9,719 research outputs found
Decomposable Theories
We present in this paper a general algorithm for solving first-order formulas
in particular theories called "decomposable theories". First of all, using
special quantifiers, we give a formal characterization of decomposable theories
and show some of their properties. Then, we present a general algorithm for
solving first-order formulas in any decomposable theory "T". The algorithm is
given in the form of five rewriting rules. It transforms a first-order formula
"P", which can possibly contain free variables, into a conjunction "Q" of
solved formulas easily transformable into a Boolean combination of
existentially quantified conjunctions of atomic formulas. In particular, if "P"
has no free variables then "Q" is either the formula "true" or "false". The
correctness of our algorithm proves the completeness of the decomposable
theories.
Finally, we show that the theory "Tr" of finite or infinite trees is a
decomposable theory and give some benchmarks realized by an implementation of
our algorithm, solving formulas on two-partner games in "Tr" with more than 160
nested alternated quantifiers
Transformations among large c conformal field theories
We show that there is a set of transformations that relates all of the 24
dimensional even self-dual (Niemeier) lattices, and also leads to non-lattice
objects that however cannot be interpreted as a basis for the construction of
holomorphic conformal field theory. In the second part of this paper, we extend
our observations to higher dimensional conformal field theories build on
extremal partition functions, where we generate c=24k theories with spectra
decomposable into the irreducible representations of the Fischer-Griess
Monster. We observe interesting periodicities in the coefficients of extremal
partition functions and characters of the extremal vertex operator algebras.Comment: 14 pages, minor corrections, new references adde
Pre-multisymplectic constraint algorithm for field theories
We present a geometric algorithm for obtaining consistent solutions to
systems of partial differential equations, mainly arising from singular
covariant first-order classical field theories. This algorithm gives an
intrinsic description of all the constraint submanifolds.
The field equations are stated geometrically, either representing their
solutions by integrable connections or, what is equivalent, by certain kinds of
integrable m-vector fields. First, we consider the problem of finding
connections or multivector fields solutions to the field equations in a general
framework: a pre-multisymplectic fibre bundle (which will be identified with
the first-order jet bundle and the multimomentum bundle when Lagrangian and
Hamiltonian field theories are considered). Then, the problem is stated and
solved in a linear context, and a pointwise application of the results leads to
the algorithm for the general case. In a second step, the integrability of the
solutions is also studied.
Finally, the method is applied to Lagrangian and Hamiltonian field theories
and, for the former, the problem of finding holonomic solutions is also
analized.Comment: 30 pp. Presented in the International Workshop on Geometric Methods
in Modern Physics (Firenze, April 2005
A new multisymplectic unified formalism for second-order classical field theories
We present a new multisymplectic framework for second-order classical field
theories which is based on an extension of the unified Lagrangian-Hamiltonian
formalism to these kinds of systems. This model provides a straightforward and
simple way to define the Poincar\'e-Cartan form and clarifies the construction
of the Legendre map (univocally obtained as a consequence of the constraint
algorithm). Likewise, it removes the undesirable arbitrariness in the solutions
to the field equations, which are analyzed in-depth, and written in terms of
holonomic sections and multivector fields. Our treatment therefore completes
previous attempt to achieve this aim. The formulation is applied to describing
some physical examples; in particular, to giving another alternative
multisymplectic description of the Korteweg-de Vries equation.Comment: 52 pp. Revision of our previous paper. Minor corrections on the
statement of some results. A new example is added (Section 6.1). Conclusions
and bibliography have been enlarged, and some comments on the higher-order
case have been adde
Multivector Fields and Connections. Setting Lagrangian Equations in Field Theories
The integrability of multivector fields in a differentiable manifold is
studied. Then, given a jet bundle , it is shown that integrable
multivector fields in are equivalent to integrable connections in the
bundle (that is, integrable jet fields in ). This result is
applied to the particular case of multivector fields in the manifold and
connections in the bundle (that is, jet fields in the repeated jet
bundle ), in order to characterize integrable multivector fields and
connections whose integral manifolds are canonical lifting of sections. These
results allow us to set the Lagrangian evolution equations for first-order
classical field theories in three equivalent geometrical ways (in a form
similar to that in which the Lagrangian dynamical equations of non-autonomous
mechanical systems are usually given). Then, using multivector fields; we
discuss several aspects of these evolution equations (both for the regular and
singular cases); namely: the existence and non-uniqueness of solutions, the
integrability problem and Noether's theorem; giving insights into the
differences between mechanics and field theories.Comment: New sections on integrability of Multivector Fields and applications
to Field Theory (including some examples) are added. The title has been
slightly modified. To be published in J. Math. Phy
A geometrical analysis of the field equations in field theory
In this review paper we give a geometrical formulation of the field equations
in the Lagrangian and Hamiltonian formalisms of classical field theories (of
first order) in terms of multivector fields. This formulation enables us to
discuss the existence and non-uniqueness of solutions, as well as their
integrability.Comment: 14 pages. LaTeX file. This is a review paper based on previous works
by the same author
Quasi-chemical Theories of Associated Liquids
It is shown how traditional development of theories of fluids based upon the
concept of physical clustering can be adapted to an alternative local
clustering definition. The alternative definition can preserve a detailed
valence description of the interactions between a solution species and its
near-neighbors, i.e., cooperativity and saturation of coordination for strong
association. These clusters remain finite even for condensed phases. The
simplest theory to which these developments lead is analogous to quasi-chemical
theories of cooperative phenomena. The present quasi-chemical theories require
additional consideration of packing issues because they don't impose lattice
discretizations on the continuous problem. These quasi-chemical theories do not
require pair decomposable interaction potential energy models. Since
calculations may be required only for moderately sized clusters, we suggest
that these quasi-chemical theories could be implemented with computational
tools of current electronic structure theory. This can avoid an intermediate
step of approximate force field generation.Comment: 20 pages, no figures replacement: minor typographical corrections,
four references added, in press Molec. Physics 199
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