39,232 research outputs found
The algebraic structure of geometric flows in two dimensions
There is a common description of different intrinsic geometric flows in two
dimensions using Toda field equations associated to continual Lie algebras that
incorporate the deformation variable t into their system. The Ricci flow admits
zero curvature formulation in terms of an infinite dimensional algebra with
Cartan operator d/dt. Likewise, the Calabi flow arises as Toda field equation
associated to a supercontinual algebra with odd Cartan operator d/d \theta -
\theta d/dt. Thus, taking the square root of the Cartan operator allows to
connect the two distinct classes of geometric deformations of second and fourth
order, respectively. The algebra is also used to construct formal solutions of
the Calabi flow in terms of free fields by Backlund transformations, as for the
Ricci flow. Some applications of the present framework to the general class of
Robinson-Trautman metrics that describe spherical gravitational radiation in
vacuum in four space-time dimensions are also discussed. Further iteration of
the algorithm allows to construct an infinite hierarchy of higher order
geometric flows, which are integrable in two dimensions and they admit
immediate generalization to Kahler manifolds in all dimensions. These flows
provide examples of more general deformations introduced by Calabi that
preserve the Kahler class and minimize the quadratic curvature functional for
extremal metrics.Comment: 54 page
A Context-theoretic Framework for Compositionality in Distributional Semantics
Techniques in which words are represented as vectors have proved useful in
many applications in computational linguistics, however there is currently no
general semantic formalism for representing meaning in terms of vectors. We
present a framework for natural language semantics in which words, phrases and
sentences are all represented as vectors, based on a theoretical analysis which
assumes that meaning is determined by context.
In the theoretical analysis, we define a corpus model as a mathematical
abstraction of a text corpus. The meaning of a string of words is assumed to be
a vector representing the contexts in which it occurs in the corpus model.
Based on this assumption, we can show that the vector representations of words
can be considered as elements of an algebra over a field. We note that in
applications of vector spaces to representing meanings of words there is an
underlying lattice structure; we interpret the partial ordering of the lattice
as describing entailment between meanings. We also define the context-theoretic
probability of a string, and, based on this and the lattice structure, a degree
of entailment between strings.
We relate the framework to existing methods of composing vector-based
representations of meaning, and show that our approach generalises many of
these, including vector addition, component-wise multiplication, and the tensor
product.Comment: Submitted to Computational Linguistics on 20th January 2010 for
revie
A Bayesian framework for functional time series analysis
The paper introduces a general framework for statistical analysis of
functional time series from a Bayesian perspective. The proposed approach,
based on an extension of the popular dynamic linear model to Banach-space
valued observations and states, is very flexible but also easy to implement in
many cases. For many kinds of data, such as continuous functions, we show how
the general theory of stochastic processes provides a convenient tool to
specify priors and transition probabilities of the model. Finally, we show how
standard Markov chain Monte Carlo methods for posterior simulation can be
employed under consistent discretizations of the data
Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra
Recently, it was noticed by us that the nonlinear holomorphic supersymmetry
of order , (-HSUSY) has an algebraic origin. We show that the
Onsager algebra underlies -HSUSY and investigate the structure of the former
in the context of the latter. A new infinite set of mutually commuting charges
is found which, unlike those from the Dolan-Grady set, include the terms
quadratic in the Onsager algebra generators. This allows us to find the general
form of the superalgebra of -HSUSY and fix it explicitly for the cases of
. The similar results are obtained for a new, contracted form of
the Onsager algebra generated via the contracted Dolan-Grady relations. As an
application, the algebraic structure of the known 1D and 2D systems with
-HSUSY is clarified and a generalization of the construction to the case of
nonlinear pseudo-supersymmetry is proposed. Such a generalization is discussed
in application to some integrable spin models and with its help we obtain a
family of quasi-exactly solvable systems appearing in the -symmetric
quantum mechanics.Comment: 18 pages, refs updated; to appear in Nucl. Phys.
New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum
quasi-Yang-Baxter algebras being simple but non-trivial deformations of
ordinary algebras of monodromy matrices realize a new type of quantum dynamical
symmetries and find an unexpected and remarkable applications in quantum
inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter
algebras the standard procedure of QISM one obtains new wide classes of quantum
models which, being integrable (i.e. having enough number of commuting
integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic
Bethe ansatz solution for arbitrarily large but limited parts of the spectrum).
These quasi-exactly solvable models naturally arise as deformations of known
exactly solvable ones. A general theory of such deformations is proposed. The
correspondence ``Yangian --- quasi-Yangian'' and `` spin models ---
quasi- spin models'' is discussed in detail. We also construct the
classical conterparts of quasi-Yang-Baxter algebras and show that they
naturally lead to new classes of classical integrable models. We conjecture
that these models are quasi-exactly solvable in the sense of classical inverse
scattering method, i.e. admit only partial construction of action-angle
variables.Comment: 49 pages, LaTe
- …