eigenvalue for L in the exterior dom...

In these solution I avoid using conditional expectations. But do try to give alternative proofs once we learnt conditional expectations. Exercise 1 For x, y ∈ R d define p t (x, y) = (2πt) − d 2 e − |x−y|2 2t. Prove that P t (x, dy) = p t (x, y)dy satisfies the Chapman-Kolmogorov equation: for Γ Borel subset of R d ,

We investigate the existence of principal eigenvalues (i.e., eigenval-ues corresponding to positive eigenfunctions) for the boundary value problem ?u(x) = g(x)u(x) on D; @u @n (x)+u(x) = 0 on @D where D is a bounded region in R N , g is an indeenite weight function and 2 R may be positive, negative or zero.