22,794 research outputs found
Non-Universal Critical Behaviour of Two-Dimensional Ising Systems
Two conditions are derived for Ising models to show non-universal critical
behaviour, namely conditions concerning 1) logarithmic singularity of the
specific heat and 2) degeneracy of the ground state. These conditions are
satisfied with the eight-vertex model, the Ashkin-Teller model, some Ising
models with short- or long-range interactions and even Ising systems without
the translational or the rotational invariance.Comment: 17 page
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
On the absence of chiral fermions in interacting lattice theories
We consider interacting theories with a compact internal symmetry group on a
regular lattice. We show that the spectrum is necessarily vector-like provided
the following conditions are satisfied: (a)~weak form of locality,
(b)~relativistic continuum limit without massless bosons, and (c)~pole-free
effective vertex functions for conserved currents.
The proof exploits the zero frequency inverse retarded propagator of an
appropriate set of interpolating fields as an effective quadratic hamiltonian,
to which the Nielsen-Ninomiya theorem is applied.
The main results of this paper have been reported in WIS-93/56-JUNE-PH,
hep-lat/9306023.Comment: WIS-93/57-JULY-PH, LaTeX, 24 page
Multi-locality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence
Using the fusion rules hypothesis for three-dimensional and two-dimensional
Navier-Stokes turbulence, we generalize a previous non-perturbative locality
proof to multiple applications of the nonlinear interactions operator on
generalized structure functions of velocity differences. We shall call this
generalization of non-perturbative locality to multiple applications of the
nonlinear interactions operator "multilocality". The resulting cross-terms pose
a new challenge requiring a new argument and the introduction of a new fusion
rule that takes advantage of rotational symmetry. Our main result is that the
fusion rules hypothesis implies both locality and multilocality in both the IR
and UV limits for the downscale energy cascade of three-dimensional
Navier-Stokes turbulence and the downscale enstrophy cascade and inverse energy
cascade of two-dimensional Navier-Stokes turbulence. We stress that these
claims relate to non-perturbative locality of generalized structure functions
on all orders, and not the term by term perturbative locality of diagrammatic
theories or closure models that involve only two-point correlation and response
functions.Comment: 25 pages, 24 figures, resubmitted to Physical Review
Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues
We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a
bounded domain \Omega\subset\RR^n with piecewise smooth boundary. We bound
the distance between an arbitrary parameter and the spectrum
in terms of the boundary -norm of a normalized trial solution of the
Helmholtz equation . We also bound the -norm of the
error of this trial solution from an eigenfunction. Both of these results are
sharp up to constants, hold for all greater than a small constant, and
improve upon the best-known bounds of Moler--Payne by a factor of the
wavenumber . One application is to the solution of eigenvalue
problems at high frequency, via, for example, the method of particular
solutions. In the case of planar, strictly star-shaped domains we give an
inclusion bound where the constant is also sharp. We give explicit constants in
the theorems, and show a numerical example where an eigenvalue around the
2500th is computed to 14 digits of relative accuracy. The proof makes use of a
new quasi-orthogonality property of the boundary normal derivatives of the
eigenmodes, of interest in its own right.Comment: 18 pages, 3 figure
Spinor gravity and diffeomorphism invariance on the lattice
The key ingredient for lattice regularized quantum gravity is diffeomorphism
symmetry. We formulate a lattice functional integral for quantum gravity in
terms of fermions. This allows for a diffeomorphism invariant functional
measure and avoids problems of boundedness of the action. We discuss the
concept of lattice diffeomorphism invariance. This is realized if the action
does not depend on the positioning of abstract lattice points on a continuous
manifold. Our formulation of lattice spinor gravity also realizes local Lorentz
symmetry. Furthermore, the Lorentz transformations are generalized such that
the functional integral describes simultaneously euclidean and Minkowski
signature. The difference between space and time arises as a dynamical effect
due to the expectation value of a collective metric field. The quantum
effective action for the metric is diffeomorphism invariant. Realistic gravity
can be obtained if this effective action admits a derivative expansion for long
wavelengths.Comment: 13 pages, proceedings 6th Aegean Summer School, Naxos 201
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