Partial regularity of weak solutions of the liquid crystal equilibrium system

For a parameter, we consider the modified relaxed energy of the liquid crystal system. Each minimizer of the modified relaxed energy is a weak solution to the liquid crystal equilibrium system. We prove the partial regularity of minimizers of the modified relaxed energy. We also prove the existence of infinitely many weak solutions for the special boundary value x.

Partial regularity of minimizers of a functional involving forms and maps

We discuss the partial regularity of minimizers of energy functionals such as (1)/(p)integral(Omega)[sigma(u)dA(p) + (1)/(2)delu(2p)]dx, where u is a map from a domain Omega is an element of R-n into the m-dimensional unit sphere of Rm+1 and A is a differential one-form in Omega.

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For n >= 5 and k >= 4, we show that any minimizing biharmonic map from Omega subset of R-n to S-k is smooth off a closed set whose Hausdorff dimension is at most n - 5. When n = 5 and k = 4, for a parameter lambda is an element of [0, 1] we introduce lambda-relaxed energy H-lambda of the Hessian energy for maps in W-2,W-2 (Omega; S-4) so that each minimizer u(lambda) of H-lambda is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of H-lambda for lambda is an element of [0, 1).

The collision branching process

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

Some new examples for nonuniqueness of the evolution problem of harmonic maps

We find some new examples to show nonuniquence for the heat flow of harmonic maps where weak solutions satisfy the same monotonicity property.

On the Jager-Kaul theorem concerning harmonic maps

In 1983, Jager and Kaul proved that the equator map u*(x) = (x/\x\,0) : B-n --> S-n is unstable for 3 less than or equal to n less than or equal to 6 and a minimizer for the energy functional E(u, B-n) = integral B-n \del u\(2) dx in the class H-1,H-2(B-n, S-n) with u = u* on partial derivative B-n when n greater than or equal to 7. In this paper, we give a new and elementary proof of this Jager-Kaul result. We also generalize the Jager-Kaul result to the case of p-harmonic maps.

The advantages of theft over toil: The design inference and arguing from ignorance

Intelligent design theorist William Dembski has proposed an explanatory filter for distinguishing between events due to chance, lawful regularity or design. We show that if Dembski's filter were adopted as a scientific heuristic, some classical developments in science would not be rational, and that Dembski's assertion that the filter reliably identifies rarefied design requires ignoring the state of background knowledge. If background information changes even slightly, the filter's conclusion will vary wildly. Dembski fails to overcome Hume's objections to arguments from design.

Progressive dysgraphia in a case of posterior cortical atrophy

Dysgraphia (agraphia) is a common feature of posterior cortical atrophy (PCA). However, detailed analyses of these spelling and writing impairments are infrequently conducted. LM is a 59-year-old woman with dysgraphia associated with PCA. She presented with a two-year history of decline in her writing and dressmaking skills. A 3D T-1-weighted MRI scan confirmed selective bi-parietal atrophy, with relative sparing of the hippocampi and other cortical regions. Analyses of LM's preserved and impaired spelling abilities indicated mild physical letter distortions and a significant spelling deficit characterised by letter substitutions, insertions, omissions, and transpositions that was systematically sensitive to word length while insensitive to real word versus nonword category, word frequency, regularity, imagery, grammatical class and ambiguity. Our findings suggest a primary graphemic buffer disorder underlies LM's spelling errors, possibly originating from disruption to the operation of a fronto-parietal network implicated in verbal working memory.

Petrodentine in derived dipnoan tooth plates

Most lungfish tooth plates, that are arranged in radiating ridges derived from the fusion of separate cusps in young juveniles, are based on a framework of enamel, mantle dentine and bone that encloses a mass of specialized dentines forming the occlusal surface. In most taxa, the specialized dentines are interdenteonal and circumdenteonal dentine, but a few derived genera have petrodentine as well. Petrodentine, as originally defined, describes a specific form of hypermineralized dentine in adult tooth plates of the Recent African lungfish Protopterus. The ontogeny of fossil and Recent lungfish tooth plates demonstrates that petrodentine is derived by continuous enhancement of the hard tissue of the primary core of the initially isolated cusps of the tooth plate, and that interdenteonal dentine with denteons of circumdenteonal dentine is a secondary development in the tooth plate around and below the first formed cusps of the ridges. In dipnoans that lack petrodentine in adults the primary core of the cusps is not enhanced, but is removed by wear. The hard tissues of the dipnoan tooth plate provide useful characters for defining dipnoan taxa, as do the differing arrangements of the tissues in each species. Details of the arrangement of the enclosed specialized dentines are surprisingly variable among genera, and are significant for the structure and function of the tooth plate. Little regularity of structure is discernible in the histology of tooth plates of early dipnoans, but derived genera have more predictable structure. Consistent with other uniquely dipnoan characters, like the composition of the dermal skull, an evolutionary progression is evident within the group in the fine structure of the dentition, and, as with the bones of the dermal skull, little similarity is demonstrable between the dentines of dipnoans and tetrapods.

An electron microscopic study of nickel sulfide inclusions in toughened glass

It has been known since the early sixties that nickel sulfide inclusions cause spontaneous fracture of toughened (thermally tempered) glass, but despite the considerable amount of work done on this problem in the last four decades, failures still occur in the field with regularity. In this study we have classified (by viewing through a 60x optical microscope) inclusions into two groups, which are classic and atypical nickel sulfides. The classics look like the nickel sulfide inclusions found at the initiation-of-fracture of windows that have broken spontaneously. We have compared the structure and composition of the atypical inclusions with the structure and composition of the classics. All of the classic and atypical nickel sulfide inclusions studied in this work were found to have a composition in the range of Ni52S48 to Ni48S52. Inclusions on the nickel rich side of stoichiometric NiS were found to be two-phase assemblies, and inclusions on the sulphur rich side of NiS were single phase. It had been proposed that the atypicals were passive, and of a different composition to the classics. However, we found that the difference between passive and dangerous nickel sulfide inclusions was not a difference in composition but rather a difference in the type of material in the internal pore space. The passive's had carbon char in their internal pore space, whereas the pore space of dangerous inclusions contained Na2O. The presence of Na2O and carbon char with the inclusions indicates that the formation of the inclusions results from a reaction of a nickel-rich phase with sodium sulphate and carbon. (C) 2001 Kluwer Academic Publishers.

Estimating economic relationships subject to firm- and time-varying equality and inequality constraints

Applied econometricians often fail to impose economic regularity constraints in the exact form economic theory prescribes. We show how the Singular Value Decomposition (SVD) Theorem and Markov Chain Monte Carlo (MCMC) methods can be used to rigorously impose time- and firm-varying equality and inequality constraints. To illustrate the technique we estimate a system of translog input demand functions subject to all the constraints implied by economic theory, including observation-varying symmetry and concavity constraints. Results are presented in the form of characteristics of the estimated posterior distributions of functions of the parameters. Copyright (C) 2001 John Wiley Sons, Ltd.