These problems are inspired by a careful study of the papers of concerning bipartite distance-regular graphs. Throughout these notes we let Γ = (X, R) denote a bipartite distance-regular graph with diameter D ≥ 3 and standard module V = C X. We fix a vertex x ∈ X and let E denote the corresponding dual primitive idempotents. We define the matrices R = D i=0 E * i+1 AE * i , L= D i=0 E * i−1 AE ...

Let (M, θ) be a strictly pseudoconvex pseudo-Hermitian compact hypersurface in C in the sense of Webster [34] with a pseudo-Hermitian real oneform θ on M . Let Rθ be the Webster pseudo scalar curvature for M with respect to θ. By the solution of the CR Yamabe problem given by Jerison and Lee [18], Gamara and Yacoub [10] and Gamara [9] (for n = 1), there is a pseudo-Hermitian real one-form θ so ...

Let G be a connected reductive algebraic group with Lie algebra g. The ground field k is algebraically closed and of characteristic zero. Fundamental results in invariant theory of the adjoint representation ofG are primarily associated with C. Chevalley and B. Kostant. Especially, one should distinguish the ”Chevalley restriction theorem” and seminal article of Kostant [5]. Later, Kostant and ...

Let Γ denote a distance-regular graph with diameter D ≥ 3, intersection numbers ai, bi, ci and Bose-Mesner algebra M. For θ ∈ C ∪∞ we define a 1 dimensional subspace of M which we call M(θ). If θ ∈ C then M(θ) consists of those Y in M such that (A−θI)Y ∈ CAD, where A (resp. AD) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ = ∞ then M(θ) = CAD. By a pseudo primitive idempotent f...

Let Θ(n, k) be the set of digraphs of order n that have at most one walk of length k with the same endpoints. Let θ(n, k) be the maximum number of arcs of a digraph in Θ(n, k). We prove that if n ≥ 5 and k ≥ n − 1 then θ(n, k) = n(n−1)/2 and this maximum number is attained at D if and only if D is a transitive tournament. θ(n, n− 2) and θ(n, n− 3) are also determined.

We show that for every fixed A > 0 and θ > 0 there is a θ = θ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q1 = Q2 := n 1/2(log n)−θ and Q3 := (log n) θ. Then for all q3 ≤ Q3, all reduced residues a3 mod q3, almost all q2 ≤ Q2, all admissible residues a2 mod q2, almost all q1 ≤ Q1 and all admissible residues a1 mod q1, there exists a representation n = p1+...

This supplementary appendix contains additional materials for the paper “Robustness and Separation in Multidimensional Screening.” Section B contains proofs of auxiliary results not included in the main paper. Section C details how the generalized virtual values coincide with traditional (ironed) virtual values in the single-good monopoly problem. Theorems, equations, and sections in the main p...

Let Θ = Θ0 ∪Θ1 be a parameter space. Consider a parametric family {f(x|θ), θ ∈ Θ}. Suppose we want to test the null hypothesis, H0, that θ ∈ Θ0 against the alternative, Ha, that θ ∈ Θ1. Let C be some critical set. Then the probability that the null hypothesis is rejected is given by β(θ) = Pθ{X ∈/ C}. Recall that the test based on C has level α if α ≥ supθ Θ0 β(θ). The restriction of β(·) on Θ1...

The following proposition is true (1) Let G, H be non empty groupoids and x be an element of ∏ 〈G,H〉. Then there exists an element g of G and there exists an element h of H such that x = 〈g, h〉. Let G1, G2, H1, H2 be non empty groupoids, let f be a map from G1 into H1, and let g be a map from G2 into H2. The functor Gr2Iso(f, g) yields a map from ∏ 〈G1, G2〉 into ∏ 〈H1,H2〉 and is defined by the ...

Let X be a smooth curve of genus g. For any vector bundles E , F on X , let μE,F : H 0(X, E) ⊗ H 0(X, F) → H 0(X, E ⊗ F) be the multiplication map. Here we study the injectivity of μE,F when E , F are general stable bundles with h1(X, E) = h1(X, F) = 0.