<a href=”https://math.hawaii.edu/home/depart/theses/PhD_2023_Kunwar.pdf“>Dissertation draft</a>

Date: Monday, April 17

Time: 2:30 – 3:20

Room: Keller 313

Title: Suppes–style natural deduction system for probability logic

Abstract:

An elegant way to work with probabilized sentences was proposed by P. Suppes. According to his approach we develop a natural deduction system $\mathbf{NKprob}(\varepsilon)$ for probability logic, inspired by Gentzen’s natural deduction system $\mathbf{NK}$ for classical propositional logic. We use a similar approach as in defining general probability natural deduction system $\mathbf{NKprob}$ (see M. Bori\v ci\’c, Publications de l’Institut Mathematique, Vol. 100(114) (2016), pp. 77–86). Our system will be suitable for manipulating sentences of the form $A^n$, where $A$ is any propositional formula and $n$ a natural number, with the intended meaning ‘the probability of truthfulness of $A$ is greater than or equal to $1-n\varepsilon$’, for some small $\varepsilon >0$.

For instance, the rules dealing with conjunction looks as follows:

$$\frac{A^m\quad B^n}{(A\wedge B)^{m+n}}(I\wedge)\qquad\frac{A^m\quad (A\wedge B)^n}{B^n}(E\wedge)$$

and with implication:

$$\frac{(\neg A)^m\quad B^n}{(A\to B)^{\min\{m,n\}}}(I\to)\qquad\frac{A^m\quad (A\to B)^n}{B^{m+n}}(E\to)$$

The system $\mathbf{NKprob}(\varepsilon)$ will be a natural counterpart of our sequent calculus $\mathbf{LKprob}(\varepsilon)$ (see M. Bori\v ci\’c, Journal of Logic and Computation 27 (4), 2017, pp. 1157–1168).

We prove that our system is sound and complete with respect to the traditional Carnap–Popper type probability semantics.

Title: On the geometric aspects behind classical mechanics.Abstract: In this talk about physics, we will not talk about physics, but the applications of mathematical tools used in that field. From a functional to symplectic geometry, passing by abstract algebra and using differential forms, we will explore some of the foundations of classical mechanics and apply them to the famous example of the2D harmonic oscillator. No knowledge of any physics is required.

Title: Mathematical modeling of mosquito-borne disease transmission in wildlife

Abstract:

Mosquito-borne diseases (MBDs) cause enormous losses of human lives and health throughout the world. Recent environmental changes, including global climatic warming and increased human encroachment on natural areas, are predicted to increase the spread of MBDs. The emergence of novel zoonoses (diseases originating in wildlife and capable of spreading in human populations) is also expected to accelerate, stretching thin our ability to respond effectively to future epidemics. These emerging crises call for substantial advancement in our understanding of how ecological processes drive MBD transmission within and between wildlife populations. But understanding the underlying factors driving MBD outbreak risk is a substantial challenge. For these diseases, transmission occurs during blood feeding, an essential step in the mosquito reproductive cycle. Blood feeding itself is an interactive ecological process: the animals upon which mosquitoes feed may defend themselves, leading to a back-and-forth struggle that can significantly alter transmission rates.

Mathematical models have long been used to study MBD transmission, dating back to the late 19th century when it was first discovered that mosquitoes vectored malaria. However, recent advances in our understanding of mosquito ecology call into question some long-held assumptions underlying these classical models. I will discuss my work which focuses on incorporating ecological interactions into mathematical models of MBD transmission, including resource competition between host animals and defensive behaviors against mosquito biting. These interactions will be further connected to recent work incorporating the effect of ambient temperature on mosquito biology into MBD transmission models.