1,314 research outputs found
Weierstrass Structure and Eigenvalue Placement of Regular Matrix Pencils under Low Rank Perturbations
[EN] We solve the problem of determining the Weierstrass structure of a regular matrix pencil obtained by a low rank perturbation of another regular matrix pencil. We apply the result to find necessary and sufficient conditions for the existence of a low rank perturbation such that the perturbed pencil has prescribed eigenvalues and algebraic multiplicities. The results hold over fields with sufficient number of elements.The work of the first and second authors was partially supported by MINECO grants MTM2017-83624-P and MTM2017-90682-REDT. The work of the first author was also partially supported by UPV/EHU grant GIU16/42.Baragana Garate, I.; Roca Martinez, A. (2019). Weierstrass Structure and Eigenvalue Placement of Regular Matrix Pencils under Low Rank Perturbations. SIAM Journal on Matrix Analysis and Applications. 40(2):440-453. https://doi.org/10.1137/18M120024544045340
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained
systems of equations and we prove that the proposed method is the Gauss-Newton
iteration to find the nearest system which has at least common roots and
which is obtained via a perturbation of prescribed structure. In the univariate
case we show the connection of our method to the optimization problem
formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate
case we generalize the expressions of Karmarkar and Lakshman, and give
explicitly several iteration functions to compute the optimum.
The arithmetic complexity of the iterations is detailed
Zero modes in a system of Aharonov-Bohm fluxes
We study zero modes of two-dimensional Pauli operators with Aharonov--Bohm
fluxes in the case when the solenoids are arranged in periodic structures like
chains or lattices. We also consider perturbations to such periodic systems
which may be infinite and irregular but they are always supposed to be
sufficiently scarce
An N=1 duality cascade from a deformation of N=4 SUSY Yang-Mills
We study relevant deformations of an N=1 superconformal theory which is an
exactly marginal deformation of U(N) N=4 SUSY Yang-Mills. The resulting theory
has a classical Higgs branch that is a complex deformation of the orbifold
C^3/Z_n x Z_n that is a non-compact Calabi-Yau space with isolated conifold
singularities. At these singular points in moduli space the theory exhibits a
duality cascade and flows to a confining theory with a mass gap. By exactly
solving the corresponding holomorphic matrix model we compute the exact quantum
superpotential generated at the end of the duality cascade and calculate
precisely how quantum effects deform the classical moduli space by replacing
the conifold singularities with three-cycles of finite size. Locally the
structure is that of the deformed conifold, but the global geometry is
different. This desingularized quantum deformed geometry is the moduli space of
probe D3-branes at the end of a duality cascade realized on the worldvolume of
(fractional) D3-branes placed at the isolated conifold singularities in the
deformation of the orbifold C^3/Z_n x Z_n with discrete torsion.Comment: Uses Latex, JHEP.cls, 43 pages, 3 figure
Lectures on F-theory compactifications and model building
These lecture notes are devoted to formal and phenomenological aspects of
F-theory. We begin with a pedagogical introduction to the general concepts of
F-theory, covering classic topics such as the connection to Type IIB
orientifolds, the geometry of elliptic fibrations and the emergence of gauge
groups, matter and Yukawa couplings. As a suitable framework for the
construction of compact F-theory vacua we describe a special class of
Weierstrass models called Tate models, whose local properties are captured by
the spectral cover construction. Armed with this technology we proceed with a
survey of F-theory GUT models, aiming at an overview of basic conceptual and
phenomenological aspects, in particular in connection with GUT breaking via
hypercharge flux.Comment: Invited contribution to the proceedings of the CERN Winter School on
Supergravity, Strings and Gauge Theory 2010, to appear in Classical and
Quantum Gravity; 63 pages; v2: references added, typos correcte
Duality and Modularity in Elliptic Integrable Systems and Vacua of N=1* Gauge Theories
We study complexified elliptic Calogero-Moser integrable systems. We
determine the value of the potential at isolated extrema, as a function of the
modular parameter of the torus on which the integrable system lives. We
calculate the extrema for low rank B,C,D root systems using a mix of analytical
and numerical tools. For so(5) we find convincing evidence that the extrema
constitute a vector valued modular form for a congruence subgroup of the
modular group. For so(7) and so(8), the extrema split into two sets. One set
contains extrema that make up vector valued modular forms for congruence
subgroups, and a second set contains extrema that exhibit monodromies around
points in the interior of the fundamental domain. The former set can be
described analytically, while for the latter, we provide an analytic value for
the point of monodromy for so(8), as well as extensive numerical predictions
for the Fourier coefficients of the extrema. Our results on the extrema provide
a rationale for integrality properties observed in integrable models, and embed
these into the theory of vector valued modular forms. Moreover, using the data
we gather on the modularity of complexified integrable system extrema, we
analyse the massive vacua of mass deformed N=4 supersymmetric Yang-Mills
theories with low rank gauge group of type B,C and D. We map out their
transformation properties under the infrared electric-magnetic duality group as
well as under triality for N=1* with gauge algebra so(8). We find several
intriguing properties of the quantum gauge theories.Comment: 35 pages, many figure
Generic change of the partial multiplicities of regular matrix pencils under low-rank perturbations
We describe the generic change of the partial multiplicities at a given eigenvalue lambda(0) of a regular matrix pencil A(0) + lambda A(1) under perturbations with low normal rank. More precisely, if the pencil A(0) + lambda A(1) has exactly g nonzero partial multiplicities at lambda(0), then for most perturbations B-0 + lambda B-1 with normal rank r < g the perturbed pencil A(0) + B-0 + lambda(A(1) + B-1) has exactly g - r nonzero partial multiplicities at lambda(0), which coincide with those obtained after removing the largest r partial multiplicities of the original pencil A(0) + A(1) at lambda(0). Though partial results on this problem had been previously obtained in the literature, its complete solution remained open
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