Mathematics Commons^™

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.

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.

Localized Learning: A Possible Alternative To Current Deep Learning Techniques, Javier Viana, Kelly Cohen, Anca Ralescu, Stephan Ralescu, Vladik Kreinovich

Departmental Technical Reports (CS)

At present, the most efficient deep learning technique is the use of deep neural networks. However, recent empirical results show that in some situations, it is even more efficient to use "localized" learning -- i.e., to divide the domain of inputs into sub-domains, learn the desired dependence separately on each sub-domain, and then "smooth" the resulting dependencies into a single algorithm. In this paper, we provide theoretical explanation for these empirical successes.

Freedom Of Will, Physics, And Human Intelligence: An Idea, Miroslav Svitek, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Among the main fundamental challenges related to physics and human intelligence are: How can we reconcile the free will with the deterministic character of physical equations? What is the physical meaning of extra spatial dimensions needed to make quantum physics consistent? and Why are we often smarter than brain-simulating neural networks? In this paper, we show that while each of these challenges is difficult to resolve on its own, it may be possible to resolve all three of them if we consider them together. The proposed possible solution is that human reasoning uses the extra spatial dimensions. This may sound …

What Is A Reasonable Way To Make Predictions?, Leonardo Orea Amador, Vladik Kreinovich

Departmental Technical Reports (CS)

Predictions are usually based on what is called laws of nature: many times, we observe the same relation between the states at different moments of time, and we conclude that the same relation will occur in the future. The more times the relation repeats, the more confident we are that the same phenomenon will be re-peated again. This is how Newton's laws and other laws came into being. This is what is called inductive reasoning. However, there are other reasonable approaches. For example, assume that a person speeds and is not caught. This may be repeated two times, three times …

How The Pavement's Lifetime Depends On The Stress Level: An Explanation Of The Empirical Formula, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva, Hoang Phuong Nguyen

Departmental Technical Reports (CS)

We show that natural invariance ideas explain the empirical dependence on the pavement's lifetime on the stress level.

Why Rectified Linear Activation Functions? Why Max-Pooling? A Possible Explanation, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

At present, the most successful machine learning technique is deep learning, that uses rectified linear activation function (ReLU) s(x) = max(x,0) as a non-linear data processing unit. While this selection was guided by general ideas (which were often imprecise), the selection itself was still largely empirical. This leads to a natural question: are these selections indeed the best or are there even better selections? A possible way to answer this question would be to provide a theoretical explanation of why these selections are -- in some reasonable sense -- the best. This paper provides a possible theoretical explanation for this …

Why Normalized Difference Vegetation Index (Ndvi)?, Francisco Zapata, Eric Smith, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Plants play a very important role in ecological systems -- they transform CO2 into oxygen. It is therefore very important to be able to estimate the overall amount of live green vegetation in a given area. The most efficient way to provide such a global analysis is to use remote sensing, i.e., multi-spectral photos taken from satellites, drones, planes, etc. At present, one of the most efficient ways to detect, based on remote sensing data, how much live green vegetation an area contains is to compute the value of the normalized difference vegetation index (NDVI). In this paper, we provide …

Shall We Be Foxes Or Hedgehogs: What Is The Best Balance For Research?, Miroslav Svitek, Olga Kosheleva, Shahnaz Shahbazova, Vladik Kreinovich

Departmental Technical Reports (CS)

Some researchers have few main ideas that they apply to many different problems -- they are called hedgehogs. Other researchers have many ideas but apply them to fewer problems -- they are called foxes. Both approaches have their advantages and disadvantages. What is the best balance between these two approaches? In this paper, we provide general recommendations about this balance. Specifically, we conclude that the optimal productivity is when the time spent on generating new ideas is equal to the time spent on understanding new applications. So, if for a researcher, understanding a new problem is much easier than generating …

As Complexity Rises, Meaningful Statements Lose Precision -- But Why?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the motivations for Zadeh's development of fuzzy logic -- and one of the explanations for the success of fuzzy techniques -- is the empirical observation that as complexity rises, meaningful statements lose precision. In this paper, we provide a possible explanation for this empirical phenomenon.

Finding Similar Stocks By Detecting Cliques In Market Graphs, Sudhashree Sayenju

Symposium of Student Scholars

The stock market provides an abundant source of data. However, when the amount of raw data becomes overwhelming it grows increasingly difficult to know how the stocks interact with each other. Stock data visualization as a market graph serves as one of the most popular way of summarizing important information. When modelling the data as a graph, vertices correspond to stocks and edges correspond to strong correlation in their pricing in a certain period of time. This project presents a technique to find stocks that behave very similarly. Such information helps investors make decisions on which stocks to purchase next. …

On Non-Linear Network Embedding Methods, Huong Yen Le

Dissertations

As a linear method, spectral clustering is the only network embedding algorithm that offers both a provably fast computation and an advanced theoretical understanding. The accuracy of spectral clustering depends on the Cheeger ratio defined as the ratio between the graph conductance and the 2nd smallest eigenvalue of its normalizedLaplacian. In several graph families whose Cheeger ratio reaches its upper bound of Theta(n), the approximation power of spectral clustering is proven to perform poorly. Moreover, recent non-linear network embedding methods have surpassed spectral clustering by state-of-the-art performance with little to no theoretical understanding to back them.

The dissertation includes work …

High Speed Hill Climbing Algorithm For Portfolio Optimization, Collether John

Tanzania Journal of Science

Portfolio can be defined as a collection of investments. Portfolio optimization usually is about maximizing expected return and/or minimising risk of a portfolio. The mean-variance model makes simplifying assumptions to solve portfolio optimization problem. Presence of realistic constraints leads to a significant different and complex problem. Also, the optimal solution under realistic constraints cannot always be derived from the solution for the frictionless market. The heuristic algorithms are alternative approaches to solve the extended problem. In this research, a heuristic algorithm is presented and improved for higher efficiency and speed. It is a hill climbing algorithm to tackle the extended …

An Adaptive Cryptosystem On A Finite Field, Awnon Bhowmik, Unnikrishnan Menon

Publications and Research

Owing to mathematical theory and computational power evolution, modern cryptosystems demand ingenious trapdoor functions as their foundation to extend the gap between an enthusiastic interceptor and sensitive information. This paper introduces an adaptive block encryption scheme. This system is based on product, exponent, and modulo operation on a finite field. At the heart of this algorithm lies an innovative and robust trapdoor function that operates in the Galois Field and is responsible for the superior speed and security offered by it. Prime number theorem plays a fundamental role in this system, to keep unwelcome adversaries at bay. This is a …

Predicting (Economic) Trends: Why Signature Method In Machine Learning, Vladik Kreinovich, Chon Van Le

Departmental Technical Reports (CS)

In many practical situations, we can predict the trend -- i.e., how the system will change -- but we cannot predict the exact timing of this change: this timing may depend on many unpredictable factors. For example, we may be sure that the economy will recover, but how fast it will recover may depend on the status of the pandemic, on the weather-affected agriculture input, etc. In such trend predictions, one of the most efficient methods is signature method, which is based on applying machine learning techniques to several special characteristics of the corresponding time series. In this paper, we …

How To Work? How To Study? Shall We Cram For The Exams? And How Is This Related To Life On Earth?, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

If we follow the same activity for a long time, our productivity decreases. To increase productivity, a natural idea is therefore to switch to a different activity, and then to switch back and resume the current task. On the other hand, after each switch, we need some time to get back to the original productivity. As a result, too frequent switches are also counterproductive. Natural questions are: shall we switch? if yes, when? In this paper, we use a simple model to provide approximate answers to these questions.

Correcting Interval-Valued Expert Estimates: Empirical Formulas Explained, Laura A. Berrout Ramos, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

Experts' estimates are approximate. To make decisions based on these estimates, we need to know how accurate these estimate are. Sometimes, experts themselves estimate the accuracy of their estimates -- by providing the interval of possible values instead of a single number. In other cases, we can gauge the accuracy of the experts' estimates by asking several experts to estimates the same quantity and using the interval range of these values. In both situations, sometimes the interval is too narrow -- e.g., if an expert is overconfident. Sometimes, the interval is too wide -- if the expert is too cautious. …

Why Moving Fast And Breaking Things Makes Sense?, Francisco Zapata, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional approach to engineering system design, engineers usually come up with several possible designs, each improving on the previous ones. In coming up with these designs, they try their best to make sure that their designs stay within the safety and other constraints, to avoid potential catastrophic crashes. The need for these safety constraints makes this design process reasonably slow. Software engineering at first followed the same pattern, but then realized that since in most cases, failure of a software test does not lead to a catastrophe, it is much faster to first ignore constraints and then adjust …

Why 70/100 Is Satisfactory? Why Five Letter Grades? Why Other Academic Conventions?, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Why 70/100 is usually a threshold for a student's satisfactory performance? Why there are usually only five letter grades? Why the usual arrangement of research, teaching, and service is 40-40-20? We show that all these arrangements -- and other similar academic arrangements -- can be explained by two ideas: the Laplace Indeterminacy Principle and the seven plus minus two law.

Blessings, God, Sacrifices: Possible Rational Explanations Of Biblical Ideas, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that many seemingly irrational Biblical ideas can actually be rationally interpreted: that God is everywhere, that we can only say what God is not, that God's name is holy, why cannot you bless as many people as you want, etc. We do not insist on our interpretations, there probably are many others, our sole objective was to show that many Biblical ideas can be rationally explained.

How To Gauge Students' Ability To Collaborate?, Christian Servin, Olga Kosheleva, Shahnaz Shahbazova, Vladik Kreinovich

Departmental Technical Reports (CS)

Usually, we mostly gauge individual students' skills. However, in the modern world, problems are rarely solved by individuals, it is usually a group effort. So, to make sure that students are successful, we also need to gauge their ability to collaborate. In this paper, we describe when it is possible to gauge the students' ability to collaborate; in situations when such a determination is possible, we explain how exactly we can estimate these abilities.

Applying Deep Learning To The Ice Cream Vendor Problem: An Extension Of The Newsvendor Problem, Gaffar Solihu

Electronic Theses and Dissertations

The Newsvendor problem is a classical supply chain problem used to develop strategies for inventory optimization. The goal of the newsvendor problem is to predict the optimal order quantity of a product to meet an uncertain demand in the future, given that the demand distribution itself is known. The Ice Cream Vendor Problem extends the classical newsvendor problem to an uncertain demand with unknown distribution, albeit a distribution that is known to depend on exogenous features. The goal is thus to estimate the order quantity that minimizes the total cost when demand does not follow any known statistical distribution. The …

Multilateration Index., Chip Lynch

Electronic Theses and Dissertations

We present an alternative method for pre-processing and storing point data, particularly for Geospatial points, by storing multilateration distances to fixed points rather than coordinates such as Latitude and Longitude. We explore the use of this data to improve query performance for some distance related queries such as nearest neighbor and query-within-radius (i.e. “find all points in a set P within distance d of query point q”). Further, we discuss the problem of “Network Adequacy” common to medical and communications businesses, to analyze questions such as “are at least 90% of patients living within 50 miles of a covered emergency …

Probability Distributions For Elliptic Curves In The Cgl Hash Function, Dhruv Bhatia, Kara Fagerstrom, Max Watson

Mathematical Sciences Technical Reports (MSTR)

Hash functions map data of arbitrary length to data of predetermined length. Good hash functions are hard to predict, making them useful in cryptography. We are interested in the elliptic curve CGL hash function, which maps a bitstring to an elliptic curve by traversing an inputdetermined path through an isogeny graph. The nodes of an isogeny graph are elliptic curves, and the edges are special maps betwixt elliptic curves called isogenies. Knowing which hash values are most likely informs us of potential security weaknesses in the hash function. We use stochastic matrices to compute the expected probability distributions of the …

The Mystery Of The Dancing Men, Manmohan Kaur

Journal of Humanistic Mathematics

In this paper I describe an activity based on a 1903 Sherlock Holmes murder mystery, in which a substitution cipher is used to encrypt secret messages. The story provides a fun and interesting way to talk about frequency analysis, and can be used as a segue into mathematical constructs such as modular arithmetic and computation. The activity is accessible to ages twelve and above, and has been successfully used in mathematics outreach and popularization efforts as well as in general education and mathematics courses.

Amplification Of Hidden Periodic Motions In 3d Videos, Thomas Boccuto, Seraiah Kutai, Kristen Mosley, Samuel Kirk

Mathematics Summer Fellows

Ordinary videos capture a surprising amount of hidden, visually imperceptible information. For instance, videos of peoples' faces may capture color changes in the skin and artery motion from heartbeats, while videos of mechanical systems can capture subtle vibrations indicating imminent failure. Algorithms can extract and exaggerate these signals for visualization on top of the original videos. In particular, Eulerian magnification algorithms sidestep the need to track hidden motions directly and instead devise multiscale bandpass filters to amplify signals in local spatial regions. In this work, we extend these techniques beyond color videos to geometric video data captured by 3D depth …

On Communication For Distributed Babai Point Computation, Maiara F. Bollauf, Vinay A. Vaishampayan, Sueli I.R. Costa

Publications and Research

We present a communication-efficient distributed protocol for computing the Babai point, an approximate nearest point for a random vector X∈Rn in a given lattice. We show that the protocol is optimal in the sense that it minimizes the sum rate when the components of X are mutually independent. We then investigate the error probability, i.e. the probability that the Babai point does not coincide with the nearest lattice point, motivated by the fact that for some cases, a distributed algorithm for finding the Babai point is sufficient for finding the nearest lattice point itself. Two different probability models for X …

Graph-Theoretic Partitioning Of Rnas And Classification Of Pseudoknots-Ii, Louis Petingi

Publications and Research

Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In previous works, a linear-time algorithm was introduced to partition dual graphs into maximally connected components called blocks and determine whether each block contains a pseudoknot or not. As pseudoknots can not be contained into two different blocks, this characterization allow us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Moreover we have extended the partitioning algorithm by classifying a pseudoknot as either recursive or non-recursive in order to continue with our research …

Infinite Peirce Distribution In The Algebra Of Compact Operators And Description Of Its Local Au-Tomorphisms, Farhodjon N. Arzikulov, Rejabboy Qo’Shaqov

Scientific Bulletin. Physical and Mathematical Research

In the present paper the infinite Peirce decomposition of the algebra 𝐾(𝐻) of com-pact operators on an infinite dimensional separable Gilbert space 𝐻 is constructed, using the norm of the algebra 𝐾(𝐻) and a maximal family of mutually or-thogonal minimal projections, i.e., self-adjoint,idempotent elements. The infinite Peirce decompo-sition on the norm of a 𝐶∗-algebra is also con-structed in 2012 by the first author. But, it turns, the condition, applied then, is not sufficient for the infi-nite Peirce decomposition on the norm constructed in 2012 to be an algebra. Therefore, in the present paper, the infinite Peirce decomposition on the norm …