Mendelian Dynamics and Sturtevant ’ s Paradigm
This a brief introduction to the formal genetics centered around two mathematical ideas going back to Gregor Mendel and Alfred Sturtevant. Preambule. The road from biology (or from any branch of science on the fundamental level) to mathematics goes in several ( often Brownian rather than straight) paths in parallel. • Identifying a class of phenomena–”particular trees in a forest”–that appear with a regularity suggesting an underlying (mathematical) structure. (Are there non-mathematical structures?) •• Designing and performing experiments/observations purifying and amplifying what is seen by the naked eye (e.g. by planting our ”trees” to an ”artificial soil”). •••Making (often implicitly) ad hoc hypotheses, (e.g. continuity, symmetry, functoriality)– that provide a logical framework for the experimental data. For example, the ”theory of coin tossing” derives its mathematical beauty and the (probabilistic) predictive power not from such ”definitions” as ”the probability is a measure of uncertainty” but from the assumption that the the probability distribution on the space Z2 of the imaginary outcomes (binary nsequences) equals the (normalized) Haar measure that is, moreover, invariant under the permutation group Sn. Such hypotheses (assumptions), fragments of the grammer of the language in which Nature delievers her messages, are what a mathematician is primerly interested in, while a scientist is concerned with the ”meaning” of a message–the structure that is harder to formilize. Desiphering the grammer of Nature, or, biologically speaking, guessing the design of a seed by looking at (sample branches of) the grown tree, is rarely (if ever) done by mathematicians (Newtons do not count), even when the experimental data are abundant (as in the present day molecular biology). The past mathematical experience channals your imagination toward the old rather than new mathematical concepts. Even rigorously reformulating such hypotheses is not a straighforward task. For example, the Dirac δ-function needed the theory of distributions to be accepted by mathematicians and the functoriality of the (derivation of the) Boltzmann equation was recognized (albeit not much exploited till now) only with the advent of the ”functoriality paradigm” within pure mathematics. When a ”seed” is cultivated in a ”mathematical soil”, what grows out of it– mathematician’s tree–might look not quite as the real one. But a math-
The Poincaré Conjecture
It is a great pleasure for me to report on the recent spectacular developments concerning the Poincare Conjecture. Grigory Perelman has solved the Poincare Conjecture. He has shown that, as Poincare conjectured, any closed, simply connected 3-manifold is homeomorphic to the 3-sphere. The paper in which Poincare posed this problem in 1904 ([14]) marked, in my view, the founding of topology as an independent discipline within pure mathematics. Over the intervening 100 years, the problem has been much studied and generalized, and many related problems have been solved. It has been linked, in one way or another, with most of the progress in topology in the last 100 years. While related problems have been solved, the original conjecture stood untouched, resisting all attempts. Before Perelman’s work, there had been no progress on toward solving the Poincare Conjecture, and many viewed it as the siren song of Topology, for many a boat had foundered on the rocks trying to reach it. There have been innumerable proposed proofs and proposed counter-examples, but none, before Perelman’s, withstood scrutiny. Solving the Poincare Conjecture is a signal achievement for Perelman, but it is also a signal achievement for all of mathematics, for it gives a measure of how far our understanding of the subject has advanced in the last 100 years. To paraphrase Newton, Perelman has seen far, but to do so he stood on the shoulders of giants who came before him. One giant, in particular, stands out. He is Richard Hamilton. Over a period of 25 years, Hamilton painstakingly built the solid and elaborate foundation upon which Perelman constructed the edifice of his proof. Without Hamilton’s work, Perelman’s would not have been possible. One of the most interesting aspects of the resolution of the Poincare Conjecture is the nature of the solution. While the problem is purely topological in its formulation, the proof is not. The proof uses deep techniques and results from other areas of mathematics, namely analysis and differential geometry. It is not at all clear a priori that these ideas have any relevance to the Poincare Conjecture, but in the end they turn out to be the only way (so far) to approach this question successfully. My goal in this article is to give you a sense of the importance, centrality, and the depth of the Poincare Conjecture. Then I will discuss surfaces and 3-dimensional spaces and describe how topologists think about them. Next, I will formulate and
L1 Adaptive Controller for Attitude Control of Multirotors
In this work we show an application of L1 Adaptive control theory for attitude control of UAVs. We implement the flight control system on a multirotor to show robustness and precise attitude tracking in the presence of modeling uncertainties and environmental disturbances. We choose backstepping control architecture, since the kinematics and dynamics of multirotors in most cases can be written in strict feedback form. We further exploit the fact that the kinematics of the plant, while free of uncertainties, is nonlinear, which makes it highly suitable for dynamic inversion control at each level of backstepping. On the other hand, plant dynamics is uncertain and is affected by environmental disturbances such as wind gusts, unmodeled dynamics etc. Therefore, we consider 3 variants of the control architecture. The first method uses backstepping to determine the moment demand and augments it with the L1 adaptive controller to account for uncertainties and provide robustness with guaranteed transient performance. The second architecture apply the concept of L1 adaptive backstepping to the same problem; and the third architecture uses L1 backstepping for quaternion representation of the system dynamics, which helps to avoid the singularities associated with Euler angles.