Theory and Algorithms Commons^™

On The Total Set Chromatic Number Of Graphs, Mark Anthony C. Tolentino, Gerone Russel J. Eugenio, Mari-Jo P. Ruiz

Theory & Applications of Graphs

Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χ_s(G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = (V, E) …

Analysis Of A Quantum Attack On The Blum-Micali Pseudorandom Number Generator, Tingfei Feng

Mathematical Sciences Technical Reports (MSTR)

In 2012, Guedes, Assis, and Lula proposed a quantum attack on a pseudorandom number generator named the Blum-Micali Pseudorandom number generator. They claimed that the quantum attack can outperform classical attacks super-polynomially. However, this paper shows that the quantum attack cannot get the correct seed and provides another corrected algorithm that is in exponential time but still faster than the classical attack. Since the original classical attacks are in exponential time, the Blum-Micali pseudorandom number generator would be still quantum resistant.

The Primitive Root Problem: A Problem In Bqp, Shixin Wu

Mathematical Sciences Technical Reports (MSTR)

Shor’s algorithm proves that the discrete logarithm problem is in BQP. Based on his algorithm, we prove that the primitive root problem, a problem that verifies if some integer g is a primitive root modulo p where p is the largest prime number smaller than 2n for a given n, which is assumed to be harder than the discrete logarithm problem, is in BQP by using an oracle quantum Turing machine.

Data And Algorithmic Modeling Approaches To Count Data, Andraya Hack

Honors College Theses

Various techniques are used to create predictions based on count data. This type of data takes the form of a non-negative integers such as the number of claims an insurance policy holder may make. These predictions can allow people to prepare for likely outcomes. Thus, it is important to know how accurate the predictions are. Traditional statistical approaches for predicting count data include Poisson regression as well as negative binomial regression. Both methods also have a zero-inflated version that can be used when the data has an overabundance of zeros. Another procedure is to use computer algorithms, also known as …

A Super Fast Algorithm For Estimating Sample Entropy, Weifeng Liu, Ying Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

: Sample entropy, an approximation of the Kolmogorov entropy, was proposed to characterize complexity of a time series, which is essentially defined as − log(B/A), where B denotes the number of matched template pairs with length m and A denotes the number of matched template pairs with m + 1, for a predetermined positive integer m. It has been widely used to analyze physiological signals. As computing sample entropy is time consuming, the box-assisted, bucket-assisted, x-sort, assisted sliding box, and kd-tree-based algorithms were proposed to accelerate its computation. These algorithms require O(N²) or …

Provably Weak Instances Of Plwe Revisited, Again, Katherine Mendel

CSB and SJU Distinguished Thesis

Learning with Errors has emerged as a promising possibility for postquantum cryptography. Variants known as RLWE and PLWE have been shown to be more efficient, but the increased structure can leave them vulnerable to attacks for certain instantiations. This work aims to identify specific cases where proposed cryptographic schemes based on PLWE work particularly poorly under a specific attack.

The Nature Of Numbers: Real Computing, Bradley J. Lucier

Journal of Humanistic Mathematics

While studying the computable real numbers as a professional mathematician, I came to see the computable reals, and not the real numbers as usually presented in undergraduate real analysis classes, as the natural culmination of my evolving understanding of numbers as a schoolchild. This paper attempts to trace and explain that evolution. The first part recounts the nature of numbers as they were presented to us grade-school children. In particular, the introduction of square roots induced a step change in my understanding of numbers. Another incident gave me insight into the brilliance of Alan Turing in his paper introducing both …

Games For One, Games For Two: Computationally Complex Fun For Polynomial-Hierarchical Families, Kye Shi

HMC Senior Theses

In the first half of this thesis, we explore the polynomial-time hierarchy, emphasizing an intuitive perspective that associates decision problems in the polynomial hierarchy to combinatorial games with fixed numbers of turns. Specifically, problems in �� are thought of as 0-turn games, ���� as 1-turn “puzzle” games, and in general ��ₖ�� as ��-turn games, in which decision problems answer the binary question, “can the starting player guarantee a win?” We introduce the formalisms of the polynomial hierarchy through this perspective, alongside definitions of ��-turn CIRCUIT SATISFIABILITY games, whose ��ₖ��-completeness is assumed from prior work (we briefly justify this assumption …

Finding Optimal Cayley Map Embeddings Using Genetic Algorithms, Jacob Buckelew

Honors Program Theses

Genetic algorithms are a commonly used metaheuristic search method aimed at solving complex optimization problems in a variety of fields. These types of algorithms lend themselves to problems that can incorporate stochastic elements, which allows for a wider search across a search space. However, the nature of the genetic algorithm can often cause challenges regarding time-consumption. Although the genetic algorithm may be widely applicable to various domains, it is not guaranteed that the algorithm will outperform other traditional search methods in solving problems specific to particular domains. In this paper, we test the feasibility of genetic algorithms in solving a …

Decoding Cyclic Codes Via Gröbner Bases, Eduardo Sosa

Honors Theses

In this paper, we analyze the decoding of cyclic codes. First, we introduce linear and cyclic codes, standard decoding processes, and some standard theorems in coding theory. Then, we will introduce Gr¨obner Bases, and describe their connection to the decoding of cyclic codes. Finally, we go in-depth into how we decode cyclic codes using the key equation, and how a breakthrough by A. Brinton Cooper on decoding BCH codes using Gr¨obner Bases gave rise to the search for a polynomial-time algorithm that could someday decode any cyclic code. We discuss the different approaches taken toward developing such an algorithm and …

Lasso: Listing All Subset Sums Obediently For Evaluating Unbounded Subset Sums, Christopher N. Burgoyne, Travis J. Wheeler

Graduate Student Theses, Dissertations, & Professional Papers

In this study we present a novel algorithm, LASSO, for solving the unbounded and bounded subset sum problem. The LASSO algorithm was designed to solve the unbounded SSP quickly and to return all subsets summing to a target sum. As speed was the highest priority, we benchmarked the run time performance of LASSO against implementations of some common approaches to the bounded SSP, as well as the only comparable implementation for solving the unbounded SSP that we could find. In solving the bounded SSP, our algorithm had a significantly faster run time than the competing algorithms when the target sum …

Matrix Interpretations And Tools For Investigating Even Functionals, Benjamin Stringer

Theses and Dissertations--Computer Science

Even functionals are a set of polynomials evaluated on the terms of hollow symmetric matrices. Their properties lend themselves to applications such as counting subgraph embeddings in generic (weighted or unweighted) host graphs and computing moments of binary quadratic forms, which occur in combinatorial optimization. This research focuses primarily on counting subgraph embeddings, which is traditionally accomplished with brute-force algorithms or algorithms curated for special types of graphs. Even functionals provide a method for counting subgraphs algebraically in time proportional to matrix multiplication and is not restricted to particular graph types. Counting subgraph embeddings can be accomplished by evaluating a …

Rankings Of Mma Fighters, Michael Schaefer

All Graduate Theses, Dissertations, and Other Capstone Projects

Ranking is an essential process that allows sporting authorities to determine the relative performance of athletes. While ranking is straightforward in some sports, it is more complicated in MMA (mixed martial arts), where competition is often fragmented. This paper describes the mathematics behind four existing ranking algorithms: Elo’s System, Massey’s Method, Colley’s Method, and Google’s PageRank, and shows how to adapt them to rank MMA fighters in the UFC (Ultimate Fighting Championship). We also provide a performance analysis for each ranking method.

Computer Program Simulation Of A Quantum Turing Machine With Circuit Model, Shixin Wu

Mathematical Sciences Technical Reports (MSTR)

Molina and Watrous present a variation of the method to simulate a quantum Turing machine employed in Yao’s 1995 publication “Quantum Circuit Complexity”. We use a computer program to implement their method with linear algebra and an additional unitary operator defined to complete the details. Their method is verified to be correct on a quantum Turing machine.

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 …

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.

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 …

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 …

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 …

The “Knapsack Problem” Workbook: An Exploration Of Topics In Computer Science, Steven Cosares

Open Educational Resources

This workbook provides discussions, programming assignments, projects, and class exercises revolving around the “Knapsack Problem” (KP), which is widely a recognized model that is taught within a typical Computer Science curriculum. Throughout these discussions, we use KP to introduce or review topics found in courses covering topics in Discrete Mathematics, Mathematical Programming, Data Structures, Algorithms, Computational Complexity, etc. Because of the broad range of subjects discussed, this workbook and the accompanying spreadsheet files might be used as part of some CS capstone experience. Otherwise, we recommend that individual sections be used, as needed, for exercises relevant to a course in …

The Generalized Riemann Hypothesis And Applications To Primality Testing, Peter Hall

University Scholar Projects

The Riemann Hypothesis, posed in 1859 by Bernhard Riemann, is about zeros
of the Riemann zeta-function in the complex plane. The zeta-function can be repre-
sented as a sum over positive integers n of terms 1/ns when s is a complex number
with real part greater than 1. It may also be represented in this region as a prod-
uct over the primes called an Euler product. These definitions of the zeta-function
allow us to find other representations that are valid in more of the complex plane,
including a product representation over its zeros. The Riemann Hypothesis says that
all …

Optimizing Networking Topologies With Shortest Path Algorithms, Jordan Sahs

UNO Student Research and Creative Activity Fair

Communication networks tend to contain redundant devices and mediums of transmission, thus the need to locate, document, and optimize networks is increasingly becoming necessary. However, many people do not know where to start the optimization progress. What is network topology? What is this “Shortest Path Problem”, and how can it be used to better my network? These questions are presented, taught, and answered within this paper. To supplement the reader’s understanding there are thirty-eight figures in the paper that are used to help convey and compartmentalize the learning process needed to grasp the materials presented in the ending sections.

In …

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.

The Complexity Of Symmetry, Matthew Lemay

HMC Senior Theses

One of the main goals of theoretical computer science is to prove limits on how efficiently certain Boolean functions can be computed. The study of the algebraic complexity of polynomials provides an indirect approach to exploring these questions, which may prove fruitful since much is known about polynomials already from the field of algebra. This paper explores current research in establishing lower bounds on invariant rings and polynomial families. It explains the construction of an invariant ring for whom a succinct encoding would imply that NP is in P/poly. It then states a theorem about the circuit complexity partial …

An Update On The Computational Theory Of Hamiltonian Period Functions, Bradley Joseph Klee

Graduate Theses and Dissertations

Lately, state-of-the-art calculation in both physics and mathematics has expanded to include the field of symbolic computing. The technical content of this dissertation centers on a few Creative Telescoping algorithms of our own design (Mathematica implementations are given as a supplement). These algorithms automate analysis of integral period functions at a level of difficulty and detail far beyond what is possible using only pencil and paper (unless, perhaps, you happen to have savant-level mental acuity). We can then optimize analysis in classical physics by using the algorithms to calculate Hamiltonian period functions as solutions to ordinary differential equations. The simple …

Controlling Aircraft Yaw Movement By Interval Type-2 Fuzzy Logic, Yamama Shafeek, Laith Majeed, Rasha Naji

Emirates Journal for Engineering Research

Aircraft yaw movement is essential in maneuvering; it has been controlled by some methods which achieved tracking but not fast enough. This paper performs the dynamic modeling of aircraft yaw movement and develops PI and PI-like interval type-2 fuzzy logic controller for the model. The mathematical model is derived by inserting the parameters values of single-engine Navion aircraft into standard equations. Using Matlab/ Simulink platform, the controllers' effectivity is tested and verified in two different cases; system without disturbance and when system is disturbed by some wind gust to investigate the system robustness. Simulation results show that PI controller response …

Espade: An Efficient And Semantically Secure Shortest Path Discovery For Outsourced Location-Based Services, Bharath K. Samanthula, Divyadharshini Karthikeyan, Boxiang Dong, K. Anitha Kumari

Department of Computer Science Faculty Scholarship and Creative Works

With the rapid growth of smart devices and technological advancements in tracking geospatial data, the demand for Location-Based Services (LBS) is facing a constant rise in several domains, including military, healthcare and transportation. It is a natural step to migrate LBS to a cloud environment to achieve on-demand scalability and increased resiliency. Nonetheless, outsourcing sensitive location data to a third-party cloud provider raises a host of privacy concerns as the data owners have reduced visibility and control over the outsourced data. In this paper, we consider outsourced LBS where users want to retrieve map directions without disclosing their location information. …

A 3d Image-Guided System To Improve Myocardial Revascularization Decision-Making For Patients With Coronary Artery Disease, Haipeng Tang

Dissertations

OBJECTIVES. Coronary artery disease (CAD) is the most common type of heart disease and kills over 360,000 people a year in the United States. Myocardial revascularization (MR) is a standard interventional treatment for patients with stable CAD. Fluoroscopy angiography is real-time anatomical imaging and routinely used to guide MR by visually estimating the percent stenosis of coronary arteries. However, a lot of patients do not benefit from the anatomical information-guided MR without functional testing. Single-photon emission computed tomography (SPECT) myocardial perfusion imaging (MPI) is a widely used functional testing for CAD evaluation but limits to the absence of anatomical information. …

Optimal Control Of A Rumor Propagation Model With Different Propagation Degrees In Social Network, Yingfeng Tang

Student Works (2020-2029)

Rumor is a social interaction of information, and its development is of great significance to human beings. In this paper, by studying the D K model and a rumor model spreading with rumor latent period, deduces the rumor model with differe nt propagation degrees of the spreaders. The two equilibrium points in the system are found through derivation. In real life, enterprises often ignore the reasonable planning of the cost of rumor control. By means of public education and media technology u sing by the authorities to debunk rumors, an optimal control problem i s established. The Pontryagin’s maximum principle …

The Theory Of Cryptography In Bitcoin, Can Hong

Mathematics Senior Capstone Papers

Bitcoin is a well known virtual currency, or cryptocurrency. It was created by a group of people using the name Satoshi Nakamoto in 2008. Currently, many people are utilizing Bitcoin for personal gains and transactions. To keep transactions secure requires techniques from modern cryptography. In this paper, we explain certain aspects of the cryptography of Bitcoin. We are going to discuss two components of the cryptography of Bitcoin—hash functions and signatures. We will describe what the hash function and signature are, give some examples of hash functions, and discuss certain criteria that good hash functions should satisfy.