Generalized nonlinear master equations for local fluctuations: Dimensionality expansions and augmented mean field theory

By application of a projection operator technique we derive a formally exact generalization of the nonlinear mean field master equation introduced recently for the study of local fluctuations in a reacting medium. Our starting point is a phenomenological cell master equation. The results of our theory are applicable to the theory of a fluctuating hydrodynamic reacting system. The mean field equation is placed on a firm theoretical foundation by showing it to be the lowest order approximation in an expansion in the dimensionality of the physical space keeping the product of the number of nearest neighbors (an increasing function of dimensionality) and the typical diffusion coefficient constant. A more accurate nonlinear master equation that allows for the correlation and fluctuations in the environment of a given volume element is derived in the form of an augmented mean field equation.Work supported in part by a grant from the National Science Foundation.     

Perturbations of subalgebras of type II1 factors

In this paper we consider two von Neumann subalgebras and of a type II₁ factor . For a map φ on , we define

     

When Is a Sum of Partial Reflections Equal to a Scalar Operator?

We describe the set of values of the parameter for which there exists a Hilbert space H and n partial reflections A ₁,...,A _n (self-adjoint operators such that A _k ³ =A_k or, which is the same, self-adjoint operators whose spectra belong to the set {-1,0,1}) whose sum is equal to the scalar operator I _H .     

On the existence of solutions of variational inequalities in Banach spaces

In this paper, we study the existence of the solution of the variational inequality 〈Tx−ξ,y−x〉?0 by applying the generalized projection operator , where B is a Banach space with dual space B∗ and by using the well-known FanKKM Theorem.     

Derivatives of the Spectral Function and Sobolev Norms of Eigenfunctions on a Closed Riemannian Manifold

Let e(x, y, ) be the spectral function and the unit spectral projection operator, with respect to the Laplace–Beltrami operator on a closed Riemannian manifold M. We generalize the one-term asymptotic expansion of e(x, x, ) by Hörmander (Acta Math. 88 (1968), 341–370) to that of _{x y} e(x,y,)| _x=y for any multiindices , in a sufficiently small geodesic normal coordinate chart of M. Moreover, we extend the sharp (L ²,L ^p) (2 p) estimates of by Sogge (J. Funct. Anal. 77 (1988), 123–134; London Math. Soc. Lecture Note Ser. 137, Cambridge University Press, Cambridge, 1989; Vol. 1, pp. 416–422) to the sharp (L ², Sobolev L ^p) estimates of .     

On the error estimates for the rotational pressure-correction projection methods

In this paper we study the rotational form of the pressure-correction method that was proposed by Timmermans, Minev, and Van De Vosse. We show that the rotational form of the algorithm provides better accuracy in terms of the -norm of the velocity and of the -norm of the pressure than the standard form.

     

A ring of instantons inducing a monopole loop

We consider the superposition of infinitely many instantons on a circle in . The construction yields a self-dual solution of the Yang-Mills equations with action density concentrated on the ring. We show that this configuration is reducible in which case magnetic charge can be defined in a gauge invariant way. Indeed, we find a unit charge monopole (worldline) on the ring. This is an analytic example of the correlation between monopoles and action/topological density, however with infinite action. We show that both the Maximal Abelian Gauge and the Laplacian Abelian Gauge detect the monopole, while the Polyakov gauge does not. We discuss the implications of this configuration.     

Generalized Sasaki Projections and Riesz Ideals in Pseudoeffect Algebras

A noncommutative version of generalized Sasaki projections in pseudoeffect algebras is introduced. It is proved that an ideal in a pseudoeffect algebra is Riesz if and only if it is closed under the right and left Sasaki projections. In lattice ordered pseudoeffect algebras, it is shown that generalized Sasaki projections are one-element sets, and their explicit form is found. It is shown that if a supremum of a normal Riesz ideal in a lattice ordered pseudoeffect algebra exists, it is a central element. These results extend those obtained recently by Avallone and Vitolo for effect algebras.     

Boosted leave-many-out cross-validation: the effect of training and test set diversity on PLS statistics

It is becoming increasingly common in quantitative structure/activity relationship (QSAR) analyses to use external test sets to evaluate the likely stability and predictivity of the models obtained. In some cases, such as those involving variable selection, an internal test set – i.e., a cross-validation set – is also used. Care is sometimes taken to ensure that the subsets used exhibit response and/or property distributions similar to those of the data set as a whole, but more often the individual observations are simply assigned `at random.' In the special case of MLR without variable selection, it can be analytically demonstrated that this strategy is inferior to others. Most particularly, D-optimal design performs better if the form of the regression equation is known and the variables involved are well behaved. This report introduces an alternative, non-parametric approach termed `boosted leave-many-out' (boosted LMO) cross-validation. In this method, relatively small training sets are chosen by applying optimizable k-dissimilarity selection (OptiSim) using a small subsample size (k = 4, in this case), with the unselected observations being reserved as a test set for the corresponding reduced model. Predictive errors for the full model are then estimated by aggregating results over several such analyses. The countervailing effects of training and test set size, diversity, and representativeness on PLS model statistics are described for CoMFA analysis of a large data set of COX2 inhibitors.     

Galerkin Approximations for the Linear Parabolic Equation with the Third Boundary Condition

We solve a linear parabolic equation in ^d , d 1, with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the -method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.