Mathematics Commons^™

(R2165) Two Different Stages With Two Distinct Harvesting Processes, S. Vijaya, Krishika M

Applications and Applied Mathematics: An International Journal (AAM)

This study extends the classical Lotka-Volterra prey-predator model by incorporating a stage structured prey population consisting of juvenile and adult stages, while considering a single predator species governed by a Holling type II functional response. The growth of the juvenile prey depends on the adult prey population and since the juvenile prey has no reproduction capability. Two distinct harvesting strategies are introduced in the model. Constant-yield harvesting is applied to the juvenile stage of the prey population and represents a fixed amount of harvest regardless of population size. Constant effort harvesting is applied to the adult stage of the prey …

Shadowable Diffeomorphisms With Dominated Splitting, Naoya Sumi

Journal of Stochastic Analysis

We study the shadowability for diffeomorphisms with dominated

splitting. In this paper we show that if a diffeomorphism f has a periodic

saddle q whose unstable manifold is compatible with dominated splitting in

some sense and if f is shadowable, then any other periodic point p has a large

local stable set. Our result does not require the hyperbolicity of p.

Data-Driven Discovery Of Canonical Models In Undergraduate Differential Equations Education, Devon L. Garls, Raina Majumder, Scott A. Strong

CODEE Journal

Roughly a decade ago, Brunton, Proctor, and Kutz built upon the re-

sults of classical control theory, compressed sensing, and symbolic regression

to create a method for Sparse Identification of Nonlinear Dynamics (SINDy),

allowing one to infer a dynamical system from time series and bring mathe-

matical dynamics into the modern era of large data. This paper is designed for

a reader concurrently enrolled in a first course in ordinary differential equa-

tions and begins with a technical overview of least-squares regression and

finite-difference approximations to derivatives, which is enough to perform

an example hand-calculation on three data points. The …

A Biased Variant Of The Ladybug Clock Problem, Jacob Vogelpohl, Russell May

Communications on Number Theory and Combinatorial Theory

Instead of the common assumption of uniform probabilities for steps in a random walk on a graph, we generalize the steps to be biased and compute several statistics related to biased random walks on a cycle graph, more fancifully known as the ``ladybug clock problem." We obtain new explicit formulas for the distribution of probabilities for a walk to end on a particular node, the expected cover time for the cycle, and other similar statistics. We carry out these computations solely with generating functions related to the classic gambler's ruin problem.

Computing The Greatest Common Divisor Of Binomial Coefficients C(Mn, Mk), Chai Wah Wu

Communications on Number Theory and Combinatorial Theory

The greatest common divisor (GCD) of C(2n,2k) for 1 ≤ k < n is known to be some power of 2 times the product of all odd primes p such that 2n = pⁱ + p^j. We complete the analysis of this GCD by showing that its 2-adic valuation is at most 1 and relates it to Mersenne primes. We also show how to efficiently compute GCD {C(mn,mk) : 1 ≤ k < n} when n and m satisfy certain conditions.

On A Remark Of De Gennes Concerning Three-Dimensional Polyelectrolytes, Shiquan Li, Carl Mueller

Journal of Stochastic Analysis

This work is inspired by a remark of de Gennes [3] about polyelectrolytes, which are charged polymers. A common model for a polymer is a self-avoiding or self-repelling random walk or Brownian motion. For polyelectrolytes, the repelling potential is the Coulomb potential arising from pairs of charged particles. We show that in the continuous case of a Brownian motion in three dimensions, the spread of the polymer, in particular the radius of gyration of a polyelectrolyte of length T grows linearly with T, up to logarithmic corrections.

A Note On The Voronoi Congruences And The Residue Of The Fermat Quotient, Claire Levaillant

Communications on Number Theory and Combinatorial Theory

We prove a congruence on the residue of the Fermat quotient in base a which arises from a generalization of the Voronoi congruences and other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we recover a congruence for the generalized harmonic number H_2,(p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known for the harmonic number H_(p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruences that is different from the one of …

On Graphs With Chromatic Number And Maximum Degree Both Equal To Nine, Rachel Galindo, Jessica Mcdonald

Theory & Applications of Graphs

An equivalent version of the Borodin-Kostochka Conjecture, due to Cranston and Rabern, says that any graph with $\chi = \Delta = 9$ contains $K_3 \lor E_6$ as a subgraph. Here we prove several results in support of this conjecture, where vertex-criticality and forbidden substructure conditions get us either close or all the way to containing $K_3 \lor E_6$.

Expressing Matrix Powers With Pell Numbers, Walby Encarnacion Cena

Rose-Hulman Undergraduate Mathematics Journal

Abstract. Motivated by the theory of Cayley hash functions, we aim to write a matrix as a product of another set of given matrices. Riley gave such an algorithm using Fibonacci numbers. We extend Riley’s work to the Pell numbers. We compare our algorithm to Riley’s method and find that it is an improvement in certain cases.

Beyond Intent: Teacher Capacity, Opportunity To Learn, And The Persistence Of Math Equity Gaps In Tennessee (2000-2025), Soso Dede

The Journal of the Research Association of Minority Professors

For a quarter-century (2000–2025), Tennessee pursued mathematics reforms aiming for equity, yet stark racial and socioeconomic achievement gaps endure. Framed by Opportunity to Learn (OTL) and Mathematical Knowledge for Teaching (MKT), this study investigates the intersection of equity, teacher capacity, and policy. By synthesizing assessment data, program evaluations, policy documents, and research literature, this study reveals that systemic OTL inequities—disparate access to quality instruction, adequate resources, and teachers possessing strong MKT—are primary drivers of these enduring gaps. Such inequities deny marginalized students a fair chance at mathematical proficiency. The interplay between OTL and MKT is critical: inadequate OTL constrains knowledgeable …

Exact Formulas For Restricted Coprime Representations Of Even Integers, Andres Mauricio Salazar

Communications on Number Theory and Combinatorial Theory

Let p ≥ 5 be prime, and let g(2n,p) denote the number of unordered representations by positive integers 2n = h + k, with h ≤ k and gcd(h,6p) = gcd(k,6p) = 1. Every positive integer coprime to 6 belongs to exactly one of the families 6z + 1 and 6z + 5. In each family, divisibility by p excludes one explicit residue class of z modulo p. We use these two excluded classes to derive an exact formula for g(2n,p) from finite residue counts and nonnegative lift counts. The formula is valid for every positive integer n, including all …

(Si17-06) Green-Naghdi Theory Of Two-Temperature Fractional Thermoelasticity Under Thermal Shock And Ramp-Type Heating, Sagar Sankeshwari, Vinayak Kulkarni

Applications and Applied Mathematics: An International Journal (AAM)

The linear thermoelasticity theory without energy dissipation developed by Green-Naghdi has been reconstructed by using two temperature theory in the context of a time fractional derivative. In half-space, the mathematical model is discussed for one-dimensional wave propagation subject to two different problems, viz., thermal shock and ramp-type heating at the bounding surface. Assume the bounding surface is traction-free. The analytical solutions are obtained in the Laplace domain. The Gaver-Stehfest technique is numerically used to perform time domain inversion of the Laplace transform, satisfying Kuznetsov’s convergence condition. The impact of fractional order parameters on the variations of various fields inside the …

(Si17-15) On Becker’S Conjecture, Jitender ., Shiv Datt Kumar

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we explore identities that involve higher-degree versions of widely recognized composition formulae, specifically focusing on expressions of the form

(x^k₁ + x^k₂ + · · · + x^k_r ) · (y^k₁ + y^k₂ + · · · + y^k_s ) = (z^k₁ + z^k₂ + · · · + z^k_n),

where k ≥ 2, y = (y₁, y₂, . . . , y_s), x = (x₁, x₂, . …

(Si17-05) Properties Of The Fermatean Fuzzy Matrix And Its Applications, Vineet Kumar, Anjana Gupta, H. C. Taneja

Applications and Applied Mathematics: An International Journal (AAM)

All real-world problems involve a significant amount of imprecision. To deal with the unpredictable environment, various methods and tools have been introduced. The Fermatean fuzzy matrix (FFM) is one of the important concepts for dealing with imprecision. In this paper, we explore the concept of Fermatean fuzzy matrix and introduce certain significant definitions. Some propositions and theorems of Fermatean fuzzy matrix are introduced with their proofs. The adjoint and determinant methods of Fermatean fuzzy matrices are also introduced. Finally, we have proposed a similarity measure and illustrated it via an application of Fermatean fuzzy matrix in decision-making problems.

Implementation Of The Problem-Based Learning Model On The Topic Of The Immune System To Enhance The Critical Thinking Skills Of Grade Xi B Students At Sman 1 Tojo, Nani Sofyanti, Sutrisnawati Sutrisnawati, Vita Indri Febriani, Mohammad Jamhari, Bustamin Bustamin, Raya Agni

Jurnal Pendidikan Sains

Critical-thinking skills are essential competencies for students in 21st-century learning. Preliminary classroom observations in Class XI B at SMAN 1 Tojo indicated that instruction remained predominantly teacher-centered and provided limited opportunities for students to engage in activities involving interpretation, analysis, evaluation, and inference. This study aimed to improve students’ critical-thinking skills through the implementation of Problem-Based Learning (PBL) on the immune system topic. The study employed a Classroom Action Research (CAR) design conducted over two cycles, involving 28 students in Class XI B. Data were collected using a teacher activity observation sheet and critical-thinking skills tests assessing four indicators: interpretation, …

Modelling Acute Haemorrhagic Conjunctivitis In East Africa: An Integrated Approach To Transmission Dynamics And Outbreak Response Strategies, Geofrey Wingi Sikazwe

Tanzania Journal of Science

Acute hemorrhagic conjunctivitis (AHC) poses a significant global health risk, with the potential for rapid spread during outbreaks. For instance, a localized outbreak in Yunnan Province, China, recorded infection rates of up to 48%, illustrating how transmission can intensify under favourable epidemiological and environmental conditions. Existing mathematical models of AHC transmission overlook seasonal variations, a critical factor in epidemic spread. This study combines species distribution models (SDMs) and epidemiological modelling to analyse key drivers of AHC in East Africa. We use the SEIR type (Susceptible-Exposed-Infected-Recovered - incorporating waning immunity) framework with temperature-dependent infectivity rate as the key driver of an …

The Use Of Gadgets And Learning Motivation Among Grade Xi Students In Biology Learning At Sma Negeri 5 Palu, Yulia Citra, Mohammad Jamhari, Masrianih Masrianih, Lestari M.P Alibasyah, Mursito S. Bialangi, Yulia Windarsih

Jurnal Pendidikan Sains

This study aims to describe the profile of gadget use and learning motivation among Grade XI students in biology learning at SMA Negeri 5 Palu and to explore students’ and teachers’ experiences of gadgets as learning support tools and potential sources of digital distraction. A qualitative descriptive design was employed, supported by descriptive questionnaire profiling. A total of 74 students completed a 50-item questionnaire on gadget use and learning motivation, while 10 students and two biology teachers participated in semi-structured interviews. Questionnaire data were analyzed using frequencies and percentages, while interview data were analyzed thematically. The descriptive findings showed that …

The Application Of The Group Investigation (Gi) Model Of Cooperative Learning To Improve Students’ Understanding Of Biodiversity In Year 10 At Sman 2 Mori Atas, Nadia Rosmita Lumako, Mohammad Jamhari, Yulia Windarsih, Lilies Lilies, Rafiqa Rafiqa, Amalia Buntu

Jurnal Pendidikan Sains

Biology instruction at SMAN 2 Mori Atas was predominantly conducted through conventional teaching practices. Preliminary observations revealed that students had difficulty understanding biodiversity concepts, and several students did not meet the minimum passing criterion of 70. These conditions highlighted the need for learning activities that promote active investigation, discussion, and conceptual connections. This classroom action research examined the use of the Group Investigation (GI) cooperative learning model to improve Grade 10 students’ classroom engagement and conceptual understanding of biodiversity, as assessed through Higher-Order Thinking Skills (HOTS) questions based on the Structure of Observed Learning Outcomes (SOLO) taxonomy. The study involved …

The Effect Of The Problem-Based Learning Model On Biology Learning Outcomes In The 8th Grade Science Class At Smp Negeri 19 Palu, Delfi Srieviyani, Mursito S. Bialangi, Masrianih Masrianih, Abd Hakim Laenggeng, Vita Indri Febriani, Bustamin Bustamin

Jurnal Pendidikan Sains

Research background: The low student achievement in biology at SMP Negeri 19 Palu specifically regarding the human respiratory system is attributed to the conventional teaching methods still employed by teachers. Purpose and scope of the article: This study examines the effect of the Problem-Based Learning (PBL) model on students’ learning outcomes in biology, specifically regarding the human respiratory system. Method: This study used a quasi-experimental design with a quantitative approach. The research design employed was a nonequivalent control group design using two groups: an experimental group (Class VIII D) consisting of 25 students who received the PBL model, and a …

Differential Equation Modeling For Sustainable Resource Management: A Steady-State Optimal Harvesting Approach, Iordanka N. Panayotova, Aleksei Talonov

CODEE Journal

Mathematical models based on differential equations provide a powerful framework for connecting real-world data to informed decision-making. In this work, we present a student-accessible project that uses an optimal-control framework to study the sustainable management of biological resources.

Motivated by fisheries management, we examine a predator--prey system in which harvesting decisions must balance ecological and economic considerations. The model is formulated as an optimal control problem that seeks to maximize the total discounted net revenue from harvesting. Rather than solving for the complete time-dependent harvesting trajectory, we restrict the analysis to positive controlled coexistence equilibria and characterize an interior stationary …

How Euler Could Have Done It: Euler And An Integral Of Ramanujan, Alexander Aycock

Euleriana

We present a derivation of a definite integral discovered by Ramanujan (1887–1920) in his 1915 paper "Some Definite Integrals," from formulas and ideas already developed by Euler (1707–1783). Additionally, it is argued why Euler did not discover Ramanujan's integral himself, although he had all tools required for this task at his disposal.

A Note On Euler's Definition Of The Sum Of A Divergent Series, Alexander Aycock

Euleriana

We outline Euler's theory of divergent series and present another approach he could have taken to sum his ``factorial series". Finally, we address the contradictions that arise from his concept of a sum of a divergent series.

Euler's Explorations Of Extremal Ellipses, Jonathan David Evans

Euleriana

In the 1770s, Euler wrote a series of papers (E563, E691 and E692) about finding the ellipse with minimal area or perimeter in the family of all ellipses passing through a fixed set of points. This is a translation of all three papers from the original Latin, together with a commentary which discusses Euler's results and an appendix which addresses a question from E691 which Euler left for others to consider.

Arithmetic Theorems, Demonstrated By A New Method: An English Translation Of E271, Brendon Lasell

Euleriana

After defending in general terms the value of arithmetic demonstrations, where, according to Euler, the force of genius shines more brightly than in any other kind of demonstration, he goes on to let the force of his own genius shine forth in demonstrating the fact that, for any pair of relatively prime numbers a and N, a^ϕ(N) − 1 is always divisible by N, where we anachronistically denote by ϕ(N) the number of numbers less than N that are relatively prime to it. He approaches this theorem by considering the remainders that result when the numbers in an arithmetic …

On Double Integral Formulas: An English Translation Of E391, Thomas W. Polaski

Euleriana

In this paper, Euler investigates the theory of double intergrals to uses them to find areas, volumes and surface areas. He considers transformations of variables in double integrals. After showing that simple multiplication of the transformed differentials is not appropriate, he derives the formula for a change of variables in double integrals. Euler then uses such changes of variables to investigate areas and volumes, and then turns his attention to the so-called ``Florentine problem.'' This problem requires one to remove four equal windows from a hemispherical surface so that the remaining area is equal to the area of a square. …

Eulerian Problem Solving, Christopher Goff, Erik Tou

Euleriana

An introduction to the contents in Volume 6 (Issue 2) of Euleriana.

On The Existence, Uniqueness And Stability Of Solutions Of Sdes With State-Dependent Variable Exponent, Mustafa Avci

Journal of Stochastic Analysis

We study a time-inhomogeneous nonlinear SDE with drift and diffusion governed by state-dependent variable exponents. This framework generalizes models like the geometric Brownian motion (GBM) and the constant elasticity of variance (CEV), offering flexibility to capture complex dynamics while posing analytical challenges. Using a fixed-point approach, we prove existence and uniqueness, analyze higher-order moments, derive asymptotic estimates, and assess stability. Finally, we illustrate an application where Poisson’s equation admits a probabilistic representation via a timehomogeneous nonlinear SDE with state-dependent variable exponents.

Companion Matrices Associated To Stochastic Matrices, Andreas Boukas, Philip Feinsilver

Journal of Stochastic Analysis

Starting with a stochastic matrix, we study the behavior of powers of an associated companion matrix, which has the same characteristic polynomial as the original matrix. In general the companion matrix will have negative entries while maintaining rowsums equal to 1. We will find the growth rate even if the Ces`aro limit of the sums of the companion matrix diverge. Surprisingly, in the irreducible aperiodic case the powers of the companion matrix will converge even though the norm of the matrix exceeds 1 and it has possibly negative entries.

Z+-Valued Additive Processes In Law With Nonnegative Increments, Nadjib Bouzar

Journal of Stochastic Analysis

The goal of this article is to study in depth the subclass of Z+- valued additive processes in law with nonnegative increments. We establish key distributional properties of these processes and obtain some existence results under a variety of conditions. We show that they form a subclass of the family of inhomogeneous Markov chains with spatial homogeneity. We extend the notion of factoring introduced by Sato (2004, [9]) for Rd-valued additive processes in law to their Z+-valued counterparts with nonnegative increments. We give a sufficient condition for the existence of a factoring for these processes. Lastly, we obtain their representation …

Motivating Second-Order Differential Equation Models Beyond The Spring–Mass System, Katherine E. Rutherford, John C. Borger

CODEE Journal

Students in introductory differential equations courses often learn to associate specific equation types with familiar physical systems. While this approach supports procedural fluency, it can inhibit transfer learning: students recognize familiar forms rather than construct models from first principles. This paper presents a modeling-first instructional approach designed to help students recognize the shared structure of second-order differential equations across diverse contexts. Students derive governing equations from fundamental physical laws for multiple systems, including an Inductor, Resistor, and Capacitor (LRC) electrical circuit, a simple pendulum, and viscous pipe flow. Through guided comparison, students identify common components such as inertia, damping, restoring …