A cluster expansion approach to renormalization group transformations

The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work treats the rigorous definition of the RG map for classical Ising-type lattice systems in the infinite volume limit at high temperature. A cluster expansion is used to justify the existence of the partial derivatives of the renormalized interaction with respect to the original interaction. This expansion is derived from the formal expressions, but it is itself well-defined and convergent. Suppose in addition that the original interaction is finite-range and translation-invariant. We will show that the matrix of partial derivatives in this case displays an approximate band property. This in turn gives an upper bound for the RG linearization.Comment: 13 page

Counting Labelled Trees with Given Indegree Sequence

For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on $[n]$ with indegree sequence corresponding to a partition $\lambda$. In this paper we give two proofs of Cotterill's conjecture: one is `semi-combinatorial" based on induction, the other is a bijective proof.Comment: 10 page

Computer aided manual tracking

A scheme was developed to assist the human operator by augmenting an optic sight manual tracking loop with target rate estimates from a computer control algorithm which can either be a Kalman Filter or an alpha, beta, gamma filter. The idea is for the computer to provide rate tracking while the human operator is responsible for nullifying the tracking error. A simple schematic is shown to illustrate the implementation of this concept. A hybrid real-time man-in-loop simulation was used to compare the tracking performance of the same flight trajectory with or without this form of computer-aided track. Preliminary results show the advantage of computer-aided track against high speed aircraft at close range. However, good tracking before target state estimator maturity becomes more critical for aided track than without. Results are presented for a constant velocity flight trajectory

Functional Renormalization for Chiral and U_A(1) Symmetries at Finite Temperature

We investigated the chiral symmetry and U_A(1) anomaly at finite temperature by applying the functional renormalization group to the SU(3) linear sigma model. Expanding the local potential around the classical fields, we derived the flow equations for the renormalization parameters. In chiral limit, the flow equation for the chiral condensate is decoupled from the others and can be analytically solved. The Goldstone theorem is guaranteed in vacuum and at finite temperature, and the two phase transitions for the chiral and U_A(1) symmetry restoration happen at the same critical temperature. In general case with explicit chiral symmetry breaking, the two symmetries are partially and slowly restored, and the scalar and pseudoscalar meson masses are controlled by the restoration in the limit of high temperature.Comment: 9 pages, 9figure

A Matlab Implementation of a Flat Norm Motivated Polygonal Edge Matching Method using a Decomposition of Boundary into Four 1-Dimensional Currents

We describe and provide code and examples for a polygonal edge matching method.Comment: Contains Matlab code and 4 figure

Origin of Low Thermal Conductivity in Nuclear Fuels

Using a novel many-body approach, we report lattice dynamical properties of UO2 and PuO2 and uncover various contributions to their thermal conductivities. Via calculated Grueneisen constants, we show that only longitudinal acoustic modes having large phonon group velocities are efficient heat carriers. Despite the fact that some optical modes also show their velocities which are extremely large, they do not participate in the heat transfer due to their unusual anharmonicity. Ways to improve thermal conductivity in these materials are discussed.Comment: 4 pages, 3 figures, 1 tabl