Let E be a number field and let G be the group GLn considered as reductive algebraic group over E. Let A be the ring of adèles of E and let G(A) be the group of points of G with values in A. Let G(A) be the intersection of the kernels of the maps x 7→ |χ(x)|, x ∈ G(A), where χ ranges over the group X(G)E of characters of G defined over E. Then the (noninvariant) trace formula of Arthur is an id...

It is the purpose of this paper to prove Theorems 1 and 2 below which relate the existence, uniqueness, and general behavior of the solution, y(x, e), for small e>0, of the two-point boundary value problem <y" + f(x, y)y'+ g(x, y) = 0, y(0) = yo, y(i) = yu with the solution m(x) of the corresponding "degenerate" initial value problem f(x, u)u' + g(x, u) = 0, m(1) = yi. Theorem 1 (Existence). Le...

We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [2]. For a graph G = (V, E), n := |V |, e := |E|, and G := (V, E), where E := { xy : x ∈ V, y ∈ V, x = y, xy ∈ E }. For a bipartite graph GBPT = (X, Y ; E), GBPT := (X, Y ; E ), where E := { xy : x ∈ X, y ∈ Y, xy ∈ E }. The degree of vertex vi is denoted by di. The...

Let S∞ be the group of all permutations of N and X be a standard Borel space. Then the space XN of functions from N to X is a standard Borel space, and S∞ acts on this space by permutation where given x ∈ XN and g ∈ S∞, g · x(n) = x(g−1(n)). Given any countable subgroup G of S∞, we can likewise restrict this action to G and consider the induced orbit equivalence relation on XN, which we will no...

Let G be a finite graph with p vertices and x its chromatic polynomial. A combinatorial interpretation is given to the positive integer (-l)px(-A), where h is a positive integer, in terms of acyclic orientations of G. In particular, (-l)Px(-1) is the number of acyclic orientations of G. An application is given to the enumeration of labeled acyclic digraphs. An algebra of full binomial type, in ...

The sets of vertices and edges of an undirected, simple, finite, connected graph G are denoted by V (G) and E(G), respectively. An arbitrary nonempty finite subset of consecutive integers is called an interval. An injective mapping φ : E(G) → {1, 2, . . . , |E(G)|} is called a labeling of the graph G. If G is a graph, x is its arbitrary vertex, and φ is its arbitrary labeling, then the set SG(x...

Divergence between populations adapting to different environments may be facilitated when the populations differ in their sexual traits. We tested whether colonizing a novel environment may, through phenotypic plasticity, change sexual traits in a way that could alter the dynamics of sexual selection. This hypothesis has two components: changes in mean phenotypes across environments, and change...

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 ...

We conducted an experiment for 112 d with yearling beef heifers to evaluate the effects of cottonseed meal (CSM) fed with various concentrations of vitamin E on hematological and tissue components. Heifers were assigned randomly to four treatments, with eight heifers per treatment. The treatments consisted of the following dietary supplements: 1) CON, based on soybean meal with 30 IU vitamin E/...

Let G be a graph with vertex set V (G) and edge set E(G), and let g, f be two nonnegative integer-valued functions defined on V (G) such that g(x) ≤ f(x) for every vertex x of V (G). We use dG(x) to denote the degree of a vertex x of G. A (g, f)-factor of G is a spanning subgraph F of G such that g(x) ≤ dF (x) ≤ f(x) for every vertex x of V (F ). In particular, G is called a (g, f)-graph if G i...