Multilinear Time Invariant System Theory

In biological and engineering systems, structure, function and dynamics are highly coupled. Such interactions can be naturally and compactly captured via tensor based state space dynamic representations. However, such representations are not amenable to the standard system and controls framework which requires the state to be in the form of a vector. In order to address this limitation, recently a new class of multiway dynamical systems has been introduced in which the states, inputs and outputs are tensors. We propose a new form of multilinear time invariant (MLTI) systems based on the Einstein product and even-order paired tensors. We extend classical linear time invariant (LTI) system notions including stability, reachability and observability for the new MLTI system representation by leveraging recent advances in tensor algebra.Comment: 8 pages, SIAM Conference on Control and its Applications 2019, accepted to appea

The Einstein-Vlasov system/Kinetic theory

The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on nonrelativistic and special relativistic physics, {\it i.e.} to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models ({\it i.e.} fluid models). This paper gives introductions to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental to good comprehension of kinetic theory in general relativity.Comment: 40 pages, updated version, to appear in Living Reviews in Relativit

Effective theory for wall-antiwall system

We propose a useful method for deriving the effective theory for a system where BPS and anti-BPS domain walls coexist. Our method respects an approximately preserved SUSY near each wall. Due to the finite width of the walls, SUSY breaking terms arise at tree-level, which are exponentially suppressed. A practical approximation using the BPS wall solutions is also discussed. We show that a tachyonic mode appears in the matter sector if the corresponding mode function has a broader profile than the wall width.Comment: LaTeX file, 30 page, 5 eps figures, references adde

Gauge theory of self-similar system

On the basis of a dilatation invariant Lagrangian, governed equations are determined for probability density and gauge potential of the non-stationary self-similar stochastic system. It is shown that an automodel regime is observed at small time interval determined by the Tsallis' parameter $q>1$. An exponential falling down happens at large time where the dilatation parameter and the partial scale tend to constant values.Comment: 8 pages, LaTe

Scattering theory for the Zakharov system

We study the theory of scattering for the Zakharov system in space dimension 3. We prove in particular the existence of wave operators for that system with no size restriction on the data in larger spaces and for more general asymptotic states than were previously considered, and we determine convergence rates in time of solutions in the range of the wave operators to the solutions of the underlying linear system. We also consider the same system in space dimension 2, where we prove the existence of wave operators in the special case of vanishing asymptotic data for the wave field.Comment: latex 29 page

IMMANUEL WALLERSTEIN'S WORLD SYSTEM THEORY

World-systems analysis is not a theory, but an approach to social analysis and social change developed, among others by the Immanuel Wallerstein. Professor Wallerstein writes in three domains of world-systems analysis: the historical development of the modern world-system; the contemporary crisis of the capitalist world-economy; the structures of knowledge. The American anlyst rejects the notion of a "Third World", claiming there is only one world connected by a complex network of economic exchange relationship. Our world system is characterized by mechanisms which bring about a redistribution of resources from the periphery to the core. His analytical approach has made a significant impact and established an institutional base devoted to the general approach.World system, core, semi-periphery, periphery, external regions