950 research outputs found
Splicing Systems from Past to Future: Old and New Challenges
A splicing system is a formal model of a recombinant behaviour of sets of
double stranded DNA molecules when acted on by restriction enzymes and ligase.
In this survey we will concentrate on a specific behaviour of a type of
splicing systems, introduced by P\u{a}un and subsequently developed by many
researchers in both linear and circular case of splicing definition. In
particular, we will present recent results on this topic and how they stimulate
new challenging investigations.Comment: Appeared in: Discrete Mathematics and Computer Science. Papers in
Memoriam Alexandru Mateescu (1952-2005). The Publishing House of the Romanian
Academy, 2014. arXiv admin note: text overlap with arXiv:1112.4897 by other
author
Splicing systems and the Chomsky hierarchy
In this paper, we prove decidability properties and new results on the
position of the family of languages generated by (circular) splicing systems
within the Chomsky hierarchy. The two main results of the paper are the
following. First, we show that it is decidable, given a circular splicing
language and a regular language, whether they are equal. Second, we prove the
language generated by an alphabetic splicing system is context-free. Alphabetic
splicing systems are a generalization of simple and semi-simple splicin systems
already considered in the literature
On Hopf 2-algebras
Our main goal in this paper is to translate the diagram relating groups,
Lie algebras and Hopf algebras to the corresponding 2-objects, i.e. to
categorify it. This is done interpreting 2-objects as crossed modules and
showing the compatibility of the standard functors linking groups, Lie algebras
and Hopf algebras with the concept of a crossed module. One outcome is the
construction of an enveloping algebra of the string Lie algebra of Baez-Crans,
another is the clarification of the passage from crossed modules of Hopf
algebras to Hopf 2-algebras.Comment: 26 pages, clarification of several statement
Interaction paths promote module integration and network-level robustness of spliceosome to cascading effects
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTĂFICO E TECNOLĂGICO - CNPQFUNDAĂĂO DE AMPARO Ă PESQUISA DO ESTADO DE SĂO PAULO - FAPESPThe functionality of distinct types of protein networks depends on the patterns of protein-protein interactions. A problem to solve is understanding the fragility of protein networks to predict system malfunctioning due to mutations and other errors. Spec8111CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTĂFICO E TECNOLĂGICO - CNPQFUNDAĂĂO DE AMPARO Ă PESQUISA DO ESTADO DE SĂO PAULO - FAPESPCONSELHO NACIONAL DE DESENVOLVIMENTO CIENTĂFICO E TECNOLĂGICO - CNPQFUNDAĂĂO DE AMPARO Ă PESQUISA DO ESTADO DE SĂO PAULO - FAPESPsem informação2017/08406-7, 2017/06994-9We thank Ana Paula Assis, PĂąmela C. Santana and Leandro Giacobelli for helpful comments. PRG was supported by CNPq and FAPESP (2017/08406-7). PPC was supported by FAPESP (2017/06994-9). MC was supported by a PMP/BS postdoctoral fellowship (UFPR/UNIVALI 4
An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants
We prove an analogue of the Kotschick-Morgan conjecture in the context of
SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten
invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The
main technical difficulty in the SO(3)-monopole program relating the
Seiberg-Witten and Donaldson invariants has been to compute intersection
pairings on links of strata of reducible SO(3) monopoles, namely the moduli
spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck
compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710032].
In this monograph, we prove --- modulo a gluing theorem which is an extension
of our earlier work in [arXiv:math/9907107] --- that these intersection
pairings can be expressed in terms of topological data and Seiberg-Witten
invariants of the four-manifold. This conclusion is analogous to the
Kotschick-Morgan conjecture concerning the wall-crossing formula for Donaldson
invariants of a four-manifold with ; that wall-crossing formula and
the resulting structure of Donaldson invariants for four-manifolds with
were established, assuming the Kotschick-Morgan conjecture, by
Goettsche [arXiv:alg-geom/9506018] and Goettsche and Zagier
[arXiv:alg-geom/9612020]. In this monograph, we reduce the proof of the
Kotschick-Morgan conjecture to an extension of previously established gluing
theorems for anti-self-dual SO(3) connections (see [arXiv:math/9812060] and
references therein). Since the first version of our monograph was circulated,
applications of our results have appeared in the proof of Property P for knots
by Kronheimer and Mrowka [arXiv:math/0311489] and work of Sivek on Donaldson
invariants for symplectic four-manifolds [arXiv:1301.0377].Comment: x + 229 page
Complex surface singularities with integral homology sphere links
While the topological types of {normal} surface singularities with homology
sphere link have been classified, forming a rich class, until recently little
was known about the possible analytic structures. We proved in [Geom. Topol.
9(2005) 699-755] that many of them can be realized as complete intersection
singularities of "splice type", generalizing Brieskorn type.
We show that a normal singularity with homology sphere link is of splice type
if and only if some naturally occurring knots in the singularity link are
themselves links of hypersurface sections of the singular point.
The Casson Invariant Conjecture (CIC) asserts that for a complete
intersection surface singularity whose link is an integral homology sphere, the
Casson invariant of that link is one-eighth the signature of the Milnor fiber.
In this paper we prove CIC for a large class of splice type singularities.
The CIC suggests (and is motivated by the idea) that the Milnor fiber of a
complete intersection singularity with homology sphere link Sigma should be a
4-manifold canonically associated to Sigma. We propose, and verify in a
non-trivial case, a stronger conjecture than the CIC for splice type complete
intersections: a precise topological description of the Milnor fiber. We also
point out recent counterexamples to some overly optimistic earlier conjectures
in [Trends in Singularities, Birkhauser (2002) 181--190 and Math. Ann.
326(2003) 75--93].Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper18.abs.htm
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