From A Doodle To A Theorem: A Case Study In Mathematical Discovery, 2023 Harvard University

#### From A Doodle To A Theorem: A Case Study In Mathematical Discovery, Juan FernáNdez GonzáLez, Dirk Schlimm

*Journal of Humanistic Mathematics*

We present some aspects of the genesis of a geometric construction, which can be carried out with compass and straightedge, from the original idea to the published version (Fernández González 2016). The Midpoint Path Construction makes it possible to multiply the length of a line segment by a rational number between 0 and 1 by constructing only midpoints and a straight line. In the form of an interview, we explore the context and narrative behind the discovery, with first-hand insights by its author. Finally, we discuss some general aspects of this case study in the context of philosophy of mathematical …

Where Does Mathematics Come From? Really, Where?, 2023 Claremont McKenna College

#### Where Does Mathematics Come From? Really, Where?, Mark Huber, Gizem Karaali

*Journal of Humanistic Mathematics*

No abstract provided.

Front Matter, 2023 Claremont Colleges

Translation Of: Familles De Surfaces Isoparamétriques Dans Les Espaces À Courbure Constante, Annali Di Mat. 17 (1938), 177–191, By Élie Cartan., 2023 College of the Holy Cross

#### Translation Of: Familles De Surfaces Isoparamétriques Dans Les Espaces À Courbure Constante, Annali Di Mat. 17 (1938), 177–191, By Élie Cartan., Thomas E. Cecil

*Mathematics Department Faculty Scholarship*

This is an English translation of the article "Familles de surfaces isoparamétriques dans les espaces à courbure constante" which was originally published in Annali di Matematica 17, 177–191 (1938), by Élie Cartan.

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in French. All references should be made to the original paper.

**Mathematics Subject Classification Numbers: 53C40, 53C42, 53B25**

Combinatorial Identities Associated With A Bivariate Generating Function For Overpartition Pairs, 2023 The University of Texas Rio Grande Valley

#### Combinatorial Identities Associated With A Bivariate Generating Function For Overpartition Pairs, Atul Dixit, Ankush Goswami

*School of Mathematical and Statistical Sciences Faculty Publications and Presentations*

We obtain a three-parameter q-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with N(r,s,m,n), a function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy and Osburn. For example, one of our identities gives a closed-form evaluation of a double series in terms of Chebyshev polynomials of the second kind, thereby resulting in an analogue of Euler's pentagonal number theorem. Another of our results expresses a multi-sum involving N(r,s,m,n) in terms of just the partition function p(n). Using a result of Shimura we also relate …

Coloring Complexes And Combinatorial Hopf Monoids, 2023 The University of Texas Rio Grande Valley

#### Coloring Complexes And Combinatorial Hopf Monoids, Jacob A. White

*School of Mathematical and Statistical Sciences Faculty Publications and Presentations*

We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the quasisymmetric function invariants associated to the combinatorial Hopf monoid. We show that the collection of all such coloring complexes forms a linearized combinatorial Hopf monoid, which is the terminal object in the category of combinatorial Hopf monoids with convex characters. We also study several examples of combinatorial Hopf monoids.

A Stronger Strong Schottky Lemma For Euclidean Buildings, 2023 The Graduate Center, City University of New York

#### A Stronger Strong Schottky Lemma For Euclidean Buildings, Michael E. Ferguson

*Dissertations, Theses, and Capstone Projects*

We provide a criterion for two hyperbolic isometries of a Euclidean building to generate a free group of rank two. In particular, we extend the application of a Strong Schottky Lemma to buildings given by Alperin, Farb and Noskov. We then use this extension to obtain an infinite family of matrices that generate a free group of rank two. In doing so, we also introduce an algorithm that terminates in finite time if the lemma is applicable for pairs of certain kinds of matrices acting on the Euclidean building for the special linear group over certain discretely valued fields.

Exact Parallel Waves In General Relativity, 2023 Massachusetts Institute of Technology

#### Exact Parallel Waves In General Relativity, Cian Roche, Amir Babak Aazami, Carla Cederbaum

*Mathematics*

We conduct a review of the basic definitions and the principal results in the study of wavelike spacetimes, that is spacetimes whose metric models massless radiation moving at the speed of light, focusing in particular on those geometries with parallel rays. In particular, we motivate and connect their various definitions, outline their coordinate descriptions and present some classical results in their study in a language more accessible to modern readers, including the existence of “null coordinates” and the construction of Penrose limits. We also present a thorough summary of recent work on causality in pp-waves, and describe progress in addressing …

Completeness Of Nominal Props, 2023 Runtime Verication Inc.

#### Completeness Of Nominal Props, Samuel Balco, Alexander Kurz

*Engineering Faculty Articles and Research*

We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams.

Transcription And Translation Of Nicole Oresme: Quaestiones Super Geometricam Euclidis: Questio 2, 2023 Xavier University

#### Transcription And Translation Of Nicole Oresme: Quaestiones Super Geometricam Euclidis: Questio 2, Daniel E. Otero

*2023, February 10-11 ORESME Reading Group Meeting*

No abstract provided.

Introducing Systems Via Laplace Transforms, 2023 Bradley University

#### Introducing Systems Via Laplace Transforms, Ollie Nanyes

*CODEE Journal*

The purpose of this note is to show how to move from Laplace Transforms to a brief introduction to two dimensional systems of linear differential equations with only basic matrix algebra.

Determining The Proportionality Of Ischemic Stroke Risk Factors To Age, 2023 Technological University Dublin

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

*Articles*

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 …

Supplementary Files For "Adaptive Mapping Of Design Ground Snow Loads In The Conterminous United States", 2023 University of Utah

#### Supplementary Files For "Adaptive Mapping Of Design Ground Snow Loads In The Conterminous United States", Jadon Wagstaff, Jesse Wheeler, Brennan Bean, Marc Maguire, Yan Sun

*Browse all Datasets*

Recent amendments to design ground snow load requirements in ASCE 7-22 have reduced the size of case study regions by 91% from what they were in ASCE 7-16, primarily in western states. This reduction is made possible through the development of highly accurate regional generalized additive regression models (RGAMs), stitched together with a novel smoothing scheme implemented in the R software package remap, to produce the continental- scale maps of reliability-targeted design ground snow loads available in ASCE 7-22. This approach allows for better characterizations of the changing relationship between temperature, elevation, and ground snow loads across the Conterminous United …

Counting Power Domination Sets In Complete M-Ary Trees, 2023 Gonzaga University

#### Counting Power Domination Sets In Complete M-Ary Trees, Hays Whitlatch, Katharine Shultis, Olivia Ramirez, Michele Ortiz, Sviatlana Kniahnitskaya

*Theory and Applications of Graphs*

Motivated by the question of computing the probability of successful power domination by placing k monitors uniformly at random, in this paper we give a recursive formula to count the number of power domination sets of size k in a labeled complete m-ary tree. As a corollary we show that the desired probability can be computed in exponential with linear exponent time.

Function Spaces Via Fractional Poisson Kernel On Carnot Groups And Applications, 2023 Clark University

#### Function Spaces Via Fractional Poisson Kernel On Carnot Groups And Applications, Ali Maalaoui, Andrea Pinamonti, Gareth Speight

*Mathematics*

We provide a new characterization of homogeneous Besov and Sobolev spaces in Carnot groups using the fractional heat kernel and Poisson kernel. We apply our results to study commutators involving fractional powers of the sub-Laplacian. © 2022, The Hebrew University of Jerusalem.

From Mirrors To Wallpapers: A Virtual Math Circle Module On Symmetry, 2023 Central New Mexico Community College

#### From Mirrors To Wallpapers: A Virtual Math Circle Module On Symmetry, Nicole A. Sullivant, Christina L. Duron, Douglas T. Pfeffer

*Journal of Math Circles*

Symmetry is a natural property that children see in their everyday lives; it also has deep mathematical connections to areas like tiling and objects like wallpaper groups. The Tucson Math Circle (TMC) presents a 7-part module on symmetry that starts with reflective symmetry and culminates in the deconstruction of wallpapers into their ‘generating tiles’. This module utilizes a scaffolded, hands-on approach to cover old and new mathematical topics with various interactive activities; all activities are made available through free web-based platforms. In this paper, we provide lesson plans for the various activities used, and discuss their online implementation with Zoom, …

Hs-Integral And Eisenstein Integral Mixed Circulant Graphs, 2023 Indian Institute of Technology Guwahati

#### Hs-Integral And Eisenstein Integral Mixed Circulant Graphs, Monu Kadyan, Bikash Bhattacharjya

*Theory and Applications of Graphs*

A mixed graph is called \emph{second kind hermitian integral} (\emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of the second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set $S$ for which a mixed circulant graph $\text{Circ}(\mathbb{Z}_n, S)$ is HS-integral. We also show that a mixed circulant graph is Eisenstein integral if and only if it is HS-integral. Further, we express the eigenvalues and the HS-eigenvalues of unitary oriented circulant graphs in terms of generalized M$\ddot{\text{o}}$bius function.

Spectral Sequences And Khovanov Homology, 2023 Dartmouth College

#### Spectral Sequences And Khovanov Homology, Zachary J. Winkeler

*Dartmouth College Ph.D Dissertations*

In this thesis, we will focus on two main topics; the common thread between both will be the existence of spectral sequences relating Khovanov homology to other knot invariants. Our first topic is an invariant *MKh(L)* for links in thickened disks with multiple punctures. This invariant is different from but inspired by both the Asaeda-Pryzytycki-Sikora (APS) homology and its specialization to links in the solid torus. Our theory will be constructed from a *Z^n*-filtration on the Khovanov complex, and as a result we will get various spectral sequences relating *MKh(L)* to *Kh(L)*, *AKh(L)*, and *APS(L)*. Our …

Slices Of C_2, Klein-4, And Quaternionic Eilenberg-Mac Lane Spectra, 2023 University of Kentucky

#### Slices Of C_2, Klein-4, And Quaternionic Eilenberg-Mac Lane Spectra, Carissa Slone

*Theses and Dissertations--Mathematics*

We provide the slice (co)towers of \(\Si{V} H_{C_2}\ul M\) for a variety of \(C_2\)-representations \(V\) and \(C_2\)-Mackey functors \(\ul M\). We also determine a characterization of all 2-slices of equivariant spectra over the Klein four-group \(C_2\times C_2\). We then describe all slices of integral suspensions of the equivariant Eilenberg-MacLane spectrum \(H\ulZ\) for the constant Mackey functor over \(C_2\times C_2\). Additionally, we compute the slices and slice spectral sequence of integral suspensions of $H\ulZ$ for the group of equivariance $Q_8$. Along the way, we compute the Mackey functors \(\mpi_{k\rho} H_{K_4}\ulZ\) and $\mpi_{k\rho} H_{Q_8}\ulZ$.

Pell-Lucas Collocation Method For Solving A Class Of Second Order Nonlinear Differential Equations With Variable Delays, 2023 TÜBİTAK

#### Pell-Lucas Collocation Method For Solving A Class Of Second Order Nonlinear Differential Equations With Variable Delays, Şuayi̇p Yüzbaşi, Gamze Yildirim

*Turkish Journal of Mathematics*

In this study, the approximate solution of the nonlinear differential equation with variable delays is investigated by means of a collocation method based on the truncated Pell-Lucas series. In the first stage of the method, the assumed solution form (the truncated Pell-Lucas polynomial solution) is expressed in the matrix form of the standard bases. Next, the matrix forms of the necessary derivatives, the nonlinear terms, and the initial conditions are written. Then, with the help of the equally spaced collocation points and these matrix relations, the problem is reduced to a system of nonlinear algebraic equations. Finally, the obtained system …