The group Am of automophisms of a one-rooted m-ary tree admits a diagonal monomorphism which we denote by x. Let A be an abelian state-closed (or self-similar) subgroup of Am. We prove that the recurrence and tree-topological closure A∗ of A is additively a finitely presented Zm [[x]]module where Zm is the ring of m-adic integers. Moreover, if A∗ is torsion-free then it is a finitely generated ...

We will need a general fact relating intersection products and the diagonal. The setup is as follows: Let X ρ −→ S be a (locally trivial) fiber bundle, where X and S are oriented smooth manifolds, and the fibers are compact ndimensional manifolds. (In our application, S will be an approximation space for BG.) Then we have a Gysin map ρ∗ : H X → H S. Assume HX is a free module over HS, and let {...

Let D be a division algebra for a non-Archimedean local field. Given an irreducible representation π of GL2(D), we describe its restriction to the diagonal subgroup D∗ × D∗. The description is in terms of the structure of the twisted Jacquet module of the representation π. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on re...

The Hilbert series of the Garsia-Haiman module can be written as a generating function of standard fillings of Ferrers diagrams. It is conjectured by Haglund and Loehr that the Hilbert series of the diagonal harmonics can be written as a generating function of parking functions. In this paper we present a weight-preserving injection from standard fillings to parking functions for certain cases....

For an n-by-n complex matrix A in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of A. In this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine ...

An iterative formula for the Green polynomial is given using vertex operator realization of Hall-Littlewood function. Based on this, (1) a general combinatorial given; (2) several compact formulas are Green's polynomials associated with upper partitions length $\leq 3$ and diagonal lengths 3$; (3) Murnaghan-Nakayama type obtained; (4) an derived bitrace finite linear group $G$ Iwahori-Hecke alg...

The linear layer is one of the most pervasive modules in deep learning representations. However, it requiresO(N) parameters andO(N) operations. These costs can be prohibitive in mobile applications or prevent scaling in many domains. Here, we introduce a deep, differentiable, fully-connected neural network module composed of diagonal matrices of parameters, A and D, and the discrete cosine tran...

High-throughput expression profiling allows simultaneous measure of tens of thousands of genes at once. These data have motivated the development of reliable biomarkers for disease subtypes identification and diagnosis. Many methods have been developed in the literature for analyzing these data, such as diagonal discriminant analysis, support vector machines, and k-nearest neighbor methods. The...