Celulární automat a CML systémy

The main aim of this thesis is the study of cellular automata and discrete dynamical systems on a lattice. Both tools, cellular automata as well as dynamical systems on a lattice are introduced and elementary properties described. The relation between cellular automata and dynamical system on lattice is derived. The main goal of the thesis is also the use of the cellular automata as that mathematical tool of evolution visualization of discrete dynamical systems. The theory of cellular automata is applied to the discrete dynamical systems on a lattice Laplacian type and implemented in Java language.Hlavním cílem práce je studium vztahu celulárních automatů a diskrétních dynamických systémů na mřížce. Oba nástroje, jak celulární automat tak dynamický systém na mřížce, jsou zavedeny a jejich základní vlastnosti popsány. Vztah mezi celulárními automaty a dynamickými systémy na mřížce je podrobně popsán. Hlavním cílem práce je dále použití nástroje celulárního automatu jako matematického vizualizačního prostředku evoluce diskrétních dynamických systémů. Teorie celulárních automatů je použita na dynamické systémy na mřížce Lamplaceova typu a implementována v prostředí Java.470 - Katedra aplikované matematikyvelmi dobř

Cellular Automata

Modelling and simulation are disciplines of major importance for science and engineering. There is no science without models, and simulation has nowadays become a very useful tool, sometimes unavoidable, for development of both science and engineering. The main attractive feature of cellular automata is that, in spite of their conceptual simplicity which allows an easiness of implementation for computer simulation, as a detailed and complete mathematical analysis in principle, they are able to exhibit a wide variety of amazingly complex behaviour. This feature of cellular automata has attracted the researchers' attention from a wide variety of divergent fields of the exact disciplines of science and engineering, but also of the social sciences, and sometimes beyond. The collective complex behaviour of numerous systems, which emerge from the interaction of a multitude of simple individuals, is being conveniently modelled and simulated with cellular automata for very different purposes. In this book, a number of innovative applications of cellular automata models in the fields of Quantum Computing, Materials Science, Cryptography and Coding, and Robotics and Image Processing are presented

Studying many-body physics through quantum coding theory

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 133-140).The emerging closeness between correlated spin systems and error-correcting codes enables us to use coding theoretical techniques to study physical properties of many-body spin systems. This thesis illustrates the use of classical and quantum coding theory in classifying quantum phases arising in many-body spin systems via a systematic study of stabilizer Hamiltonians with translation symmetries. In the first part, we ask what kinds of quantum phases may arise in gapped spin systems on a D-dimensional lattice. We address this condensed matter theoretical question by giving a complete classification of quantum phases arising in stabilizer Hamiltonians at fixed points of RG transformations for D = 1; 2; 3. We found a certain dimensional duality on geometric shapes of logical operators where m-dimensional and (D m)-dimensional logical operators always form anti-commuting pairs (m is an integer). We demonstrate that quantum phases are completely classified by topological characterizations of logical operators where topological quantum phase transitions are driven by non-analytical changes of geometric shapes of logical operators. As a consequence, we argue that topological order is unstable at any nonzero temperature and self-correcting quantum memory in a strict sense may not exist where the memory time is upper bounded by some constant at a fixed temperature, regardless of the system size. Our result also implies that topological field theory is the universal theory for stabilizer Hamiltonians with continuous scale symmetries. In the second part, we ask the fundamental limit on information storage capacity of discrete spin systems. There is a well-known theoretical limit on the amount of information that can be reliably stored in a given volume of discrete spin systems. Yet, previously known systems were far below this theoretical limit. We propose a construction of classical stabilizer Hamiltonians which asymptotically saturate this limit. Our model borrows an idea from fractal geometries arising in the Sierpinski triangle, and is a rare manifestation of limit cycle behaviors with discrete scale symmetries in real-space RG transformations, which may be beyond descriptions of topological field theory.by Beni Yoshida.Ph.D