# Bi-Annual Mini-Workshops on Applied and Computational Mathematics

Nineteenth workshop       Bar-Ilan University, December 29, 2016

Eighteenth workshop        Session at IMU annual meeting, June 2-5, 2016

Seventeenth workshop        Weizmann Institute, December 28, 2015

Sixteenth workshop            Technion, December 30, 2014

Fifteenth workshop             Ort Braude College, December 27, 2012

Fourteenth workshop          Tel-Aviv University July 3, 2012

Thirteenth workshop           Bar-Ilan University, December 29, 2011

Twelfth workshop               Ben Gurion University, December 31, 2009.

Eleventh workshop              Technion, July 2, 2009

Tenth workshop                  Tel-Aviv University, December 30, 2008

Ninth workshop                  Hebrew University, July 3, 2008

Eightth workshop                Bar-Ilan, December 27, 2007

Seventh workshop             Weizmann, June 14, 2007

Sixth workshop                  Ben Gurion University, Jan 2007

Fifth workshop                  Technion, June 2006

Fourth workshop              Tel-Aviv University, January  3, 2006.

Third workshop                Hebrew University,  June 7, 2005.

Second Workshop           Bar-Ilan University, December 23, 2004.

First workshop                Weizmann Institute, December 23, 2003.

# Mini-Workshop on Applied Mathematics

Tuesday, December 23, 2003
Weizmann Institute, Zyskind 1

Few of us have thought about initiating informal one day  meetings in Applied Mathematical Analysis and Computations. The idea is to have 2-3 such meeting per year. The suggested format is to have two one-hour plenary talks which are meant to be more of a review of a field of research and four half-hour talks by local participants (a balanced mixture of young and senior researchers). It will be fine if some speakers decide to report on some of their preliminary results. After all these are supposed to be informal meetings and the idea is to promote discussion and collaborations. In this first meeting we included several visitors who came for winter vacation to Israel - in the future we hope there will be more opportunity to local participants.

The first meeting includes a disscussion on the present and future directions of Applied mathematics in Israel.

We also intend to have a preprint/reprint table where people can display their recent works.

The participation in the mini-workshop is FREE.  However, please register: Send e-mail to Carol, preferably now, but in any case before 18/12/03  if you plan to attend.
Please include details for a name tag and let us know if you plan to have lunch with us.

 9:30-10:00 Coffee 10:00-11:00 Prof. G. Zaslavsky, NYU Complexity of trajectories in Hamiltonian dynamics 11:00-11:30 Prof. N. Paldor, Hebrew U. The Shallow Water Equations on the Rotating Spherical Earth: recent Advances 11:30-12:00 Dr. Boaz Ilan, U. of Colorado at Boulder Optical light bullets in a pure Kerr medium 12:00-14:00 Lunch 14:00-15:00 Discussion on Applied Mathematics in Israel:  wwww (what, why, where, where to) Confirmed participants: Zvika Artstein (WIS), Matania Ben-Artzi (HU),                                      Gadi Fibich (TAU), Ilan Kozma (Elta). 15:00-15:30 Prof. D. Turaev, BGU On the richness of Hamiltonian chaos. 15:30-16:00 Coffee 16:00-16:30 Arkady Poliakovsky , Technion. Singular perturbation problems and liftings in BV 16:30-17:30 Prof. G. Golub, Stanford Numerical Methods for Solving Least Squares Problems  with Constraints

We thank the Faculty of mathematics and computer science at Weizmann and the Belfer institute for their support.

# Abstracts:

Prof. G. Zaslavsky, NYU

### Complexity of trajectories in Hamiltonian dynamics

It is natural to introduce such a definition of the complexity that, from one side it provides " more complexity"
the more unpredictable is a trajectory, and, from another side, the complexity notion would be related to
such physical characteristics as entropy and transport. The existed notion of epsilon-complexity is insufficient
for this goal since it can't be applied to Hamiltonian dynamics with nonuniform hyperbolicity or with zero Lyapunov exponents.
We provide some examples of this difficulty and introduce new notions of the complexity function and entropy function.
Different features of the new notions will be described and the applications will be demonstrated.

Dr. Boaz Ilan, U. of Colorado at Boulder

### Optical light bullets in a pure Kerr medium

Using a recent result on critical exponents of anisotropic NLS
equations, as well as numerical simulations, we show that small negative
fourth-order dispersion can arrest spatiotemporal collapse of ultrashort
laser pulses propagating in a planar waveguide having a pure Kerr
nonlinearity and anomalous dispersion, resulting in (2+1)D "optical
bullets". Similarly to solitons, these bullets undergo elastic
collisions. Since these bullets can self-trap from noisy Gaussian input
beams and propagate without power loss, this result may be used to
realize experimentally stable, non-dissipative optical bullets.
This is joint work with Gadi Fibich.

Prof. D. Turaev
Ben-Gurion University

### On the richness of Hamiltonian chaos.

Abstract. We call a symplectic diffeomorphism of $R^{2n}$ universal, if
its iterations can be made arbitrarily close to any other symplectic
diffeomorphism by a smooth coordinate transformation. By definition, any
dynamical phenomenon, robust in the class of smooth 2n-simensional
symplectic diffeomorphisms, can be encountered in any universal symplectic
map of the same dimension. We prove that universal maps are dense in the
class of synmplectic maps which have at least one elliptic periodic
orbit. These results strongly suggest to expect an ultimate richness of
dynamics from any given Hamiltonian system with 2 or more degrees
of freedom without a uniform partially-hyperbolic structure.

### Singular perturbation problems and liftings in BV

The study of the asymptotic behavior of the minimizers $\{u_\e\}$, as $\e$
goes to zero, for the energy
$$E_\e(u)=\int_\Omega \bigl\{|\nabla u|^2+ \frac{1}{\e2}W(u)\bigr\}\,dx\,,$$
under a mass constraint $\int_\Omega u=m$, where $W$ is a double-well potential, is motivated
by the Van der Waals-Cahn-Hilliard theory of phase transitions.
We apply a generalization of the techniques developed by
Modica and Sternberg to a lifting problem in BV-spaces.
A function $u\in BV(\Omega,S1)$ has many liftings in $BV(\Omega,\R)$ (by a {\em lifting}
we mean a function $\phi$ satisfying $u=e^{i\phi}$ a.e.~in $\Omega$), so it is natural
to look for an optimal'' lifting. Our approach consists of studying
the asymptotic behavior of the minimizers $\{\phi_\e\}$ for the energy
$$E_\e(\phi)=\int_\Omega \bigl\{|\nabla\phi|^2+ \frac{1}{\e2}|u-e^{i\phi}|^2\bigr\}\,dx\,,$$
for which we show the convergence to a lifting of $u$ with minimal BV-seminorm.

Prof. Gene H. Golub, Stanford University

### Numerical Methods for Solving Least Squares Problems  with Constraints

In this talk, we discuss the problem of solving linear least squares
problems and Total Least Squares problems with linear constraints
and/or a quadratic constraint. We are particularly interested in
developing stable numerical methods when the data  matrix is singular
or near singular. Of particular interest are matrices which are large
and sparse and for which iterative methods must be employed.  The
quadratically constrained problems arise in problems where
regularization is required. For such problems, a Lagrange multiplier
is required and that calculation may be quite intensive. The method we
propose will quickly yield an estimate of the parameter and allow for
finding the least squares solution.

Gene H. GOLUB, Computer Science Department, Stanford
University, Stanford, CA 94305, USA
golub@stanford.edu

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The Fifth Israeli Applied Math MiniWorkshop

Wednesday 7.6.06 at the Technion

********************************************************

This is the second announcement for the MiniWorkshop. We are pleased to

have a stellar lineup of speakers:

Ehud Yariv (Technion)

Avy Soffer (Rutgers)

Philip Rosenau (Tel Aviv)

Vered Rom-Kedar and Eliezer Shochat (Weizmann)

Alex and Michael Bronstein (Technion)

Matania Ben-Artzi (Jerusalem)

Avi Levy (Intel)

Dani Givoli (Technion)

A. As usual the MiniWorkshop will consists of 30 minutes talks.

B. Although there is no registration fee for this meeting, we ask all

those who plan to participate to inform the organizer (Koby Rubinstein

koby@math.technion.ac.il) by 20.5.06 in order to help make

suitable local arrangements.

C. The meeting will conclude at an Apre-Math party at the Camel beach