Rings of quantum integrals for generalised Calogero–Moser problems

The rings of quantum integrals for generalised Calogero–Moser problems are studied in the special case when all the parameters are integers. The problem is reduced to the description of the rings of polynomials satisfying a certain quasi-invariance property (quasi-invariants). The quasi-invariants of dihedral groups are fully described. It is shown that they form a free module over invariants generated by m-harmonic polynomials. The m-harmonic polynomials for general Coxeter group are introduced and investigated. For the non-Coxeter generalisations of Calogero–Moser problems related to the systems An(m), Cn+1(m, l), the rings of quantum integrals are considered. The Poincare series for the quasi-invariants of two-dimensional deformations are computed. It is shown that the rings of quasi-invariants are Gorenstein like in the Coxeter case

The application of the theory of fibre bundles to differential geometry

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The Polynomial Eigenvalue Problem is Well Conditioned for Random Inputs

We compute the exact expected value of the squared condition number for the polynomial eigenvalue problem, when the input matrices have entries coming from the standard complex Gaussian distribution, showing that in general this problem is quite well conditioned.The first author's work was partially supported by Agencia Nacional de Investigación e Innovación (ANII), Uruguay, and by CSIC group 618, Universidad de La República, Uruguay. The second author's work was partially supported by MTM2017-83816-P and MTM2017-90682-REDT from Spanish Ministry of Science MICINN and by 21.SI01.64658 from Universidad de Cantabria and Banco de Santander

Transceiver design and interference alignment in wireless networks: complexity and solvability

University of Minnesota M.S. thesis. November 2013. Major: Mathematics. Advisor: Gennady Lyubeznik. 1 computer file (PDF); vi, 58 pages.This thesis aims to theoretically study a modern linear transceiver design strategy, namely interference alignment, in wireless networks. We consider an interference channel whereby each transmitter and receiver are equipped with multiple antennas. The basic problem is to design optimal linear transceivers (or beamformers) that can maximize the system throughput. The recent work [1] suggests that optimal beamformers should maximize the total degrees of freedom through the interference alignment equations. In this thesis, we first state the interference alignment equations and study the computational complexity of solving these equations. In particular, we prove that the problem of maximizing the total degrees of freedom for a given interference channel is NP-hard. Moreover, it is shown that even checking the achievability of a given tuple of degrees of freedom is NP-hard when each receiver is equipped with at least three antennas. Interestingly, the same problem becomes polynomial time solvable when each transmit/receive node is equipped with no more than two antennas.The second part of this thesis answers an open theoretical question about interference alignment on generic channels: What degrees of freedom tuples (d1, d2, ..., dK) are achievable through linear interference alignment for generic channels? We partially answer this question by establishing a general condition that must be satisfied by any degrees of freedom tuple (d1, d2, ..., dK) achievable through linear interference alignment. For a symmetric system with dk = d for all k, this condition implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than K(M + N)/(K + 1), where M and N are the number of transmit and receive antennas, respectively. We also show that this bound is tight when the number of antennas at each transceiver is divisible by the number of data streams

Harmonic Analysis as the Exploitation of Symmetry- A Historical Survey

Paper by George W. Macke