Mathematics Commons^™

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.

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.

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. …

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 …

Some Topics In Leavitt Path Algebras And Their Generalizations., Mohan R. Dr.

Doctoral Theses

The purpose of this section is to motivate the historical development of Leavitt algebras, Leavitt path algebras and their various generalizations and thus provide a context for this thesis. There are two historical threads which resulted in the definition of Leavitt path algebras. The first one is about the realization problem for von Neumann regular rings and the second one is about studying algebraic analogs of graph C ∗ -algebras. In what follows we briefly survey these threads and also introduce important concepts and terminology which will recur throughout.

Comparing Leibniz With Newton, Peter C. Kaemingk

Agora

This essay attempts to analyze the differences between the calculus systems of Newton and Leibniz, mainly regarding the foundations and justifications of their art of calculus.

Scaling Limits Of Some Random Interface Models., Biltu Dan Dr.

Doctoral Theses

In this thesis, we study some probabilistic models of random interfaces. Interfaces between different phases have been topic of considerable interest in statistical physics. These interfaces are described by a family of random variables, indexed by the ddimensional integer lattice, which are considered as a height configuration, namely they indicate the height of the interface above a reference hyperplane. The models are defined in terms of an energy function (Hamiltonian), which defines a Gibbs measure on the set of height configurations. More formally, letÏ• = {Ï•x}x∈Z dbe a collection of real numbers indexed by the d-dimensional integer lattice Z d. …

On The Inertia Conjecture And Its Generalizations., Soumyadip Das Dr.

Doctoral Theses

This thesis concerns problems related to the ramification behaviour of the branched Galois covers of smooth projective connected curves defined over an algebraically closed field of positive characteristic. Our first main problem is the Inertia Conjecture proposed by Abhyankar in 2001. We will show several new evidence for this conjecture. We also formulate a certain generalization of it which is our second problem, and we provide evidence for it. We give a brief overview of these problems in this introduction and reserve the details for Chapter 4.Let k be an algebraically closed field, and U be a smooth connected affine …

Perron Theorem For Delay Differential Equations, William Barker

Theses and Dissertations

This note discusses results furthering the study of bounded solutions of non-homogeneous second-order differential equations with delay. The equation of interest has already been shown to possess unique bounded solutions in accordance to Perron's Theorem. We will determine the negativity of the solutions given a positive forcing term by establishing a uniform bound regarding small time delay.

Generalized Grassmann Algebras And Applications To Stochastic Processes, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we present the groundwork for an Itô/Malliavin stochastic calculus and Hida's white noise analysis in the context of a supersymmetry with Z₃-graded algebras. To this end, we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.

Novel Theorems And Algorithms Relating To The Collatz Conjecture, Michael R. Schwob, Peter Shiue, Rama Venkat

Mathematical Sciences Faculty Research

Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and algorithms that explore relationships and properties between the natural numbers, their peak values, and the conjecture. These contributions primarily analyze the number of Collatz iterations it takes for a given integer to reach 1 or a number less than itself, or the relationship between a starting number and its peak value.

The Structure Of Finite Commutative Idempotent Involutive Residuated Lattices, Peter Jipsen, Olim Tuyt, Diego Valota

Mathematics, Physics, and Computer Science Faculty Articles and Research

We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction to prove that the fusion reduct of any finite member is a distributive semilattice, and to show that this variety is not locally finite.

Controlled Branching Processes With Continuous Time, Miguel Gonzalez, Manuel Molina, Ines Del Puerto, Nikolay Yanev, George Yanev

School of Mathematical & Statistical Sciences Faculty Publications

A class of controlled branching processes with continuous time is introduced and some limiting distributions are obtained in the critical case. An extension of this class as regenerative controlled branching processes with continuous time is proposed and some asymptotic properties are considered.

Generalized Girsanov Transform Of Processes And Zakai Equation With Jumps, Masatoshi Fujisaki, Takashi Komatsu

Journal of Stochastic Analysis

No abstract provided.

On The Solutions Of Three-Variable Frobenius-Related Problems Using Order Reduction Approach, Tian-Xiao He, Peter J.-S. Shiue, Rama Venkat

Mathematical Sciences Faculty Research

This paper presents a new approach to determine the number of solutions of three-variable Frobenius-related problems and to find their solutions by using order reducing methods. Here, the order of a Frobenius-related problem means the number of variables appearing in the problem. We present two types of order reduction methods that can be applied to the problem of finding all nonnegative solutions of three-variable Frobenius-related problems. The first method is used to reduce the equation of order three from a three-variable Frobenius-related problem to be a system of equations with two fixed variables. The second method reduces the equation of …

Expansion Of The Department Of Mathematics At Princeton University And The Founding Of The School Of Mathematics At The Institute For Advanced Study: 1900-1950, Andrew Beman-Cavallaro

Great Plains Graduate Review

From 1900 to 1950 Princeton, New Jersey, hosted two of the most prestigious institutions and provided a location for the expansive mathematical investigations taking place just before, during, and immediately after the Second World War. Princeton University and the Institute for Advanced Study housed, fed, and provided workspaces for an array of Mathematicians uncovering new research methodologies (resulting in the defeat of both the Nazi Party and the Empire of Nippon), the foundation for modern experimental Mathematics, and expansions of Theoretical and Applied Physics.

Conservative Unconditionally Stable Decoupled Numerical Schemes For The Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq System, Wenbin Chen, Daozhi Han, Xiaoming Wang, Yichao Zhang

Mathematics and Statistics Faculty Research & Creative Works

We propose two mass and heat energy conservative, unconditionally stable, decoupled numerical algorithms for solving the Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq system that models thermal convection of two-phase flows in superposed free flow and porous media. The schemes totally decouple the computation of the Cahn-Hilliard equation, the Darcy equations, the heat equation, the Navier-Stokes equations at each time step, and thus significantly reducing the computational cost. We rigorously show that the schemes are conservative and energy-law preserving. Numerical results are presented to demonstrate the accuracy and stability of the algorithms.

Cohomological Blow Ups Of Graded Artinian Gorenstein Algebras Along Surjective Maps, Anthony Iarrobino, Pedro Macias Marques, Chris Mcdaniel, Alexandra Seceleanu, Junzo Watanabe

Department of Mathematics: Faculty Publications

We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are …