Solvability in the large for a class of complex vector fields on the cylinder

This work deals with global solvability of a class of complex vector fields of the form L = partial derivative/partial derivative t + (a(x, t)+ ib(x, t))partial derivative/partial derivative x, where a and b are real-valued C-infinity functions, defined on the cylinder Omega = R x S-1. Relatively compact (Sussmann) orbits are allowed. The connection with Malgrange's notion of L-convexity for supports is investigated. (C) 2011 Elsevier Masson SAS. All rights reserved.

Correction to “The Importance of Convexity in Learning With Squared Loss” [Sep 98 1974-1980]

The paper "the importance of convexity in learning with squared loss" gave a lower bound on the sample complexity of learning with quadratic loss using a nonconvex function class. The proof contains an error. We show that the lower bound is true under a stronger condition that holds for many cases of interest.

Loop transformations: convexity, pruning, and optimization

High-level loop transformations are a key instrument in mapping computational kernels to effectively exploit the resources in modern processor architectures. Nevertheless, selecting required compositions of loop transformations to achieve this remains a significantly challenging task; current compilers may be off by orders of magnitude in performance compared to hand-optimized programs. To address this fundamental challenge, we first present a convex characterization of all distinct, semantics-preserving, multidimensional affine transformations. We then bring together algebraic, algorithmic, and performance analysis results to design a tractable optimization algorithm over this highly expressive space. Our framework has been implemented and validated experimentally on a representative set of benchmarks running on state-of-the-art multi-core platforms.

Calcification and ossification of the convexity of the falx cerebri and related subdural space in human cadavers

OBJECTIVE: The present work was planned to report the incidence of calcification and ossification of an isolated cranial dural fold. The form, degree of severity and range of extension of such changes will be described. Involvement of the neighboring brain tissue and blood vessels, whether meningeal or cerebral, will also be determined. The results of this study might highlight the occasional incidence of intracranial calcification and ossification in images of the head and their interpretation, by radiologists and neurologists, to be of dural or vascular origin.

METHODS: Two human formalin-fixed cadavers, one middle-aged female another older male, were investigated at the Anatomy Laboratory, College of Medicine, King Faisal University, Dammam, Kingdom of Saudi Arabia during the period from 2000 to 2003. In each cadaver, the skullcap was removed and the convexity of the cranial dura mater, as well as the individual dural folds, were carefully examined for any calcification or ossification. The meningeal and cerebral blood vessels together with the underlying brain were grossly inspected for such structural changes. Calcified or ossified tissues, when identified, were subjected to histological examination to confirm their construction.

RESULTS: The female cadaver showed a calcified parietal emissary vein piercing the skullcap and projecting into the scalp. The latter looked paler and deficient in hair on its right side. The base of the stump was surrounded by a granular patch of calcification. The upper convex border of the falx cerebri was hardened and it presented granules, plaques and a cauliflower mass, which all proved to be osseous in structure. The meningeal and right cerebral vessels were mottled with calcium granules. The underlying temporal and parietal lobes of the right cerebral hemisphere were degenerated. The male cadaver also revealed a calcified upper border of the falx cerebri and superior sagittal sinus. Osseous granules and plaques, similar to those of the first specimen, were also identified but without gross changes in the underlying brain.

CONCLUSION: Calcification or ossification of an isolated site of the cranial dura mater and the intracranial blood vessels might occur. These changes should be kept in mind while interpreting images of the skull and brain. Clinical assessment and laboratory investigations are required to determine whether these changes are idiopathic, traumatic, or as a manifestation of a generalized disease such as hyperparathyroidism, vitamin D-intoxication, or chronic renal failure.

Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials

Let C-n(lambda)(x), n = 0, 1,..., lambda > -1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal. in (-1, 1) with respect to the weight function (1 - x(2))(lambda-1/2). Denote by X-nk(lambda), k = 1,....,n, the zeros of C-n(lambda)(x) enumerated in decreasing order. In this short note, we prove that, for any n is an element of N, the product (lambda + 1)(3/2)x(n1)(lambda) is a convex function of lambda if lambda greater than or equal to 0. The result is applied to obtain some inequalities for the largest zeros of C-n(lambda)(x). If X-nk(alpha), k = 1,...,n, are the zeros of Laguerre polynomial L-n(alpha)(x), also enumerated in decreasing order, we prove that x(n1)(lambda)/(alpha + 1) is a convex function of alpha for alpha > - 1. (C) 2002 Published by Elsevier B.V. B.V.

Sets of probability distributions, independence, and convexity

This paper analyzes concepts of independence and assumptions of convexity in the theory of sets of probability distributions. The starting point is Kyburg and Pittarelli's discussion of "convex Bayesianism" (in particular their proposals concerning E-admissibility, independence, and convexity). The paper offers an organized review of the literature on independence for sets of probability distributions; new results on graphoid properties and on the justification of "strong independence" (using exchangeability) are presented. Finally, the connection between Kyburg and Pittarelli's results and recent developments on the axiomatization of non-binary preferences, and its impact on "complete" independence, are described.

A pointwise constrained version of the Liapunov convexity theorem for vectorial linear first-order control systems

We generalize the Liapunov convexity theorem's version for vectorial control systems driven by linear ODEs of first-order p = 1 , in any dimension d ∈ N , by including a pointwise state-constraint. More precisely, given a x ‾ ( ⋅ ) ∈ W p , 1 ( [ a , b ] , R d ) solving the convexified p-th order differential inclusion L p x ‾ ( t ) ∈ co { u 0 ( t ) , u 1 ( t ) , … , u m ( t ) } a.e., consider the general problem consisting in finding bang-bang solutions (i.e. L p x ˆ ( t ) ∈ { u 0 ( t ) , u 1 ( t ) , … , u m ( t ) } a.e.) under the same boundary-data, x ˆ ( k ) ( a ) = x ‾ ( k ) ( a ) & x ˆ ( k ) ( b ) = x ‾ ( k ) ( b ) ( k = 0 , 1 , … , p − 1 ); but restricted, moreover, by a pointwise state constraint of the type 〈 x ˆ ( t ) , ω 〉 ≤ 〈 x ‾ ( t ) , ω 〉 ∀ t ∈ [ a , b ] (e.g. ω = ( 1 , 0 , … , 0 ) yielding x ˆ 1 ( t ) ≤ x ‾ 1 ( t ) ). Previous results in the scalar d = 1 case were the pioneering Amar & Cellina paper (dealing with L p x ( ⋅ ) = x ′ ( ⋅ ) ), followed by Cerf & Mariconda results, who solved the general case of linear differential operators L p of order p ≥ 2 with C 0 ( [ a , b ] ) -coefficients. This paper is dedicated to: focus on the missing case p = 1 , i.e. using L p x ( ⋅ ) = x ′ ( ⋅ ) + A ( ⋅ ) x ( ⋅ ) ; generalize the dimension of x ( ⋅ ) , from the scalar case d = 1 to the vectorial d ∈ N case; weaken the coefficients, from continuous to integrable, so that A ( ⋅ ) now becomes a d × d -integrable matrix; and allow the directional vector ω to become a moving AC function ω ( ⋅ ) . Previous vectorial results had constant ω, no matrix (i.e. A ( ⋅ ) ≡ 0 ) and considered: constant control-vertices (Amar & Mariconda) and, more recently, integrable control-vertices (ourselves).

Electronic states of trans-polyacetylene, poly(p-phenylene vinylene) and sp-hybridised carbon species in amorphous hydrogenated carbon probed by resonant Raman scattering

Inclusions of sp-hybridised, trans-polyacetylene [trans-(CH)x] and poly(p-phenylene vinylene) (PPV) chains are revealed using resonant Raman scattering (RRS) investigation of amorphous hydrogenated carbon (a-C:H) films in the near IR – UV range. The RRS spectra of trans-(CH)x core Ag modes and the PPV CC-H phenylene mode are found to transform and disperse as the laser excitation energy ћωL is increased from near IR through visible to UV, whereas sp-bonded inclusions only become evident in UV. This is attributed to ћωL probing of trans-(CH)x chain inhomogeneity and the distribution of chains with varying conjugation length; for PPV to the resonant probing of phelynene ring disorder; and for sp segments, to ћωL probing of a local band gap of end-terminated polyynes. The IR spectra analysis confirmed the presence of sp, trans-(CH)x and PPV inclusions. The obtained RRS results for a-C:H denote differentiation between the core Ag trans-(CH)x modes and the PPV phenylene mode. Furthermore, it was found that at various laser excitation energies the changes in Raman spectra features for trans-(CH)x segments included in an amorphous carbon matrix are the same as in bulk trans-polyacetylene. The latter finding can be used to facilitate identification of trans-(CH)x in the spectra of complex carbonaceous materials.

Are combination forecasts of S&P 500 volatility statistically superior?

Forecasting volatility has received a great deal of research attention, with the relative performances of econometric model based and option implied volatility forecasts often being considered. While many studies find that implied volatility is the pre-ferred approach, a number of issues remain unresolved, including the relative merit of combining forecasts and whether the relative performances of various forecasts are statistically different. By utilising recent econometric advances, this paper considers whether combination forecasts of S&P 500 volatility are statistically superior to a wide range of model based forecasts and implied volatility. It is found that a combination of model based forecasts is the dominant approach, indicating that the implied volatility cannot simply be viewed as a combination of various model based forecasts. Therefore, while often viewed as a superior volatility forecast, the implied volatility is in fact an inferior forecast of S&P 500 volatility relative to model-based forecasts.

The jump component of S&P 500 volatility and the VIX index

Much research has investigated the differences between option implied volatilities and econometric model-based forecasts. Implied volatility is a market determined forecast, in contrast to model-based forecasts that employ some degree of smoothing of past volatility to generate forecasts. Implied volatility has the potential to reflect information that a model-based forecast could not. This paper considers two issues relating to the informational content of the S&P 500 VIX implied volatility index. First, whether it subsumes information on how historical jump activity contributed to the price volatility, followed by whether the VIX reflects any incremental information pertaining to future jump activity relative to model-based forecasts. It is found that the VIX index both subsumes information relating to past jump contributions to total volatility and reflects incremental information pertaining to future jump activity. This issue has not been examined previously and expands our understanding of how option markets form their volatility forecasts.