We study the statistics of the number of connected components and the volume of a random real algebraic hypersurface in RP^n defined by a
real Fubini-Study distributed homogeneous polynomial of degree d.
We prove that the expectation of the number of connected components of such hypersurface has order d^n, the asymptotic being in d for n fixed.
We do not restrict ourselves to the random homogeneous case and we consider more generally random polynomials belonging to a window of eigenspaces of the
Laplacian on the sphere S^n, proving that the same asymptotic holds. As for the volume, we prove its expectation is of order d.
Both these behaviors exhibit expectation of maximal order in light of Milnor's bound and the a priori bound for the volume.
The problem of mapping the interior of a Jordan polygon univalently
by the Poisson integral of a step function was posed by T. Sheil-Small (1989). We
describe a simple solution using "ear clipping" from computational geometry.
We investigate a problem posed by L. Hauswirth, F. H\'elein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function",
i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data.
As they suggested, we show, under some a priori assumptions, that there are only three exceptional domains:
the exterior of a disk, a halfplane, and a nontrivial example.
We show however that one cannot obtain any axially symmetric analogues of the nontrivial example in four dimensions.
For a hyponormal operator, C. R. Putnam's inequality gives an upper bound on the norm of its self-commutator.
In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain,
there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality.
For a nontrivial domain, we compare these estimates to exact results.
Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain.
When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants).
We conjecture an improved version of Putnam's inequality within this restricted setting.
In the study of permutations, generalized patterns extend classical patterns by allowing
the requirement that certain adjacent integers in a pattern must be adjacent in the permutation.
For any generalized pattern p of length k with b blocks, we prove that
the number of occurrences of p in a random permutation of large size is exponentially concentrated about its mean.
We also give a lower bound on avoidance of the generalized pattern 12-34, which answers a question
of S. Elizalde (2006).
It was recently noticed [Khavinson, Mineev, Putinar, Teodorescu] (2010) that lemniscates do not survive Laplacian growth.
This raises the question: Is there a growth process for which polynomial lemniscates are solutions?
The answer is "yes", and the law governing the boundary velocity is reciprocal to that of Laplacian
growth.
Viewing lemniscates as solutions to a moving-boundary problem gives
a new perspective on results from classical potential theory,
and interesting properties emerge while comparing lemniscate growth to Laplacian
growth.
It is well known that, in the plane, the boundary of any quadrature domain (in the classical sense) coincides with the zero set of a polynomial.
We show, by explicitly constructing some four-dimensional examples, that this is not always the case.
This confirms, in dimension 4, a conjecture of the second author.
Our method is based on the Schwarz potential and involves elliptic integrals of the third kind.
In a recent paper [Bayer, Dyer, Giang] (2006), the authors considered a certain class of gravitational lenses consisting of n non-overlapping objects with radial densities. They
concluded that there can be at most 6(n - 1) + 1 lensed images of a single light source.
The only assumption made on the projected mass density of each object is that it is
radial and does not diverge faster than 1/r, where r is the distance to the center of the
object. We show that this is too general a class of densities to consider while imposing
a bound of 6(n - 1) + 1. We also provide an example to emphasize that the general problem of finding the correct hypothesis to obtain sharp
bounds for the maximal number of images inside the region occupied by masses with
radial densities is wide open.
The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the
Laplacian growth (or ``Hele-Shaw``) problem in the plane.
The guiding principle in this connection is the fact that ``non-physical'' singularities in the ``oil domain'' of the Schwarz function are stationary,
and the ``physical'' singularities obey simple dynamics.
We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential,
introduced by D. Khavinson and H. S. Shapiro \cite{KhSh} (1989).
A generalization is also given for the so-called ``elliptic growth'' problem by defining a generalized Schwarz potential.
New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow.
We demonstrate, by example, how $\mathbb{C}^n$-techniques can be used to locate the singularity set of the Schwarz potential.
One of our methods is to prolong available local extension theorems by constructing ``globalizing families''.
We make three conjectures in potential theory relating to our investigation.
The main result of the paper states the following: Let psi be a polynomial in n variables. Suppose that there exists a constant C > 0 such that any polynomial f has a
polynomial decomposition f = psi q_f + h_f with \Delta^k h_f = 0 and the degree of q_f does not exceed the degree
of f plus C. Then the degree of psi is at most 2k. Here \Delta^k is the kth iterate of the Laplace operator
\Delta. As an application, new classes of domains in R^n are identified for which the
Khavinson-Shapiro conjecture holds.
Elliptical galaxies with density constant on confocal
ellipses can produce at most four "bright" images of a single source.
A more physically interesting example has density that is constant
on homothetic ellipses. In that case bright images can be seen to
correspond to zeros of a certain transcendental harmonic mapping.
We give a bound on the total number of such zeros.
This is a survey article based on an invited talk delivered by Dmitry Khavinson at the CRM workshop on Hilbert Spaces
of Analytic Functions held at CRM, Universite de Montreal, December 8-12, 2008.
A conjecture in astronomy was recently resolved as an accidental corollary to a theorem regarding zeros of certain planar harmonic maps. As a step towards extending the Fundamental Theorem of Algebra, the theorem gave a bound of 5n-5 for the number of zeros of a function of the form r(z) - z-, where r(z) is rational of degree n. In this paper, we will investigate the case when r(z) is a Blaschke product. The resulting (sharp) bound is n+3 and the proof is simple. We discuss an application to gravitational lenses consisting of collinear point masses.
We consider the Dirichlet problem in the plane with entire data on algebraic curves.
More specifically are interested in where singularities develop when a solution is continued analytically.
Our approach relies on annihilating measures supported on a complex version of the finite sets called lightning bolts,
which were used by Kolmogorov and Arnold to solve Hilbert's 13th problem.
A special case of Sharkovsky's theorem says that if a continuous function has a period-3 point then it has periodic points of every order.
In this note we investigate how often orbit types of period-n guarantee a period-3 point. An informal statement of our result is that as n goes to infinity the probability that a period-n orbit guarantees a period-3 point converges to 1.