In 1974, Harry Ruderman proposed the following problem in the problem section of this MONTHLY: if m > n ≥ 0 are integers such that 2m − 2n divides 3m − 3n , then show that 2m − 2n divides xm − xn for all natural numbers x . This problem is still open. We prove that there are only finitely many pairs of natural numbers m, n such that 2m − 2n divides 3m − 3n . Since the proof involves the Schmidt...

We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of W infinity algebras (W∞, W1+∞, W 1,1 ∞ , W M ∞ , W N 1+∞, W M,N ∞ ) with central charges (2, 1, 3, 2M , N , 2M +N). The characters of these representations are computed. We construct a new extended superalgebra W ∞ , whose bosonic sector is W ∞ ⊕ W N 1+∞. Its representations obta...

Let m,n ∈ N. In this paper we study the right permutation action of the symmetric group S2n on the set of all the Brauer n-diagrams. A new basis for the free Z-module Bn spanned by these Brauer n-diagrams is constructed, which yields Specht filtrations for Bn. For any 2m-dimensional vector space V over a field of arbitrary characteristic, we give an explicit and characteristic-free description ...

Recall we were considering the problem minz∈Rn p(z) where p(z) is a degree 2m polynomial such that p(z) = ∑ |α|≤2m pαz , where α = (α1, . . . , αn) is a vector of exponents, using the notation that z = z1 1 . . . z αn n . One equivalent way of expressing this problem is to notice this is the same as minimizing ∫ p(z)μ(dz) over the set of all probability distributions μ and note that it’s equiva...

Suppose a graph G have n vertices, m edges, and t triangles. Letting λn (G) be the largest eigenvalue of the Laplacian of G and μn (G) be the smallest eigenvalue of its adjacency matrix, we prove that λn (G) ≥ 2m2 − 3nt m (n2 − 2m) , μn (G) ≤ 3n3t− 4m3 nm (n2 − 2m) , with equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, then λn (G) ≥ 2mn (r − 1) (...

These groups form an intriguing set of invariants. Despite the vast literature on this subject, the groups have been calculated only in very few cases, such as for the projective spaces. One of the goals of this note is to understand these invariants in case the variety has “few cycles” in the sense that rational and homological equivalence coincide for all m-cycles up to a certain rank m = s. ...

so we take the incoming and the outgoing electrons to be on-shell, p2 = p′2 = m2, but the photon is off-shell, q2 6= 0. Moreover, we put the vertex in the context of the complete electron line — including the external line factors, thus ū(p′)× ieΓμ × u(p). As discussed in class, this simplifies the Lorentz and Dirac structure of the vertex and allows us to write it as Γ(p, p) = Fel(q )× (p′ + p...

In this paper, we consider the following Yamabe type problem of polyharmonic operator : { Dmu = |u| 4m N−2m u on S u ∈ H(S ), (P ) where N ≥ 2m + 1,m ∈ N+, S is the unit sphere with the induced Riemannian metric g = gSN , and Dm is the elliptic differential operator of 2m order given by Dm = m ∏ k=1 (−∆g + 14(N − 2k)(N + 2k − 2)) where ∆g is the Laplace-Beltrami operator on S . We will show tha...

beta(2)-Microglobulin-related (A beta 2M) amyloidosis is a frequent and serious complication in patients on long-term dialysis, and beta(2)-microglobulin is a major structural component of A beta 2M amyloid fibrils. Several biologic molecules inhibiting the depolymerization of A beta 2M amyloid fibrils at a neutral pH were found recently. The effect of trifluoroethanol and glycosaminoglycans (G...

A number of investigations have established the critical role of interleukin 4 (IL-4) in mediating the development of T helper (Th)2 effector cells in vitro and in vivo. Despite intensive study, the origin of the IL-4 required for Th2 priming and differentiation remains unclear. Natural killer (NK)1.1+ alpha/beta T cell receptor+ T(NT) cells, a unique lineage of cells capable of producing large...