Seminars, 2018

  • May 22, 2018
    Nonlinear Schrödinger equations, Lotka-Volterra models, and control of soliton collisions in broadband optical waveguide systems
    Dr. Avner Peleg

    Transmission rates in broadband optical waveguide systems are significantly enhanced by launching many pulse sequences through the same waveguide. Since pulses from different sequences propagate with different group velocities, intersequence pulse collisions are very frequent, and can lead to severe transmission degradation. On the other hand, the energy exchange in pulse collisions can be beneficially used for realizing fast control of the transmission.
    In the current work we develop a general approach for exploiting the energy exchange in intersequence collisions for transmission stabilization and switching, using solitons as the optical pulses. The approach is based on showing that collision-induced amplitude dynamics of N sequences of solitons of the nonlinear Schrödinger (NLS) equation in the presence of dissipative perturbations can be described by N-dimensional Lotka-Volterra (LV) models. To derive the LV models, we first analyze the effects of a single two-soliton collision, using a perturbative expansion in the eigenmodes of the linear operator describing small perturbations about the NLS soliton. We then use stability and bifurcation analysis for the equilibrium points of the LV models to develop ways for achieving robust transmission stabilization and switching that work well for a variety of optical waveguides. Further enhancement of transmission stability is enabled by suppression of resonant emission of radiation in nonlinear waveguide couplers with frequency dependent linear gain-loss. Furthermore, we show that supercritical Hopf bifurcations of the equilibrium points of the LV models can be used to induce large stable oscillations of soliton amplitudes along ultra-long distances. The latter finding is an important step towards realizing spatio-temporal chaos with multiple sequences of colliding solitons in nonlinear optical waveguides.
  • May 15, 2018
    Pseudo-tropical curves
    Dr. Sergei Lanzat, Technion

    We consider a generalization of tropical curves, removing requirements ofrationality of slopes and integrality and discuss the resulting theory andits interrelations with other areas. Balancing conditions are interpretedas criticality of a certain action functional. A generalized Bezout theoreminvolves Minkowsky sum and mixed area. A problem of counting curves passing through an appropriate collection of points turns out to be related to the Plucker relations of Grassmanians. We also discuss new recursive relations for this count.
    This is the joint work with Michael Polyak.
  • May 8, 2018
    Quasi-normal surfaces and approximations of least area surfaces in 3-maniflods
    Dr. Eliezer Appleboim, Technion

    This work presents an approximation method by simple pieces of least area surfaces in 3-manifolds.
    In [JR] a piecewise linear (PL) version of minimal and least area surfaces is presented. This PL version is based on normal surfaces. A normal surface in a 3-manifold is a surface that intersects a given triangulation of the manifold in a special simple manner. It was shown in [JR] that PL-minimal and least area surfaces share many of the properties of classical differential geometric least area surfaces. However, the extent at which this analogy holds if far from being fully understood. For instance, the following question is very natural in this context, yet it is still open:
    Question: Can every minimal surface in the smooth sense, be presented as a limit surface of a sequence of PL-minimal surfaces, appropriately constructed?
    Addressing the above question is the main theme of this work. Specifically, a method of approximating a least area surface by simple pieces will be given. It will be shown that there exist obstructions that prevent the approximating simple pieces from being the PL- minimal surfaces as defined in [JR]. A modified version of PL-minimal surfaces will be defined, and it will be shown that this family of surfaces yields a good approximation of least area surfaces, i.e., a least area surface is a limit surface, in an appropriate sense, of a sequence of simple approximating surfaces, and the area of the given least area surface is also the limit of the areas of the approximating surfaces.
    This study is of major interest of its own sake, since it gives a new topological insight and understanding of known geometric results about the global behavior of least area surfaces. Potential applications of this study in the areas of Image Processing and Computer Graphics will also be discussed. Examples are the problem of removing noise from an image, and finding the shortest path on a triangulated surface. Some preliminary experimental results will be shown.
    Reference:  [JR] W. Jaco and H. Rubinstein, PL-Minimal Surfaces in 3-Manifolds, J. Differntial Geometry, 27, 1988.
  • May 2, 2018
    Spaces of real places of ordered fields
    Dr. Michael Machura, ORT Braude College

    We consider a famous open problem: Does every compact Hausdorff space is  a space of real places of some ordered field? 
    Until now not many examples of spaces of real places are known. What about so simple spaces as interval, sphere, torus? Are all spaces of real places metrizable?
    How algebraic and transtendental extention affects of space of real places? 
    We shall provide partiall answers to these questions. The fields of rationals functions of one and more variable shall be analized.
  • April 24, 2018
    Umbral calculus and identities for orthogonal polynomials
    Dr. Orli Herscovici, Haifa University

    The roots of Umbral calculus go back to the XVII century. It attracted an attention of many scientists in the second half of the XVIII century, but a rigorous study of the Umbral calculus belongs to 70-80 years of the XIX century. Since then the Umbral calculus is a powerful technique for study of orthogonal polynomials of the Sheffer type among them are the Bernoulli, Euler, Hermite, Laguerre, and many other polynomials.
    By using these techniques, one can obtain new identities for Sheffer orthogonal polynomials and establish connections between different kind of polynomials.
  • April 17, 2018
    How we can partition the integers into arithmetic progressions and related questions
    Dr. Ofir Schnabel, Haifa University

    The evens and odds form a partition of the integers into arithmetic progressions. It is natural to try to describe in general how the integers can be partitioned into arithmetic progressions. For example, a classic result from the 1950's shows that if a set of arithmetic progressions partitions the integers, there must be two arithmetic progressions with the same difference. Another direction is to try to determine when such a partition is a proper refinements of another non-trivial partition.
    In my talk I will give some of the more interesting results on this subject, report some (relatively) new results and present two generalizations of partitioning the integers by arithmetic progressions, namely:
    1. Partitions of the integers by Beatty sequences (will be defined).
    2. Coset partition of a group.
    The main conjecture in the first topic is due to A. Fraenkel and describes all the partitions having distinct moduli.
    The main conjecture in the second topic, due to M. Herzog and J. Schonheim, claims that in every coset partition of a group there must be two cosets of the same index.
    Again, we will briefly discuss the history of these conjectures, recall some of the main results and report some new results.
  • March 27, 2018
    A probabilistic model of thermal explosion in polydisperese fuel spray
    Dr. Ophir Nave and Dr. Vladimir Gol’dshtein, Ben-Gurion University

    Our research concerned with an analysis of polydisperse spray droplets distribution on the thermal explosion processes. In many engineering applications it is usual to relate to the practical polydisperse spray as a monodisperse spray. The Sauter Mean Diameter (SMD) and its variations are frequently used for this purpose Lefebvere (1989). The SMD and its modifications depend only on``integral'' characterization of polydisperse sprays and can be the same for very different types of polydisperse spray distributions.
    We presents a new, simplified model of the thermal explosion in a combustible gaseous mixture containing vaporizing fuel droplets of different radii (polydisperse). The polydispersity is modeled using a Probability Density Function (PDF) that corresponds to the initial distribution of fuel droplets size. This approximation of polydisperse spray is more accurate than the traditional 'parcel' approximation and permits an analytical treatment of the simplified model. Since the system of the governing equations represents a multi-scale problem, the method of invariant (integral) manifolds is applied.
    An explicit expression of the critical condition for thermal explosion limit is derived analytically. Numerical simulations demonstrate an essential dependence of these thermal explosion conditions on the PDF type and represent a natural generalization of the thermal explosion conditions of the classical Semenov theory.
  • March 20, 2018
    On the Eisenbud-Green-Harris Conjecture
    Dr. Abed Abedelfatah, Haifa University