We give a new proof of the existence of Klyachko models for unitary representations of @math over a non-archimedean local field @math . Our methods are purely local and are based on studying distinction within the class of ladder representations introduced by Lapid and Minguez. We classify those ladder representations that are distinguished with respect to Klyachko models. We prove the hereditary property of these models for induced representations from arbitrary finite length representations. Finally, in the other direction and in the context of admissible representations induced from ladder, we study the relation between distinction of the parabolic induction with respect to the symplectic groups and distinction of the inducing data.
The present work began as an attempt to extend the distinction results of @cite_1 and @cite_11 from the unitary dual of @math to the admissible dual of @math
@cite_1 @cite_11
2132069633 2052956617 2215703121 2858362441
Given two documents A and B we define two mathematical notions: their resemblance r(A, B) and their containment c(A, B) that seem to capture well the informal notions of "roughly the same" and "roughly contained." The basic idea is to reduce these issues to set intersection problems that can be easily evaluated by a process of random sampling that can be done independently for each document. Furthermore, the resemblance can be evaluated using a fixed size sample for each document. This paper discusses the mathematical properties of these measures and the efficient implementation of the sampling process using Rabin (1981) fingerprints. Given topological spaces X1, ..., Xn with product space X, probability measures μi on Xi together with a real function h on X define a marginal problem as well as a dual problem. Using an extended version of Choquet's theorem on capacities, an analogue of the classical duality theorem of linear programming is established, imposing only weak conditions on the topology of the spaces Xi and the measurability resp. boundedness of the function h. Applications concern, among others, measures with given support, stochastic order and general marginal problems. Whereas formal category theory is classically considered within a @math -category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of hypervirtual double categories, a notion extending that of virtual double category by adding cells with nullary target. [...] After this the notion of weak' Kan extension within a hypervirtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in a hypervirtual double category, and compared to that of a good yoneda structure on a @math -category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding @math defines @math as the free small cocompletion of @math , in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad @math on a hypervirtual double category @math several hypervirtual double categories @math of @math -algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors @math . The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in @math lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in @math . As a first example we apply the previous to monoidal structures on categories, hence recovering Day convolution of presheaves and Im-Kelly's result on free monoidal cocompletion, as well as obtaining a "monoidal Yoneda lemma". In this paper, we study codes correcting @math duplications of @math consecutive symbols. These errors are known as tandem duplication errors, where a sequence of symbols is repeated and inserted directly after its original occurrence. Using sphere packing arguments, we derive non-asymptotic upper bounds on the cardinality of codes that correct such errors for any choice of parameters. Based on the fact that a code correcting insertions of @math zero-blocks can be used to correct @math tandem duplications, we construct codes for tandem duplication errors. We compare the cardinalities of these codes with their sphere packing upper bounds. Finally, we discuss the asymptotic behavior of the derived codes and bounds, which yields insights about the tandem duplication channel.