The Sixth Vasil Popov Prize awarded to
Joel A. Tropp of California Institute of Technology
A B S T R A C T S
SOME FIXED POINT THEOREMS FOR KANNAN MAPPINGS
Shavetambry Tejpal
shwetambry@gmail.com,
T. D. Narang
tdnarang1948@yahoo.co.in
2000 Mathematics Subject Classification:
Primary: 47H10; Secondary: 54H25.
Key words:
Kannan mappings, uniform normal structure, admissible sets,
convex metric spaces.
Some results on the existence and uniqueness of fixed points for Kannan mappings on admissible subsets of bounded metric spaces and on bounded closed convex subsets of complete convex metric spaces having uniform normal structure are proved in this paper. These results extend and generalize some results of Ismat Beg and Akbar Azam [Ind. J. Pure Appl. Math. 18 (1987), 594-596], A. A. Gillespie and B. B. Williams [J. Math. Anal. Appl. 74 (1980), 382-387] and of Yoichi Kijima and Wataru Takahashi [Kodai Math Sem. Rep. 21 (1969), 326-330].
ASYMPTOTIC ANALYSIS OF A SCHRÖDINGER-POISSON SYSTEM WITH QUANTUM WELLS AND MACROSCOPIC NONLINEARITIES IN DIMENSION 1
A. Faraj
ali.faraj@univ-rennes1.fr
2000 Mathematics Subject Classification:
35Q02, 35Q05, 35Q10, 35B40.
Key words:
Schrödinger-Poisson system, Asymptotic analysis, Semi-classical analysis, Spectral theory.
We consider the stationary one dimensional Schrödinger-Poisson system on a bounded interval with a background potential describing a quantum well. Using a partition function which forces the particles to remain in the quantum well, the limit h®0 in the nonlinear system leads to a uniquely solved nonlinear problem with concentrated particle density. It allows to conclude about the convergence of the solution.
SEMI-SYMMETRIC ALGEBRAS: GENERAL CONSTRUCTIONS. PART II
Valentin Vankov Iliev
viliev@math.bas.bg
2000 Mathematics Subject Classification:
15A69, 15A78.
Key words:
Semi-symmetric power, semi-symmetric algebra, coalgebra structure, inner product.
In [3] we present the construction of the semi-symmetric
algebra [χ](E) of a module E over a commutative ring K with unit, which generalizes the tensor algebra T(E), the symmetric algebra S(E), and the exterior algebra ∧(E), deduce some of its functorial properties, and prove a classification theorem. In the present paper we continue the study of the semi-symmetric algebra and discuss its graded dual, the corresponding canonical bilinear form, its coalgebra structure, as well as left and right inner products. Here we present a unified treatment of these topics whose exposition in [2, A.III] is
made simultaneously for the above three particular (and, without a shadow of doubt - most important) cases.
A NOTE ON THE L^{2}-NORM OF THE SECOND FUNDAMENTAL FORM OF ALGEBRAIC MANIFOLDS
Andrea Loi
loi@unica.it,
Michela Zedda
michela.zedda@gmail.com
2000 Mathematics Subject Classification:
53C42, 53C55.
Key words:
Kähler metrics, holomorphic maps into projective space, algebraic manifolds, degree.
Let M
CP^{n}
be an algebraic manifold of complex dimension d and let σ_{f} be its second fundamental form.
In this paper we address the following conjecture, which is the analogue of the one stated by M. Gromov for smooth immersions:
if ||σ_{f} || ^{2}_{L2} < 2 d vol(CP^{d}) then M is totally geodesic and equality holds iff f is congruent to the standard embedding of the complex quadric Q_{d} into CP^{n}.
We prove the conjecture in the following three cases: (i) d = 1; (ii) M is a complete intersection;
(iii) the scalar curvature of M is constant.
NONSTANDARD FINITE DIFFERENCE SCHEMES WITH APPLICATION TO FINANCE: OPTION PRICING
Mariyan Milev
mariyan.milev@unive.it,
Aldo Tagliani
tagliani@unitn.it
2000 Mathematics Subject Classification:
65M06, 65M12.
Key words:
Black-Scholes equation, finite difference schemes, Jacobi matrix;
M-matrix, nonsmooth initial conditions, positivity-preserving.
The paper is devoted to pricing options characterized by discontinuities in the initial conditions of the respective Black-Scholes partial differential equation. Finite difference schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We analyze the drawbacks of the Crank-Nicolson scheme that is most frequently used numerical method in Finance because of its second order accuracy. We propose an alternative scheme that is free of spurious oscillations and satisfy the positivity requirement, as it is demanded for the financial solution of the Black-Scholes equation.
ON THE ASYMPTOTIC DISTRIBUTION OF CLOSED ORBITS FOR A CLASS OF OPEN BILLIARDS
Julien Giol
julien.giol@bucknell.edu
2000 Mathematics Subject Classification:
37D40.
Key words:
billiard, symbolic dynamics, periodic orbits.
We explain why the lengths of the closed orbits in a certain class of open billiards are asymptotically equidistributed with respect to the number of their reflections.
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