Digital Commons Network^™

(R1956) Neutrosophic Soft E-Compact Spaces And Application Using Entropy Measure, P. Revathi, K. Chitirakala, A. Vadivel

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the concept of neutrosophic soft e-compactness is presented on neutrosophic soft topological spaces using the definition of e-open cover and its types. In addition, neutrosophic soft e-compactness and neutrosophic soft e-separation axioms are associated. Also, the concept of neutrosophic soft locally e-compactness is introduced in neutrosophic soft topological spaces and some of its properties are discussed. Added to that, an application in decision making problem is given using entropy.

Efficient Algorithms For Data Processing Under Type-3 (And Higher) Fuzzy Uncertainty, Vladik Kreinovich, Olga Kosheleva, Patricia Melin, Oscar Castillo

Departmental Technical Reports (CS)

It is known that to more adequately describe expert knowledge, it is necessary to go from the traditional (type-1) fuzzy techniques to higher order ones: type-2, probably type-3 and even higher. Until recently, only type-1 and type-2 fuzzy sets were used in practical applications. However, lately, it turned out that type-3 fuzzy sets are also useful in some applications. Because of this practical importance, it is necessary to design efficient algorithms for data processing under such type-3 (and higher order) fuzzy uncertainty. In this paper, we show how we can combine known efficient algorithms for processing type-1 and type-2 uncertainty …

Why Quantiles Are A Good Description Of Volatility In Economics: An Alternative Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

In econometrics, volatility of an investment is usually described by its Value-at-Risk (VaR), i.e., by an appropriate quantile of the corresponding probability distribution. The motivations for selecting VaR are largely empirical: VaR provides a more adequate description of what people intuitively perceive as risk. In this paper, we analyze this situation from the viewpoint of decision theory, and we show that this analysis naturally leads to the Value-at-Risk, i.e., to a quantile.

Interestingly, this analysis also naturally leads to an optimization problem related to quantile regression.

Why Decreased Gaps Between Brain Cells Cause Severe Headaches: A Symmetry-Based Geometric Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

When we analyze biological tissue under the microscope, cells are directly neighboring each other, with no gaps between them. However, a more detailed analysis shows that in vivo, there are small liquid-filled gaps between the cells, and these gaps are important: e.g., in abnormal situations, when the size of the gaps between brain cells decreases, this leads to severe headaches and other undesired effects. At present, there is no universally accepted explanation for this phenomenon. In this case, we show that the analysis of correpsonding geometric symmetries empirical phenomenon leads to a natural explanation for this effect.

How To Represent Uncertainty Via Qudits: Probability Distributions, Regular, Intuitionistic, And Picture Fuzzy Sets, F-Transforms, Etc., Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While modern computers are fast, there are still many important practical situations in which we need even faster computations. It turns out that, due to the fact that the speed of all communications is limited by the speed of light, the only way to make computers drastically faster is to drastically decrease the size of computer's components. When we decrease their size to sizes comparable with micro-sizes of individual molecules, it becomes necessary to take into account specific physics of the micro-world -- known as quantum physics. Traditional approach to designing quantum computers -- i.e., computers that take effect of …

Why Rarity Score Is A Good Evaluation Of A Non-Fungible Token, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the new forms of investment is investing in so-called non-fungible tokens -- unique software objects associated with different real-life objects like songs, painting, photos, videos, characters in computer games, etc. Since these tokens are a form of financial investment, investors would like to estimate the fair price of such tokens. For tokens corresponding to objects that have their own price -- such as a song or a painting -- a reasonable estimate is proportional to the price of the corresponding object. However, for tokens corresponding to computer game characters, we cannot estimate their price this way. Based on …

Towards Better Ways To Compute The Overall Grade For A Class, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional way to compute the overall grade for the class is to use the weighted sum of the grades for all the assignments and exams, including the final exam. In terms of encouraging students to study hard throughout the semester, this grading scheme is better than an alternative scheme, in which all that matters is the grade on the final exam: in contrast to this alternative scheme, in the weighted-sum approach, students are penalized if they did not do well in the beginning of the semester. In practice, however, instructors sometimes deviate from the weighted-sum scheme: indeed, if the weighted …

How To Combine Expert Estimates? How To Estimate Probability In The Intersection Of Two Populations?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we consider two different practical problems that turned to be mathematically similar: (1) how to combine two expert-provided probabilities of some event and (2) how to estimate the frequency of a certain phenomenon (e.g., illness) in an intersection of two populations if we know the frequencies in each of these populations. In both cases, we use the maximum entropy approach to come up with a solution.

Why Quantile Regression Works Well In Economics: A Partial Explanation, Olga Kosheleva, Vassilis G. Kaburlasos, Vladik Kreinovich, Roengchai Tansuchat

Departmental Technical Reports (CS)

To get a better picture of the future behavior of different economics-related quantities, we need to be able to predict not only their mean values, but also their distribution. For example, it is desirable not only to predict future average income, but also to predict the future distribution of income. One of the convenient ways to describe a probability distribution is by using alpha-quantiles such as medians (corresponding to alpha = 0.5), quartiles (corresponding to alpha = 0.25 and alpha = 0.75), etc. In principle, an alpha-quantile of the desired future quantity can depend on beta-quantiles of current distributions corresponding …

Invariance-Based Approach Explains Empirical Formulas From Pavement Engineering To Deep Learning, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many application areas, there are effective empirical formulas that need explanation. In this paper, we focus on two such challenges: neural networks, where a so-called softplus activation function is known to be very efficient, and pavement engineering, where there are empirical formulas describing the dependence of the pavement strength on the properties of the underlying soil. We show that similar scale-invariance ideas can explain both types of formulas -- and, in the case of pavement engineering, invariance ideas can lead to a new formula that combines the advantages of several known ones.

How To Detect (And Analyze) Independent Subsystems Of A Black-Box (Or Grey-Box) System, Saeid Tizpaz-Niari, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Often, we deal with black-box or grey-box systems where we can observe the overall system's behavior, but we do not have access to the system's internal structure. In many such situations, the system actually consists of two (or more) independent components: a) how can we detect this based on observed system's behavior? b) what can we learn about the independent subsystems based on the observation of the system as a whole? In this paper, we provide (partial) answers to these questions.

Hawthorne Effect: An Explanation Based On Decision Theory, Sofia Holguin, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

It is known that people feel better (and even work better) if someone pays attention to them; this is known as the Hawthorne effect. At first glance, it sounds counter-intuitive: this attention does not bring you any material benefits, so why would you feel better? If you are sick and someone gives you medicine, this will make you feel better, but if someone just pays attention, why does that make you feel better? In this paper, we use the general ideas of decision theory to explain this seemingly counterintuitive phenomenon.

Why Should Exactly 1/4 Be Returned To The Original Owner: An Economic Explanation Of An Ancient Recommendation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

What if someone bought a property in good faith, not realizing that this property was unjustly confiscated from the previous owner? In such situations, if the new owner decided to sell this property, Talmud recommended that a fair way is to return 1/4 of the selling price to the original owner. However, it does not provide any explanation of why exactly 1/4 -- and not any other portion -- is to be returned. In this paper, we provide an economic explanation for this recommendation, an explanation that fits well with other ancient recommendations about debts.

Why Convex Combination Is An Effective Crossover Operation In Continuous Optimization: A Theoretical Explanation, Kelly Cohen, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When evolutionary computation techniques are used to solve continuous optimization problems, usually, convex combination is used as a crossover operation. Empirically, this crossover operation works well, but this success is, from the foundational viewpoint, a challenge: why this crossover operation works well is not clear. In this paper, we provide a theoretical explanation for this empirical success.

Estimating Skewness And Higher Central Moments Of An Interval-Valued Fuzzy Set, Juan Carlos Figueroa-Garcia, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A known relation between membership functions and probability density functions allows us to naturally extend statistical characteristics like central moments to the fuzzy case. In case of interval-valued fuzzy sets, we have several possible membership functions consistent with our knowledge. For different membership functions, in general, we have different values of the central moments. It is therefore desirable to compute, in the interval-valued fuzzy case, the range of possible values for each such moment. In this paper, we provide efficient algorithms for this computation.

Exploring The Numerical Range Of Block Toeplitz Operators, Brooke Randell

Master's Theses

We will explore the numerical range of the block Toeplitz operator with symbol function \(\phi(z)=A_0+zA_1\), where \(A_0, A_1 \in M_2(\mathbb{C})\). A full characterization of the numerical range of this operator proves to be quite difficult and so we will focus on characterizing the boundary of the related set, \(\{W(A_0+zA_1) : z \in \partial \mathbb{D}\}\), in a specific case. We will use the theory of envelopes to explore what the boundary looks like and we will use geometric arguments to explore the number of flat portions on the boundary. We will then make a conjecture as to the number of flat …

On The Numerical Range Of Compact Operators, Montserrat Dabkowski

Master's Theses

One of the many characterizations of compact operators is as linear operators which
can be closely approximated by bounded finite rank operators (theorem 25). It is
well known that the numerical range of a bounded operator on a finite dimensional
Hilbert space is closed (theorem 54). In this thesis we explore how close to being
closed the numerical range of a compact operator is (theorem 56). We also describe
how limited the difference between the closure and the numerical range of a compact
operator can be (theorem 58). To aid in our exploration of the numerical range of
a compact …

Periodic Functions Related To The Gompertz Difference Equation, Tom Cuchta, Nick Wintz

Faculty Scholarship

We investigate periodicity of functions related to the Gompertz difference equation. In particular, we derive difference equations that must be satisfied to guarantee periodicity of the solution.

A Generalized Family Of Exponentiated Composite Distributions, Bowen Liu, Malwane Ananda

Mathematical Sciences Faculty Research

In this paper, we propose a new family of distributions, by exponentiating the random variables associated with the probability density functions of composite distributions. We also derive some mathematical properties of this new family of distributions, including the moments and the limited moments. Specifically, two special models in this family are discussed. Three real datasets were chosen, to assess the performance of these two special exponentiated-composite models. When fitting to these three datasets, these three special exponentiated-composite distributions demonstrate significantly better performance, compared to the original composite distributions.

The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza

Dissertations, Theses, and Capstone Projects

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf …

Thickened Surfaces, Checkerboard Surfaces, And Quantum Link Invariants, Joseph W. Boninger

Dissertations, Theses, and Capstone Projects

This dissertation has two parts, each motivated by an open problem related to the Jones polynomial. The first part addresses the Volume Conjecture of Kashaev, Murakami, and Murakami. We define a polynomial invariant, J^T_n, of links in the thickened torus, which we call the nth toroidal colored Jones polynomial, and we show J^T_n satisfies many properties of the original colored Jones polynomial. Most significantly, J^T_n exhibits volume conjecture behavior. We prove a volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions …

(R1961) On Fuzzy Upper And Lower Theta Star Semicontinuous Multifunctions, A. Mughil, A. Vadivel, O. Uma Maheswari

Applications and Applied Mathematics: An International Journal (AAM)

This work introduces the concepts of fuzzy upper and lower theta star (respectively theta)- semicontinuous multifunction on fuzzy topological spaces in the Sostak sense. In L-fuzzy topological spaces, the mutual relationships of these fuzzy upper (resp. fuzzy lower) theta star (resp. theta)-semicontinuous multifunctions are established, as well as several characterizations and properties. Later, researchers looked at the composition and union of these multifunctions.

An Investigation Into Crouzeix's Conjecture, Timothy T. Royston

Master's Theses

We will explore Crouzeix’s Conjecture, an upper bound on the norm of a matrix after the application of a polynomial involving the numerical range. More formally, Crouzeix’s Conjecture states that for any n × n matrix A and any polynomial p from C → C,
∥p(A)∥ ≤ 2 sup_{z∈W (A)} |p(z)|.
Where W (A) is a set in C related to A, and ∥·∥ is the matrix norm. We first discuss the conjecture, and prove the simple case when the matrix is normal. We then explore a proof for a class of matrices given by Daeshik Choi. We expand …

General Rogue Wave Solutions To The Sasa-Satsuma Equation, Chengfa Wu, Guangxiong Zhang, Changyan Shi, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

General rogue wave solutions to the Sasa-Satsuma equation are constructed by the Kadomtsev-Petviashvili (KP) hierarchy reduction method. These solutions are presented in three different forms. The first form is expressed in terms of recursively defined differential operators while the second form shares a similar solution structure except that the differential operators are no longer recursively defined. Instead of using differential operators, the third form is expressed by Schur polynomials.

Solution Of The Reconstruction-Of-The-Measure Problem For Canonical Invariant Subspaces, Raul E. Curto, Sang H. Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable weighted shifts W(α,β), when the initial data are given as the Berger measure of the restriction of W(α,β) to a canonical invariant subspace, together with the marginal measures for the 0–th row and 0–th column in the weight diagram for W(α,β). We prove that the natural necessary conditions are indeed sufficient. When the initial data correspond to a soluble problem, we give a concrete formula for the Berger measure of W(α,β). Our strategy is to build on previous results for back-step extensions and onestep extensions. A key new theorem allows us …

Dynamical Systems And Matching Symmetry In Beta-Expansions, Karl Zieber

Master's Theses

Symbolic dynamics, and in particular β-expansions, are a ubiquitous tool in studying more complicated dynamical systems. Applications include number theory, fractals, information theory, and data storage.

In this thesis we will explore the basics of dynamical systems with a special focus on topological dynamics. We then examine symbolic dynamics and β-transformations through the lens of sequence spaces. We discuss observations from recent literature about how matching (the property that the itinerary of 0 and 1 coincide after some number of iterations) is linked to when T_β,⍺ generates a subshift of finite type. We prove the set of ⍺ in …

An Introduction To Fröberg's Conjecture, Caroline Semmens

Master's Theses

The goal of this thesis is to make Fröberg's conjecture more accessible to the average math graduate student by building up the necessary background material to understand specific examples where Fröberg's conjecture is true.

Unique Signed Minimal Wiring Diagrams And The Stanley-Reisner Correspondence, Vanessa Newsome-Slade

Master's Theses

Biological systems are commonly represented using networks consisting of interactions between various elements in the system. Reverse engineering, a method of mathematical modeling, is used to recover how the elements in the biological network are connected. These connections are encoded using wiring diagrams, which are directed graphs that describe how elements in a network affect one another. A signed wiring diagram provides additional information about the interactions between elements relating to activation and inhibition. Due to cost concerns, it is optimal to gain insight into biological networks with as few experiments and data as possible. Minimal wiring diagrams identify the …

Van Kampen Diagrams And Small Cancellation Theory, Kelsey N. Lowrey

Master's Theses

Given a presentation of G, the word problem asks whether there exists an algorithm to determine which words in the free group, F(A), represent the identity in G. In this thesis, we study small cancellation theory, developed by Lyndon, Schupp, and Greendlinger in the mid-1960s, which contributed to the resurgence of geometric group theory. We investigate the connection between Van Kampen diagrams and the small cancellation hypotheses. Groups that have a presentation satisfying the small cancellation hypotheses C'(1/6), or C'(1/4) and T(4) have a nice solution to the word problem known as Dehn’s Algorithm.

Joint Control Of Manufacturing And Onsite Microgrid System Via Novel Neural-Network Integrated Reinforcement Learning Algorithms, Jiaojiao Yang, Zeyi Sun, Wenqing Hu, Louis Steinmeister

Mathematics and Statistics Faculty Research & Creative Works

Microgrid is a promising technology of distributed energy supply system, which consists of storage devices, generation capacities including renewable sources, and controllable loads. It has been widely investigated and applied for residential and commercial end-use customers as well as critical facilities. In this paper, we propose a joint state-based dynamic control model on microgrids and manufacturing systems where optimal controls for both sides are implemented to coordinate the energy demand and supply so that the overall production cost can be minimized considering the constraint of production target. Markov Decision Process (MDP) is used to formulate the decision-making procedure. The main …