An Accumulative Model for Quantum Theories

For a general quantum theory that is describable by a path integral formalism, we construct a mathematical model of an accumulation-to-threshold process whose outcomes give predictions that are nearly identical to the given quantum theory. The model is neither local nor causal in spacetime, but is both local and causal is in a non-observable path space. The probabilistic nature of the squared wavefunction is a natural consequence of the model. We verify the model with simulations, and we discuss possible discrepancies from conventional quantum theory that might be detectable via experiment. Finally, we discuss the physical implications of the model.Comment: 14 pages, 3 figure

Analysis of Malaria Control Measures Effectiveness Using Multi-Stage Vector Model

We analyze an epidemiological model to evaluate the effectiveness of multiple means of control in malaria-endemic areas. The mathematical model consists of a system of several ordinary differential equations, and is based on a multicompartment representation of the system. The model takes into account the mutliple resting-questing stages undergone by adult female mosquitos during the period in which they function as disease vectors. We compute the basic reproduction number $\mathcal R_0$, and show that that if $\mathcal R_0<1$, the disease free equilibrium (DFE) is globally asymptotically stable (GAS) on the non-negative orthant. If $\mathcal R_0>1$, the system admits a unique endemic equilibrium (EE) that is GAS. We perform a sensitivity analysis of the dependence of $\mathcal R_0$ and the EE on parameters related to control measures, such as killing effectiveness and bite prevention. Finally, we discuss the implications for a comprehensive, cost-effective strategy for malaria control.Comment: 34 pages , 3 figure

Chemical Impacts of the Microbiome Across Scales Reveal Novel Conjugated Bile Acids

A mosaic of cross-phyla chemical interactions occurs between all metazoans and their microbiomes. In humans, the gut harbors the heaviest microbial load, but many organs, particularly those with a mucosal surface, associate with highly adapted and evolved microbial consortia. The microbial residents within these organ systems are increasingly well characterized, yielding a good understanding of human microbiome composition, but we have yet to elucidate the full chemical impact the microbiome exerts on an animal and the breadth of the chemical diversity it contributes. A number of molecular families are known to be shaped by the microbiome including short-chain fatty acids, indoles, aromatic amino acid metabolites, complex polysaccharides, and host lipids; such as sphingolipids and bile acids. These metabolites profoundly affect host physiology and are being explored for their roles in both health and disease. Considering the diversity of the human microbiome, numbering over 40,000 operational taxonomic units, a plethora of molecular diversity remains to be discovered. Here, we use unique mass spectrometry informatics approaches and data mapping onto a murine 3D-model to provide an untargeted assessment of the chemical diversity between germ-free (GF) and colonized mice (specific-pathogen free, SPF), and report the finding of novel bile acids produced by the microbiome in both mice and humans that have evaded characterization despite 170 years of research on bile acid chemistry

Proving Taylor's Theorem from the Fundamental Theorem of Calculus by Fixed-point Iteration

Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most general form can be proved simply as an immediate consequence of the Fundamental Theorem of Calculus (FTOC). The proof shows the deep connection between the Taylor expansion and fixed-point iteration, which is a foundational concept in numerical and functional analysis. One elegant variant of the proof also demonstrates the use of combinatorics and symmetry in proofs in mathematical analysis. Since the proof emphasizes concepts and techniques that are widely used in current science and industry, it can be a valuable addition to the undergraduate mathematics curriculum.Comment: 10 page