Pacific Northwest 2026 Section Meeting Program of events

This panel will highlight the ways we can bring our expertise off campus and advocate for mathematics and education more broadly in both regional and national settings. Building off Dr. Marano’s morning presentation, this panel will begin with a presentation on opportunities for national advocacy with groups like AAAS, followed by a discussion of state-based local policy fellowships. Then, we hope to have speakers present more public scholarship, describing how they use mathematical and data science skills to analyze and provide summaries shared with campuses or regional communities to better understand the state of the world.

Getting started with undergraduate research can seem daunting, both for faculty and students. The Center for Undergraduate Research in Mathematics (CURM) funds mini-grants for undergraduate research groups and provides supportive, scaffolded research mentoring to get started including how to promote effective collaborations, how to support and affirm students of all backgrounds, and how to select a good research project for even early-career students. This panel will feature faculty who have received CURM mini-grants and others who have been involved in undergraduate research.

As a high-impact practice, undergraduate research experiences help students clarify their career interests and post-graduate plans and foster a sense of belonging by building a learning community with peers and mentors. Studies show that when research experiences are earlier in the academic career, they are more likely to ignite an interest in STEM, which is key for students who may not enter college seeing themselves as successful STEM students. Evidence suggests a pronounced effect of high-impact practices on the experiences of underserved students. In addition, research experiences enhance students’ resumes and also prepare them with transferable skills that are valued by employers (independence as well as team-working, critical thinking, problem-solving, time management, communication, and tech skills, among others), making them more competitive in the job market.

Mosaic Knot theory is a field where mathematical knots are placed on grid diagrams called mosaics. This was established by Dr. Lomonaco and Dr. Kauffman in Quantum Knots and Mosaics. In this work, we place improved lower and upper bounds on the size of a mosaic given only its crossing number or using crossing, braid index, and braid length. This is carried out by constructing an arbitrary diagram of a braid on a mosaic.

The weighted spanning tree enumerator (WSTE) is the sum of weights of all spanning trees of a graph. A recent result expressed the WSTE of multi-partite graphs in terms of the WSTE of its partition graph, which greatly simplifies the computation of the WSTE of the original graph. The proof to this result is algebraic, using the Weighted Matrix Tree Theorem. In this talk, we will present the new result and the work we have done in deriving a graph theoretic justification for the result.

We develop new geometric spaces by blending the half-plane model for the hyperbolic plane with taxicab geometry. Starting with a taxicab norm on each tangent space, the lengths of curves are analyzed, allowing us to determine geodesics, and from these, a distance function. Then we show that these spaces are hyperbolic in the sense of Gromov.

Traditional baseball wisdom says that a batting lineup composed of players with very diverse offensive skill sets will produce more runs than one composed of similar players (all other factors being equal). However, very little has been done to empirically validate or challenge this theory, partially due to a lack of tools worthy of answering the question. Within the emerging mathematical field of Topology, a computational tool in Topological Data Analysis (TDA) called Persistent Homology presents a robust method for measuring the diversity of a dataset. Using historical data from Major League Baseball teams and a python script designed to calculate the diversity of a dataset at the H_0 and H_1 levels of Persistent Homology, this centuries-old question can be addressed in a meaningful way.

The Steele Polynomial is a graph polynomial that arises in the study of minimal spanning trees and gives insights about the connectedness of the graph. In this talk, we present our research on the behavior of the limits of zeroes in the complex plane of the Steele polynomial for various infinite sequences of graphs. Our results use a recent generalization of the Beraha-Kahane-Weiss theorem that extends to the case of repeated roots of the characteristic polynomial of the recursion relationship satisfied by the Steele Polynomials.

Graphs can be encoded into a matrix according to some rule. The eigenvalues of the matrix are used to understand the structural properties of graphs. If two graphs share a set of eigenvalues, they are called cospectral. A tree is a graph with no cycles, and for most matrix representations, almost all trees have a cospectral mate.

The distance Laplacian matrix is found by subtracting the distance matrix from the diagonal transmission matrix.

There is an open conjecture that trees, for their distance Laplacian matrices, do not have a cospectral mate and are therefore spectrally determined. We show that a family of trees of diameter 4 are determined by their spectrum.

Sangaku is a unique mathematical sub-branch of geometry and Japanese mathematics wasan. This topic stands out due to its mathematical results and its prevalence in the Edo period. This paper aims to provide the reader with a general overview of the history and mathematics of wasan and sangaku, along with an understanding of the Gion shrine problem (a famously difficult sangaku). We begin by covering historical factors that influenced the trajectory of Japanese mathematics. Following that, we describe several impressive results stemming from wasan that Japan developed independently from contemporary cultures. We then describe what sangaku is and its history, while providing several visual examples. Finally, we introduce and analyze a solution of the Gion shrine problem, whose original solution was supposedly a 1,024 degree polynomial, but eventually a 10th degree polynomial solution was found.

The United States holds the record for the longest standing democracy, but in this democracy, some representation is more representative than others. Gerrymandering refers to the manipulation of electoral district boundaries in order for political parties to maximize their political advantage in the House of Representatives. we study a simplified version of electoral maps via a grid-based puzzle commonly known as Gerrymandering Puzzles. These rectangular puzzles represent our states holding cells representing voters colored based on their political affiliation. The goal of the game is to draw contiguous regions along these voters in order to maximize the number of districts won by a chosen political party.

We examine how strategies analogous to real-world gerrymandering practices arise in the maximization of Gerrymandering Puzzles. We also explore what makes a valid Gerrymandering Puzzle through the analysis of the given specific puzzle dimensions, district size, and proportionality of political affiliation.

Finally, we investigate algorithmic approaches to constructing optimal districts with the use of combinatorics. These models provide a framework to understand the mechanisms and political games underlying the real-world applications of gerrymandering while also serving as an accessible educational tool for illustrating its effects.

As math students, we have all wondered about the set of all complex numbers, \(\mathbb{C}\text{.}\) In particular, we have all been amazed by the imaginary unit \(i\text{,}\) which we are first taught to be the square root of minus one, that is \(i=\sqrt{-1}\text{.}\) Later, we are taught that \(i\) is more carefully defined as the complex square root of \(-1\) with a minimum positive angular argument.

Algebraists, such as Galois, long searched for answers regarding the \textit{algebraic structures} of the real numbers \(\mathbb{R}\) and the complex numbers, \(\mathbb{C}\text{.}\) One particular question that arose was: How doe these structures relate to each other? The same questions of structural relationships can be extended beyond relating the complex numbers to the reals, exploring how the structure of the complex numbers can be approached in terms of structures such as groups, rings, and fields.

In this talk, we will explore some structures that are isomorphic to \(\mathbb{C}\text{.}\) We will provide an overview of some basic facts about structures in group theory and ring theory, and what isomorphism means in those contexts. We will emphasize the inter-workings between certain algebraic structures and the set of all complex numbers.

This research was prompted by an introductory mathematics research focused class. Specifically, it involves series, infinite sums of numbers which may or may not add to finite ones. In this instance, we explore series that do in fact converge. To do so, we use an arctangent subtraction identity and telescoping properties of series, finding the specific convergence values of two arctangent series and attempt to generalize thrice; First successfully with a general arctangent series that converges to a real number in a bounded set, the second time unsuccessfully, and the third successfully with a general series that converges to any real number in a real, unbounded function.

The show Dancing With The Stars (DWTS) has gone through many changes to the way it combines judge scores and fan voting to decide which couple to eliminate each week. The show is in need of a fair voting system that still allows for fan votes to contribute to the results in a meaningful way. We took a look at the different voting systems the show has used in the past and attempted to understand how the fan votes contribute in order to build our own voting system that is fairer to the contestants. Afterwards, we used our model to analyze outside factors such as which pro dancers were with each celebrity, the gender of the dancers, and the effect of social media.

We will predict men’s and women’s brackets for March Madness using multiple machine learning algorithms including random forests, gradient descent, long-short term memory, and multi-layered perceptrons. By combining data from both Kaggle’s March Machine Learning Mania and KenPom’s season stats, we create a comprehensive dataset that we use to accurately predict the ideal bracket using the Brier score as our metric.

AI is revving up the speed of accessing information, but is it pushing the brakes on student learning? How can we guide students to harness the power of technology while deepening their own understanding? Spurred by changes to our major involving linear algebra and numerical analysis, we present innovative uses of generative AI to enhance experiential learning across the mathematics curriculum. Argumentation skills for proof-writing, like many others, are best honed via dialogue. Why not include AI in the conversation? While these conversations may look like prolonged chatting or computer-guided programming, they can boost curiosity, logical reasoning, and character development. The results of classroom implementations are presented alongside educational best practices and inquiry-based resources (including the SIGMAA IBL community).

Standards-based grading (SBG) allows students to demonstrate mastery through multiple attempts at well-defined learning outcomes. In my experience, this has worked well, but in recent years students have struggled to schedule reassessments on their own, which defeats much of the purpose of SBG.

To address this, I’ve been experimenting with what I call ""n-k grading."" Each week, students are assessed on current material and also have the opportunity to reattempt problems from the previous k weeks. For example, in week 8, an exam might include four questions on new learning outcomes plus one question each from outcomes covered in weeks 7, 6, 5, etc. Everyone completes the new material; the reassessment questions are optional and allow students to improve earlier marks without scheduling separate attempts.

By embedding reassessment opportunities into every weekly exam, the barrier to retaking problems essentially disappears. Students seem to engage more consistently with the iterative process that makes SBG effective. I’ll share my experience implementing n-k grading across several courses, discuss practical considerations, and invite feedback on this approach.

The current syllabus for your first course in linear algebra is already stressed by squeezing the content of your favorite year-long text into one semester or even one quarter. Client disciplines such as data science, economics and engineering use powerful tools from linear algebra, and those disciplines want us to introduce them to students. How do we teach our students about those tools without blowing up the first (and often only) linear algebra course that students take?

I argue that what students really need is to better understand bases, and how special choices of bases lead to the powerful tools (matrix factorizations and algorithms to construct them) that our clients use. In order to accommodate this new emphasis, I will suggest what we should reduce or remove to accommodate it.

Hyperbolic space arose in the early 19th century as a consequence of attempts to prove the Parallel Postulate in Euclidean geometry and the realization that negating the Parallel Postulate did not result in an inconsistent geometry. Since then, notions of hyperbolicity have evolved from specific spaces, through curvature and symmetry to metric structure. The metric approach, via Gromov hyperbolicity has enjoyed a great deal of consideration since its introduction in 1987, and a number of equivalent formulations have developed, some more intuitive than others. In this talk, I offer a perspective on Gromov hyperbolicity that aligns with his original definition, and I describe some new spaces that can be shown to be Gromov hyperbolic without relying on alternate characterizations.

The Frobenius coin problem asks us for the largest number that cannot be made from coins of fixed denominations. The denumerant \(r(n)\) of a numerical monoid \(M=\langle a_1,\dots,a_d\rangle\) counts the number of ways to represent \(n\) in \(M\text{,}\) and its maximal representable function \(R(k)\) is the largest integer with at most k representations. We show that finitely many values of \(R(k)\) uniquely determine the generators of \(M\text{,}\) and present a reconstruction algorithm.

Knot theory provides a great tool kit for exploration from many angles. The research I will focus on in this talk began with using ideas of mosaic knot theory to define a new variation of the unknotting number. We worked on this project as a summer research project during the summer of 2025. As part of the program we also led groups of high school students in a STEM summer immersion program to learn about our research. In this talk I’ll describe both the deeper dive my students took in knot theory research and the shallow wading pool we created to let the high school students take a dip.

The kropki variant of sudoku adds white or black dots between cells to indicate their digits are consecutive or in 1:2 ratio, respectively. In this talk we will explore whether having kropki dots is enough to uniquely specify a sudoku puzzle, even with no digits initially given. For 4x4 sudoku grids, this is always the case. But looking at 6x6 puzzles we show, using Ramsey Theory like arguments, that in a very strong way, kropki dots are not enough to always guarantee a unique solution.

At the Seattle Universal Math Museum (SUMM), our mission is to spark a love of math in every person. Supported by MAA, SUMM has launched a program designed to support the development of positive math identities in middle school students who are most adversely affected by inequities in math education, helping them see themselves as capable, creative, and engaged doers of mathematics.

The program, called SUMMIT (Students Using Mathematics to Inspire Tomorrow), runs throughout the school year in three distinct phases. Phase 1 has students explore a myriad of math topics through hands-on lessons including games and activities. Phase 2 brings in mentors from SUMM’s community to guide students in researching a math topic of their choice; Phase 3 offers students the opportunity to to help facilitate public math-focused events at libraries, community centers, schools, and more throughout the summer.

By removing competition, and encouraging cooperation and teamwork, SUMMIT centers the community aspect of math and provides a space for students to collaborate and interact with mathematics in a fun and stress-free environment.

In this talk we will discuss our ongoing experience piloting SUMMIT at a middle school in Seattle, Washington, including challenges and successes, student comments, and project progress.

Maternal mortality is an issue affecting lives around the world. We are interested in seeing if the overall wealth disparities in a country impact the rates of maternal mortality, and we cross reference that information with the healthcare access and quality score in said country. We represent these wealth disparities using the Gini Index and Palma Ratio, and then observe the correlation coefficient between those and the maternal mortality rates and healthcare access and quality score. We found that there is indeed a strong positive correlation between maternal mortality rates, and a strong negative correlation between healthcare access and quality scores and maternal mortality rates.

We use Kepler Mapper, a python implementation of the mapper algorithm in order to analyze and visualize high-dimensional data of Alaskan fresh water pollutants through created topological maps. These maps show us relationships in pollutants, geographic position, pH, and type of hydro-logical source. We look at the distinction between burnt streams and streams surrounded by healthy forest as a potential source of pollutants and pH changes, as well as a similar comparison between lakes, streams, and precipitation run-off.

We use Kepler Mapper as a way to topologically analyze a dataset found from the UC Irvine repository. The data contains records on adults and their obesity levels, age, height, and several other parameters concerning the individual’s health. We analyze the relationship between adults, their obesity level, and other factors such as whether the individual smokes or drinks.

Myocardial infarction, better known as a heart attack, is the leading cause of death worldwide and in the United States. Finding ways to reduce this death rate is essential for ensuring the sustainability of life. We utilized the Mapper algorithm to analyze a date set of Myocardial Infarction (MI) patient information. We explore topological relationships found within the data and present our findings.

We give an elementary proof that \(C^2\) expanding maps on \([0,1]\) where each branch is surjective are mixing with respect to their absolutely continuous invariant measure, without the spectral theory of transfer operators.

The existence of infinite families of \(N \times N\) Jacobi matrices representing the Hamiltonians of quantum spin chains with and without early state exclusion (ESE) has been shown to exist for any even \(N \geq 4\text{.}\) However, their existence for odd \(N \geq 7\) has remained an open problem. In this talk, we consider a chain of qubits experiencing nearest-neighbor interactions with environmental effects and present infinite families of \(7 \times 7\) Jacobi matrices with and without ESE.

Despite being the top concentration of medical research worldwide, little is known about the origin and progression of cancer. Characterized by abnormal and uncontrolled reproduction of cells, cancer is one of the deadliest diseases faced by modern medicine. In this talk, we form differential equation models to describe the growth rates of various types of healthy and cancerous cells in a tumor founded on the cancer stem cell (CSC) hypothesis. In this assumption of growth dynamics, a small population of CSCs primarily drive tumor growth, while the bulk of the tumor is made up of their progeny, nonCSCs (differentiated cancer cells). We propose three stratified models to test this hypothesis and analyze these dynamical systems for bifurcation over biologically realistic ranges for each variable in order to describe the interaction between both healthy and cancerous stem and differentiated cells. The first model describes growth coming solely from the CSCs, whereas the second further includes growth from nonCSCs as well as their death rate, and the third extends to include interactions with healthy stem and differentiated cells. We will discuss the results of analyzing these models, their biological implications, and future research possibilities.

Be it light, sound, seismic events, even the very subatomic particles composing matter, our reality is fundamentally underpinned by the mathematical structure of waves. In this talk, I explore one of the most valuable tools we have for making sense of these waves; Fourier Analysis. I start by defining the Fourier transform in an abstract sense as a tool to decompose a function into component sinusoids, then use the orthogonality of sine and cosine to define it rigorously on real functions, before using Euler’s formula to extend this definition to complex functions. I provide and explain evocative demonstrations of various functions decomposed into their component complex sinusoids in the form of assorted Manim animations, as well as a physical model composed of 3D printed gear mechanisms acting as said sinusoids. I then discuss further applications of Fourier Analysis, including its application in many PDEs, deconvolution, seismology, signal processing, file compression, and its role in the derivation of Heisenberg’s uncertainty principle.

This presentation investigates orthogonality as a unifying principle in the solution of linear partial differential equations, particularly the wave and heat equations, while highlighting its connections to Fourier analysis and linear algebra. Motivated by the decomposition of musical signals into constituent frequencies, we view functions as elements of an inner product space and expand them in orthogonal bases of trigonometric functions.

From a linear algebraic perspective, these expansions parallel eigenvector decompositions, with Fourier modes arising as eigenfunctions of differential operators. Through separation of variables, solutions to PDEs are expressed as infinite series of orthogonal modes, each evolving independently over time.

Using audio demonstrations and mathematical framing, this talk reveals how the superposition principle and orthogonality underpin both the analysis of sound and the structure of PDE solutions, highlighting a shared mathematical language across music, vibration, and diffusion.

In this talk, the author demonstrates the generalized Stokes’ Theorem in higher dimensions through a mathematical physics perspective. Additionally, we consider how differential geometry on abstract mathematical spaces shapes our understanding of Stokes’ Theorem through rigorous mathematical proof. Then, we aim to visualize these ideas, leading to a discussion on further directions in this mathematical and frontier physics study.

We present a comparison of two sets of discrete-time, host--parasitoid models. The first set was previously analyzed by Marcinko and Kot (2020) and assumed that density-dependence precedes parasitism in the life-cycle of the host. In this talk, we present our analysis of a parallel set of models, now specifying that parasitism precedes density-dependence. Each of the four models in this second set includes a particular combination of standard functional forms for density-dependent growth of the host species and for parasitism. We focus on comparing the two sets of models to gain insight into the impacts of the order of events in the host species’ life-cycle. We include consideration of bifurcations, as well as comparing equilibria and other stable attractors.

In 1940, Pillai proved that every sequence of 16 or fewer consecutive rational integers contains an integer that is coprime to all the others. He also conjectured that for every \(n\geq 17\text{,}\) there exists a sequence of n consecutive integers that does not have this property. The following year, Brauer proved Pillai’s conjecture. Various authors have extended Pillai’s problem to other sets of numbers and call a sequence a Pillai sequence if there exists a lower bound \(B\) such that for all \(n\geq B\) the sequence contains \(n\) consecutive integers with none coprime to all the others. In this spirit, we prove that every arithmetic progression of algebraic integers is a Pillai sequence.

An edge-magic labeling of a graph is an assignment of integers to the vertices and edges such that the sum of the labels on an edge and its vertices is constant throughout the graph. A long standing conjecture has been that all finite trees have an edge-magic labeling.

As often happens with graph labelings, the analogous question for infinite trees is easier to work with. For many classes of infinite trees, there is always an edge-magic labeling. However, in the case of infinite trees with only one vertex of infinite degree and only finitely many vertices not adjacent to it, the question becomes about a different sort of labeling of the finite parts of the graph, which appears to be difficult to classify. In this talk we will share some initial work toward this goal.

Project UNSOLVED is a seminar-style, elective mathematics class for middle and high schoolers that ran from August 2025 to March 2026. Co-teaching the course with two university professors and two undergraduate mathematics majors, the first part of the course introduced students to unsolved problems in mathematics through hands-on activities and exploration. In the second part of the course, students conducted a deep dive into a particular unsolved problem in mathematics that interested them. They explored the background of their problem and played with their problems. Many students conducted mini experiments related to their problem to conduct and collected data from. Students then analyzed their results and generated a poster sharing their work. This session will be run like lightning talks. We will project the poster and give the students 2-3 minutes to share what they did. For example, one group of students explored the illumination problem of lighting a room with a single light source and what shapes of rooms allow for this illumination. There will be time for questions from the audience.

This research project explores the formal intersection between abstract algebra and functional programming, specifically utilizing the Haskell language as a medium for mathematical expression. This will be done through historical lens with one of the historical events being the Curry-Howard correspondence. The primary objective is to show how Isomorphism and Cayley’s Theorem are connected with Haskell. We explore Isomorphism by illustrating how abstract group properties can be mapped onto functional paradigms found in programming. With the idea of Isomorphism, we will show and prove Cayley’s Theorem, by representing every group G as a subgroup of the symmetric group acting on G. The presentation will showcase how Haskell’s type classes and functions are directly related to Isomorphism theorems, especially with Cayley’s Theorem. Overall showcasing how mathematical theory and functional code are two sides of the same coin and how programming and proofs are one.

The Cantor set, named after the famed mathematician Georg Cantor, has many fascinating topological properties, such as being closed, nowhere dense, and uncountable. To construct the Cantor set, one begins by removing the middle open interval of size 1/3 from the closed interval [0,1]. Then, remove the middle open interval of the resulting two closed intervals. Repeating this process infinitely and taking the intersection of all iterations yields the Cantor set. In contrast, a fat Cantor set is constructed by removing the middle open interval of a size smaller than 1/3. Although fat Cantor sets have many of the same properties as standard Cantor sets, there are surprising differences. This presentation introduces the construction and properties of the Cantor set and a fat Cantor set, and briefly explores their applications.

Not all functions are continuous. Classic examples include functions that jump or behave differently on rationals and irrationals. For instance, Dirichlet’s function is discontinuous everywhere, but small modifications can make it continuous at a single point, showing how continuity can be adjusted.

We can go further. Thomae’s function is continuous at every irrational number but discontinuous at every rational number, showing that discontinuities can be dense yet structured.

This leads to the deeper question: how complicated can the set of discontinuities be? The key result is that for any function, the set of discontinuities is an \(F_\sigma\) set, meaning it can be written as a countable union of closed sets. However, not every subset of \(\mathbb{R}\) has this form, for example, the irrationals do not, highlighting a fundamental limitation and guiding the study of discontinuity.

The topologist’s sine curve is a well known function that shows us the differences between Path connectedness and connectedness. Intuition at first might have us believe these two concepts may often seem to say the same thing, that a space is one piece. The topologist’s sine curve however, shows us that they are in fact two different ways to understand what it means to be one piece. An interesting problem that shows us the importance of path connectedness and connectedness is Smale’s Paradox. Smale’s Paradox, that a sphere can turn inside out in a 3D space through continuous deformation, allowing self-intersections but no creases, tears, or punctures, finds that connectedness and local path connectedness in this case leads to path-connectedness. This is what allows us to resolve the paradox. When looking at cases such as the topologist’s sine curve and Smale’s Paradox, we see how important understanding definitions and their applications can often deepen our understanding and allow us to solve problems that may intuitively seem impossible.

The movie Tron (1982) was the first feature length film to put a scene that was entirely computer-generated imagery (CGI). It created a believable world. Modern open world video games such as Minecraft and No Man’s Sky create these types of worlds at the press of a button. It may be surprising (or not) that good ol’ calculus and vectors are involved in creating some of these procedurally generated worlds and environments. In this talk, we will explore the mathematics behind procedural generation and get to know about how these worlds are created.

One-way layout of count data having over/under dispersion arises in many practical situations. However, as in the continuous and some other discrete data situations, some observations might be missing in the one way layout of count data. In this presentation I will talk about estimation procedures for the parameters involved in the one way layout of count data under different missing data scenarios, the comparative behaviour of the score tests developed by Barnwal and Paul (1988, Biometrika, 75(2), 215–222) and score type statistic developed by Saha (2008, J. Stat. Plan. Inference, 138(7), 2067–2081) for complete data, and the comparative effect of missing data on the score and score-type statistic under different missing data scenarios.

The Moscow Papyrus is an ancient mathematical text written about 3700 years ago by an unknown Egyptian scribe. In it are some of the oldest geometry problems that have survived to the present day. In this talk we’ll take a tour of these geometric problems and their solutions, and reflect on how the ancient Egyptians’ conception of both shape and proof differ from our own.

Written in 1850, Herman Melville’s classic novel is one of the most widely studied texts of American literature. Nevertheless, its use of mathematical language and ideas is often overlooked. This talk aims to illuminate some of these remarkable aspects of the novel, offering a quantitative investigation of Melville’s familiarity of mathematical terms using simple tools of text analysis. Additionally, we will look at Melville’s striking use of the tautochrone curve.

In insurance data analytics and actuarial practice, distortion risk measures are widely used to assess the riskiness of the distribution tail. Point and interval estimates of these measures are applied to price extreme events, determine reserves, design risk transfer strategies, and allocate capital. However, the computation of such estimates often relies on Monte Carlo simulation which, depending on the complexity of the problem, can be computationally expensive and require significant expertise.

This study investigates analytic and numerical evaluation of distortion risk measures with the goal of reducing computational burden. Specifically, we consider several commonly used measures—value-at-risk (VaR), conditional tail expectation (CTE), proportional hazards transform (PHT), Wang transform (WT), and Gini shortfall (GS). These measures are evaluated for loss severity distributions including shifted exponential, Pareto I, and shifted lognormal distributions, chosen to represent common shapes of insurance losses.

While explicit formulas exist for VaR and CTE, analytic evaluation of PHT, WT, and GS is not always feasible. For such cases, we establish rigorous conditions for finiteness and derive two-sided bounds. The accuracy of these bounds is examined through numerical evaluation and simulation-based estimation.

We present a method of estimating model parameters for linear autonomous ODEs using least-squares regression. The coefficient of determination can be used as a measure of model fit. The method is demonstrated using US population data to fit a logistic growth model. Also, a completing species model is used to describe the interaction of two different species of yeast.