116,273 research outputs found
Ways to restrict the differential path
People had developed some attack methods to attack hash function. These methods need to choose some differential pattern [Dau05]. We present a way to restrict the collisions that hold the differential pattern . At the same time, to build a hash function that meet the different needs, we propose a construction
Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics
I extract some philosophical morals from some aspects of Lagrangian
mechanics. (A companion paper will present similar morals from Hamiltonian
mechanics and Hamilton-Jacobi theory.) One main moral concerns methodology:
Lagrangian mechanics provides a level of description of phenomena which has
been largely ignored by philosophers, since it falls between their accustomed
levels--``laws of nature'' and ``models''. Another main moral concerns
ontology: the ontology of Lagrangian mechanics is both more subtle and more
problematic than philosophers often realize.
The treatment of Lagrangian mechanics provides an introduction to the subject
for philosophers, and is technically elementary. In particular, it is confined
to systems with a finite number of degrees of freedom, and for the most part
eschews modern geometry. But it includes a presentation of Routhian reduction
and of Noether's ``first theorem''.Comment: 106 pages, no figure
Supersymmetry, homology with twisted coefficients and n-dimensional knots
Let be any natural number. Let be any -dimensional knot in
. We define a supersymmetric quantum system for with the following
properties. We firstly construct a set of functional spaces (spaces of
fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric
infinitesimal transformations) in an explicit way. Thus we obtain a set of the
Witten indexes for . Our Witten indexes are topological invariants for
-dimensional knots. Our Witten indexes are not zero in general. If is
equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten
indexes restrict the Alexander polynomials of -knots. If one of our Witten
indexes for an -knot is nonzero, then one of the Alexander polynomials
of is nontrivial. Our Witten indexes are connected with homology with
twisted coefficients. Roughly speaking, our Witten indexes have path integral
representation by using a usual manner of supersymmetric theory.Comment: 10pages, no figure
Symmetry Protected Topological phases and Generalized Cohomology
We discuss the classification of SPT phases in condensed matter systems. We
review Kitaev's argument that SPT phases are classified by a generalized
cohomology theory, valued in the spectrum of gapped physical systems. We
propose a concrete description of that spectrum and of the corresponding
cohomology theory. We compare our proposal to pre-existing constructions in the
literature.Comment: 27 pages, 10 figures. v2: citation updat
Symmetry Protected Topological phases and Generalized Cohomology
We discuss the classification of SPT phases in condensed matter systems. We
review Kitaev's argument that SPT phases are classified by a generalized
cohomology theory, valued in the spectrum of gapped physical systems. We
propose a concrete description of that spectrum and of the corresponding
cohomology theory. We compare our proposal to pre-existing constructions in the
literature.Comment: 27 pages, 10 figures. v2: citation updat
On Conformal Field Theory and Stochastic Loewner Evolution
We describe Stochastic Loewner Evolution on arbitrary Riemann surfaces with
boundary using Conformal Field Theory methods. We propose in particular a CFT
construction for a probability measure on (clouded) paths, and check it against
known restriction properties. The probability measure can be thought of as a
section of the determinant bundle over moduli spaces of Riemann surfaces.
Loewner evolutions have a natural description in terms of random walk in the
moduli space, and the stochastic diffusion equation translates to the Virasoro
action of a certain weight-two operator on a uniformised version of the
determinant bundle.Comment: 24 pages, 4 figures, LaTeX; v2: added section 4.1, references and
minor clarifications, version to appear in NP
Multiple scattering in random mechanical systems and diffusion approximation
This paper is concerned with stochastic processes that model multiple (or
iterated) scattering in classical mechanical systems of billiard type, defined
below. From a given (deterministic) system of billiard type, a random process
with transition probabilities operator P is introduced by assuming that some of
the dynamical variables are random with prescribed probability distributions.
Of particular interest are systems with weak scattering, which are associated
to parametric families of operators P_h, depending on a geometric or mechanical
parameter h, that approaches the identity as h goes to 0. It is shown that (P_h
-I)/h converges for small h to a second order elliptic differential operator L
on compactly supported functions and that the Markov chain process associated
to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L
are selfadjoint (densely) defined on the space L2(H,{\eta}) of
square-integrable functions over the (lower) half-space H in R^m, where {\eta}
is a stationary measure. This measure's density is either (post-collision)
Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes
with infinitesimal generator L respectively correspond to what we call MB
diffusion and (generalized) Legendre diffusion. Concrete examples of simple
mechanical systems are given and illustrated by numerically simulating the
random processes.Comment: 34 pages, 13 figure
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