5,020 research outputs found
Schur Partial Derivative Operators
A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice
cells. The corresponding lattice diagram determinant is \Delta_L(X;Y)=\det \|
x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the
study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space
M_L is the space spanned by all partial derivatives of \Delta_L(X;Y). The
``shift operators'', which are particular partial symmetric derivative
operators are very useful in the comprehension of the structure of the M_L
spaces. We describe here how a Schur function partial derivative operator acts
on lattice diagrams with distinct cells in the positive quadrant.Comment: 8 pages, LaTe
Schur correlation functions on
The Schur limit of the superconformal index of four-dimensional superconformal field theories has been shown to equal the supercharacter
of the vacuum module of their associated chiral algebra. Applying localization
techniques to the theory suitably put on , we obtain a direct
derivation of this fact. We also show that the localization computation can be
extended to calculate correlation functions of a subset of local operators,
namely of the so-called Schur operators. Such correlators correspond to
insertions of chiral algebra fields in the trace-formula computing the
supercharacter. As a by-product of our analysis, we show that the standard lore
in the localization literature stating that only off-shell supersymmetrically
closed observables are amenable to localization, is incomplete, and we
demonstrate how insertions of fermionic operators can be incorporated in the
computation.Comment: 49 page
Operator and commutator moduli of continuity for normal operators
We study in this paper properties of functions of perturbed normal operators
and develop earlier results obtained in \cite{APPS2}. We study operator
Lipschitz and commutator Lipschitz functions on closed subsets of the plane.
For such functions we introduce the notions of the operator modulus of
continuity and of various commutator moduli of continuity. Our estimates lead
to estimates of the norms of quasicommutators in terms of
, where and are normal operator and is a
bounded linear operator. In particular, we show that if 0<\a<1 and is a
H\"older function of order \a, then for normal operators and ,
\|f(N_1)R-Rf(N_2)\|\le\const(1-\a)^{-2}\|f\|_{\L_\a}\|N_1R-RN_2\|^\a\|R\|^{1-\a}.
In the last section we obtain lower estimates for constants in operator
H\"older estimates.Comment: 33 page
Time-parallel iterative solvers for parabolic evolution equations
We present original time-parallel algorithms for the solution of the implicit
Euler discretization of general linear parabolic evolution equations with
time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory
of parabolic problems, we show that the standard nonsymmetric time-global
system can be equivalently reformulated as an original symmetric saddle-point
system that remains inf-sup stable with respect to the same natural parabolic
norms. We then propose and analyse an efficient and readily implementable
parallel-in-time preconditioner to be used with an inexact Uzawa method. The
proposed preconditioner is non-intrusive and easy to implement in practice, and
also features the key theoretical advantages of robust spectral bounds, leading
to convergence rates that are independent of the number of time-steps, final
time, or spatial mesh sizes, and also a theoretical parallel complexity that
grows only logarithmically with respect to the number of time-steps. Numerical
experiments with large-scale parallel computations show the effectiveness of
the method, along with its good weak and strong scaling properties
Multiple operator integrals and higher operator derivatives
In this paper we consider the problem of the existence of higher derivatives
of the function t\mapsto\f(A+tK), where \f is a function on the real line,
is a self-adjoint operator, and is a bounded self-adjoint operator. We
improve earlier results by Sten'kin. In order to do this, we give a new
approach to multiple operator integrals. This approach improves the earlier
approach given by Sten'kin. We also consider a similar problem for unitary
operators.Comment: 24 page
Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods
The Immersed Boundary method is a simple, efficient, and robust numerical
scheme for solving PDE in general domains, yet it only achieves first-order
spatial accuracy near embedded boundaries. In this paper, we introduce a new
high-order numerical method which we call the Immersed Boundary Smooth
Extension (IBSE) method. The IBSE method achieves high-order accuracy by
smoothly extending the unknown solution of the PDE from a given smooth domain
to a larger computational domain, enabling the use of simple Cartesian-grid
discretizations (e.g. Fourier spectral methods). The method preserves much of
the flexibility and robustness of the original IB method. In particular, it
requires minimal geometric information to describe the boundary and relies only
on convolution with regularized delta-functions to communicate information
between the computational grid and the boundary. We present a fast algorithm
for solving elliptic equations, which forms the basis for simple, high-order
implicit-time methods for parabolic PDE and implicit-explicit methods for
related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat,
Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise
convergence for Dirichlet problems and third-order pointwise convergence for
Neumann problems
On the Lipschitz continuity of spectral bands of Harper-like and magnetic Schroedinger operators
We show for a large class of discrete Harper-like and continuous magnetic
Schrodinger operators that their band edges are Lipschitz continuous with
respect to the intensity of the external constant magnetic field. We generalize
a result obtained by J. Bellissard in 1994, and give examples in favor of a
recent conjecture of G. Nenciu.Comment: 15 pages, accepted for publication in Annales Henri Poincar
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