Seminar on Applied Mathematics
Michael Burt, Prof. Emeritus Technion, Israel Institute of Technology (Technion, Izrael)
UNIFORM NETWORKS. SPONGE SURFACES AND UNIFORM SPONGE POLYHE- DRA IN 3-D SPACE
Abstract: After two and half millennia of uncontested supremacy of Pythagorean and Euclidean geometry and R.B.Fuller inspired spherical geodesics, of the last seven decades or so, the stage of morphological exploration and design science is ripe for considering an all pervading new imagery, that of hyperbolic sponge phenomenology and related art forms and configurations. Nature is saturated with sponge structures on every possible scale of physical-biological reality. The expressions: "sponge", "spongeous" were adopted to describe a physical phenomenon which is character- ized by porosity and visual permeability. It transpires that the phenomenon is by far the most abundant in nature and is central to the physical-morphological nature of the human urban habitat as well, and represents its de?ning imagery. "Our study of natural form"?the essence of morphology?"is part of that wider science of form which deals with the forms assumed by nature, under all aspects and conditions, and in a still wider sense, with forms which are theoretically imaginable"? (On growth and form? D'Arcy Thompson). The amount of morphological insights into the sponge phenomenon, as accumulated over the last millen- nia, is incredibly meager. Only in the last 200 years and mostly in the last 60 years we have witnessed the emergence of basic concepts, insights and imagery. Lately, after confronting the prevalent definitions and allowing for polyhedral maps with curved edge-lines and face-surfaces, the amount of Uniform Sponge Polyhedra exploded, to reveal a multitude of new polyhedral sponge configurations; spherical, toroidal and hyperbolical and their governing hierarchical order. Since Periodic Sponge Surfaces and their tessellations (as sponge polyhedra) are strongly related to periodic space networks, as their (dual) tunnel systems, their exhaustive search appears to be one and the same research issue. It came as a sobering surprise to realize that no systematic research effort was invested in the 3-D space networks, in spite of their critical importance and relevance to organic and inorganic structural chemistry and crystallography in particulate. What is "theoretically imaginable" is at the core of this presentation; a 'tour de force ' of the Uniform Networks and Periodic Sponge Surfaces and Uniform Sponge Polyhedra, mostly new imagery which might play a signi?cant role in the morphological research of natural bio-forms and physical nano-structures, in°uence the way we perceive our expanding urban habitat, and, maybe even promote images and ideas of innovative space-structures, and specially those, critically dependent on the way we manipulate space.
Gradimir Milovanović , Univerzitet MEGATREND i Matematički institut SANU
WELL-CONDITIONED MATRICES FOR NUMERICAL TREATMENT OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND
Abstract: In the cases A = [?1; 1] and A = [0;+1), with w(x) = (1 ? x)Ž(1 + x)?, Ž; ? > ?1, and w(x) = xŽe?x? , Ž > ?1, ? > 1=2, respectively, in this paper We consider Fredholm integral equations of the second kind on f(y) + ? ZA k(x; y)f(x)w(x) dx = g(y); y 2 A; in the spaces of continuous functions equipped with certain uniform weighted norms, where A = [?1; 1] and A = [0;+1). The corresponding weight functions are w(x) = (1 ? x)Ž(1 + x)?, Ž; ? > ?1, and w(x) = xŽe?x? , Ž > ?1, ? > 1=2, respectively. Assuming the continuity of the kernel k(x; y) we use NystrAom methods and prove the stability, the convergence and the well-conditioning of the corresponding matrices. The last property can be derived only from the continuity of the kernel and not from its special form. We give the error estimates and some numerical examples.
Duško Katić, Institut "Mihailo Pupin"
REINFORCEMENT LEARNING IN HUMANOID ROBOTICS
Abstract: Reinforcement learning offers one of the most general framework to humanoid robotics towards true autonomy and versatility. Humanoid Robotics is a very challenging domain for reinforcement learn- ing, however, since robots cannot perceive the underlying state of their environment and because training time is usually quite limited. This talk discusses on what must be a optimal solution for application of reinforcement learning in humanoid robotics. The importance of hybrid approach is emphasized through- out the talk. The hybrid aspect is connected with application of model-based and model free approaches as well as with combination of different paradigms of computational intelligence. The general goal in synthesis of reinforcement learning control algorithms is the development of methods which scale into the dimensionality of humanoid robots and can generate actions for biped with many degrees of freedom. In this talk, we will consider specially that control of walking of active and passive dynamic walkers by using of reinforcement learning can be eąciently solved. Various straightforward and hybrid intelligent control algorithms based RL for active and passive biped locomotion will be presented. The proposed reinforcement learning algorithms is based on two diŽerent learning structures: actor-critic architecture and Q-learning structures. Also, RL algorithms can use numerical and fuzzy evaluative feedback informa- tion for external reinforcement. The proposed RL algorithms use the learning elements that consists of various types of neural networks, fuzzy logic nets or fuzzy-neuro networks with focus on fast convergence properties and small number of learning trials.In this talk, we give an overview on learning with policy gradient methods for humanoid robotics with a strong focus on recent advances in the field. We outline previous applications to robotics and show how the most recently developed methods can significantly improve learning performance. The main goal of is to review which policy gradient methods are appli- cable to robotics and which issues matter. We focus on the most useful methods and discuss several algorithms in-depth.