Measures of axial symmetry for ovals

A measure of axial symmetry for ovals is defined, and eleven particular measures are studied. Lower bounds for these measures are determined on the classes of arbitrary ovals, centrally symmetric ovals, and ovals of constant breadth. The proofs of these results make use only of elementary geometry and the properties of convexity.     

Remarks on the Gieseker-Petri divisor in genus eight

In the moduli space of curves of genus 8, M ₈, denote by GP ₈ the locus of curves that do not satisfy the Gieseker-Petri theorem. In this short note we study the projective plane models of curves of genus 8 that do not satisfy the Gieseker-Petri theorem. We use these projective models to exhibit an irreducible divisorial component in GP ₈ and we show that GP ₈ is an irreducible divisor.     

ASYMPTOTIC APPROXIMATION BYBERNSTEIN-DURRMEYER OPERATORS AND THEIR DERIVATIVES

The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators M_n. We derive the complete asymptotic expansion of the operators M_n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Finally, we obtain a Voronovskaja-type formula for simultaneous approximation by linear combinations of M_n,     

Universal problem for Kähler differentials in A-modules: Non-commutative and commutative cases

Let A be an associative and unital K-algebra sheaf, where K is a commutative ring sheaf, and ε an (A, A)-bimodule, that is, a sheaf of (A, A)-bimodules. We construct an (A, A)-bimodulc which is K-isomorphic with the K-module D _K (A, ε) of germs of K-derivations. A similar isomorphism is obtained, this time around with respect to A, between the K-module D _K (A, ε) with the A-module Hom _A (Ω _K (A), ε). where A, in addition of being associative and unital, is assumed to be commutative, and Ω _K (A) denotes the A-module of germs of Kähler differentials. Finally, we expound on functoriality of Kähler differentials.     

Some new BIBDs with block size 7

For λ ≥ 3 the existence problem for(ν,7,λ) BIBDs has been completely solved, except when ν = 253. In this paper, 3 new direct (ν,7,λ) BIBD constructions are given for (ν,λ) = (259,1), (106,2), and (253,2). Some features of the known (175,7,1) BIBD are also discussed. Finally, a few (ν,7,1) BIBDs are obtained recursively. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 146–150, 2000     

Doubly Resolvable Nearly Kirkman Triple Systems

Necessary conditions for existence of a resolvable group divisible design (GDD) with block size 3 and type (a nearly Kirkman triple system, NKTS(v)), are and (mod 6). In this paper, we look at doubly resolvable NKTS(v)s; here we find that these necessary conditions are sufficient, except possibly for 64 values of v.     

Assessment of maximum likelihood PCA missing data imputation

Maximum likelihood principal component analysis (MLPCA) was originally proposed to incorporate measurement error variance information in principal component analysis (PCA) models. MLPCA can be used to fit PCA models in the presence of missing data, simply by assigning very large variances to the non‐measured values. An assessment of maximum likelihood missing data imputation is performed in this paper, analysing the algorithm of MLPCA and adapting several methods for PCA model building with missing data to its maximum likelihood version. In this way, known data regression (KDR), KDR with principal component regression (PCR), KDR with partial least squares regression (PLS) and trimmed scores regression (TSR) methods are implemented within the MLPCA method to work as different imputation steps. Six data sets are analysed using several percentages of missing data, comparing the performance of the original algorithm, and its adapted regression‐based methods, with other state‐of‐the‐art methods. Copyright © 2016 John Wiley & Sons, Ltd.