Mathematics Commons^™

A Central Limit Theorem For The Number Of Excursion Set Components Of Gaussian Fields, Dmitry Beliaev, Michael Mcauley, Stephen Muirhead

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For a smooth stationary Gaussian field f on R^d and level ℓ ∈ R, we consider the number of connected components of the excursion set {f ≥ ℓ} (or level set {f = ℓ}) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for ‘additive’ geometric functionals of the excursion/level sets (e.g. the volume or …

Signatures Of Time Interval Reproduction In The Human Electroencephalogram (Eeg), Harvey Mccone, John. S. Butler, Redmond. G. O'Connell

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No abstract provided.

Neural Correlates Of Audiovisual Narrative Speech Perception In Children And Adults On The Autism Spectrum: A Functional Magnetic Resonance Imaging Study, Lars A. Ross, Sophie Molholm, John S. Butler, Victor A. Del Bene, Tufikameni Brima, John J. Foxe

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Autistic individuals show substantially reduced benefit from observing visual articulations during audiovisual speech perception, a multisensory integration deficit that is particularly relevant to social communication. This has mostly been studied using simple syllabic or word-level stimuli and it remains unclear how altered lower-level multisensory integration translates to the processing of more complex natural multisensory stimulus environments in autism. Here, functional neuroimaging was used to examine neural correlates of audiovisual gain (AV-gain) in 41 autistic individuals to those of 41 age-matched non-autistic controls when presented with a complex audiovisual narrative. Participants were presented with continuous narration of a story in auditory-alone, …

Probing A Neural Unreliability Account Of Auditory Sensory Processing Atypicalities In Rett Syndrome, Tufikameni Brima, Shlomit Beker, Kevin D. Prinsloo, John Butler, Aleksandra Djukic, Edward G. Freedman, Sophie Molholm, John J. Foxe

Articles

Background: In the search for objective tools to quantify neural function in Rett Syndrome (RTT), which are crucial in the evaluation of therapeutic efficacy in clinical trials, recordings of sensory-perceptual functioning using event-related potential (ERP) approaches have emerged as potentially powerful tools. Considerable work points to highly anomalous auditory evoked potentials (AEPs) in RTT. However, an assumption of the typical signal-averaging method used to derive these measures is “stationarity” of the underlying responses – i.e. neural responses to each input are highly stereotyped. An alternate possibility is that responses to repeated stimuli are highly variable in RTT. If so, this …

Renormalized Stress-Energy Tensor For Scalar Fields In Hartle-Hawking, Boulware, And Unruh States In The Reissner-Nordström Spacetime, Julio Arrechea, Cormac Breen, Adrian Ottewill, Peter Taylor

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In this paper, we consider a quantum scalar field propagating on the Reissner-Nordström black hole spacetime. We compute the renormalized stress-energy tensor for the field in the Hartle-Hawking, Boulware and Unruh states. When the field is in the Hartle-Hawking state, we renormalize using the recently developed “extended coordinate” prescription. This method, which relies on Euclidean techniques, is very fast and accurate. Once, we have renormalized in the Hartle-Hawking state, we compute the stress-energy tensor in the Boulware and Unruh states by leveraging the fact that the difference between stress-energy tensors in different quantum states is already finite. We consider a …

Determining The Proportionality Of Ischemic Stroke Risk Factors To Age, Elizabeth Hunter, John D. Kelleher

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While age is an important risk factor, there are some disadvantages to including it in a stroke risk model: age can dominate the risk score and lead to over-or under-predictions in some age groups. There is evidence to suggest that some of these disadvantages are due to the non-proportionality of other risk factors with age, eg, risk factors contribute differently to stroke risk based on an individual’s age. In this paper, we present a framework to test if risk factors are proportional with age. We then apply the framework to a set of risk factors using Framingham heart study data …

The Pierce Decomposition And Pierce Embedding Of Endomorphism Rings Of Abelian P-Groups, Brendan Goldsmith, Luigi Salce

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We prove that more classes of groups than those described by Pierce have the property that the map Ψ is surjective, and we furnish examples of groups which do not have this property. Several results connecting the Pierce decomposition and the Pierce embedding of End(G) are obtained that allow one to derive general conditions on a group G which ensure that the Pierce embedding of End(G) is not surjective.

Duality For Spaces Of Polynomials On Tree Spaces, Christopher Boyd, Costas Poulios, Milena Venkova

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In this paper we study the duality of certain spaces of polynomials on a class of tree spaces. In particular, we are interested in establishing isometric isomorphisms between the spaces of integral and nuclear polynomials and in determining when all polynomials are weakly continuous on bounded sets. This will provide us with new examples of what are known as Q-reflexive spaces.

On The Cubic Equation With Its Siebeck–Marden–Northshield Triangle And The Quartic Equation With Its Tetrahedron, Emil Prodanov

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The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck–Marden–Northshield equilateral triangle and regular tetrahedron. The Viète trigonometric formulæ for the roots of the cubic are established through the rotation of the triangle by variation of the free term of the cubic. A very detailed complete root classification for the quartic 𝑥4 + 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 is proposed for which the conditions are imposed on the individual coefficients 𝑎, 𝑏, 𝑐, and 𝑑. The maximum and minimum lengths of the interval containing the four real roots of …

On The Modelling Of Short And Intermediate Water Waves, Rossen I. Ivanov

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The propagation of water waves of finite depth and flat bottom is studied in the case when the depth is not small in comparison to the wavelength. This propagation regime is complementary to the long-wave regime described by the famous KdV equation. The Hamiltonian approach is employed in the derivation of a model equation in evolutionary form, which is both nonlinear and nonlocal, and most likely not integrable. Possible implications for the numerical solutions are discussed.

Nonlinear Two-Dimensional Water Waves With Arbitrary Vorticity, Delia Ionescu-Kruse, Rossen Ivanov

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We consider the two-dimensional water-wave problem with a general non-zero vorticity field in a fluid volume with a flat bed and a free surface. The nonlinear equations of motion for the chosen surface and volume variables are expressed with the aid of the Dirichlet-Neumann operator and the Green function of the Laplace operator in the fluid domain. Moreover, we provide new explicit expressions for both objects. The field of a point vortex and its interaction with the free surface is studied as an example. In the small-amplitude long-wave Boussinesq and KdV regimes, we obtain appropriate systems of coupled equations for …

Turbulence Phenomena In Magnetohydrodynamic Phase Transitions, Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

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The model developed in (Fabrizio in J. Eng. Math., 2023) and (Fabrizio in Int. J. Eng. Sci. 44:529–539, 2006), involving the use of a local Reynolds number, is applied to describe phase transitions in a fluid. Specifically, it is applied in a magnetohydrodynamics context to study the evolution of turbulence in certain phenomena. The relevant equations describing the system are those of Navier-Stokes, Ginzburg-Landau and the magnetohydrodynamic equations, all suitably interconnected.

A Covariance Formula For The Number Of Excursion Set Components Of Gaussian Fields And Applications, Dmitry Beliaev, Michael Mcauley, Stephen Muirhead

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We derive a covariance formula for the number of excursion or level set components of a smooth stationary Gaussian field on R^d contained in compact domains. We also present two applications of this formula: (1) for fields whose correlations are integrable we prove that the variance of the component count in large domains is of volume order and give an expression for the leading constant, and (2) for fields with slower decay of correlation we give an upper bound on the variance which is of optimal order if correlations are regularly varying, and improves on best-known bounds if correlations …

On The Modelling Of Short And Intermediate Water Waves, Rossen Ivanov

Articles

Most of the model equations for water waves are approximations for the long-wave propagation regimes, since most of the energy of the wave motion is concentrated in these waves. Long waves (or shallow-water waves) are defined usually as the depth to wavelength ratio δ = h/λ < 0.05. Several famous integrable nonlinear equations, like the K d V equation [1,2], are models for long waves of small amplitude. The short waves (or waves over deep water) are usually defined with δ > 0.5, and the intermediate waves (or transitional waves) - with 0.05 < δ < 0.5. The intermediate and short waves received a lot less attention, and one reason is perhaps the fact that the corresponding approximations lead to more complicated, nonlinear and nonlocal equations. In [3] an integral equation for surface waves has been proposed for arbitrary wavelengths and finite depth. The problem has been studied in [4] and model equations both for long and short waves are derived from the governing equations as well. The short-wave effects usually compete with the capillarity effects and then resonances can be observed — these have been studied quite a lot, see for example [5–12]. For the intermediate long waves or for waves on deep water the so-called Benjamin–Ono (BO) [13–15] and the Intermediate Long Wave Equation (ILWE) [16–18] are derived for the internal waves below a flat surface, which leads to some simplifications and these models are in fact integrable.

The Lagrangian Formulation For Wave Motion With A Shear Current And Surface Tension, Conor Curtin, Rossen Ivanov

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The Lagrangian formulation for the irrotational wave motion is straightforward and follows from a Lagrangian functional which is the difference between the kinetic and the potential energy of the system. In the case of fluid with constant vorticity, which arises for example when a shear current is present, the separation of the energy into kinetic and potential is not at all obvious and neither is the Lagrangian formulation of the problem. Nevertheless, we use the known Hamiltonian formulation of the problem in this case to obtain the Lagrangian density function, and utilising the Euler-Lagrange equations we proceed to derive some …

A Mode-Sum Prescription For The Renormalized Stress Energy Tensor On Black Hole Spacetimes, Peter Taylor, Cormac Breen, Adrian Ottewill

Articles

In this paper, we describe an extremely efficient method for computing the renormalized stress-energy tensor of a quantum scalar field in spherically symmetric black hole spacetimes. The method applies to a scalar field with arbitrary field parameters. We demonstrate the utility of the method by computing the renormalized stress-energy tensor for a scalar field in the Schwarzschild black hole spacetime, applying our results to discuss the null energy condition and the semiclassical backreaction.

Corner’S Theorem On Modules With Anti-Isomorphic Endomorphism Algebras, Brendan Goldsmith, Noel White

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We present a version of an unpublished result of A.L.S. Corner on p-adic modules with anti-isomorphic endomorphism algebras. The result gives a complete description of necessary conditions for two such modules to have anti-isomorphic endomorphism algebras and a sufficient condition is also given. A main difference in the current version is that extensive use is made of our ability to describe certain homomorphism groups.

Hamiltonian Approach To Modelling Interfacial Internal Waves Over Variable Bottom, Rossen Ivanov, Calin Martin, Michail Todorov

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We study the effects of an uneven bottom on the internal wave propagation in the presence of stratification and underlying non-uniform currents. Thus, the presented models incorporate vorticity (wave–current interactions), geophysical effects (Coriolis force) and a variable bathymetry. An example of the physical situation described above is well illustrated by the equatorial internal waves in the presence of the Equatorial Undercurrent (EUC). We find that the interface (physically coinciding with the thermocline and the pycnocline) satisfies in the long wave approximation a KdV–mKdV type equation with variable coefficients. The soliton propagation over variable depth leads to effects such as soliton …

Neuromatch Academy: A 3-Week, Online Summer School In Computational Neuroscience, Bernard Marius 'T Hart, Titipat Achakulvisut, Ayoade Adeyemi, Athena Akrami, Bradly Alicea, Alicia Alonso-Andres, Diego Alzate-Correa, Arash Ash, Jesus J. Ballesteros, Aishwarya Balwani, Eleanor Batty, Ulrik Beierholm, Ari S. Benjamin, Upinder Bhalla, Gunnar Blohm, Joachim C. H. Blohm, Kathryn Bonnen, Marco Brigham, Bingni W. Brunton, John Butler, Brandon Caie, N Alex Cayco Gajic, Sharbatanu Chatterjee, Spyridon Chavlis, Ruidong Chen, You Cheng, H. M. Chow, Raymond Chua, Yunwei Dai, Isaac David, Eric E. J. Dewitt, Julien Denis, Alish Dipani, Arianna Dorschel, Jan Drugowitsch, Kshitij Dwivedi, Sean Escola, Haoxue Fan, Roozbeh Farhoodi, Yicheng Fei, Pierre-Étienne Fiquet, Lorenzo Fontolan, Jeremy Forest, Yuki Fujishima, Byron V. Galbraith, Mario Galdamez, Richard Gao, Julijana Gjorgjieva, Alexander Gonzalez, Qinglong Gu, Yueqi Guo, Ziyi Guo, Pankaj K. Gupta, Busra Tugce Gurbuz, Caroline Haimerl, Jordan B. Harrod, Alexandre Hyafil, Martin Irani, Daniel Jacobson, Michelle Johnson, Ilenna Simone Jones, Gili Karni, Robert E. Kass, Hyosub Edward Kim, Andreas M. Kist, Randal Koene, Konrad Kording, Matthew R. Krause, Arvind Kumar, Norma K. Kühn, Ray Lc, Matthew L. Laporte, Junseok Lee, Songting Li, Sikun Lin, Yang Lin, Shuze Liu, Tony Liu, Jesse A. Livezey, Linlin Lu, Jakob H. Macke, Kelly Mahaffy, A. Lucas Martins, Nicolás Martorell, Manolo Martínez, Marcelo G. Mattar, Jorge Aurelio Menendez, Kenneth D. Miller, Patrick J. Mineault, Nosratullah Mohammadi, Yalda Mohsenzadeh, Elenor Morgenroth, Taha Morshedzadeh, Alice Claudia Mosberger, Madhuvanthi Muliya, Marieke Mur, John D. Murray, Yashas Nd, Richard Naud, Prakriti Nayak, Anushka Oak, Itzel Olivos Castillo, Seyedmehdi Orouji, Jorge Otero-Millan, Marius Pachitariu, Biraj Pandey, Renato Paredes, Jesse Parent, Il Memming Park, Megan A. K. Peters, Xaq Pitkow, Panayiota Poirazi, Haroon Popal, Sandhya Prabhakaran, Tian Qiu, Srinidhi Ragunathan, Raul Rodriguez-Cruces, David Rolnick, Ashish Kumar Sahoo, Saeed Salehinajafabadi, Cristina Savin, Shreya Saxena, Paul Schrater, Karen Schroeder, Alice C. Schwarze, Madineh Sedigh-Sarvestani, K. Yuvaraj Sekhar, Reza Shadmehr, Maryam M. Shanechi, Siddhant Sharma, Eric Shea-Brown, Krishna V. Shenoy, Carolina L. Shimabukuro, Sergey Shuvaev, Man Ching Alison Sin, Maurice Smith, Nicholas A. Steinmetz, Karolina Stosio, Elizabeth Straley, Gabrielle Strandquist, Carsen Stringer, Rimjhim Tomar, Ngoc Tran, Sofia Triantafillou, Lawrence Udeigwe, Davide Valeriani, Vincent Valton, Maryam Vaziri-Pashkam, Peter Vincent, Gal Vishne, Pascal Wallisch, Peiyuan Wang, Claire Ward, Michael Waskom, Kunlin Wei, Anqi Wu, Zhengwei Wu, Brad Wyble, Lei Zhang, Daniel Zysman, Federico D’Oleire Uquillas, Tara Van Viegen

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No abstract provided.

Mathematical Analysis Of The Van Der Waals Equation, Emil Prodanov

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The parametric cubic van der Waals polynomial $p V^3 - (R T + b p) V^2 + a V - a b$ is analysed mathematically and some new generic features (theoretically, for any substance) are revealed: the temperature range for applicability of the van der Waals equation, $T > a/(4Rb)$, and the isolation intervals, at any given temperature between $a/(4Rb)$ and the critical temperature $8a/(27Rb)$, of the three volumes on the isobar--isotherm: $3b/2 < V_A \le 3b$, $ 2b < V_B < 4b/(3 - \sqrt{5})$, and $3b < V_C < b + RT/p$. The unstable states of the van der Waals model have also been generically localized: they lie in an interval within the isolation interval of $V_B$. In the case of unique intersection point of an isotherm with an isobar, the isolation interval of this unique volume is also determined. A discussion on finding the volumes $V_{A, B, C}$, on the premise of Maxwell's hypothesis, is also presented.

Viscoelastic And Electromagnetic Materials With Nonlinear Memory, Claudio Giorgi, John Murrough Golden

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A method is presented for generating free energies relating to nonlinear constitutive equations with memory from known free energies associated with hereditary linear theories. Some applications to viscoelastic solids and hereditary electrical conductors are presented. These new free energies are then used to obtain estimates for nonlinear integro-differential evolution problems describing the behavior of nonlinear plasmas with memory.

Free Energies For Nonlinear Materials With Memory, John Murrough Golden

Articles

An exploration of representations of free energies and associated rates of dissipation for a broad class of nonlinear viscoelastic materials is presented in this work. Also included are expressions for the stress functions and work functions derivable from such free energies. For simplicity, only the scalar case is considered. Certain standard formulae are generalized to include higher power terms.

It is shown that the correct initial procedure in this context is to specify the rate of dissipation as a positive semi-definite functional and then to determine the free energy from this, rather than the other way around, which would be …

Data Driven Bayesian Network To Predict Critical Alarm, Joseph Mietkiewicz, Anders Madsen

Articles

Modern industrial plants rely on alarm systems to ensure their safe and effective functioning. Alarms give the operator knowledge about the current state of the industrial plants. Trip alarms indicating a trip event indicate the shutdown of systems. Trip events in power plants can be costly and critical for the running of the operation.This paper demonstrates how trips events based on an alarm log from an offshore gas production can be reliably predicted using a Bayesian network. If a trip event is reliably predicted and the main cause of it is identified, it will allow the operator to prevent it. …

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

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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

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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

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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

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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

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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

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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

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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.