S-convergence in the theory of homogenization of problems of optimal control

We give a definition of homogenized problems of optimal control in terms of variational S-limits and establish sufficient conditions for a family of problems of optimal control described by nonlinear operator relations to be compact with respect to the operation of homogenization.     

Progress and bottlenecks in simulating lattice gauge theory with fermions

We present a progress report in lattice gauge theory computer simulations which includes the effects of light, dynamical fermions. Microcanonical and hybrid microcanonical-Langevin alogrithms are presented and discussed. A method for “accelerating” stochastic differential equations and defeating critical slowing down is reviewed. Physics applications such as the thermodynamics of quantum chromodynamics, hierarchal energy scales in unified gauge theories, and the phase diagram of theories with many fermion species are discussed. Prospects for future research are assessed.     

Studies of chiral symmetry breaking in SU(2) lattice gauge theory

We study chiral symmetry breaking (_χSB) in SU(2) lattice gauge theory with quarks in the $\text{l =}\text{1}\text{2}\text{, l = 1, l =}\text{3}\text{2}$, and l = 2 representations of the color group. We perform Monte Carlo evaluations of $\text{〈}\text{ψ}\text{ψ〉}$ in the quenched approximation and extract the relevant length scales for _χSB. We revise a previous estimate for the ratio between the chiral symmetry restoration temperatures for fundamental and adjoint quarks and obtain $\text{T}{}_{\mathrm{l\; =\; 1}}\text{/T}{}_{\mathrm{l}}\text{=}\text{1}\text{2}\text{～ 8}$. Our results for the higher representations, $\text{l =}\text{3}\text{2}\text{and}\text{l = 2}$, are consistent with Casimir scaling and give C₂g_mom² ～ 4. Many aspects of our calculational method are explained in detail. The issues discussed include the relation between _χSB in the quenched approximation and the spectrum of the Dirac operator, the flavor symmetries of euclidean staggered fermions, estimates of finite-size effects and the reliability of m → 0 extrapolations on finite lattices.     

Comparison of recent massive muon pair data with the asymptotically free parton model

We present calculations of vW₂ and of massive muon pair production cross sections in the kinematic ranges of recent experiments. These calculations test the asymptotically free parton model and excellent agreement with the data is found. Estimates of the transverse momentum 〈q_⊥〉_μμ expected in massive dimuon experiments are made. For s = 750 GeV² and M_μμ ≈ 10 GeV, we predict 〈q_⊥〉_μμ ≈ 1.4 GeV/c.     

The renormalization group and the ϵ expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ? = 4?d is explained [d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.     

On weak compactness of bounded sets in Banach and locally convex spaces

We investigate the compactness of one class of bounded subsets in Banach and locally convex spaces. We obtain a generalization of the Banach-Alaoglu theorem to a class of subsets that are not polars of convex balanced neighborhoods of zero. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp 731–739, June. 2000.