We develop techniques to compute higher loop string amplitudes for twisted N = 2 theories with ĉ = 3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a

Stochastic quantization is used to derive exact equations, connecting multilocal field correlators in the ϕ 3 theory and gluodynamics. Perturbative expansion of the obtained equations in the lowest orders is presented. 1

This paper defines the beta function and other linear orbit pa-rameters using the exact equations of motion. The fl, fi and ˆ functions are redefined using the exact equations. Expressions are found for the transfer matrix and the emittance. The dif-ferential equations for · D x=fl1=2 is found. New

This paper defines the beta function and other linear orbit parameters using the exact equations of motion. The #, # and # functions are redefined using the exact equations. Expressions are found for the transfer matrix and the emittance. The differential equations for # = x/# 1/2 is found

simulation methods to simulate trajectories of discrete, stochastic systems, (methods that are rigorously equivalent to the Master Equation approach) but these do not scale well to systems with many reaction pathways. This paper presents the Next Reaction Method, an exact algorithm to simulate coupled

Several numerical schemes for the solution of hyperbolic conservation laws are based on exploiting the information obtained by considering a sequence of Riemann problems. It is argued that in existing schemes much of this information is degraded, and that only certain features of the exact solution

or differential equations, which are verified through computer simulations. The analysis provides convenient approximate or exact solutions as well as useful convergence time and growth ratio estimates. The paper recommends practical application of the analyses and suggests a number of paths for more detailed

about Y through a ‘bottleneck ’ formed by a limited set of codewords ˜X. This constrained optimization problem can be seen as a generalization of rate distortion theory in which the distortion measure d(x, ˜x) emerges from the joint statistics of X and Y. This approach yields an exact set of self

to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the ℓ1-minimization problem (‖x‖ℓ1:= i |xi|) min g∈R n ‖y − Ag‖ℓ1 provided that the support of the vector of errors is not too large, ‖e‖ℓ0: = |{i: ei ̸= 0} | ≤ ρ · m

to be related to appropriate sets of salient behavioral, normative, and control beliefs about the behavior, but the exact nature of these relations is still uncertain. Expectancy — value formulations are found to be only partly successful in dealing with these relations. Optimal rescaling of expectancy