Mathematics Commons^™

Reconstruction Of Sparse Recurrent Connectivity And Inputs From The Nonlinear Dynamics Of Neuronal Networks, Victor J. Barranca

Mathematics & Statistics Faculty Works

Reconstructing the recurrent structural connectivity of neuronal networks is a challenge crucial to address in characterizing neuronal computations. While directly measuring the detailed connectivity structure is generally prohibitive for large networks, we develop a novel framework for reverse-engineering large-scale recurrent network connectivity matrices from neuronal dynamics by utilizing the widespread sparsity of neuronal connections. We derive a linear input-output mapping that underlies the irregular dynamics of a model network composed of both excitatory and inhibitory integrate-and-fire neurons with pulse coupling, thereby relating network inputs to evoked neuronal activity. Using this embedded mapping and experimentally feasible measurements of the firing rate …

Functional Implications Of Dale's Law In Balanced Neuronal Network Dynamics And Decision Making, Victor J. Barranca, Asha Bhuiyan , '23, Max Sundgren , '22, Fangzhou Xing , '22

Mathematics & Statistics Faculty Works

The notion that a neuron transmits the same set of neurotransmitters at all of its post-synaptic connections, typically known as Dale's law, is well supported throughout the majority of the brain and is assumed in almost all theoretical studies investigating the mechanisms for computation in neuronal networks. Dale's law has numerous functional implications in fundamental sensory processing and decision-making tasks, and it plays a key role in the current understanding of the structure-function relationship in the brain. However, since exceptions to Dale's law have been discovered for certain neurons and because other biological systems with complex network structure incorporate individual …

An Affine Approach To Peterson Comparison, Linda Chen, E. Milićević, J. Morse

Mathematics & Statistics Faculty Works

The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov’s strange duality isomorphism.

Topological Speedups Of ℤD-Actions, Aimee S.A. Johnson, D. M. Mcclendon

Mathematics & Statistics Faculty Works

We study minimal ℤ^d-Cantor systems and the relationship between their speedups, their collections of invariant Borel measures, their associated unital dimension groups, and their orbit equivalence classes. In the particular case of minimal ℤ^d-odometers, we show that their bounded speedups must again be odometers but, contrary to the 1-dimensional case, they need not be conjugate, or even isomorphic, to the original. Furthermore, we give examples of speedups of ℤ^d-odometers which show the significant role played by a choice of ‘cone’ associated to the speedup.

The Complexity Threshold For The Emergence Of Kakutani Inequivalence, V. Cyr, Aimee S.A. Johnson, B. Kra, A. Şahi̇n

Mathematics & Statistics Faculty Works

We show that linear complexity is the threshold for the emergence of Kakutani inequivalence for measurable systems supported on a minimal subshift. In particular, we show that there are minimal subshifts of arbitrarily low superlinear complexity that admit both loosely Bernoulli and non-loosely Bernoulli ergodic measures and that no minimal subshift with linear complexity can admit inequivalent measures.

Maximal And Minimal Weak Solutions For Elliptic Problems With Nonlinearity On The Boundary, S. Bandyopadhyay, M. Chhetri, B. B. Delgado, Nsoki Mavinga, R. Pardo

Mathematics & Statistics Faculty Works

This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.

The Topological Slice Genus Of Satellite Knots, P. Feller, Allison N. Miller, J. Pinzón-Caicedo

Mathematics & Statistics Faculty Works

We present evidence supporting the conjecture that, in the topological category, the slice genus of a satellite knot P(K) is bounded above by the sum of the slice genera of K and P(U). Our main result establishes this conjecture for a variant of the topological slice genus, the ℤ–slice genus. Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern P. This stands in stark contrast with the smooth category, where, for example, there are many genus 1 knots whose (n,1)–cables have arbitrarily large smooth 4–genera. …

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. …

Jacobi's Four Squares Theorem, Arman Yagci

Honors Papers

Jacobi’s Four Squares Theorem is a celebrated result of number theory that provides a formula for the number of ways a positive integer n can be written as a sum of four integral squares. In this paper, we prove this theorem using the theory of modular forms.

Contractible 4-Manifolds, Alexandra Du

Honors Papers

Compact, contractible 4-manifolds distinct from D^4 were first constructed by Mazur and Poénaru. Sparks defined a collection of compact, contractible 4-manifolds called Jester manifolds. We study Mazur and Jester 4-manifolds. In particular, we show that all Jester manifolds are not homeomorphic to D^4, and that a collection of them are pairwise nonhomeomorphic.

Mathematics For Liberal Arts (Ma301), Lecture Notes, Lyubomir I. Boyadzhiev

Open Educational Resources

These lecture notes are designed to provide the students majoring in the liberal arts with an understanding and appreciation of mathematics as a lively, interesting, and surprisingly rich human activity with many fascinating applications. The text, emphasizing strongly intuitive thinking and visualization, is a collection of topics chosen to show the open-minded readers that:

• The connection between the mathematics presented in the course and down-to-earth, concrete real-life problems is transparent and immediate.
• Modern mathematical discoveries do not have to be the exclusive province of professional mathematicians.
• There is an important aesthetic component in mathematics, and just as in art and …

A Contraction Based Approach To Tensor Isomorphism, Anh Kieu

Honors Theses

Tensor isomorphism is a hard problem in computational complexity theory. Tensor isomorphism arises not just in mathematics, but also in other applied fields like Machine Learning, Cryptography, and Quantum Information Theory (QIT). In this thesis, we develop a new approach to testing (non)-isomorphism of tensors that uses local information from "contractions" of a tensor to detect differences in global structures. Specifically, we use projective geometry and tensor contractions to create a labelling data structure for a given tensor, which can be used to compare and distinguish tensors. This contraction labelling isomorphism test is quite general, and its practical potential remains …

On Generalized Fibonacci And Lucas Hybrid Polynomials, N. Rosa Ait-Amrane, Hacene Belbachir, Eli̇f Tan

Turkish Journal of Mathematics

In this paper, we introduce a new generalization of Fibonacci and Lucas hybrid polynomials. We investigate some basic properties of these polynomials such as recurrence relations, the generating functions, the Binet formulas, summation formulas, and a matrix representation. We derive generalized Cassini's identity and generalized Honsberger formula for generalized Fibonacci hybrid polynomials by using their matrix representation.

Collected Papers (On Neutrosophics Theory And Applications), Volume Xii, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This twelfth volume of Collected Papers includes 86 papers comprising 976 pages on Neutrosophic Theory and Applications, published between 2013-2021 in the international journal and book series NSS by the author alone or in collaboration with the following 112 co-authors (alphabetically ordered) from 21 countries.

Polychromatic Colorings Of Certain Subgraphs Of Complete Graphs And Maximum Densities Of Substructures Of A Hypercube, Ryan Tyler Hansen

Graduate Theses, Dissertations, and Problem Reports (ETD)

If G is a graph and H is a set of subgraphs of G, an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, poly_HG, is the largest number of colors in an H-polychromatic coloring. We determine poly_HG exactly when G is a complete graph on n vertices, q a fixed nonnegative integer, and H is the family of one of: all matchings spanning n-q vertices, all 2-regular graphs spanning at least n-q vertices, or all cycles of length precisely n-q. …

On Generalizations Of Supereulerian Graphs, Sulin Song

Graduate Theses, Dissertations, and Problem Reports (ETD)

A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let $\kappa'(G)$ and $\delta(G)$ be the edge-connectivity and the minimum degree of a graph $G$, respectively. For integers $s \ge 0$ and $t \ge 0$, a graph $G$ is $(s,t)$-supereulerian if for any disjoint edge sets $X, Y \subseteq E(G)$ with $|X|\le s$ and $|Y|\le t$, $G$ has a spanning closed trail that contains $X$ and avoids $Y$. This dissertation is devoted to providing some results on $(s,t)$-supereulerian graphs and …

Structure-Dependent Characterizations Of Multistationarity In Mass-Action Reaction Networks, Galyna Voitiuk

Graduate Theses, Dissertations, and Problem Reports (ETD)

This project explores a topic in Chemical Reaction Network Theory. We analyze networks with one dimensional stoichiometric subspace using mass-action kinetics. For these types of networks, we study how the capacity for multiple positive equilibria and multiple positive nondegenerate equilibria can be determined using Euclidian embedded graphs. Our work adds to the catalog of the class of reaction networks with one-dimensional stoichiometric subspace answering in the affirmative a conjecture posed by Joshi and Shiu: Conjecture 0.1 (Question 6.1 [26]). A reaction network with one-dimensional stoichiometric subspace and more than one source complex has the capacity for multistationarity if and only …

On Complete, Horizontal And Vertical Lifts From A Manifold With Fλ (6, 4) Structure To Its Cotangent Bundle, Manisha M. Kankarej, Jai Pratap Singh

All Works

Manifolds with fλ(6, 4) structure was defined and studied in the past. Later the geometry of tangent and cotangent bundles in a differentiable manifold with fλ(6, 4) structure was studied. The aim of the present paper is to study complete, horizontal and vertical lifts from a manifold with fλ (6, 4)-structure to its cotangent bundle.

On The Frequency Module Of The Hull Of A Primitive Substitution Tiling, April Lynne D. Say-Awen, Dirk Frettlöh, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal -module, where is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a …

On Twin Edge Colorings In M-Ary Trees, Jayson D. Tolentino, Reginaldo M. Marcelo, Mark Anthony C. Tolentino

Mathematics Faculty Publications

Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from ℤ_k and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in ℤ_k) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by χ′_t(G …

Deployment Of Mathematical Resources To A Philippine High School Through A Community Lte Network, Ma. Louise Antonette N. De Las Peñas, Maria Alva Q. Aberin, Agnes Garciano, Juan Carlo F. Mallari, Jumela F. Sarmiento, Mark Anthony C. Tolentino, Debbie Marie Verzosa

Mathematics Faculty Publications

In the Philippines, one challenge that continues to be faced by the Department of Education in bringing educational content in a blended learning modality is the lack of internet access of the learners. This paper discusses the distribution, through a community LTE network, of mathematical resources for Grades 7 to 10 to teachers and students of a particular high school in the Philippines. It also gives details on particular technological tools (mathematical applications) that were created to help the mathematical learning of students in a remote setting.

Sigma Chromatic Numbers Of The Middle Graph Of Some Families Of Graphs, Jay-R Manamtam, Agnes Garciano, Mark Anthony C. Tolentino

Mathematics Faculty Publications

Let G be a nontrivial connected graph and let c : V (G) → be a vertex coloring of G, where adjacent vertices may have the same color. For a vertex υ of G, the color sum σ(υ) of υ is the sum of the colors of the vertices adjacent to υ. The coloring c is said to be a sigma coloring of G if σ(u) ≠ σ(υ) whenever u and υ are adjacent vertices in G. The minimum number of colors that can be …

Development Of An App And Videos To Support The Fraction Learning Trajectory From Grades 1-7, Debbie Marie Verzosa, Ma. Louise Antonette N. De Las Peñas, Maria Alva Q. Aberin, Agnes Garciano, Jumela F. Sarmiento, Juan Carlo F. Mallari, Mark Anthony C. Tolentino

Mathematics Faculty Publications

Lack of procedural fluency in fractions impedes access to advanced mathematical courses and limits opportunities for entry into STEM-related fields. This paper describes the design and pedagogical basis of the Moving Fractions app and supplementary fraction videos for promoting fraction learning. Moving Fractions utilizes game-design factors to draw students through a trajectory of fraction learning from part-whole comparisons to a more robust understanding of the measurement concept of fractions. The supplementary video immerses students in a broad range of fraction representations. The app and video are intended to form a fraction learning package for distribution in Philippine schools. Future work …

Benchmarking Graph Neural Networks, Thomas Laurent, Vijay Prakash Dwivedi, Chaitanya K. Joshi, Anh Tuan Luu, Yoshua Bengio, Xavier Bresson

Mathematics, Statistics and Data Science Faculty Works

In the last few years, graph neural networks (GNNs) have become the standard toolkit for analyzing and learning from data on graphs. This emerging field has witnessed an extensive growth of promising techniques that have been applied with success to computer science, mathematics, biology, physics and chemistry. But for any successful field to become mainstream and reliable, benchmarks must be developed to quantify progress. This led us in March 2020 to release a benchmark framework that i) comprises of a diverse collection of mathematical and real-world graphs, ii) enables fair model comparison with the same parameter budget to identify key …

Online Algorithms With Advice For The 𝒌-Search Problem, Jhoirene B. Clemente, Henry N. Adorna, Proceso L. Fernandez Jr

Department of Information Systems & Computer Science Faculty Publications

In the online search problem, a seller seeks to find the maximum price from a sequence of prices p1, p2,…, pn that is revealed in a piece-wise manner. The bound for all prices is well known in advance with m ≤ pί ≤ M. In the online k-search problem, the seller seeks to find the k maximum out of the n prices. In this paper, we present a tight bound of [Formula Presented] on the advice complexity of optimal online algorithms for online k-search. We also provide online algorithms with advice that use less than the required number of bits …

An Introduction To Pascal’S Triangle, Daniel Chenoweth

Masters Essays

No abstract provided.

Exploration Of Triangles In Euclidean, Spherical, And Hyperbolic Spaces For The High School Geometry Classroom, John Rowell

Masters Essays

No abstract provided.

A Comparison Of Stacking Methods To Estimate Survival Using Residual Lifetime Data From Prevalent Cohort Studies, Zhaoheng Li

Mathematics, Statistics, and Computer Science Honors Projects

Prevalent cohort studies are widely used for their cost-efficiency and convenience. However, in such studies, only the residual lifetime can be observed. Traditionally, researchers rely on self-reported onset times to infer the underlying survival distribution, which may introduce additional bias that confounds downstream analysis. This study compares two stacking procedures and one mixture model approach that uses only residual lifetime data while leveraging the strengths of different estimators. Our simulation results show that the two stacked estimators outperform the nonparametric maximum likelihood estimator (NPMLE) and the mixture model, allowing robust and accurate estimations for underlying survival distributions.

P-Adic Analysis: A Quick Introduction, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

These notes aim to provide a fast introduction to p-adic analysis assuming basic knowledge in algebra and analysis. The text corresponds to the lecture notes for a Mini-Course in the L. Santal´o Research Summer School 2019. Palacio de la Magdalena, Santander, Spain, June 24-28, 2019.

The Octagon At Large M, Riccardo Argurio, Matteo Bertolini, Sebastían Franco, Eduardo García-Valdecasas, Shani Meynet, Antoine Pasternak, Valdo Tatitscheff

Publications and Research

Recently, the first instance of a model of D-branes at Calabi-Yau singularities where supersymmetry is broken dynamically into stable vacua has been proposed. This construction was based on a system of N regular and M = 1 fractional branes placed at the tip of the so-called (orientifolded) Octagon singularity. In this paper we show that this model admits a large M generalization, having the same low energy effective dynamics. This opens up the possibility that the effect on geometry is smooth, and amenable to describing the gauge theory all along the RG flow, including the deep IR, in terms of …