2,499 research outputs found
Cocycles over interval exchange transformations and multivalued Hamiltonian flows
We consider interval exchange transformations of periodic type and construct
different classes of recurrent ergodic cocycles of dimension over this
special class of IETs. Then using Poincar\'e sections we apply this
construction to obtain recurrence and ergodicity for some smooth flows on
non-compact manifolds which are extensions of multivalued Hamiltonian flows on
compact surfaces.Comment: 45 pages, 2 figure
Finite Resolution Dynamics
We develop a new mathematical model for describing a dynamical system at
limited resolution (or finite scale), and we give precise meaning to the notion
of a dynamical system having some property at all resolutions coarser than a
given number. Open covers are used to approximate the topology of the phase
space in a finite way, and the dynamical system is represented by means of a
combinatorial multivalued map. We formulate notions of transitivity and mixing
in the finite resolution setting in a computable and consistent way. Moreover,
we formulate equivalent conditions for these properties in terms of graphs, and
provide effective algorithms for their verification. As an application we show
that the Henon attractor is mixing at all resolutions coarser than 10^-5.Comment: 25 pages. Final version. To appear in Foundations of Computational
Mathematic
Therapeutic target discovery using Boolean network attractors: improvements of kali
In a previous article, an algorithm for identifying therapeutic targets in
Boolean networks modeling pathological mechanisms was introduced. In the
present article, the improvements made on this algorithm, named kali, are
described. These improvements are i) the possibility to work on asynchronous
Boolean networks, ii) a finer assessment of therapeutic targets and iii) the
possibility to use multivalued logic. kali assumes that the attractors of a
dynamical system, such as a Boolean network, are associated with the phenotypes
of the modeled biological system. Given a logic-based model of pathological
mechanisms, kali searches for therapeutic targets able to reduce the
reachability of the attractors associated with pathological phenotypes, thus
reducing their likeliness. kali is illustrated on an example network and used
on a biological case study. The case study is a published logic-based model of
bladder tumorigenesis from which kali returns consistent results. However, like
any computational tool, kali can predict but can not replace human expertise:
it is a supporting tool for coping with the complexity of biological systems in
the field of drug discovery
Frobenius structures on double Hurwitz spaces
We construct Frobenius structures of "dual type" on the moduli space of
ramified coverings of with given ramification type over two
points, generalizing a construction of Dubrovin. A complete hierarchy of
hydrodynamic type is obtained from the corresponding deformed flat connection.
This provides a suitable framework for the Whitham theory of an enlarged class
of integrable hierarchies; we treat as examples the q-deformed Gelfand-Dickey
hierarchy and the sine-Gordon equation, and compute the corresponding solutions
of the WDVV equations.Comment: 28 page
Semantic Unification A sheaf theoretic approach to natural language
Language is contextual and sheaf theory provides a high level mathematical
framework to model contextuality. We show how sheaf theory can model the
contextual nature of natural language and how gluing can be used to provide a
global semantics for a discourse by putting together the local logical
semantics of each sentence within the discourse. We introduce a presheaf
structure corresponding to a basic form of Discourse Representation Structures.
Within this setting, we formulate a notion of semantic unification --- gluing
meanings of parts of a discourse into a coherent whole --- as a form of
sheaf-theoretic gluing. We illustrate this idea with a number of examples where
it can used to represent resolutions of anaphoric references. We also discuss
multivalued gluing, described using a distributions functor, which can be used
to represent situations where multiple gluings are possible, and where we may
need to rank them using quantitative measures.
Dedicated to Jim Lambek on the occasion of his 90th birthday.Comment: 12 page
Wigner Functions versus WKB-Methods in Multivalued Geometrical Optics
We consider the Cauchy-problem for a class of scalar linear dispersive
equations with rapidly oscillating initial data. The problem of high-frequency
asymptotics of such models is reviewed,in particular we highlight the
difficulties in crossing caustics when using (time-dependent) WKB-methods.
Using Wigner measures we present an alternative approach to such asymptotic
problems. We first discuss the connection of the naive WKB solutions to
transport equations of Liouville type (with mono-kinetic solutions) in the
prebreaking regime. Further we show that the Wigner measure approach can be
used to analyze high-frequency limits in the post-breaking regime, in
comparison with the traditional Fourier integral operator method. Finally we
present some illustrating examples.Comment: 38 page
Operator Formalism on General Algebraic Curves
The usual Laurent expansion of the analytic tensors on the complex plane is
generalized to any closed and orientable Riemann surface represented as an
affine algebraic curve. As an application, the operator formalism for the
systems is developed. The physical states are expressed by means of creation
and annihilation operators as in the complex plane and the correlation
functions are evaluated starting from simple normal ordering rules. The Hilbert
space of the theory exhibits an interesting internal structure, being splitted
into ( is the number of branches of the curve) independent Hilbert
spaces. Exploiting the operator formalism a large collection of explicit
formulas of string theory is derived.Comment: 34 pages of plain TeX + harvmac, With respect to the first version
some new references have been added and a statement in the Introduction has
been change
Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in Life Sciences
Complex-dynamical fractal is a hierarchy of permanently, chaotically changing versions of system structure, obtained as the unreduced, causally probabilistic general solution of arbitrary interaction problem (physics/0305119, physics/9806002). Intrinsic creativity of this extension of usual fractality determines its exponentially high operation efficiency, which underlies many specific functions of living systems, such as autonomous adaptability, "purposeful" development, intelligence and consciousness (at higher complexity levels). We outline in more detail genetic applications of complex-dynamic fractality, demonstrate the dominating role of genome interactions, and show that further progressive development of genetic research, as well as other life-science applications, should be based on the dynamically fractal structure analysis of interaction processes involved. The obtained complex-dynamical fractal of a living organism specifies the intrinsic unification of its interaction dynamics at all levels, from genome structure to higher brain functions. We finally summarise the obtained extension of mathematical concepts and approaches closely related to their biological applications
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