Mathematics Commons^™

Knots And Links In Overtwisted Contact Manifolds, Rima Chatterjee

LSU Doctoral Dissertations

Suppose $(\M,\xi)$ be an overtwisted contact 3-manifold. We prove that any Legendrian and transverse link in $(\M,\xi)$ having overtwisted complement can be coarsely classified by their classical invariants. Next, we defined an invariant called the support genus for transverse links and extended the definition of support genus of Legendrian knots to Legendrian links and prove that any coarse equivalence class of Legendrian and transverse loose links has support genus zero. Further, we show that the converse is not true by explicitly constructing an example. We also find a relationship between the support genus of the transverse link and its Legendrian …

Stochastic Navier-Stokes Equations With Markov Switching, Po-Han Hsu

LSU Doctoral Dissertations

This dissertation is devoted to the study of three-dimensional (regularized) stochastic Navier-Stokes equations with Markov switching. A Markov chain is introduced into the noise term to capture the transitions from laminar to turbulent flow, and vice versa. The existence of the weak solution (in the sense of stochastic analysis) is shown by studying the martingale problem posed by it. This together with the pathwise uniqueness yields existence of the unique strong solution (in the sense of stochastic analysis). The existence and uniqueness of a stationary measure is established when the noise terms are additive and autonomous. Certain exit time estimates …

Nonlinear Potential Analysis On Sobolev Multiplier Spaces, Keng Hao Ooi

LSU Doctoral Dissertations

We characterize preduals and Kothe duals to a class of Sobolev multiplier type spaces. Our results fit in well with the modern theory of function spaces of harmonic analysis and are also applicable to nonlinear partial differential equations. As a maneuver, we make use of several tools from nonlinear potential theory, weighted norm inequalities, and the theory of Banach function spaces to obtain our results. After characterizing the preduals, we establish a capacitary strong type inequality which resolves a special case of a conjecture by David R. Adams. As a consequence, we obtain several equivalent norms for Choquet integrals associated …

Determining Biases In The Card-Chameleon Cryptosystem, Isaac Reiter, Eric Landquist

Communications on Number Theory and Combinatorial Theory

Throughout history, spies, soldiers, and others have relied on so-called {\em hand ciphers} to send encrypted messages. Since the creation of Pontifex (also known as Solitaire) by Bruce Schneier in 1999, a number of hand ciphers utilizing a standard deck of playing cards have emerged. Since there are $52! \approx 2^{225.58}$ possible ways to order a deck of cards, there are over 225 bits of entropy in a well-shuffled deck of cards. Theoretically, this can provide enough security to rival modern computer-based cryptosystems. In this paper, we describe and analyze one such playing card cipher, Card-Chameleon, created by Matthew McKague. …

New Representations For A Semi-Markov Chain And Related Filters, Robert J. Elliott, W. P. Malcolm

Journal of Stochastic Analysis

No abstract provided.

Designing Efficient Algorithms For Sensor Placement, Gabriel Loos

Honors College Theses

Sensor placement has many applications and uses that can be seen everywhere you go.These include, but not limited to, monitoring the structural health of buildings and bridgesand navigating Unmanned Aerial Vehicles(UAV).We study ways that leads to efficient algorithms that will place as few as possible sen-sors to cover an entire area. We will tackle the problem from both 2-dimensional and3-dimensional points of view. Two famous related problems are discussed: the art galleryproblem and the terrain guarding problem. From the top view an area presents a 2-D im-age which will enable us to partition polygonal shapes and use graph theoretical results …

Math Escape Rooms: A Novel Approach For Engaging Learners In Math Circles, Janice F. Rech, Paula Jakopovic, Hannah Seidl, Greg Lawson, Rachel Pugh

Journal of Math Circles

Engaging middle and high school students in Math Circles requires time, planning and creativity. Finding novel approaches to maintain the interest of a variety of learners can be challenging. This paper outlines a model for developing and implementing math escape rooms as a unique structure for facilitating collaborative problem solving in a Math Circle. These escape rooms were designed and hosted by undergraduate secondary mathematics education majors. We provide possible structures for hosting escape rooms that could translate to a range of settings, as well as reflections and lessons learned through our experiences that could inform practitioners in other settings.

Wild Randomness And The Application Of Hyperbolic Diffusion In Financial Modelling, Will Hicks

Journal of Stochastic Analysis

No abstract provided.

An Efficient Algorithm To Test Potential Bipartiteness Of Graphical Degree Sequences, Kai Wang

Theory & Applications of Graphs

As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in polynomial time, at the expense of possibly producing an erroneous output on some ``yes'' instances but with very low error rate.

Towards A General Framework For Practical Quantum Network Protocols, Sumeet Khatri

LSU Doctoral Dissertations

The quantum internet is one of the frontiers of quantum information science. It will revolutionize the way we communicate and do other tasks, and it will allow for tasks that are not possible using the current, classical internet. The backbone of a quantum internet is entanglement distributed globally in order to allow for such novel applications to be performed over long distances. Experimental progress is currently being made to realize quantum networks on a small scale, but much theoretical work is still needed in order to understand how best to distribute entanglement and to guide the realization of large-scale quantum …

Linear Decomposition And Anticipating Integral For Certain Random Variables, Ching-Tang Wu, Ju-Yi Yen

Journal of Stochastic Analysis

No abstract provided.

Baudelaire's Ideas Of Vagueness And Uniqueness In Art: Algorithm-Based Explanations, Luc Longpre, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

According to the analysis by the French philosopher Jean-Paul Sartre, the famous French poet and essayist Charles Baudelaire described (and followed) two main -- somewhat unusual -- ideas about art: that art should be vague, and that to create an object of art, one needs to aim for uniqueness. In this paper, we provide an algorithm-based explanation for these seemingly counter-intuitive ideas, explanation related to Kolmogorov complexity-based formalization of Garrett Birkhoff's theory of beauty.

How Void Ratio Depends On Grain Size In Soil Mechanics: Theoretical Explanation, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

When designing a road, it is important to know how many voids are in the underlying soil -- since these voids will affect the road stiffness. It is difficult to measure the voids ratio directly, so instead, we need to estimate it based on easier-to-measure characteristics such as grain size. There are empirical formulas for such estimation. In this paper, we provide a possible theoretical explanation for these empirical formulas.

Why Base-20, Base-40, And Base-60 Number Systems?, Sean R. Aguilar, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Historically, to describe numbers, some cultures used bases much larger than our usual base 10, namely, bases 20, 40, and 60. There are explanations for base 60, there is some explanation for base 20, but base 40 -- used in medieval Russia -- remains largely a mystery. In this paper, we provide a possible explanation for all these three bases, an explanation based on the natural need to manage large groups of people. We also speculate why different cultures used different bases.

Dimension Compactification Naturally Follows From First Principles, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

According to modern physics, space-time originally was of dimension 11 or higher, but then additional dimensions became compactified, i.e., size in these directions remains small and thus, not observable. As a result, at present, we only observed 4 dimensions of space-time. There are mechanisms that explain how compactification may have occurred, but the remaining question is why it occurred. In this paper, we provide two first-principles-based explanations for space-time compactification: based on Second Law of Thermodynamics and based on geometry and symmetries.

Additional Spatial Dimensions Can Help Speed Up Computations, Luc Longpre, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While we currently only observe 3 spatial dimensions, according to modern physics, our space is actually at least 10-dimensional. In this paper, on different versions of the multi-D spatial models, we analyze how the existence of the additional spatial dimensions can help computations. It turns out that in all the versions, there is some speed up -- moderate when the extra dimensions are actually compactified, and drastic if extra dimensions are separated by a potential barrier.

What Is The True Formula For Soil Permeability? Not Clear, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

To design and maintain pavements, it is important to know how fast water will penetrate the underlying soil. The speed of this penetration is determined by a quantity called permeability. There are several seemingly very different empirical and semi-empirical formulas that predict permeability. A recent attempt to select the formula that best fits the experimental data ended up in an unexpected conclusion that all three formula provide a good fit for the data. But these formulas are very different, how come that all three of them fit the same data? In this paper, we explain this somewhat paradoxical result.

Low-Complexity Zonotopes Can Enhance Uncertainty Quantification (Uq), Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, the only information that we know about the measurement error is the upper bound D on its absolute value. In this case, once we know the measurement result X, the only information that we have about the actual value x of the corresponding quantity is that this value belongs to the interval [X − D, X + D]. How can we estimate the accuracy of the result of data processing under this interval uncertainty? In general, computing this accuracy is NP-hard, but in the usual case when measurement errors are relatively small, we can linearize the …

Covid-19 And Knowledge Based Computation, Rohit J. Parikh

Publications and Research

The problem of dealing with Covid-19, until a vaccine is universally administered, is to decrease the rate of transmission while getting some social and economic activity going.

Infection passes from one person A to another person B when A is infected and B is susceptible. That is to say that B is not infected and not yet immune.

Social activity also takes place when one person interacts with another. Perhaps A is a taxpayer and B is a tax consultant. Then filing the tax return may take the form of the two of them meeting. Much can be done electronically …

Adaptive Kriging Method For Uncertainty Quantification Of The Photoelectron Sheath And Dust Levitation On The Lunar Surface, Xinpeng Wei, Jianxun Zhao, Xiaoming He, Zhen Hu, Xiaoping Du, Daoru Han

Mechanical and Aerospace Engineering Faculty Research & Creative Works

This paper presents an adaptive Kriging based method to perform uncertainty quantification (UQ) of the photoelectron sheath and dust levitation on the lunar surface. The objective of this study is to identify the upper and lower bounds of the electric potential and that of dust levitation height, given the intervals of model parameters in the one-dimensional (1D) photoelectron sheath model. To improve the calculation efficiency, we employ the widely used adaptive Kriging method (AKM). A task-oriented learning function and a stopping criterion are developed to train the Kriging model and customize the AKM. Experiment analysis shows that the proposed AKM …