FI-modules and stability for representations of symmetric groups

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.Comment: 54 pages. v4: new title, paper completely reorganized; final version, to appear in Duke Math Journa

Phase behaviour of block copolymer melts with arbitrary architecture

The Leibler theory [L. Leibler, Macromolecules, v.13, 1602 (1980)] for microphase separation in AB block copolymer melts is generalized for systems with arbitrary topology of molecules. A diagrammatic technique for calculation of the monomeric correlation functions is developed. The free energies of various mesophases are calculated within the second-harmonic approximation. Model highly-branched tree-like structures are considered as an example and their phase diagrams are obtained. The topology of molecules is found to influence the spinodal temperature and asymmetry of the phase diagrams, but not the types of phases and their order. We suggest that all model AB block-copolymer systems will exhibit the typical phase behaviour.Comment: Submitted to J. Chem. Phys., see also http://rugmd4.chem.rug.nl/~morozov/research.htm

XML-based genetic rules for scene boundary detection in a parallel processing environment

Genetic programming is based on Darwinian evolutionary theory that suggests that the best solution for a problem can be evolved by methods of natural selection of the fittest organisms in a population. These principles are translated into genetic programming by populating the solution space with an initial number of computer programs that can possibly solve the problem and then evolving the programs by means of mutation, reproduction and crossover until a candidate solution can be found that is close to or is the optimal solution for the problem. The computer programs are not fully formed source code but rather a derivative that is represented as a parse tree. The initial solutions are randomly generated and set to a certain population size that the system can compute efficiently. Research has shown that better solutions can be obtained if 1) the population size is increased and 2) if multiple runs are performed of each experiment. If multiple runs are initiated on many machines the probability of finding an optimal solution are increased exponentially and computed more efficiently. With the proliferation of the web and high speed bandwidth connections genetic programming can take advantage of grid computing to both increase population size and increasing the number of runs by utilising machines connected to the web. Using XML-Schema as a global referencing mechanism for defining the parameters and syntax of the evolvable computer programs all machines can synchronise ad-hoc to the ever changing environment of the solution space. Another advantage of using XML is that rules are constructed that can be transformed by XSLT or DOM tree viewers so they can be understood by the GP programmer. This allows the programmer to experiment by manipulating rules to increase the fitness of a rule and evaluate the selection of parameters used to define a solution

Translational groups as generators of gauge transformations

We examine the gauge generating nature of the translational subgroup of Wigner's little group for the case of massless tensor gauge theories and show that the gauge transformations generated by the translational group is only a subset of the complete set of gauge transformations. We also show that, just like the case of topologically massive gauge theories, translational groups act as generators of gauge transformations in gauge theories obtained by extending massive gauge noninvariant theories by a Stuckelberg mechanism. The representations of the translational groups that generate gauge transformations in such Stuckelberg extended theories can be obtained by the method of dimensional descent. We illustrate these with the examples of Stuckelberg extended first class versions of Proca, Einstein-Pauli-Fierz and massive Kalb-Ramond theories in 3+1 dimensions. A detailed analysis of the partial gauge generation in massive and massless 2nd rank symmetric gauge theories is provided. The gauge transformations generated by translational group in 2-form gauge theories are shown to explicitly manifest the reducibility of gauge transformations in these theories.Comment: Latex, 20 pages, no figures, Version to appear in Physical Review