Variational conditions with smooth constraints: structure and analysis

 This is an expository paper about the analysis of variational conditions over sets defined in finite-dimensional spaces by fairly smooth functions satisfying a constraint qualification. The primary focus is on results that can provide quantitative and computable sensitivity information for particular instances of the problems under study, and our objective is to give a personal view of the state of current knowledge in this area and of gaps in that knowledge that require future work. The writing style is informal, in keeping with the objective of focusing the reader's attention on the basic concepts and the relationships between them, rather than on details of the particular results themselves. Received: December 1, 2002 / Accepted: April 25, 2003 Published online: May 28, 2003 Key words. variational condition – variational inequality – complementarity – sensitivity – stability – nondegeneracy Mathematics Subject Classification (2000): Primary: 90C31. Secondary: 47J20, 49J40, 49J53, 90C33     

Distribution results for lattices in SL(2,ℚ p )

In this paper we study the ergodic properties of the linear action of lattices Γ of SL(2,ℚ_p) on ℚ_p × ℚ_p and distribution results for orbits of Γ. Following Serre, one can define a “geodesic flow” for an associated tree (actually associated to GL(2,ℚ_p)). The approach we use is based on an extension of this approach to “frame flows” which are a natural compact group extension of the geodesic flow.     

Free Groups from Fields

In the nineteenth century, field theory brilliantly resolveda number of questions that had taxed mathematicians for centuries;for example, ‘The circle cannot be squared’ by straightedge and compass, and solving polynomial equations by radicalsis not always possible. These successes have continued to beheld up as superb examples of the power of mathematical thought,and are demonstrated at an undergraduate level. The purposeof this article is to provide another such natural example whichleads to a concrete realisation of the free group on 2 generators.     

Stereospecific solid-state NaBH4-promoted reduction of 1-methylpentacyclo[5.4.0.02,6.03,10.05,9]undecane-8,11-dione

NaBH₄ reduction of a cage dione proceeds in a stereospecific fashion to give theendo,endo-diol. This reactivity is related to the crystal structure.     

The Livsic Cocycle Equation for Compact Lie Group Extensions of Hyperbolic Systems

In this article we extend well-known results of Livsic on theregularity of measurable solutions to the cocycle equation.Livsic proved a regularity result for real valued cocycles,but for compact Lie group valued cocycles his result was restrictedto coboundaries (that is, trivial or constant cocycles). Inthis article we prove the complete result. We present several useful applications of these results.     

Time-resolved synchrotron X-ray study of crystalline phase transition in poly(aryl ether ketone ketone) containing alternated terephthalic/isophthalic moieties

Unique crystallization and melting behavior in poly(aryl ether ketone ketone) containing alternated terephthalic and isophthalic moieties were studied by time-resolved synchrotron x-ray methods. Recently, this material has been shown to exhibit three polymorphs (forms I, II, and III). In this work, we further investigated their distinctive thermal properties and found that form I is the dominating and the most thermally stable phase while form II is favored by fast nucleation conditions and is the least stable phase. On the other hand, form III represents a minor intermediate phase that usually coexists with form I and can be transferred from form II and to form I. Structural and morphological changes in form I have been followed by simultaneous wide-angle x-ray diffraction (WAXD)/small-angle x-ray scattering (SAXS) measurements during cold- or melt-crystallization and subsequent melting. In all cases, a larger dimensional change was found in the crystallographic a-axis than the b-axis during heating and cooling. This may be due to the greater lateral stress variation with respect to temperature along the a direction of the primary lamellae which is induced by either the formation of secondary lamellae or the preferential chain-folding direction in poly(aryl ether ketone ketone)s. During the phase transitions of form II ← III in the cold-crystallized specimen and form III ← I in the melt-crystallized samples, lamellar variables (long period, lamellar thickness, and invariant) obtained from SAXS remain almost constant. This indicates that the density distribution in the long spacing is independent of the melting in form II or III. For melt-crystallization, the corresponding changes in unit-cell dimensions and lamellar morphology during the annealing-induced low endotherm are most consistent with the argument that these changes are due to the melting of thin lamellar population. © 1995 John Wiley & Sons, Inc.     

A Pilot Survey of Vocal Health in Young Singers

The objective of this study was to determine the incidence of vocal problems in young choir singers and to correlate vocal problems with demographic and behavioral information. A questionnaire addressing vocal habits and hygiene was offered to 571 young choir singers, up to 25 years of age, who sing at least weekly; 129 (22.6%) responded. More than one-half of the respondents had experienced vocal difficulty, particularly older adolescents. Detrimental behaviors and circumstances surveyed were not reflective of the incidence of vocal difficulty, except for morning hoarseness, chronic fatigue, insomnia, and female gender after puberty. Voice care professionals should be aware that self-reported voice difficulties are common among young choral singers, especially postpubescent girls, and children with symptoms consistent with reflux (morning hoarseness) and emotional stress (insomnia). Laryngologists should communicate with choral conductors and singing teachers to enhance early identification and treatment of children with voice complaints, and to develop choral educational strategies that help decrease their incidence.     

Outflow refreshment angiography: A bright blood, bright static tissue technique

A new “bright blood” strategy, outflow refreshment imaging, is introduced in which a number of overlapping slices are excited in rapid succession. Flowing spins that refresh each overlapped slice portion contribute a bright signal. Additionally, static tissue in each non-overlapped slice portion also yields a bright signal. However, the flow/static contrast is comparable to that produced in inflow refreshment images, and angiograms can be generated by conventional maximum intensity projection processing. The dual ability to visualize angiograms and static tissue images is a major benefit of the strategy. Computer simulations of flow sensitivities and in vivo results are presented which compare the outflow and inflow refreshment imaging strategies.