The evolution of multiple scalar fields in cosmology has been much studied, particularly when the potential is formed from a series of exponentials. For a certain subclass of such systems it is possible to get “assisted“ behaviour, where the presence of multiple terms in the potential effectively makes it shallower than the individual terms indicate. It is also known that when compactifying on ...

Reward-based learning in neural systems is challenging because a large number of parameters that affect network function must be optimized solely on the basis of a reward signal that indicates improved performance. Searching the parameter space for an optimal solution is particularly difficult if the network is large. We show that Hebbian forms of synaptic plasticity applied to synapses between...

In a 5-dimensional spacetime (M, gab) with a Killing vector field ξ a which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of ξ gives a 4-dimensional space S. The reduction of (M, gab) is studied in the geometric language, which is a generalization of Geroch’s method for the reduction of 4-dimensional spacetime. A 4dimensional gravity coupled to a vect...

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We describe the reduction from four to two dimensions of the SU(2) Donaldson-Witten theory and the dual twisted Seiberg-Witten theory, i.e. the Abelian topological field theory corresponding to the Seiberg–Witten monopole equations. NBI-HE-96-14 hep-th/9603023 ∗E-mail: [email protected]

We discuss dimensional reduction for Hamiltonian systems which possess nonconstant Poisson brackets between pairs of coordinates and between pairs of momenta. The associated Jacobi identities imply that the dimensionally reduced brackets are always constant. Some examples are given alongside the general theory.

We present a method of dimensional reduction for the general Markov model of sequence evolution on a phylogenetic tree. We show that taking certain linear combinations of the associated random variables (site pattern counts) reduces the dimensionality of the model from exponential in the number of extant taxa, to quadratic in the number of taxa, while retaining the ability to statistically iden...

Dimensional reduction occurs when the critical behavior of one system can be related to that of another system in a lower dimension. We show that this occurs for directed branched polymers (DBP) by giving an exact relationship between DBP models in D+1 dimensions and repulsive gases at negative activity in D dimensions. This implies relations between exponents of the two models: γ(D + 1) = α(D)...

This is a pedagogical account of the some of the recent results of Brydges and Imbrie, described from the point of view of Grassmann integration. Some simple extensions are pointed out. Historical Introduction The subject of dimensional reduction, in the context considered in this lecture, has a long and rather peculiar history. The strangest aspect of all is that the simplest formulation of th...