241,916 research outputs found
Computing Linear Matrix Representations of Helton-Vinnikov Curves
Helton and Vinnikov showed that every rigidly convex curve in the real plane
bounds a spectrahedron. This leads to the computational problem of explicitly
producing a symmetric (positive definite) linear determinantal representation
for a given curve. We study three approaches to this problem: an algebraic
approach via solving polynomial equations, a geometric approach via contact
curves, and an analytic approach via theta functions. These are explained,
compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in
Systems, Optimization and Control, Birkhauser, Base
Quivers from Matrix Factorizations
We discuss how matrix factorizations offer a practical method of computing
the quiver and associated superpotential for a hypersurface singularity. This
method also yields explicit geometrical interpretations of D-branes (i.e.,
quiver representations) on a resolution given in terms of Grassmannians. As an
example we analyze some non-toric singularities which are resolved by a single
CP1 but have "length" greater than one. These examples have a much richer
structure than conifolds. A picture is proposed that relates matrix
factorizations in Landau-Ginzburg theories to the way that matrix
factorizations are used in this paper to perform noncommutative resolutions.Comment: 33 pages, (minor changes
Computing generalized inverses using LU factorization of matrix product
An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and
the Moore-Penrose inverse of a given rational matrix A is established. Classes
A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R*
and T*(AT*)+, where R and T are rational matrices with appropriate dimensions
and corresponding rank. The proposed algorithm is based on these general
representations and the Cholesky factorization of symmetric positive matrices.
The algorithm is implemented in programming languages MATHEMATICA and DELPHI,
and illustrated via examples. Numerical results of the algorithm, corresponding
to the Moore-Penrose inverse, are compared with corresponding results obtained
by several known methods for computing the Moore-Penrose inverse
A matrix model for the latitude Wilson loop in ABJM theory
In ABJ(M) theory, we propose a matrix model for the exact evaluation of BPS
Wilson loops on a latitude circular contour, so providing a new weak-strong
interpolation tool. Intriguingly, the matrix model turns out to be a particular
case of that computing torus knot invariants in Chern-Simons
theory. At weak coupling we check our proposal against a three-loop
computation, performed for generic framing, winding number and representation.
The matrix model is amenable of a Fermi gas formulation, which we use to
systematically compute the strong coupling and genus expansions. For the
fermionic Wilson loop the leading planar behavior agrees with a previous string
theory prediction. For the bosonic operator our result provides a clue for
finding the corresponding string dual configuration. Our matrix model is
consistent with recent proposals for computing Bremsstrahlung functions exactly
in terms of latitude Wilson loops. As a by-product, we extend the conjecture
for the exact Bremsstrahlung function to generic
representations and test it with a four-loop perturbative computation. Finally,
we propose an exact prediction for at unequal gauge group ranks.Comment: 73 pages; v2: several improvements, JHEP published versio
Hypergeometric Functions of Matrix Arguments and Linear Statistics of Multi-Spiked Hermitian Matrix Models
This paper derives central limit theorems (CLTs) for general linear spectral
statistics (LSS) of three important multi-spiked Hermitian random matrix
ensembles. The first is the most common spiked scenario, proposed by Johnstone,
which is a central Wishart ensemble with fixed-rank perturbation of the
identity matrix, the second is a non-central Wishart ensemble with fixed-rank
noncentrality parameter, and the third is a similarly defined non-central
ensemble. These CLT results generalize our recent work to account for multiple
spikes, which is the most common scenario met in practice. The generalization
is non-trivial, as it now requires dealing with hypergeometric functions of
matrix arguments. To facilitate our analysis, for a broad class of such
functions, we first generalize a recent result of Onatski to present new
contour integral representations, which are particularly suitable for computing
large-dimensional properties of spiked matrix ensembles. Armed with such
representations, our CLT formulas are derived for each of the three spiked
models of interest by employing the Coulomb fluid method from random matrix
theory along with saddlepoint techniques. We find that for each matrix model,
and for general LSS, the individual spikes contribute additively to yield a
correction term to the asymptotic mean of the linear statistic, which we
specify explicitly, whilst having no effect on the leading order terms of the
mean or variance
BISMO: A Scalable Bit-Serial Matrix Multiplication Overlay for Reconfigurable Computing
Matrix-matrix multiplication is a key computational kernel for numerous
applications in science and engineering, with ample parallelism and data
locality that lends itself well to high-performance implementations. Many
matrix multiplication-dependent applications can use reduced-precision integer
or fixed-point representations to increase their performance and energy
efficiency while still offering adequate quality of results. However, precision
requirements may vary between different application phases or depend on input
data, rendering constant-precision solutions ineffective. We present BISMO, a
vectorized bit-serial matrix multiplication overlay for reconfigurable
computing. BISMO utilizes the excellent binary-operation performance of FPGAs
to offer a matrix multiplication performance that scales with required
precision and parallelism. We characterize the resource usage and performance
of BISMO across a range of parameters to build a hardware cost model, and
demonstrate a peak performance of 6.5 TOPS on the Xilinx PYNQ-Z1 board.Comment: To appear at FPL'1
Uniform determinantal representations
The problem of expressing a specific polynomial as the determinant of a
square matrix of affine-linear forms arises from algebraic geometry,
optimisation, complexity theory, and scientific computing. Motivated by recent
developments in this last area, we introduce the notion of a uniform
determinantal representation, not of a single polynomial but rather of all
polynomials in a given number of variables and of a given maximal degree. We
derive a lower bound on the size of the matrix, and present a construction
achieving that lower bound up to a constant factor as the number of variables
is fixed and the degree grows. This construction marks an improvement upon a
recent construction due to Plestenjak-Hochstenbach, and we investigate the
performance of new representations in their root-finding technique for
bivariate systems. Furthermore, we relate uniform determinantal representations
to vector spaces of singular matrices, and we conclude with a number of future
research directions.Comment: 23 pages, 3 figures, 4 table
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