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

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

Optimal Real Time Drone Path Planning for Harvesting Information from a Wireless Sensor Network

We consider a remote sensing system in which fixed sensors are placed in a region, and a drone flies over the region to collect information from cluster heads. We assume that the drone has a fixed maximum range, and that the energy consumption for information transmission from the cluster heads increases with distance according to a power law. Given these assumptions, we derive local optimum conditions for a drone path that either minimizes the total energy or the maximum energy required by the cluster heads to transmit information to the drone. We show how a homotopy approach can produce a family of solutions for different drone path lengths, so that a locally optimal solution can be found for any drone range. We implement the homotopy solution in python, and demonstrate the tradeoff between drone range and cluster head power consumption for several geometries. Execution time is sufficiently rapid for the computation to be performed real time, so the drone path can be recalculated on the fly. The solution is shown to be globally optimal for sufficiently long drone path lengths. For future work, we indicate how the solution can be modified to accommodate moving sensors.Comment: 20 pages, 4 figure