276 research outputs found
Topological complexity of the relative closure of a semi-Pfaffian couple
Gabrielov introduced the notion of relative closure of a Pfaffian couple as
an alternative construction of the o-minimal structure generated by
Khovanskii's Pfaffian functions. In this paper, use the notion of format (or
complexity) of a Pfaffian couple to derive explicit upper-bounds for the
homology of its relative closure.
Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti
numbers.Comment: 12 pages, 1 figure. v3: Proofs and bounds have been slightly improve
Singular perturbation of polynomial potentials in the complex domain with applications to PT-symmetric families
In the first part of the paper, we discuss eigenvalue problems of the form
-w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on
two rays in the complex plane. We give sufficient conditions for continuity of
the spectrum when the leading coefficient of P tends to 0. In the second part,
we apply these results to the study of topology and geometry of the real
spectral loci of PT-symmetric families with P of degree 3 and 4, and prove
several related results on the location of zeros of their eigenfunctions.Comment: The main result on singular perturbation is substantially improved,
generalized, and the proof is simplified. 37 pages, 16 figure
Approximation of definable sets by compact families, and upper bounds on homotopy and homology
We prove new upper bounds on homotopy and homology groups of o-minimal sets
in terms of their approximations by compact o-minimal sets. In particular, we
improve the known upper bounds on Betti numbers of semialgebraic sets defined
by quantifier-free formulae, and obtain for the first time a singly exponential
bound on Betti numbers of sub-Pfaffian sets.Comment: 20 pages, 2 figure
Elementary proof of the B. and M. Shapiro conjecture for rational functions
We give a new elementary proof of the following theorem: if all critical
points of a rational function g belong to the real line then there exists a
fractional linear transformation L such that L(g) is a real rational function.
Then we interpret the result in terms of Fuchsian differential equations whose
general solution is a polynomial and in terms of electrostatics.Comment: 21 page
On metrics of curvature 1 with four conic singularities on tori and on the sphere
We discuss conformal metrics of curvature 1 on tori and on the sphere, with
four conic singularities whose angles are multiples of pi/2. Besides some
general results we study in detail the family of such symmetric metrics on the
sphere, with angles (pi/2,3pi/2,pi/2,3pi/2).Comment: 25 pages, 5 figure
Some lower bounds in the B. and M. Shapiro conjecture for flag varieties
The B. and M. Shapiro conjecture stated that all solutions of the Schubert
Calculus problems associated with real points on the rational normal curve
should be real. For Grassmannians, it was proved by Mukhin, Tarasov and
Varchenko. For flag varieties, Sottile found a counterexample and suggested
that all solutions should be real under certain monotonicity conditions. In
this paper, we compute lower bounds on the number of real solutions for some
special cases of the B. and M. Shapiro conjecture for flag varieties, when
Sottile's monotonicity conditions are not satisfied.Comment: 21 pages, 6 figures, see also http://www.math.purdue.edu/~agabrie
- …