1,257 research outputs found
An algebro-geometric proof of Witten's conjecture
We present a new proof of Witten's conjecture. The proof is based on the
analysis of the relationship between intersection indices on moduli spaces of
complex curves and Hurwitz numbers enumerating ramified coverings of the
2-sphere.Comment: 12 pages, no figure
Towards the Intersection Theory on Hurwitz Spaces
Moduli spaces of algebraic curves and closely related to them Hurwitz spaces,
that is, spaces of meromorphic functions on the curves, arise naturally in
numerous problems of algebraic geometry and mathematical physics, especially in
relationship with the string theory and Gromov--Witten invariants. In
particular, the classical Hurwitz problem about enumeration of topologically
distinct ramified coverings of the sphere with prescribed ramification type
reduces to the study of geometry and topology of these spaces. The cohomology
rings of such spaces are complicated even in the simplest cases of rational
curves and functions. However, the cohomology classes that are the most
important from the point of view of applications (namely, the classes
Poincar\'e dual to the strata of functions with given singularities) can be
expressed in terms of relatively simple ``basic'' classes (which are, in a
sense, tautological). The aim of the present paper is to identify these basic
classes, to describe relations among them, and to find expressions for the
strata in terms of these classes. Our approach is based on R. Thom's theory of
universal polynomials of singularities, which has been extended to the case of
multisingularities by the first author. Although the general Hurwitz problem
still remains open, our approach allows one to achieve a significant progress
in its solution, as well as in the understanding of the geometry and topology
of Hurwitz spaces.Comment: 29 pages, AMSTe
Hurwitz numbers and intersections on moduli spaces of curves
This article is an extended version of preprint math.AG/9902104. We find an
explicit formula for the number of topologically different ramified coverings
of a sphere by a genus g surface with only one complicated branching point in
terms of Hodge integrals over the moduli space of genus g curves with marked
points.Comment: 30 pages (AMSTeX). Minor typos are correcte
New alphabet-dependent morphological transition in a random RNA alignment
We study the fraction of nucleotides involved in the formation of a
cactus--like secondary structure of random heteropolymer RNA--like molecules.
In the low--temperature limit we study this fraction as a function of the
number of different nucleotide species. We show, that with changing ,
the secondary structures of random RNAs undergo a morphological transition:
for as the chain length goes to infinity,
signaling the formation of a virtually "perfect" gapless secondary structure;
while , what means that a non-perfect structure with
gaps is formed. The strict upper and lower bounds are
proven, and the numerical evidence for is presented. The relevance
of the transition from the evolutional point of view is discussed.Comment: 4 pages, 3 figures (title is changed, text is essentially reworked),
accepted in PR
Random Time Forward Starting Options
We introduce a natural generalization of the forward-starting options, first
discussed by M. Rubinstein. The main feature of the contract presented here is
that the strike-determination time is not fixed ex-ante, but allowed to be
random, usually related to the occurrence of some event, either of financial
nature or not. We will call these options {\bf Random Time Forward Starting
(RTFS)}. We show that, under an appropriate "martingale preserving" hypothesis,
we can exhibit arbitrage free prices, which can be explicitly computed in many
classical market models, at least under independence between the random time
and the assets' prices. Practical implementations of the pricing methodologies
are also provided. Finally a credit value adjustment formula for these OTC
options is computed for the unilateral counterparty credit risk.Comment: 19 pages, 1 figur
Permutation combinatorics of worldsheet moduli space
52 pages, 21 figures52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published version52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published versio
Necklace-Cloverleaf Transition in Associating RNA-like Diblock Copolymers
We consider a diblock copolymer, whose links are capable
of forming local reversible bonds with each other. We assume that the resulting
structure of the bonds is RNA--like, i.e. topologically isomorphic to a tree.
We show that, depending on the relative strengths of A--A, A--B and B--B
contacts, such a polymer can be in one of two different states. Namely, if a
self--association is preferable (i.e., A--A and B--B bonds are comparatively
stronger than A--B contacts) then the polymer forms a typical randomly branched
cloverleaf structure. On the contrary, if alternating association is preferable
(i.e. A--B bonds are stronger than A--A and B--B contacts) then the polymer
tends to form a generally linear necklace structure (with, probably, some rear
side branches and loops, which do not influence the overall characteristics of
the chain). The transition between cloverleaf and necklace states is studied in
details and it is shown that it is a 2nd order phase transition.Comment: 17 pages, 9 figure
- …