On the logarithm component in trace defect formulas

In asymptotic expansions of resolvent traces \Tr(A(P-\lambda)^{-1}) for classical pseudodifferential operators on closed manifolds, the coefficient $C_0(A,P)$ of $(-\lambda)^{-1}$ is of special interest, since it is the first coefficient containing nonlocal elements from $A$; on the other hand if $A=I$ and $P=D^*D$ it gives part of the index of $D$. $C_0(A,P)$ also equals the zeta function value at 0 when $P$ is invertible. $C_0(A,P)$ is a trace modulo local terms, since $C_0(A,P)-C_0(A,P')$ and $C_0([A,A'],P)$ are local. By use of complex powers $P^s$ (or similar holomorphic families of order $s$), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from $A$, $A'$, $\log P$ and $\log P'$. The present paper has two purposes: One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of $P$ as in the original proofs. We here also give a simple direct proof of a recent residue formula of Scott for $C_0(I,P)$. The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems.Comment: 41 page

Generalization of Some Inequalities for the Ratio of Gamma Functions

access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present some monotonic functions and some generalized inequalities involving the ratios of analogues of the Gamma function. Mathematics Subject Classification: 33B15, 26A4

Split-2 Bisimilarity has a Finite Axiomatization over CCS with<br> Hennessy&#39;s Merge

This note shows that split-2 bisimulation equivalence (also known as timed equivalence) affords a finite equational axiomatization over the process algebra obtained by adding an auxiliary operation proposed by Hennessy in 1981 to the recursion, relabelling and restriction free fragment of Milner's Calculus of Communicating Systems. Thus the addition of a single binary operation, viz. Hennessy's merge, is sufficient for the finite equational axiomatization of parallel composition modulo this non-interleaving equivalence. This result is in sharp contrast to a theorem previously obtained by the same authors to the effect that the same language is not finitely based modulo bisimulation equivalence

Reduction of Late Design Changes Through Early Phase Need Analysis

Published and used by INCOSE with permission. Abstract. Contractors in the oil and gas industry are experiencing an increased pressure to deliver projects at a lower cost and at a shorter schedule. Extensive requirements combined with strict governing documents restrict project designs. Therefore, contractors need efficient governing processes that capture customers ’ needs and ensure that system requirements relates to these customer needs. In the execution phase of a project, the engineering degrees of freedom are limited. In the early project phase of concept and study, the basic design is established. The design freeze occurs in the tender phase, based on a best interpretation of customer needs. This causes any changes made in the execution phase to be costly and have the potential to impose severe subsequent consequences. Proper use of systems engineering ensures a minimal amount of late design changes. The most important systems engineering process in such a context, is the capturing of customer needs and definition of user requirements. We researched the processes used to capture system requirements today, and the potential impact of using systems engineering techniques for this purpose. We uncovered multiple gaps in the current process of capturing customer needs, which subsequently led to the definition of system requirements based on international standards and best practices, instead of actual needs. Our research also showed that with a process of capturing customer needs based on systems engineering techniques, as much as 92 % of the costs imposed by late design changes are avoidable

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Vowel normalization is a computation that is meant to account for the differences in the absolute direct (physical or psychophysical) representations of qualitatively equivalent vowel productions that arise due to differences in speaker properties such as body size types, age, gender, and other socially interpreted categories that are based on natural variation in vocal tract size and shape. In this dissertation, we address the metaphysical and epistemological aspects of vowel normalization pertaining to spoken language acquisition during early infancy. We begin by reviewing approaches to conceptualizing and modeling the phonetic components of early spoken language acquisition, forming a catalog of phenomena that serves as the basis for our discourse. We then establish the existence of a vowel normalization computation carried out by infants early in their spoken language acquisition, and put forward a conceptual and technical framework for its investigation which focuses attention on the generative nature of the computation. We then situate the acquisition of vowel normalization within a broader developmental framework encompassing a suite of vocal learning phenomena, including language-specific caretaker vocal exchanges