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William Weiss

Publications and source records attributed to William Weiss.

8 recordsLinked to original sources
arXiv2021

Loeb Extension and Loeb Equivalence II

The paper answers two open questions that were raised in by Keisler and Sun. The first question asks, if we have two Loeb equivalent spaces $(\Omega, \mathcal F, \mu)$ and $(\Omega, \mathcal G, \nu)$, does there exist an internal probability measure $P$ defined on the internal algebra $\mathcal H$ generated from $\mathcal F\cup \mathcal G$ such that $(\Omega, \mathcal H, P)$ is Loeb equivalent to $(\Omega, \mathcal F, \mu)$? The second open problem asks if the $\sigma$-product of two $\sigma$-additive probability spaces is Loeb equivalent to the product of the same two $\sigma$-additive probability spaces. Continuing work in a previous paper, we give a confirmative answer to the first problem when the underlying internal probability spaces are hyperfinite, a partial answer to the first problem for general internal probability spaces, and settle the second question negatively by giving a counter-example. Finally, we show that the continuity sets in the $\sigma$-algebra of the $\sigma$-product space are also in the algebra of the product space.

math.FA↗
arXiv2020

Finitely-additive, countably-additive and internal probability measures

We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is infinitesimal. As an application to standard probability theory, we show that every finitely-additive Borel probability measure $P$ on a separable metric space is a limit of a sequence of countably-additive Borel probability measures if and only if the space is totally bounded.

math.LO↗
arXiv2020

Loeb Extension and Loeb Equivalence

In Keisler and Sun (2004), the authors raise several open problems on Loeb equivalences between various internal probability spaces. We provide counter-examples for the first two open problems. Moreover, we reduce the third open problem to the following question: Is the internal algebra generated by the union of two Loeb equivalent internal algebras a subset of the Loeb extension of any one of the internal algebra?

math.LO↗
arXiv2018

On the tightness of $G_\delta$-modifications

The $G_\delta$-modification $X_\delta$ of a topological space $X$ is the space on the same underlying set generated by, i.e. having as a basis, the collection of all $G_\delta$ subsets of $X$. Bella and Spadaro recently investigated the connection between the values of various cardinal functions taken on $X$ and $X_\delta$, respectively. In their paper, as Question 2, they raised the following problem: Is $t(X_\delta) \le 2^{t(X)}$ true for every (compact) $T_2$ space $X$? Note that this is actually two questions. In this note we answer both questions: In the compact case affirmatively and in the non-compact case negatively. In fact, in the latter case we even show that it is consistent with ZFC that no upper bound exists for the tightness of the $G_\delta$-modifications of countably tight, even Frechet spaces.

math.GN↗
arXiv2017

Protection of hamsters from mortality by reducing fecal moxifloxacin concentration with DAV131A in a model of moxifloxacin-induced Clostridium difficile colitis

BackgroundLowering the gut exposure to antibiotics during treatments can prevent microbiota disruption. We evaluated the effect of an activated charcoal-based adsorbent, DAV131A, on fecal free moxifloxacin concentration and mortality in a hamster model of moxifloxacin-induced C. difficile infection.Methods215 hamsters receiving moxifloxacin subcutaneously (D1-D5) were orally infected at D3 with C. difficile spores. They received various doses (0-1800mg/kg/day) and schedules (BID, TID) of DAV131A (D1-D8). Moxifloxacin concentration and C. difficile counts were determined at D3, and mortality at D12. We compared mortality, moxifloxacin concentration and C. difficile counts according to DAV131A regimens, and modelled the link between DAV131A regimen, moxifloxacin concentration and mortality. ResultsAll hamsters that received no DAV131A died, but none of those that received 1800mg/kg/day. A significant dose-dependent relationship between DAV131A dose and (i) mortality rates, (ii) moxifloxacin concentration and (iii) C. difficile counts was evidenced. Mathematical modeling suggested that (i) lowering moxifloxacin concentration at D3, which was 58$\mu$g/g (95%CI=50-66) without DAV131A, to 17$\mu$g/g (14-21) would reduce mortality by 90% and (ii) this would be achieved with a daily DAV131A dose of 703mg/kg (596-809).ConclusionsIn this model of C. difficile infection, DAV131A reduced mortality in a dose-dependent manner by decreasing fecal free moxifloxacin concentration.

q-bio.QM↗
arXiv2015

Generalized symmetric systems and thin-very tall compact scattered spaces

We solve a well--known problem in the theory of compact scattered spaces and superatomic boolean algebras by showing that, under GCH and for each regular cardinal $\kappa \geq \omega$, there is a poset $\mathcal P_\kappa$ preserving all cardinals and forcing the existence of a $\kappa$--thin very tall locally compact scattered space. For $\kappa > \omega$, we conceive the poset $\mathcal P_\kappa$ as a higher analogue of the poset $\mathcal P_\omega$ originally introduced by Asper\'{o} and Bagaria in the context of an (unpublished) alternative consistency proof.

math.LO↗
arXiv2009

Definable Davies' Theorem

We prove the following analogue of a Theorem of R.O. Davies: Every $Σ^1_2$ function $f:\R\times\R\to\R$ can be represented as a sum of rectangular $Σ^1_2$ functions if and only if all reals are constructible.

math.LO↗