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Articles 1 - 30 of 138
Full-Text Articles in Mathematics
Recommendations To Internal Auditors Regarding The Auditing And Attestation Of Mathematical Programming Models, Jose Rincón, Greg Akai, Daryl Ono
Recommendations To Internal Auditors Regarding The Auditing And Attestation Of Mathematical Programming Models, Jose Rincón, Greg Akai, Daryl Ono
LMU Librarian Publications & Presentations
Mathematical programming planning models increase operational efficiency and minimize operating costs, but the underlying mathematics generally is complex. Combinatorial optimization is technically sophisticated which requires a strong quantitative background to successfully implement. Most internal auditors will not have the technical training to critically assess the underlying mathematics of mathematical programming planning models, but the internal auditor can still provide insight and attestation which can increase the efficiency of mathematical programming planning models.
Finite N-Quandles Of Twisted Double Handcuff And Complete Graph, Veronica Backer-Peral
Finite N-Quandles Of Twisted Double Handcuff And Complete Graph, Veronica Backer-Peral
Honors Thesis
The Double Handcuff and K4 graphs can be generalized to a single family of spatial graphs by adding a variable number of twists between two edges. We can identify spatial graphs by calculating a quotient of the fundamental quandle, known as an N-quandle, which is a spatial graph invariant. In this paper, we prove that the N-quandle associated with this family of spatial graphs is finite when all but two edges are given a label of 2, and the remaining two edges are assigned labels from the natural numbers. To prove that the N-quandle is finite, we produce Cayley graphs …
Quandles With Orbit Series Conditions, Alissa Crans
Quandles With Orbit Series Conditions, Alissa Crans
Mathematics Faculty Works
We introduce the notion of an orbit series in a quandle. Using this notion we define four families of quandles based on finiteness conditions on their orbit series. Intuitively, the classes tOS and tOSn correspond to finitary compositions of trivial quandles while the classes OS and OSn correspond to finitary compositions of connected quandles. We study properties of these four families of quandles and explore their relationships with several previously studied families of quandles: reductive, n-reductive, locally reductive, n-locally reductive, and solvable quandles
From Biquandle Structures To Hom-Biquandles, Alissa Crans
From Biquandle Structures To Hom-Biquandles, Alissa Crans
Mathematics Faculty Works
We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures. We also continue the study of biquandle homomorphisms into a medial biquandle begun in [Hom quandles, J. Knot Theory Ramifications 23(2) (2014)], finding biquandle analogs of results therein. We describe the biquandle structure of the Hom-biquandle, and consider the relationship between the Hom-quandle and Hom-biquandle.
Shining A Light On A Hidden Figure: Dorothy Hoover, Lily S. Khadjavi
Shining A Light On A Hidden Figure: Dorothy Hoover, Lily S. Khadjavi
Mathematics Faculty Works
No abstract provided.
A Linear Optimal Feedback Control For Producing 1,3-Propanediol Via Microbial Fermentation, Yangping Ma
A Linear Optimal Feedback Control For Producing 1,3-Propanediol Via Microbial Fermentation, Yangping Ma
Mathematics Faculty Works
In this paper, we consider a multistage feedback control strategy for the production of 1,3-propanediol(1,3-PD) in microbial fermentation. The feedback control strategy is widely used in industry, and to the best of our knowledge, this is the first time it is applied to 1,3-PD. The feedback control law is assumed to be linear of the concentrations of biomass and glycerol, and the coefficients in the controller are continuous. A multistage feedback control law is obtained by using the control parameterization method on the coefficient functions. Then, the optimal control problem can be transformed into an optimal parameter selection problem. The …
Graphs Admitting Only Constant Splines, Alissa Crans, Blake Mellor
Graphs Admitting Only Constant Splines, Alissa Crans, Blake Mellor
Mathematics Faculty Works
We study generalized graph splines, introduced by Gilbert, Tymoczko, and Viel (2016). For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.
Beauty, Bees, And God: The Fibonacci Sequence As A Theological Springboard In Secondary Mathematics, John D. Brahier
Beauty, Bees, And God: The Fibonacci Sequence As A Theological Springboard In Secondary Mathematics, John D. Brahier
Journal of Catholic Education
Catholic schools primarily should be in the business of making saints. This article identifies and explores a meaningful, engaging point of contact between mathematics and theology for high school math classes, the Fibonacci Sequence. This sequence serves as an engaging introduction to sequences and series; more importantly, the topic can be used as a springboard to theological discussions. The paper will provide a brief historical background to the Fibonacci Sequence, an explanation of how it can be used in a high school math classroom, and an exploration of three different theological touchpoints that the Fibonacci Sequence offers.
Finite N-Quandles Of Torus And Two-Bridge Links, Alissa Crans, Blake Mellor, Patrick Shanahan
Finite N-Quandles Of Torus And Two-Bridge Links, Alissa Crans, Blake Mellor, Patrick Shanahan
Mathematics Faculty Works
We compute Cayley graphs and automorphism groups for all finite n-quandles of two-bridge and torus knots and links, as well as torus links with an axis.
An Enumeration Process For Racks, Patrick Shanahan
An Enumeration Process For Racks, Patrick Shanahan
Mathematics Faculty Works
Given a presentation for a rack , we define a process which systematically enumerates the elements of . The process is modeled on the systematic enumeration of cosets first given by Todd and Coxeter. This generalizes and improves the diagramming method for -quandles introduced by Winker. We provide pseudocode that is similar to that given by Holt, Eick, and O'Brien for the Todd-Coxeter process. We prove that the process terminates if and only if is finite, in which case, the procedure outputs an operation table for the finite rack. We conclude with an application to knot theory
Remarks On Suzuki's Epimorphism Number, Patrick Shanahan
Remarks On Suzuki's Epimorphism Number, Patrick Shanahan
Mathematics Faculty Works
A partial order on prime knots can be defined by declaring 𝐽≥𝐾, if there exists an epimorphism from the knot group of 𝐽 onto the knot group of 𝐾. Suppose that 𝐽 is a 2-bridge knot that is strictly greater than 𝑚 distinct, nontrivial knots. In this paper, we determine a lower bound on the crossing number of 𝐽 in terms of 𝑚. Using this bound, we answer a question of Suzuki regarding the 2-bridge epimorphism number EK(𝑛) which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number 𝑛. We establish …
The Origins Of Spectra, An Organization For Lgbt Mathematicians, Lily S. Khadjavi
The Origins Of Spectra, An Organization For Lgbt Mathematicians, Lily S. Khadjavi
Mathematics Faculty Works
No abstract provided.
On The Structure Of Hom Quandles, Alissa Crans
On The Structure Of Hom Quandles, Alissa Crans
Mathematics Faculty Works
We continue the study of the quandle of homomorphisms into a medial quandle begun in [2]. We show that it suffices to consider only medial source quandles, and therefore the structure theorem of [11] provides a characterization of the Hom quandle. In the particular case when the target is 2-reductive this characterization takes on a simple form that makes it easy to count and determine the structure of the Hom quandle
Rudolf Laban's Dream: Re-Envisioning And Re-Scoring Ballet, Choreutics, And Simple Functional Movements With Vector Signs For Deflecting Diagonal Inclinations, Jeffrey Scott Longstaff
Rudolf Laban's Dream: Re-Envisioning And Re-Scoring Ballet, Choreutics, And Simple Functional Movements With Vector Signs For Deflecting Diagonal Inclinations, Jeffrey Scott Longstaff
Journal of Movement Arts Literacy Archive (2013-2019)
Several methods of movement notation, forerunners of modern-day Labanotation/Kinetography were published by Rudolf Laban in his 1926 book Choreographie. One of these has been referred to as vector signs because they represent movement as orientations (slopes) of lines through space. This article begins by comparing Labanotation direction symbols with Laban's earlier vector signs by looking at differences when simple sequences are scored in both formats. Concepts of space within the vector signs are examined, particularly Laban's idea of deflecting inclinations where movements are categorized as mixtures of two fundamental contrasting spatial and dynamic tendencies: dimensional stability and diagonal mobility. This …
Stochastic Maximum Principle For Partial Information Optimal Investment And Dividend Problem Of An Insurer, Yanping Ma
Stochastic Maximum Principle For Partial Information Optimal Investment And Dividend Problem Of An Insurer, Yanping Ma
Mathematics Faculty Works
We study an optimal investment and dividend problem of an insurer, where the aggregate insurance claims process is modeled by a pure jump Lévy process. We allow the management of the dividend payment policy and the investment of surplus in a continuous-time financial market, which is composed of a risk free asset and a risky asset. The information available to the insurer is partial information. We generalize this problem as a partial information regular-singular stochastic control problem, where the control variable consists of regular control and singular control. Then maximum principles are established to give sufficient and necessary optimality conditions …
The Multilinear Structure Of Relu Networks, Thomas Laurent
The Multilinear Structure Of Relu Networks, Thomas Laurent
Mathematics Faculty Works
We study the loss surface of neural networks equipped with a hinge loss criterion and ReLU or leaky ReLU nonlinearities. Any such network defines a piecewise multilinear form in parameter space. By appealing to harmonic analysis we show that all local minima of such network are non-differentiable, except for those minima that occur in a region of parameter space where the loss surface is perfectly flat. Non-differentiable minima are therefore not technicalities or pathologies; they are heart of the problem when investigating the loss of ReLU networks. As a consequence, we must employ techniques from nonsmooth analysis to study these …
Deep Linear Networks With Arbitrary Loss: All Local Minima Are Global, Thomas Laurent
Deep Linear Networks With Arbitrary Loss: All Local Minima Are Global, Thomas Laurent
Mathematics Faculty Works
We consider deep linear networks with arbitrary convex differentiable loss. We provide a short and elementary proof of the fact that all local minima are global minima if the hidden layers are either 1) at least as wide as the input layer, or 2) at least as wide as the output layer. This result is the strongest possible in the following sense: If the loss is convex and Lipschitz but not differentiable then deep linear networks can have sub-optimal local minima.
Integrating Non-Euclidean Geometry Into High School, John Buda
Integrating Non-Euclidean Geometry Into High School, John Buda
Honors Thesis
The purpose of this project is to provide the framework for integrating the study of non-Euclidean geometry into a high school math class in such a way that both aligns with the Common Core State Standards and makes use of research-based practices to enhance the learning of traditional geometry. Traditionally, Euclidean geometry has been the only strand of geometry taught in high schools, even though mathematicians have developed several other strands. The non-Euclidean geometry that I focus on in this project is what is known as taxicab geometry. With the Common Core Standards for Math Practice pushing students to “model …
Polygons, Pillars And Pavilions: Discovering Connections Between Geometry And Architecture, Sean Patrick Madden
Polygons, Pillars And Pavilions: Discovering Connections Between Geometry And Architecture, Sean Patrick Madden
Journal of Catholic Education
Crowning the second semester of geometry, taught within a Catholic middle school, the author's students explored connections between the geometry of regular polygons and architecture of local buildings. They went on to explore how these principles apply famous buildings around the world such as the monuments of Washington, D.C. and the elliptical piazza of Saint Peter's Basilica at Vatican City within Rome, Italy.
Optimization And Control Of Agent-Based Models In Biology: A Perspective, G. An, B. G. Fitzpatrick, S. Christley, P. Federico, A. Kanarek, R. Miller Neilan, M. Oremland, R. Salinas, R. Laubeanbacher, S. Lenhart
Optimization And Control Of Agent-Based Models In Biology: A Perspective, G. An, B. G. Fitzpatrick, S. Christley, P. Federico, A. Kanarek, R. Miller Neilan, M. Oremland, R. Salinas, R. Laubeanbacher, S. Lenhart
Mathematics Faculty Works
Agent-based models (ABMs) have become an increasingly important mode of inquiry for the life sciences. They are particularly valuable for systems that are not understood well enough to build an equation-based model. These advantages, however, are counterbalanced by the difficulty of analyzing and using ABMs, due to the lack of the type of mathematical tools available for more traditional models, which leaves simulation as the primary approach. As models become large, simulation becomes challenging. This paper proposes a novel approach to two mathematical aspects of ABMs, optimization and control, and it presents a few first steps outlining how one might …
On Homology Of Associative Shelves, Alissa Crans
On Homology Of Associative Shelves, Alissa Crans
Mathematics Faculty Works
Homology theories for associative algebraic structures are well established and have been studied for a long time. More recently, homology theories for selfdistributive algebraic structures motivated by knot theory, such as quandles and their relatives, have been developed and investigated. In this paper, we study associative self-distributive algebraic structures and their one-term and two-term (rack) homology groups.
The Alexander Polynominal For Virtual Twist Knots, Isaac Benioff, Blake Mellor
The Alexander Polynominal For Virtual Twist Knots, Isaac Benioff, Blake Mellor
Mathematics Faculty Works
We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial Δ0 (as defined by Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications12 (2003) 987–1000]) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from Δ0 .
Grnsight: A Web Application And Service For Visualizing Models Of Small- To Medium-Scale Gene Regulatory Networks, Kam D. Dahlquist, John David N. Dionisio, Ben G. Fitzpatrick, Nicole A. Anguiano, Anindita Varshneya, Britain J. Southwick, Mihir Samdarshi
Grnsight: A Web Application And Service For Visualizing Models Of Small- To Medium-Scale Gene Regulatory Networks, Kam D. Dahlquist, John David N. Dionisio, Ben G. Fitzpatrick, Nicole A. Anguiano, Anindita Varshneya, Britain J. Southwick, Mihir Samdarshi
Biology Faculty Works
GRNsight is a web application and service for visualizing models of gene regulatory networks (GRNs). A gene regulatory network (GRN) consists of genes, transcription factors, and the regulatory connections between them which govern the level of expression of mRNA and protein from genes. The original motivation came from our efforts to perform parameter estimation and forward simulation of the dynamics of a differential equations model of a small GRN with 21 nodes and 31 edges. We wanted a quick and easy way to visualize the weight parameters from the model which represent the direction and magnitude of the influence of …
Topological Symmetry Groups Of Complete Bipartite Graphs, Kathleen Hake, Blake Mellor, Matthew Pittluck
Topological Symmetry Groups Of Complete Bipartite Graphs, Kathleen Hake, Blake Mellor, Matthew Pittluck
Mathematics Faculty Works
The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs [7]; it is natural next to consider complete bipartite graphs. In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to A4 , S4 or A5 [12]; in this paper we determine which complete bipartite graphs have an embedding in S 3 whose topological symmetry group …
The Development Of Notation In Mathematical Analysis, Alyssa Venezia
The Development Of Notation In Mathematical Analysis, Alyssa Venezia
Honors Thesis
The field of analysis is a newer subject in mathematics, as it only came into existence in the last 400 years. With a new field comes new notation, and in the era of universalism, analysis becomes key to understanding how centuries of mathematics were unified into a finite set of symbols, precise definitions, and rigorous proofs that would allow for the rapid development of modern mathematics. This paper traces the introduction of subjects and the development of new notations in mathematics from the seventeenth to the nineteenth century that allowed analysis to flourish. In following the development of analysis, we …
Solving The Ko Labyrinth, Alissa S. Crans
Alexander And Writhe Polynomials For Virtual Knots, Blake Mellor
Alexander And Writhe Polynomials For Virtual Knots, Blake Mellor
Mathematics Faculty Works
We give a new interpretation of the Alexander polynomial Δ0 for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, Δ0 determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.
The Segal–Shale–Weil Representation, The Indices Of Kashiwara And Maslov, And Quantum Mechanics, Michael C. Berg
The Segal–Shale–Weil Representation, The Indices Of Kashiwara And Maslov, And Quantum Mechanics, Michael C. Berg
Mathematics Faculty Works
We produce a connection between the Weil 2-cocycles defining the local and adèlic metaplectic groups defined over a global field, i.e. the double covers of the attendant local and adèlic symplectic groups, and local and adèlic Maslov indices of the type considered by Souriau and Leray. With the latter tied to phase integrals occurring in quantum mechanics, we provide a formulation of quadratic reciprocity for the underlying field, first in terms of an adèlic phase integral, and then in terms of generalized time evolution unitary operators.
Colorings, Determinants And Alexander Polynomials For Spatial Graphs, Terry Kong, Alec Lewald, Blake Mellor, Vadim Pigrish
Colorings, Determinants And Alexander Polynomials For Spatial Graphs, Terry Kong, Alec Lewald, Blake Mellor, Vadim Pigrish
Mathematics Faculty Works
A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and p-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which p the graph is p-colorable, and that a p-coloring of a graph corresponds to a representation of …
Involutory Quandles Of (2,2,R)-Montesinos Links, Jim Hoste, Patrick D. Shanahan
Involutory Quandles Of (2,2,R)-Montesinos Links, Jim Hoste, Patrick D. Shanahan
Mathematics Faculty Works
In this paper we show that Montesinos links of the form L(1/2, 1/2, p/q;e), which we call (2,2,r)-Montesinos links, have finite involutory quandles. This generalizes an observation of Winker regarding the (2, 2, q)-pretzel links. We also describe some properties of these quandles.