Applied Mathematics Commons^™

On Optimal Stopping And Impulse Control With Constraint, J. L. Menaldi, M. Robin

Mathematics Faculty Research Publications

The optimal stopping and impulse control problems for a Markov-Feller process are considered when the controls are allowed only when a signal arrives. This is referred to as control problems with constraint. In [28, 29, 30], the HJB equation was solved and an optimal control (for the optimal stopping problem, the discounted impulse control problem and the ergodic impulse control problem, respectively) was obtained, under suitable conditions, including a setting on a compact metric state space. In this work, we extend most of the results to the situation where the state space of the Markov process is locally compact.

Large And Small Data Blow-Up Solutions In The Trojan Y Chromosome Model, Rana D. Parshad, Matthew Beauregard, Eric M. Takyi, Thomas Griffin, Landrey Bobo

Faculty Publications

The Trojan Y Chromosome Strategy (TYC) is an extremely well investigated biological control method for controlling invasive populations with an XX-XY sex determinism. In [35, 36] various dynamical properties of the system are analyzed, including well posedness, boundedness of solutions, and conditions for extinction or recovery. These results are derived under the assumption of positive solutions. In the current manuscript, we show that if the introduction rate of trojan fish is zero, under certain large data assumptions, negative solutions are possible for the male population, which in turn can lead to finite time blow-up in the female and male populations. …

A Deep Learning Approach To Uncertainty Quantification, Mst Afroja Akter

Mathematics & Statistics ETDs

In this thesis we consider ordinary differential equations (ODEs) with random parameters. We focus on Monte Carlo (MC) sampling for computing the statistics of some quantities of interest (QoIs) given by the solution of the ODE problems. We use the 4th order accurate Runge-Kutta (RK4) method as the deterministic ODE solver. We then develop a hybrid MC sampling method that combines RK4 with neural network models to efficiently compute the statistics of QoIs within a desired accuracy. We present several numerical examples to verify the accuracy and efficiency of the proposed hybrid method compared to classical MC sampling. The hybrid …

Graphicacy For Numeracy: Review Of Fundamentals Of Data Visualization: A Primer On Making Informative And Compelling Figures By Claus O. Wilke (2019), Christy M. Bebeau

Numeracy

Wilke, Claus O. 2019. Fundamentals of Data Visualization: A Primer on Making Informative and Compelling Figures. (Sebastopol, CA: O’Reilly Media, Inc.). 390 pp. ISBN 978-1-492-03108-6. First edition. First release: 03-15-2019.

Claus O. Wilke has authored an excellent reference about producing and understanding static figures, figures used online, in print, and for presentations. His book is neither a statistics nor programming text, but familiarity with basic statistical concepts is helpful. Written in three parts, the book presents both the math and artistic design aspects of telling a story through figures. Wilke makes extensive use of examples, labels them good, bad, …

Quenching Estimates For A Non-Newtonian Filtration Equation With Singular Boundary Conditions, Matthew Beauregard, Burhan Selcuk

Faculty Publications

In this paper, the quenching behavior of the non-Newtonian filtration equation (φ(u))t = (|ux| r−2 ux)x with singular boundary conditions, ux (0, t) = u −p (0, t), ux (a, t) = (1 − u(a, t))−q is investigated. Various conditions on the initial condition are shown to guarantee quenching at either the left or right boundary. Theoretical quenching rates and lower bounds to the quenching time are determined when φ(u) = u and r = 2. Numerical experiments are provided to illustrate and provide additional validation of the theoretical estimates to the quenching rates and times.

L^{\Infty}-Estimates Of The Solution Of The Navier-Stokes Equations For Periodic Initial Data, Santosh Pathak

Mathematics & Statistics ETDs

In this doctoral dissertation, we consider the Cauchy problem for the 3D incompressible Navier-Stokes equations. Here, we are interested in a smooth periodic solution of the problem which happens to be a special case of a paper by Otto Kreiss and Jens Lorenz. More precisely, we will look into a special case of their paper by two approaches. In the first approach, we will try to follow the similar techniques as in the original paper for smooth periodic solution. Because of the involvement of the Fourier expansion in the process, we encounter with some intriguing factors in the periodic case …

Why 7 Plus Minus 2? A Possible Geometric Explanation, Laxman Bokati, Vladik Kreinovich, Jordan Katz

Departmental Technical Reports (CS)

It is known that, in general, a person keeps in mind between 5 and 9 objects -- this is known as the 7 plus minus 2 law. In this paper, we provide a possible simple geometric explanation for this psychological feature.

Ranking-Based Voting Revisited: Maximum Entropy Approach Leads To Borda Count (And Its Versions), Olga Kosheleva, Vladik Kreinovich, Guo Wei

Departmental Technical Reports (CS)

In many practical situations, we need to make a group decision that takes into account preferences of all the participants. Ideally, we should elicit, from each participant, a full information about his/her preferences, but such elicitation is usually too time-consuming to be practical. Instead, we only elicit, from each participant, his/her ranking of different alternatives. One of the semi-heuristic methods for decision making under such information is Borda count, when for each alternative and each participant, we count how many alternatives are worse, and then select the alternatives for which the sum of these numbers is the largest. In this …

Why Matrix Factorization Works Well In Recommender Systems: A Systems-Based Explanation, Griselda Acosta, Manuel Hernandez, Natalia Villanueva-Rosales, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

Many computer-based services use recommender systems that predict our preferences based on our degree of satisfaction with the past selections. One of the most efficient techniques making recommender systems successful is matrix factorization. While this technique works well, until now, there was no general explanation of why it works. In this paper, we provide such an explanation.

Dynamic Triggering Of Earthquakes: Symmetry-Based Geometric Analysis, Laxman Bokati, Richard Alfaro, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that seismic waves from a remote earthquake can trigger a small local earthquake. Recent analysis has shown that this triggering occurs mostly when the direction of the incoming wave is orthogonal to the direction of the local fault, some triggerings occur when these directions are parallel, and very few triggerings occur when the angle between these two directions is different from 0 and 90 degrees. In this paper, we propose a symmetry-based geometric explanation for this unexpected observation.

How We Can Explain Simple Empirical Memory Rules, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Researchers have found out that normally, we remember about 30% of the information; however, if immediately after reading, we get a test, the rate increases to 45%. In this paper, we show that Zipf law can explain this empirical dependence.

Why Pink Noise Is Best For Enhancing Sleep And Memory: System-Based Explanation, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

Several researchers found out that acoustic stimulation during sleep enhances sleep and enhances memory. An interesting -- and somewhat mysterious -- part of this phenomenon is that out of all possible types of noise, the pink noise leads to the most efficient stimulation. In this paper, we use general system-based ideas to explain why in this phenomenon, pink noise works best.

Why Iq Test Scores Are Slightly Decreasing: Possible System-Based Explanation For The Reversed Flynn Effect, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

Researchers who monitor the average intelligence of human population have reasonably recently made an unexpected observation: that after many decades in which this level was constantly growing (this is known as the Flynn effect), at present, this level has started decreasing again. In this paper, we show that this reversed Flynn effect can be, in principle, explained in general system-based terms: namely, it is similar to the fact that a control system usually overshoots before stabilizing at the desired level. A similar idea may explain another unexpected observation -- that the Universe's expansion rate, which was supposed to be decreasing, …

In Alsina Et Al. Derivation Of Min-Max Fuzzy Logic From Distributivity, All Conditions Are Necessary: A Proof, Vladik Kreinovich, Ildar Batyrshin, Nailya Kubysheva

Departmental Technical Reports (CS)

In their 1983 paper, C. Alsina, E. Trillas, and L. Valverde proved that distributivity, monotonicity, and boundary conditions imply that the "and"-operation is min and the "or"-operation is max. In this paper, we show that all these conditions are necessary for Alsina et al. result to be true.

Beyond P-Boxes And Interval-Valued Moments: Natural Next Approximations To General Imprecise Probabilities, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To make an adequate decision, we need to know the probabilities of different consequences of different actions. In practice, we only have partial information about these probabilities -- this situation is known as imprecise probabilities. A general description of all possible imprecise probabilities requires using infinitely many parameters. In practice, the two most widely used few-parametric approximate descriptions are p-boxes (bounds on the values of the cumulative distribution function) and interval-valued moments (i.e., bounds on moments). In some situations, these approximations are not sufficiently accurate. So, we need more accurate more-parametric approximations. In this paper, we explain what are the …

Why Beta Priors: Invariance-Based Explanation, Olga Kosheleva, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

In the Bayesian approach, to describe a prior distribution on the set [0,1] of all possible probability values, typically, a Beta distribution is used. The fact that there have been many successful applications of this idea seems to indicate that there must be a fundamental reason for selecting this particular family of distributions. In this paper, we show that the selection of this family can indeed be explained if we make reasonable invariance requirements.

How To Gauge A Combination Of Uncertainties Of Different Type: General Foundations, Ingo Neumann, Vladik Kreinovich, Thach N. Nguyen

Departmental Technical Reports (CS)

In many practical situations, for some components of the uncertainty (e.g., of the measurement error) we know the corresponding probability distribution, while for other components, we know only upper bound on the corresponding values. To decide which of the algorithms or techniques leads to less uncertainty, we need to be able to gauge the combined uncertainty by a single numerical value -- so that we can select the algorithm for which this values is the best. There exist several techniques for gauging the combination of interval and probabilistic uncertainty. In this paper, we consider the problem of gauging the combination …

Runge–Kutta–Gegenbauer Explicit Methods For Advection-Diffusion Problems, Stephen O'Sullivan

Articles

In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of arbitrarily high order of accuracy are introduced in closed form. The stability domain of RKG polynomials extends in the the real direction with the square of polynomial degree, and in the imaginary direction as an increasing function of Gegenbauer parameter. Consequently, the polynomials are naturally suited to the construction of high order stabilized Runge-Kutta (SRK) explicit methods for systems of PDEs of mixed hyperbolic-parabolic type.

We present SRK methods composed of L ordered forward Euler stages, with complex-valued stepsizes derived from the roots of RKG stability polynomials of degree $L$. Internal stability …

How To Make A Decision Based On The Minimum Bayes Factor (Mbf): Explanation Of The Jeffreys Scale, Vladik Kreinovich, Olga Kosheleva, Nguyen Duc Trung

Departmental Technical Reports (CS)

In many practical situations, we need to select a model based on the data. It is, at present, practically a consensus that the traditional p-value-based techniques for such selection often do not lead to adequate results. One of the most widely used alternative model selection techniques is the Minimum Bayes Factor (MBF) approach, in which a model is preferred if the corresponding Bayes factor -- the ratio of likelihoods corresponding to this model and to the competing model -- is sufficiently large for all possible prior distributions. Based on the MBF values, we can decide how strong is the evidence …

Why Derivative: Invariance-Based Explanation, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To many students, the notion of a derivative seems unrelated to any previous mathematics -- and is, thus, difficult to study and to understand. In this paper, we show that this notion can be naturally derived from a more intuitive notion of invariance.