We determine the integral extension groups $Ext^1({\Delta}(h),{\Delta}(h(k)))$ and $Ext^k({\Delta}(h),{\Delta}(h(k)))$, where ${\Delta}(h),{\Delta}(h(k))$ are Weyl modules of general linear group $GL_n$ corresponding to hook partitions $h=(a,1^b)$, $h(k)=(a+k,1^{b-k})$.

This paper concerns the description of holomorphic extensions of algebraic number fields. After expanding the notion of adele class group to number fields of infinite degree over Q, a hyperbolized adele class group ŜK is assigned to every number field K/Q. The projectivization of the Hardy space PH•[K] of graded-holomorphic functions on ŜK possesses two operations ⊕ and ⊗ giving it the structur...

We point out that the notion of an unparticle, recently introduced by Georgi, can be interpreted as a particular case of a field with continuously distributed mass considered in ref.[11]. We also point out that the simplest renormalizable extension of the SU c (3)⊗SU L (2)⊗U (1) Standard Model is the extension with the replacement of the U (1) gauge propagator 1 k 2 → 1 k 2 + ∞ 0 ρ(t) −t+k 2 +i...

(Here ΛE is a suitable period lattice commensurable with the Néron lattice of E.) The function Φ, which is transcendental as a function of τ , enjoys the following notable algebraicity property, a consequence of the theory of complex multiplication: Theorem HP: Let K ⊂ C be a quadratic imaginary extension of Q, and let K denote its maximal abelian extension. If τ belongs to H ∩ K, then Φ(τ) bel...

In this paper, we consider the defect (also called ramification deficiency) of finite extensions of valued fields. For a valued field (K, v), we will denote its value group by vK and its residue field by K or by Kv. An extension of valued fields is written as (L|K, v), meaning that v is a valuation on L and K is equipped with the restriction of this valuation. Every finite extension L of a valu...

For F/K an algebraic function field in one variable over a finite field of constants K (i.e., F is a finite algebraic extension of K(x) where x ∈ F is transcendental over K), let N(F ) and g(F ) denote the number of places of degree one and the genus, respectively, of F . Let F = (F1, F2, F3, . . .) be a tower of function fields, each defined over K. Further, we will assume that F1 ⊆ F2 ⊆ F3 . ...

We point out that the notion of an unparticle, recently introduced by Georgi, can be interpreted as a particular case of a field with continuously distributed mass considered in ref.[14]. We also point out that the simplest renormalizable extension of the SU c (3)⊗SU L (2)⊗U (1) Standard Model is the extension with the replacement of the U (1) gauge propagator 1 k 2 → 1 k 2 + ∞ 0 ρ(t) −t+k 2 +i...

In this paper we prove a conjecture of Jacquet about supercuspidal representations of GLn(K ) distinguished by GLn(k), or by Un(k), for K a quadratic unramified extension of a non-Archimedean local field k.

The precoloring extension coloring problem consists in deciding, given a positive integer k, a graph G = (V,E) and k pairwise disjoint subsets V0, . . . , Vk−1 of V , if there exists a (vertex) coloring S = (S0, . . . , Sk−1) of G such that Vi ⊆ Si, for all i = 0, . . . , k − 1. In this note, we show that the precoloring extension coloring problem is NP-complete in triangle free planar graphs w...