Discrete semi-stable distributions

The purpose of this paper is to introduce and study the concepts of discrete semi-stability and geometric semi-stability for distributions with support inZ ₊. We offer several properties, including characterizations, of discrete semi-stable distributions. We establish that these distributions posses the property of infinite divisibility and that their probability generating functions admit canonical representations that are analogous to those of their continuous counterparts. Properties of discrete geometric semi-stable distributions are deduced from the results obtained for discrete semi-stability. Several limit theorems are established and some examples are constructed.     

Potentially semi-stable deformation rings

Let be a finite extension and the absolute Galois group of . For a complete local ring with finite residue and a finite free -module equipped with an action of , we show that has a maximal quotient over which the representation is semi-stable with Hodge-Tate weights in a given range. We show an analogous result for representations which are potentially semi-stable of fixed Galois type and -adic Hodge type.

If is the universal deformation of , then we compute the dimension of and we show that these rings are sometimes smooth.

Finally we apply this theory to show, in some new cases, the compatibility of the -adic Galois representation attached to a Hilbert modular form with the local Langlands correspondence at .

     

Stable and semi-stable unicyclic graphs

We show by a constructive proof, that if a unicyclic graph has a transposition in its automorphism group, then it is stable. Using a similar technique, we also determine which unicyclic graphs are not semi-stable.     

Two applications of semi-stable graphs

A Michigan graph G on a vertex set V is called semi-stable if for some υ?V, Γ(G_υ) = Γ(G)_υ. It can be shown that all regular graphs are semi-stable and this fact is used to show (i) that if Γ(G) is doubly transitive then G = K_n or K?_n, and (ii) that Γ(G) can be recovered from Γ(G_υ). The second result is extended to the case of stable graphs.     

p-Rank and semi-stable reduction of curves

Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above X_k≔ x × _rk. Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order pⁿ, we compute the p-rank of tf⁻¹ (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by pⁿ −1.     

Filtered modules corresponding to potentially semi-stable representations

We classify the filtered modules with coefficients corresponding to two-dimensional potentially semi-stable p-adic representations of the absolute Galois groups of p-adic fields under the assumptions that p is odd and the coefficients are large enough.     

A note on lattices in semi-stable representations

Let p be a prime, K a finite extension over \mathbb Q_p{{\mathbb Q}_p} and G = Gal([`(K)] /K){G = {\rm Gal}(\overline K /K)} . We extend Kisin’s theory on j{\varphi} -modules of finite E(u)-height to give a new classification of G-stable \mathbb Z1_p{{\mathbb Z}1_p} -lattices in semi-stable representations.     

GIT-equivalence and semi-stable subcategories of quiver representations

In this paper, we answer the question of when the subcategory of semi-stable representations is the same for two rational vectors for an acyclic quiver. This question has been previously answered by Ingalls, Paquette, and Thomas in the tame case in [14]. Here we take a more invariant theoretic approach, to answer this question in general. We recover the known result in the tame case.     

Stable Runge-Kutta integrations for differential systems with semi-stable equilibria

Summary. In this paper, the stability of Runge-Kutta methods for differential systems with semi-stable equilibria is analyzed. In fact, a way of circumventing the very severe unconditional stability barrier [3] is proposed. The usage of time-meshes with step-size ratios under some constant greater than one, as usual in standard numerical codes, is considered in order to recover stability for many implicit methods. The main result roughly says that any strongly A-stable Runge-Kutta method is stable for this kind of differential systems on this sort of meshes.Work supported by project BMF2001-2562Mathematical Subject Classifications (2000): 65L20     

Differential systems with semi-stable equilibria and numerical methods

Summary. In this paper the (long time) stability of numerical methods for a class of nonlinear autonomous differential systems possessing a semi-stable equilibrium, is considered. The local dynamics of the class under the H-assumptions is reviewed. Some interesting differential systems of stiff type are seen to belong to this class. It is also shown that the Implicit Euler method is the only one of practical interest which is unconditionally stable. Mathematics Subject Classification (2000):65L20, 47H10, 37G05     

Character sheaves on the semi-stable locus of a group compactification

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the “ordinary” restriction of a character sheaf on the compactification to a “semi-stable stratum” is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16, 12.6] inside the semi-stable locus of the wonderful compactification.     

Supports of semi-stable probability measures on locally convex spaces

LetE be a locally convex space. Let be an absolutely convexly tight Radom semi-stable probability measure onE with index 1<2 and Lévy measureM. The main result of this paper shows that the closed semigroup generated by the support ofM and the negative of the barycenter ofM restricted to a suitable compact subset ofE is a (closed) linear space ofE, and that the support of is a suitable translate of this linear space. This result complements a few known results concerning the supports of stable and semi-stable probability measures. In particular, it extends an analogous result proved recently for the support of -stable probability measures 1<2 (Ref. 4). Related results concerning the support of Radon semi-stable probability measures onE of index 0<<1 are also discussed.The research of this author was partially supported by AFSOR Grant No. 90-0168The research of this author was supported by KBN Grant, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.     

On a ramification bound of torsion semi-stable representations over a local field

Let p be a rational prime, k be a perfect field of characteristic p, W=W(k) be the ring of Witt vectors, K be a finite totally ramified extension of Frac(W) of degree e and r be a non-negative integer satisfying r<p−1. In this paper, we prove the upper numbering ramification group for j>u(K,r,n) acts trivially on the pn-torsion semi-stable GK-representations with Hodge-Tate weights in {0,…,r}, where u(K,0,n)=0, u(K,1,n)=1+e(n+1/(p−1)) and u(K,r,n)=1−p−n+e(n+r/(p−1)) for 1<r<p−1.     

Rank Properties in Finite Semigroups II: The Small Rank and the Large Rank

This paper is concerned with the application of two possible definitions of rank to certain well-known semigroups.AMS Subject Classification (2000), 20M10