Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions

Functional central limit theorems for triangular arrays of rowwise independent stochastic processes are established by a method replacing tail probabilities by expectations throughout. The main tool is a maximal inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by an additional inequality due to M. Ledoux and M. Talagrand. The entropy condition emerging in our theorems was introduced by K. S. Alexander, whose functional central limit theorem for so-calledmeasure-like processeswill be also regained. Applications concern, in particular, so-calledrandom measure processeswhich include function-indexed empirical processes and partial-sum processes (with random or fixed locations). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examples include nonparametric regression and intensity estimation for spatial Poisson processes.     

A Curtin–Hammett pentamethylene chain symmetrization process in the Bergman cyclization of an 11‐membered ring enediyne

Pentamethylene chain conformational effects for the Bergman cyclization of the 11‐membered ring enediyne, (3Z)‐3‐cycloundecene‐1,5‐diyne, 2, are examined theoretically with unrestricted Becke, three‐parameter, Lee–Yang–Parr/6‐31 G(d,p) calculations. A C₁ symmetric enediyne conformation was found to be the global minimum, where its nonsymmetric pentamethylene chain prevented π‐orbital alignment of the acetylene groups for C–C σ bond product formation. The Bergman cyclization of 2 was found to be conformationally dependent. In a Curtin–Hammett type process, the C₁ symmetric 2 inverts to one of the C_S or C₂ symmetric conformers required for the Bergman cyclization, which produced a C_S or C₂ symmetric 1,4‐diradical intermediate. The activation energy for the cyclization is slightly higher to reach the C₂ symmetry diradical compared with the C_S symmetry diradical. Copyright © 2012 John Wiley & Sons, Ltd.     

Symmetrization of matrix technique in a finite element context

A finite element Galerkin-based formulation of the mass conservation and momentum equations can require, if convective type terms are retained in the coefficient matrix, a non-symmetric solver. The resulting increase in core storage for efficient utilization of CPU time can be considerable. The current paper advocates a simple symmetrization of matrix technique, at element level which results in a considerable reduction in core requirement. The increase in CPU time required when solving linear systems of equations is considerable. However, for nonlinear systems the penalty can be negligible.     

Concentration of area in half-planes

For the standard class of normalized univalent functions analytic in the unit disk , we consider a problem on the minimal area of the image concentrated in any given half-plane. This question is related to a well-known problem posed by A. W. Goodman in 1949 that regards minimizing area covered by analytic univalent functions under certain geometric constraints. An interesting aspect of this problem is the unexpected behavior of the candidates for extremal functions constructed via geometric considerations.

     

Random polarizations

We derive conditions under which random sequences of polarizations (two-point symmetrizations) on S^d${\mathbb{S}}^d$, R^d${\mathbb{R}}^d$, or H^d${\mathbb{H}}^d$ converge almost surely to the symmetric decreasing rearrangement. The parameters for the polarizations are independent random variables whose distributions need not be uniform. The proof of convergence hinges on an estimate for the expected distance from the limit that yields a bound on the rate of convergence. In the special case of i.i.d. sequences, almost sure convergence holds even for polarizations chosen at random from suitable small sets. As corollaries, we find bounds on the rate of convergence of Steiner symmetrizations that require no convexity assumptions, and show that full rotational symmetry can be achieved by randomly alternating Steiner symmetrizations in a finite number of directions that satisfy an explicit non-degeneracy condition. We also present some negative results on the rate of convergence and give examples where convergence fails.     

The hydrogen bond under pressure

The hydrogen bonds are quite pervasive in several classes of materials. Its parameters are known to show systematic variations with hydrogen bond length, and pressure variable is thus a natural way for studying hydrogen bonded substances. In this article, we review the unifying features as obtained through several experimental and theoretical investigations. Amongst other things, it is examined whether the observed pressure-induced variations in parameters of hydrogen bonds are consistent with the co-relations known on different chemical substances at normal pressure. In particular, the controversies on variations of O–H and H- - -O pairs with pressure and symmetrization of hydrogen bond have been resolved. The effects of close packing promoted by pressure such as formation of muli-centered hydrogen bonds and steric repulsions and the way the hydrogen bonds counter these in different ways are also examined.     

Engel conditions and symmetric tensors

In a recent study of Engel Lie rings, Serena Cicalò and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of this fact. Our main result generalizes this in the language of tensor algebra, and describes a relatively small generating set for the module generated by all n-th tensor powers of elements of a finitely generated ?-module M, in terms of a generating set for M.     

Symmetrization of organomercury(II) chlorides using basic alumina

The symmetrization reaction of organomercury(II) chlorides (RHgCl) to R₂Hg and HgCl₂ in toluene solution under identical conditions using a basic alumina column has been studied­in order to compare the effect of the nature of the R groups on the extent of symmetrization. The efficiency of symmetrization depends markedly on the electron-withdrawing nature of­R, varying from 90–94% for R = trichlorovinyl or phenyl to 11% for R = 2,6-dimethylphenyl.© 1998 John Wiley & Sons, Ltd.