Mathematics Commons^™

The Circle Of Life: The Mathematics Of Predator-Prey Dynamics, John Butler, Rebecca Brady

Articles

Some animals hunt other animals to feed themselves; these animals are called predators. Animals who are hunted and eaten are known as prey. What do you think would happen if a predator were introduced to an ecosystem where the prey previously lived without fear of being hunted? Would the new predator eat all the prey animals until they go extinct? Actually, the relationship between predator and prey is far more interesting than this. In this article, we show what the predator-prey relationship looks like over time and explain how scientists can make predictions about future population levels, all using basic …

Profiling Mathematical Procedural And Problem-Solving Skills Of Undergraduate Students Following A New Mathematics Curriculum, Fiona Faulkner, Mark Prendergast, Cormac Breen, Michael Carr

Articles

In 2010 a mathematics curriculum was introduced in Irish second level schools entitled ‘Project Maths’ (PM). It aimed to refocus second level mathematics teaching and learning away from an over emphasis on procedural mathematics towards problem solving and real understanding [Department of Education and Skills (DES). (2010). Report of the Project Maths implementation support group. https://www.education.ie/en/Publications/Policy-Reports/Report-of-the-Project-Maths-Implementation-Group.pdf]. This paper aims to examine the performance of 1st year undergraduate students’ procedural and problem solving skills after the introduction of PM. A diagnostic test was developed to determine students’ skills in each area and findings demonstrated that students perform statistically significantly …

On The Socles Of Fully Inert Subgroups Of Abelian P-Groups, Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith

Articles

We define the so-called fully inert socle-regular and weakly fully inert socle-regular Abelian p-groups and study them with respect to certain of their numerous interesting properties. For instance, we prove that in the case of groups of length ω, these two group classes coincide, but that in the case of groups of length ω+1, they differ. Some structural and characterization results are also obtained. The work generalizes concepts which have been of interest recently in the theory of entropy in algebra and builds on recent investigations by Danchev and Goldsmith (Arch Math (3) 92:191–199, 2009; J Algebra 323:3020–3028, 2010).

On The Socles Of Characteristically Inert Subgroups Of Abelian P-Groups, Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith

Articles

We define the notion of a characteristically inert socle-regular Abelian p-group and explore such groups by focussing on their socles, thereby relating them to previously studied notions of socle-regularity. We show that large classes of p-groups, including all divisible, totally projective and torsion-complete p-groups, share this property when the prime p is odd. The present work generalizes notions of full inertia intensively studied recently by several authors and is a development of a recent work of the authors published in Mediterranean J. Math. (2021).

Metrics For Performance Quantification Of Adaptive Mesh Refinement, Nicole Beisiegel, Cristóbal E. Castro, Jörn Behrens

Articles

Non-uniform, dynamically adaptive meshes are a useful tool for reducing computational complexities for geophysical simulations that exhibit strongly localised features such as is the case for tsunami, hurricane or typhoon prediction. Using the example of a shallow water solver, this study explores a set of metrics as a tool to distinguish the performance of numerical methods using adaptively refined versus uniform meshes independent of computational architecture or implementation. These metrics allow us to quantify how a numerical simulation benefits from the use of adaptive mesh refinement. The type of meshes we are focusing on are adaptive triangular meshes that are …

Using A Hybrid Agent-Based And Equation Based Model To Test School Closure Policies During A Measles Outbreak, Elizabeth Hunter, John D. Kelleher

Articles

Background

In order to be prepared for an infectious disease outbreak it is important to know what interventions will or will not have an impact on reducing the outbreak. While some interventions might have a greater effect in mitigating an outbreak, others might only have a minor effect but all interventions will have a cost in implementation. Estimating the effectiveness of an intervention can be done using computational modelling. In particular, comparing the results of model runs with an intervention in place to control runs where no interventions were used can help to determine what interventions will have the greatest …

Multicomponent Fokas-Lenells Equations On Hermitian Symmetric Spaces, Vladimir Gerdjikov, Rossen Ivanov

Articles

Multi-component integrable generalizations of the Fokas-Lenells equation, associated with each irreducible Hermitian symmetric space are formulated. Description of the underlying structures associated to the integrability, such as the Lax representation and the bi-Hamiltonian formulation of the equations is provided. Two reductions are considered as well, one of which leads to a nonlocal integrable model. Examples with Hermitian symmetric spaces of all classical series of types A.III, BD.I, C.I and D.III are presented in details, as well as possibilities for further reductions in a general form.

On The Evolution Equation For Modelling The Covid-19 Pandemic, Jonathan Blackledge

Books/Book chapters

The paper introduces and discusses the evolution equation, and, based exclusively on this equation, considers random walk models for the time series available on the daily confirmed Covid-19 cases for different countries. It is shown that a conventional random walk model is not consistent with the current global pandemic time series data, which exhibits non-ergodic properties. A self-affine random walk field model is investigated, derived from the evolutionary equation for a specified memory function which provides the non-ergodic fields evident in the available Covid-19 data. This is based on using a spectral scaling relationship of the type 1/ωα where ω …

A Review Of The Fractal Market Hypothesis For Trading And Market Price Prediction, Jonathan Blackledge, Marc Lamphiere

Articles

This paper provides a review of the Fractal Market Hypothesis (FMH) focusing on financial times series analysis. In order to put the FMH into a broader perspective, the Random Walk and Efficient Market Hypotheses are considered together with the basic principles of fractal geometry. After exploring the historical developments associated with different financial hypotheses, an overview of the basic mathematical modelling is provided. The principal goal of this paper is to consider the intrinsic scaling properties that are characteristic for each hypothesis. In regard to the FMH, it is explained why a financial time series can be taken to be …

Isolation Intervals Of The Real Roots Of The Parametric Cubic Equation, Emil Prodanov

Articles

The isolation intervals of the real roots of the real symbolic monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c$ are found in terms of simple functions of the coefficients of the polynomial (such as: $-a$, $-a/3$, $-c/b$, $\pm \sqrt{-b}$, when $b$ is negative), and the roots of some auxiliary quadratic equations whose coefficients are also simple functions of the coefficients of the cubic. All possible cases are presented with clear and very detailed diagrams. It is very easy to identify which of these diagrams is the relevant one for any given cubic equation and …

The Siebeck-Marden-Northshield Theorem And The Real Roots Of The Symbolic Cubic Equation, Emil Prodanov

Articles

The isolation intervals of the real roots of the symbolic monic cubic polynomial x 3 ` ax2 ` bx ` c are determined, in terms of the coefficients of the polynomial, by solving the Siebeck–Marden–Northshield triangle — the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial.

On Newton's Rule Of Signs, Emil Prodanov

Articles

Analysing the cubic sectors of a real polynomial of degree n, a minor modification of Newton’s Rule of signs is proposed with which stricter upper limits on the number of real roots can be found. A new necessary condition for reality of the roots of a polynomial is also proposed. Relationship between the quadratic elements of the polynomial is established through its roots and those of its derivatives. Some aspects of polynomial discriminants are also discussed.

Semiclassical Backreaction On Asymptotically Anti–De Sitter Black Holes, Peter Taylor, Cormac Breen

Articles

We consider a quantum scalar field on the classical background of an asymptotically anti–de Sitter black hole and the backreaction the field’s stress-energy tensor induces on the black hole geometry. The backreaction is computed by solving the reduced-order semiclassical Einstein field equations sourced by simple analytical approximations for the renormalized expectation value of the scalar field stress-energy tensor. When the field is massless and conformally coupled, we adopt Page’s approximation to the renormalized stress-energy tensor, while for massive fields we adopt a modified version of the DeWitt-Schwinger approximation. The latter approximation must be modified so that it possesses the correct …

A Method For Locating The Real Roots Of The Symbolic Quintic Equation Using Quadratic Equations, Emil Prodanov

Articles

A method is proposed with which the locations of the roots of the monic symbolic quintic polynomial $x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$ can be determined using the roots of two {\it resolvent} quadratic polynomials: $q_1(x) = x^2 + a_4 x + a_3$ and $q_2(x) = a_2 x^2 + a_1 x + a_0$, whose coefficients are exactly those of the quintic polynomial. The different cases depend on the coefficients of $q_1(x)$ and $q_2(x)$ and on some specific relationships between them. The method is illustrated with the full analysis of one of …

Five-Wave Resonances In Deep Water Gravity Waves: Integrability, Numerical Simulations And Experiments, Dan Lucas, Marc Perlin, Dian-Yong Liu, Shane Walsh, Rossen Ivanov, Miguel D. Bustamante

Articles

In this work we consider the problem of finding the simplest arrangement of resonant deep water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wave vectors K₁ + K₂ = K₃ (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wave packets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction …

Bayesian Dynamic Network Actor Models With Application To South Korean Covid-19 Patient Movement Data, Antonio Arrizza, Alberto Caimo

Articles

Motivated by the ongoing COVID-19 pandemic, this article intro- duces Bayesian dynamic network actor models for the analysis of infected individuals’ movements in South Korea during the first three months of 2020. The relational event data modelling framework makes use of network statistics capturing the structure of movement events from and to several country’s municipalities. The fully probabilistic Bayesian approach allows to quantify the uncertainty associated to the relational tendencies explaining where and when movement events are established and where they are directed. The observed patient movements’ patterns at an early stage of the pandemic can provide interesting insights about …

Collaborations In Environmental Initiatives For An Effective Gover- Nance Of Social-Ecological Systems: What The Scientific Literature Suggests., Elena Andriollo, Alberto Caimo, Laura Secco, Elena Pisani

Articles

Moving from the scientific literature on evaluation of environmental projects and programs, this study identifies how and under which conditions collaborations are considered effective for adaptive gover- nance of SES. The method adopted is a systematic literature review based on the quantitative and qualitative analysis of 56 articles selected through specific queries on the SCOPUS database and published from 2004 to 2020. Results of the quantitative analysis underline conditions able to make collaborations effective for adaptive governance of SES: the importance of transdisciplinary research tackling both environmental and social sciences, the perceived urgency of stakeholders to tackle environmental challenges and …

Climate Change Impacts On Wind Energy Generation In Ireland, Eadaoin Doddy Clarke, Conor Sweeney, Frank Mcdermott, Seánie Griffin, Joao Monteiro Correia, Paul Nolan, Laura Cooke

Articles

An ensemble of high-resolution regional climate model simulation data is used to examine the impacts of climate change on offshore and onshore wind energy genera- tion in Ireland. Two Representative Concentration Pathway (RCP) scenarios (RCP 4.5 and 8.5) are analysed for the mid-term (2041–2060) and the long-term (2081–2100) future. Wind energy is projected to decrease (≤2%) overall in future climate scenarios. Changes are evident by mid-century and are more pronounced by late 21st century, particularly for RCP 8.5 offshore. Seasonally, wind energy is projected to decrease by less than 6% in summer and to increase slightly in winter (up to …

Abelian P-Groups With Minimal Full Inertia, Brendan Goldsmith, Luigi Salce

Articles

The class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also …

On The Bassian Property For Abelian Groups, Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith

Articles

We define an object (group, ring, module, algebra, etc.) to be Bassian if it is not possible to embed it in a proper homomorphic image of itself. Here we study this concept for Abelian groups and achieve a complete characterization of all such groups in terms of their associated torsion-free and p-primary ranks.

On P-Adic Modules With Isomorphic Endomorphism Algebras, Brendan Goldsmith, Noel White

Articles

We investigate pairs of modules over the ring of p-adic integers having isomorphic endomorphism algebras. In many cases this forces the modules to be isomorphic but there are two exceptional situations where isomorphism does not follow.

Neurophysiological Correlates Of Dual Tasking In People With Parkinson's Disease And Freezing Of Gait, Conor Fearon, John Butler, Saskia Waechter, Isabelle Killane, Simon Kelly, Richard B Reilly, Timothy Lynch

Articles

Freezing of gait in people with Parkinson's disease (PwP) is associated with executive dysfunction and motor preparation deficits. We have recently shown that electrophysiological markers of motor preparation, rather than decision-making, differentiate PwP with freezing of gait (FOG +) and without (FOG -) while sitting. To examine the effect of locomotion on these results, we measured behavioural and electrophysiological responses in PwP with and without FOG during a target response time task while sitting (single-task) and stepping-in-place (dual-task). Behavioural and electroencephalographic data were acquired from 18 PwP (eight FOG +) and seven young controls performing the task while sitting and …

Quotient-Transitivity And Cyclic Subgroup-Transitivity, Brendan Goldsmith, Ketao Gong

Articles

We introduce two new notions of transitivity for Abelian 𝑝-groups based on isomorphism of quotients rather than the classical use of equality of height sequences associated with Abelian 𝑝-group theory. Unlike the classical theory where “most” groups are transitive, these new notions lead to much smaller classes, but even these classes are sufficiently large to be interesting.

A Hybrid Agent-Based And Equation Based Model For The Spread Of Infectious Diseases, Elizabeth Hunter, Brian Mac Namee, John D. Kelleher

Articles

Both agent-based models and equation-based models can be used to model the spread of an infectious disease. Equation-based models have been shown to capture the overall dynamics of a disease outbreak while agent-based models are able to capture heterogeneous characteristics of agents that drive the spread of an outbreak. However, agent-based models are computationally intensive. To capture the advantages of both the equation-based and agent-based models, we create a hybrid model where the disease component of the hybrid model switches between agent-based and equation-based. The switch is determined using the number of agents infected. We first test the model at …

New Bounds On The Real Polynomial Roots, Emil M. Prodanov

Articles

The presented analysis determines several new bounds on the roots of the equation $a_n x^n + a_{n−1} x^{n−1} + · · · + a_0 = 0$ (with $a_n > 0$). All proposed new bounds are lower than the Cauchy bound max $\{ 1, sum_{j=0}^{n-1} | a_j / a_n | \}$. Firstly, the Cauchy bound formula is derived by presenting it in a new light — through a recursion. It is shown that this recursion could be exited at earlier stages and, the earlier the recursion is terminated, the lower the resulting root bound will be. Following a separate analysis, it is …

Italian Sociologists: A Community Of Disconnected Groups, Aliakbar Akbaritabar, Vincent Traag, Alberto Caimo, Flaminio Squazzoni

Articles

Examining coauthorship networks is key to study scientific collaboration patterns and structural characteristics of scientific communities. Here, we studied coauthorship networks of sociologists in Italy, using temporal and multi-level quantitative analysis. By looking at publications indexed in Scopus, we detected research communities among Italian sociologists. We found that Italian sociologists are fractured in many disconnected groups. The giant connected component of the Italian sociology could be split into five main groups with a mixture of three main disciplinary topics: sociology of culture and communication (present in two groups), economic sociology (present in three groups) and general sociology (present in three …

On The Socles Of Fully Inert Subgroups Of Abelian P-Groups, Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith

Articles

We define the so-called fully inert socle-regular and weakly fully inert socle-regular Abelian p-groups and study them with respect to certain of their numerous interesting properties. For instance, we prove that in the case of groups of length ! these two group classes coincide but that in the case of groups of length ! + 1 they differ. Some structural and characterization results are also obtained. The work generalizes concepts which have been of interest recently in the theory of entropy in algebra and builds on recent investigations by the second and third named authors in Arch. Math. Basel (2009) …

New Estimates For The Number Of Integer Polynomials With Given Discriminants, Natalia Budarina, Vasilii Bernik, Hugh O'Donnell

Articles

In this paper, we propose a new method of upper bounds for the number of integer polynomials of the fourth degree with a given discriminant. By direct calculation similar results were established by H. Davenport and D. Kaliada for polynomials of second and third degrees.

Classification Of The Real Roots Of The Quartic Equation And Their Pythagorean Tunes, Emil Prodanov

Articles

Presented is a two-tier analysis of the location of the real roots of the general quartic equation x4+ax3+bx2+cx+d=0 with real coefficients and the classification of the roots in terms of a, b, c, and d, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and …

Saperi: Approaching Gender Gap Using Spatial Ability Training Week In High-School Context, Maria Giulia Ballatore, Gavin Duffy, Sheryl Sorby, Anita Tabacco

Articles

Maria Giulia Ballatore, Gavin Duffy, Sheryl Sorby, and Anita Tabacco. 2020. SAperI: approaching gender gap using Spatial Ability training week in high-school context. In Eighth International Conference on Technological Ecosystems for Enhancing Multiculturality (TEEM’20), October 21–23, 2020, Salamanca, Spain. ACM, New York, NY, USA, 7 pages. https://doi.org/10.1145/3434780.3436577