Mathematics Commons^™

Rough Paths And Regularization, André O. Gomes, Alberto Ohashi, Francesco Russo, Alan Teixeira

Journal of Stochastic Analysis

No abstract provided.

Three Questions From Cctm Teachers About Mathematical Modeling, Robyn Stankiewicz-Van Der Zanden, Rachel Levy

Colorado Mathematics Teacher

This article shares three questions and answers about mathematical modeling in the classroom from an April 2020 online conversation with participants of a CCTM webinar. We hope that the answers to these questions will motivate teachers to embrace the value of implementing math modeling tasks, help students see the math all around them in the world, and empower future professionals to reach for the mathematical tools in their pockets to make data-driven decisions.

Why Neural Networks In The First Place: A Theoretical Explanation, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Neural networks -- specifically, deep neural networks -- are, at present, the most effective machine learning techniques. There are reasonable explanations of why deep neural networks work better than traditional "shallow" ones, but the question remains: why neural networks in the first place? why not networks consisting of non-linear functions from some other family of functions? In this paper, we provide a possible theoretical answer to this question: namely, we show that of all families with the smallest possible number of parameters, families corresponding to neurons are indeed optimal -- for all optimality criteria that satisfy some reasonable requirements: : …

Why Daubechies Wavelets Are So Successful, Solymar Ayala Cortez, Laxman Bokati, Aaron Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

In many applications, including analysis of seismic signals, Daubechies wavelets perform much better than other families of wavelets. In this paper, we provide a possible theoretical explanation for the empirical success of Daubechies wavelets. Specifically, we show that these wavelets are optimal with respect to any optimality criterion that satisfies the natural properties of scale- and shift-invariance.

Uncertainty: Ideas Behind Neural Networks Lead Us Beyond Kl-Decomposition And Interval Fields, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we know that there is a functional dependence between a quantity q and quantities a1, ..., an, but the exact form of this dependence is only known with uncertainty. In some cases, we only know the class of possible functions describing this dependence. In other cases, we also know the probabilities of different functions from this class -- i.e., we know the corresponding random field or random process. To solve problems related to such a dependence, it is desirable to be able to simulate the corresponding functions, i.e., to have algorithms that transform simple intervals or …

While, In General, Uncertainty Quantification (Uq) Is Np-Hard, Many Practical Uq Problems Can Be Made Feasible, Anderson Gray, Scott Ferson, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, many general mathematical formulations of uncertainty quantification problems are NP-hard, meaning that (unless it turned out that P = NP) no feasible algorithm is possible that would always solve these problems. In this paper, we argue that if we restrict ourselves to practical problems, then the correspondingly restricted problems become feasible -- namely, they can be solved by using linear programming techniques.

Excursions In Summation, Brock Erwin

Fall Showcase for Research and Creative Inquiry

Using polynomials from series representation of functions to approximate other functions on the closed interval from [-1,1].

Ethical Dilemma Of Self-Driving Cars: Conservative Solution, Christian Servin, Vladik Kreinovich, Shahnaz Shahbazova

Departmental Technical Reports (CS)

When designing software for self-driving cars, we need to make an important decision: When a self-driving car encounters an emergency situation in which either the car's passenger or an innocent pedestrian have a good change of being injured or even die, which option should it choose? This has been a subject of many years of ethical discussions -- and these discussions have not yet led to a convincing solution. In this paper, we propose a "conservative" (status quo) solution that does not require making new ethical decisions -- namely, we propose to limit both the risks to passengers and risks …

Discrete Hypergeometric Legendre Polynomials, Tom Cuchta

Mathematics Faculty Research

A discrete analog of the Legendre polynomials defined by discrete hypergeometric series is investigated. The resulting polynomials have qualitatively similar properties to classical Legendre polynomials. We derive their difference equations, recurrence relations, and generating function.

Infinitely Many Radial Solutions For A P-Laplacian Problem With Indefinite Weight, Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez

All HMC Faculty Publications and Research

We prove the existence of infinitely many sign changing radial solutions for a p-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem …

Introduction To Discrete Mathematics: An Oer For Ma-471, Mathieu Sassolas

Open Educational Resources

The first objective of this book is to define and discuss the meaning of truth in mathematics. We explore logics, both propositional and first-order , and the construction of proofs, both formally and human-targeted. Using the proof tools, this book then explores some very fundamental definitions of mathematics through set theory. This theory is then put in practice in several applications. The particular (but quite widespread) case of equivalence and order relations is studied with detail. Then we introduces sequences and proofs by induction, followed by number theory. Finally, a small introduction to combinatorics is …

Non-Local Approximation Properties, Kira Pierce

Fall Showcase for Research and Creative Inquiry

This project concerns the approximation properties of a given set where X is a scattered sequence and Ï•(x) = 1/x* ln(1 + x^2 ). Similar approximation sets are commonly used in interpolation problems and are especially helpful due to their Fourier representation. For our work, we will work to prove the following theorem.

A Deep Learning Improved Numerical Method For The Simulation Of Rogue Waves Of Nonlinear Schrödinger Equation, Rui-Qi Wang, Liming Ling, Delu Zeng, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

Modulation instability (MI) is a pervasive phenomenon in nonlinear science. It is inevitable for simulating rogue wave or breather solutions of the focusing nonlinear Schrödinger equation (NLSE) and other application problems with MI involved. Due to MI, the small perturbation on the boundary can lead to large and non-negligible errors for the simulation of initial-boundary problems. To deal with this challenging problem, we propose a method to modify the boundary problem through a deep learning algorithm so that the long time simulation for the rogue wave or breather solutions to the NLSE can be performed with a superior numerical errors. …

Embedding Spheres In Knot Traces, P. Feller, Allison N. Miller, M. Nagel, P. Orson, M. Powell, A. Ray

Mathematics & Statistics Faculty Works

The trace of the n-framed surgery on a knot in S³ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable 3-dimensional knot invariants. For each n, this provides conditions that imply a knot is topologically n-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

A Neural Network Ensemble Approach To System Identification, Elisa Negrini, Giovanna Citti, Luca Capogna

Mathematics Sciences: Faculty Publications

We present a new algorithm for learning unknown governing equations from trajectory data, using and ensemble of neural networks. Given samples of solutions x(t) to an unknown dynamical system x˙(t) = f(t, x(t)), we approximate the function f using an ensemble of neural networks. We express the equation in integral form and use Euler method to predict the solution at every successive time step using at each iteration a different neural network as a prior for f. This procedure yields M-1 time-independent networks, where M is the number of time steps at which x(t) is observed. Finally, we obtain a …

Predicting Circadian Phase Across Populations: A Comparison Of Mathematical Models And Wearable Devices, Yitong Huang, Caleb Mayer, Philip Cheng, Alankrita Siddula, Helen J. Burgess, Christopher Drake, Cathy Goldstein, Olivia Walch, Daniel B. Forger, Daniel B. Forger

Mathematics Sciences: Faculty Publications

From smart work scheduling to optimal drug timing, there is enormous potential in translating circadian rhythms research results for precision medicine in the real world. However, the pursuit of such effort requires the ability to accurately estimate circadian phase outside of the laboratory. One approach is to predict circadian phase noninvasively using light and activity measurements and mathematical models of the human circadian clock. Most mathematical models take light as an input and predict the effect of light on the human circadian system. However, consumer-grade wearables that are already owned by millions of individuals record activity instead of light, which …

Using Desmos Activitybuilder For Dynamic Proof Building, Eryn M. Maher, Ha Nguyen

College of Science & Mathematics: Faculty Presentations

We will demonstrate a Desmos Activity used to support proof building for middle grades Geometry concepts. Desmos ActivityBuilder is a user-friendly interactive tool designed for teaching mathematics that integrates scientific calculators, graphing abilities, virtual manipulatives, media, and more. Participants will create a task in Desmos ActivityBuilder to use in their classroom, copying and editing a dynamic proof-building task.

The Robust Inverse Scattering Method For Focusing Ablowitz–Ladik Equation On The Non-Vanishing Background, Yiren Chen, Bao-Feng Feng, Liming Ling

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we consider the robust inverse scattering method for the Ablowitz–Ladik (AL) equation on the non-vanishing background, which can be used to deal with arbitrary-order poles on the branch points and spectral singularities in a unified way. The Darboux matrix is constructed with the aid of loop group method and considered within the framework of robust inverse scattering transform. Various soliton solutions are constructed without using the limit technique. These solutions include general soliton, breathers, as well as high order rogue wave solutions.

Methodology For Solving The Radiation Problem Proposed By Planck In Physics On The Basis Of The Principle Of Consistency, Raygul Khozhanazarova

Karakalpak Scientific Journal

1. www.quantumphysics.iop.org
2. М. Planck. Verhandl. Dtsch. phys. Ges., 1900, 2, 237-245. М. Planck. Ann. Phys., 1901, 4, 553-563.
3. M. Planck, Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum, ibid., 237-245 (1900)/
4. M. Planck. Uber des Gesetz der Energieverteilung im Normalspektrum, Ann. d.Phys. 4, 553—563 (1901).
5. Maks Plank. 1858-1958: Sb. k stoletiyu so dnya rozhdeniya Maksa Planka / Pod red. A. F. Ioffe i A. T. Grigor'yana. M.: Izd-vo AN SSSR, 1958. 276 p.
6. L. Boltzmann. Populare Schriften. Leipzig 1905, p. 227.
7. M.Plank. Vvedeniye v teoreticheskuyu fiziku. Chast' pyataya. Teoriya teploty. Ob"yedinennoye nauchno-tekhnicheskoye izdatel'stvo NKTP SSSR. Glavnaya redaktsiya obshchetekhnicheskoy literatury i nomografii. …

Neural Network Learning Of Improved Compressive Sensing Sampling And Receptive Field Structure, Victor J. Barranca

Mathematics & Statistics Faculty Works

While the theory of compressive sensing (CS) in modern signal processing typically indicates that uniformly random sampling facilitates the efficient recovery of sparse signals, such measurements are infeasible in many engineering applications and are not well reflected by the constraints of natural systems, including neuronal networks in the brain. Uniformly random sampling also does not leverage the underlying structure of many classes of signals, and may therefore be suboptimal in these cases. We address these issues by formulating a novel neural network framework for learning improved CS sampling based on the intrinsic structure present in classes of training signals. Beyond …