A rigorous real time Feynman Path Integral and Propagator

We will derive a rigorous real time propagator for the Non-relativistic Quantum Mechanic $L^2$ transition probability amplitude and for the Non-relativistic wave function. The propagator will be explicitly given in terms of the time evolution operator. The derivation will be for all self-adjoint nonvector potential Hamiltonians. For systems with potential that carries at most a finite number of singularity and discontinuities, we will show that our propagator can be written in the form of a rigorous real time, time sliced Feynman path integral via improper Riemann integrals. We will also derive the Feynman path integral in Nonstandard Analysis Formulation. Finally, we will compute the propagator for the harmonic oscillator using the Nonstandard Analysis Feynman path integral formuluation; we will compute the propagator without using any knowledge of classical properties of the harmonic oscillator

A Cauchy-Dirac delta function

The Dirac delta function has solid roots in 19th century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his infinitesimal definition of delta.Comment: 24 pages, 2 figures; Foundations of Science, 201

Stevin numbers and reality

We explore the potential of Simon Stevin's numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with arXiv:1104.0375, arXiv:1108.2885, arXiv:1108.420

Serotype Distribution and Invasive Potential of Group B Streptococcus Isolates Causing Disease in Infants and Colonizing Maternal-Newborn Dyads

Serotype-specific polysaccharide based group B streptococcus (GBS) vaccines are being developed. An understanding of the serotype epidemiology associated with maternal colonization and invasive disease in infants is necessary to determine the potential coverage of serotype-specific GBS vaccines.Colonizing GBS isolates were identified by vaginal swabbing of mothers during active labor and from skin of their newborns post-delivery. Invasive GBS isolates from infants were identified through laboratory-based surveillance. GBS serotyping was done by latex agglutination. Serologically non-typeable isolates were typed by a serotype-specific PCR method. The invasive potential of GBS serotypes associated with sepsis within seven days of birth was evaluated in association to maternal colonizing serotypes.GBS was identified in 289 (52.4%) newborns born to 551 women with GBS-vaginal colonization and from 113 (5.6%) newborns born to 2,010 mothers in whom GBS was not cultured from vaginal swabs. The serotype distribution among vaginal-colonizing isolates was as follows: III (37.3%), Ia (30.1%), and II (11.3%), V (10.2%), Ib (6.7%) and IV (3.7%). There were no significant differences in serotype distribution between vaginal and newborn colonizing isolates (P = 0.77). Serotype distribution of invasive GBS isolates were significantly different to that of colonizing isolates (P<0.0001). Serotype III was the most common invasive serotype in newborns less than 7 days (57.7%) and in infants 7 to 90 days of age (84.3%; P<0.001). Relative to serotype III, other serotypes showed reduced invasive potential: Ia (0.49; 95%CI 0.31-0.77), II (0.30; 95%CI 0.13-0.67) and V (0.38; 95%CI 0.17-0.83).In South Africa, an anti-GBS vaccine including serotypes Ia, Ib and III has the potential of preventing 74.1%, 85.4% and 98.2% of GBS associated with maternal vaginal-colonization, invasive disease in neonates less than 7 days and invasive disease in infants between 7-90 days of age, respectively

Maternal colonization with Streptococcus agalactiae and associated stillbirth and neonatal disease in coastal Kenya

Streptococcus agalactiae (group B streptococcus, GBS) causes neonatal disease and stillbirth, but its burden in sub-Saharan Africa is uncertain. We assessed maternal recto-vaginal GBS colonization (7,967 women), stillbirth and neonatal disease. Whole-genome sequencing was used to determine serotypes, sequence types and phylogeny. We found low maternal GBS colonization prevalence (934/7,967, 12%), but comparatively high incidence of GBS-associated stillbirth and early onset neonatal disease (EOD) in hospital (0.91 (0.25-2.3)/1,000 births and 0.76 (0.25-1.77)/1,000 live births, respectively). However, using a population denominator, EOD incidence was considerably reduced (0.13 (0.07-0.21)/1,000 live births). Treated cases of EOD had very high case fatality (17/36, 47%), especially within 24 h of birth, making under-ascertainment of community-born cases highly likely, both here and in similar facility-based studies. Maternal GBS colonization was less common in women with low socio-economic status, HIV infection and undernutrition, but when GBS-colonized, they were more probably colonized by the most virulent clone, CC17. CC17 accounted for 267/915 (29%) of maternal colonizing (265/267 (99%) serotype III; 2/267 (0.7%) serotype IV) and 51/73 (70%) of neonatal disease cases (all serotype III). Trivalent (Ia/II/III) and pentavalent (Ia/Ib/II/III/V) vaccines would cover 71/73 (97%) and 72/73 (99%) of disease-causing serotypes, respectively. Serotype IV should be considered for inclusion, with evidence of capsular switching in CC17 strains

A coinductive semantics of the Unlimited Register Machine

We exploit (co)inductive specifications and proofs to approach the evaluation of low-level programs for the Unlimited Register Machine (URM) within the Coq system, a proof assistant based on the Calculus of (Co)Inductive Constructions type theory. Our formalization allows us to certify the implementation of partial functions, thus it can be regarded as a first step towards the development of a workbench for the formal analysis and verification of both converging and diverging computations

Cauchy's infinitesimals, his sum theorem, and foundational paradigms

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy's proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy's proof closely and show that it finds closer proxies in a different modern framework. Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc

Program analysis is harder than verification: A computability perspective

We study from a computability perspective static program analysis, namely detecting sound program assertions, and verification, namely sound checking of program assertions. We first design a general computability model for domains of program assertions and correspond- ing program analysers and verifiers. Next, we formalize and prove an instantiation of Rice\u2019s theorem for static program analysis and verifica- tion. Then, within this general model, we provide and show a precise statement of the popular belief that program analysis is a harder prob- lem than program verification: we prove that for finite domains of pro- gram assertions, program analysis and verification are equivalent prob- lems, while for infinite domains, program analysis is strictly harder than verification

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.Comment: 69 pages, 3 figure