Let G be an edge-bicolored graph where each edge is colored either red or blue. We study problems of obtaining an induced subgraph H from G that simultaneously satisfies given properties for H ’s red graph and blue graph. In particular, we considerDually Connected Induced Subgraph problem — find from G a k-vertex induced subgraph whose red and blue graphs are both connected, and Dual Separator ...

The ground field k is algebraically closed and of characteristic zero. Let g be a simple Lie algebra over k. The goal of this note is to prove a closed formula for the Dynkin index of a principal sl2-subalgebra of g, see Theorem 3.2. The key step in the proof uses the “strange formula” of Freudenthal–de Vries. As an application, we (1) compute the Dynkin index any simple g-module regarded as sl...

The local Langlands correspondence is a conjectural connection between representations of groups G(k) for connected reductive groups G over a p-adic field k and certain homomorphisms (Langlands parameters) from the Galois (or WeilDeligne group) of k into a complex Lie group Gwhich is dual, in a certain sense, to G and which encodes the splitting structure of G over k. More introductory remarks ...

Abstract. Let G be a connected complex semisimple group, assumed to have trivial center, and let K be a maximal compact subgroup of G. Then G/K, with a fixed G-invariant Riemannian metric, is a Riemannian symmetric space of the complex type. Now let Γ be a discrete subgroup of G that acts freely and cocompactly on G/K. We consider the Segal–Bargmann transform, defined in terms of the heat equat...

as a Feynman integral over an infinite dimensional moduli space of G-connections on M , in terms of additional data, namely a Lie algebra g (with dual Coxeter number čg) and a level k ∈ N. In this paper we only consider WRT invariants with g = sl2 and it is convenient to put K = k + 2. This functional integral is not well-defined in a literal sense, being an integral over an infinite dimensiona...

We obtain a classification of the irreducible (nonunitary) representations of a connected semisimple Lie group G, in terms of their restriction to a maximal compact subgroup K of G. (A classification in terms of analytic properties of the representations has been given by R. P. Langlands [(1973), mimeographed notes, Institute for Advanced Study, Princeton, NJ] for linear groups.) We first defin...

Let $K$ be a locally compact hypergroup with left Haar measure and let $L^1(K)$ be the complex Lebesgue space associated with it. Let $L^infty(K)$ be the dual of $L^1(K)$. The purpose of this paper is to present some necessary and sufficient conditions for $L^infty(K)^*$ to have a topologically left invariant mean. Some characterizations of amenable hypergroups are given.

To create a nanoscale dual switch, two responsive DNA motifs, azobenzene-modified DNAs and G-telomeric repeat sequences, were introduced together into the nanoframe system. The dual-switching behaviors controlled by photoirradiation and K(+) were successfully visualized in real time by high-speed atomic force microscopy.