Physical Sciences and Mathematics Commons^™

New Effective Transformational Computational Methods, Jun Zhang, Ruzong Fan, Fangyang Shen, Junyi Tu

Publications and Research

Mathematics serves as a fundamental intelligent theoretic basis for computation, and mathematical analysis is very useful to develop computational methods to solve various problems in science and engineering. Integral transforms such as Laplace Transform have been playing an important role in computational methods. In this paper, we will introduce Sumudu Transform in a new computational approach, in which effective computational methods will be developed and implemented. Such computational methods are straightforward to understand, but powerful to incorporate into computational science to solve different problems automatically. We will provide computational analysis and essentiality by surveying and summarizing some related recent works, …

Approaches To The Erdős–Straus Conjecture, Ivan V. Morozov

Publications and Research

The Erdős–Straus conjecture, initially proposed in 1948 by Paul Erdős and Ernst G. Straus, asks whether the equation 4/n = 1/x + 1/y + 1/z is solvable for all n ∈ N and some x, y, z ∈ N. This problem touches on properties of Egyptian fractions, which had been used in ancient Egyptian mathematics. There exist many partial solutions, mainly in the form of arithmetic progressions and therefore residue classes. In this work we explore partial solutions and aim to expand them.

Number Theoretic Arithmetic Functions And Dirichlet Series, Ivan V. Morozov

Publications and Research

In this study, we will study number theoretic functions and their associated Dirichlet series. This study lay the foundation for deep research that has applications in cryptography and theoretical studies. Our work will expand known results and venture into the complex plane.

A Result In The Theory Of Twin Primes, Nelson Carella

Publications and Research

This article determines a lower bound for the number of twin primes $p$ and $p+2$ up to a large number $x$.

Logic, Co-Ordination And The Envelope Of Our Beliefs, Rohit J. Parikh

Publications and Research

Each of us has a story which we can think of as a set of beliefs, hopefully consistent. We make our decisions in view of our beliefs which may be probabilistic, in the general case, but simple yes or no as in this paper. Our beliefs are our envelope just as the shell of a tortoise is its envelope.

Decision theory - or single agent game theory tells us when to make the best choice in a game of us against nature. But nature has no desire to further or frustrate our efforts. Nature is mysterious but not malign.

Things …

Conflict Dynamics In Scale-Free Networks With Degree Correlations And Hierarchical Structure, Eduardo Jacobo-Villegas, Bibiana Obregón-Quintana, Lev Guzmán-Vargas, Larry S. Liebovitch

Publications and Research

We present a study of the dynamic interactions between actors located on complex networks with scale-free and hierarchical scale-free topologies with assortative mixing, that is, correlations between the degree distributions of the actors. The actor’s state evolves according to a model that considers its previous state, the inertia to change, and the influence of its neighborhood. We show that the time evolution of the system depends on the percentage of cooperative or competitive
interactions. For scale-free networks, we find that the dispersion between actors is higher when all interactions are either cooperative or competitive, while a balanced presence of interactions …

Generalization Of Bi-Canonical Degrees, Joseph Brennan, Laura Ghezzi, Jooyoun Hong, Wolmer Vasconcelos

Publications and Research

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module ω. Attached to each such ring R, when ω is an ideal, there are integers–the type of R, the reduction number of ω–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where ω is not necessarily an ideal. We also discuss generalizations to rings …

Extractable Entanglement From A Euclidean Hourglass, Takanori Anegawa, Norihiro Iizuka, Daniel Kabat

Publications and Research

We previously proposed that entanglement across a planar surface can be obtained from the partition function on a Euclidean hourglass geometry. Here we extend the prescription to spherical entangling surfaces in conformal field theory. We use the prescription to evaluate log terms in the entropy of a conformal field theory in two dimensions, a conformally coupled scalar in four dimensions, and a Maxwell field in four dimensions. For Maxwell we reproduce the extractable entropy obtained by Soni and Trivedi. We take this as evidence that the hourglass prescription provides a Euclidean technique for evaluating extractable entropy in quantum field theory.

Trapped Surfaces, Topology Of Black Holes, And The Positive Mass Theorem, Lan-Hsuan Huang, Dan A. Lee

Publications and Research

No abstract provided.

Combinatorial Optimization With Photonics-Inspired Clock Models, Mostafa Honari-Latifpour, Matthew S. Mills, Mohammad-Ali Miri

Publications and Research

NP-hard combinatorial optimization problems are in general hard problems that their computational complexity grows faster than polynomial scaling with the size of the problem. Thus, over the years there has been a great interest in developing unconventional methods and algorithms for solving such problems. Here, inspired by the nonlinear optical process of q-photon down-conversion, in which a photon is converted into q degenerate lower energy photons, we introduce a nonlinear dynamical model that builds on coupled single-variable phase oscillators and allows for efficiently approximating the ground state of the classical q-state planar Potts Hamiltonian. This reduces the exhaustive search in …

Amm Problem #12279, Brad Isaacson

Publications and Research

No abstract provided.

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 …

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 …

Interfacial Dynamics And Ionic Transport Of Radiologic Contrast Media In Carbohydrate Matrix: Utility And Limits Of X-Ray Imaging, Lin Mousa, Hayley Sanchez, Subhendra Sarkar, Zoya Vinokur

Publications and Research

Hello, our names are Lin Mousa and Hayley Sanchez, this semester we participated in a research project dedicated to analyzing the interactions of contrast media with the molecular components of fruits to compare how they would react with the human brain. This project involved the injection of fruits with varying contrasts and the imaging of the diffusion and interactions of the contrast within the fruits with X-rays. With setup technical parameters on the x-ray equipment images were taken with identical setups at an hourly rate for several days. The final results of this experiment indicated that contrasts such as Gadolinium …

Application Of Randomness In Finance, Jose Sanchez, Daanial Ahmad, Satyanand Singh

Publications and Research

Brownian Motion which is also considered to be a Wiener process and can be thought of as a random walk. In our project we had briefly discussed the fluctuations of financial indices and related it to Brownian Motion and the modeling of Stock prices.

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 …

Dynamics Of The Meromorphic Families $F_\Lambda=\Lambda \Tan^Pz^Q$, Tao Chen, Linda Keen

Publications and Research

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials P_a(z) =z^d−1(z−da/(d−1)), the family f_λ=λtan^pz^q. These functions have a super-attractive fixed point, and, depending on p, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard …

Interdisciplinary Thinking: Financial Literacy Crosses Disciplinary Boundaries, Marla A. Sole

Publications and Research

Financial literacy is ideally suited to be integrated into mathematics courses and taught in an interdisciplinary manner. Students learn best and are motivated when tackling real-world meaningful questions. This article shares how elementary mathematics was applied to better understand the debate about raising the minimum wage and the United States National Debt. To serve as a guide for other teachers who wish to incorporate financial literacy into their mathematics courses and take an interdisciplinary approach, this article suggests readings, data sets, and pedagogical practices. Students were engaged and enthusiastic to work on problems that challenged their thinking about financial issues.

A Twisted Generalization Of The Classical Dedekind Sum, Brad Isaacson

Publications and Research

In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory.

Amm Problem #12219, Brad Isaacson

Publications and Research

No abstract provided.

Three Imprimitive Character Sums, Brad Isaacson

Publications and Research

We express three imprimitive character sums in terms of generalized Bernoulli numbers. These sums are generalizations of sums introduced and studied by Arakawa, Berndt, Ibukiyama, Kaneko and Ramanujan in the context of modular forms and theta function identities. As a corollary, we obtain a formula for cotangent power sums considered by Apostol.

Approximation By The K^Lambda Means Of Fourier Series And Conjugate Series Of Functions In H_{Alpha,P}, Ben Landon, Holly Carley, R. N. Mohapatra

Publications and Research

No abstract provided.

An Enticing Study Of Prime Numbers Of The Shape �� = ��^2 + ��^2, Xiaona Zhou

Publications and Research

We will study and prove important results on primes of the shape ��² + ��² using number theoretic techniques. Our analysis involves maps, actions over sets, fixed points and involutions. This presentation is readily accessible to an advanced undergraduate student and lay the groundwork for future studies.

A Differential-Algebraic Criterion For Obtaining A Small Maximal Cohen-Macaulay Module, Hans Schoutens

Publications and Research

We show how for a three-dimensional complete local ring in positive characteristic, the existence of an F-invariant, differentiable derivation implies Hochster’s small MCM conjecture. As an application we show that any three-dimensional pseudo-graded ring in positive characteristic satisfies Hochster’s small MCM conjecture.

An In-Depth Look At P-Adic Numbers, Xiaona Zhou

Publications and Research

In this study, we consider $p$-adic numbers. We will also study the $p$-adic norm representation of real number, which is defined as $\mathbb{Q}_p = \{\sum_{j=m}^{\infty }a_j p^j: a_j \in \mathbb{D}_p, m\in\mathbb{Z}, a_m\neq 0\} \cup \{0\}$, where $p$ is a prime number. We explore properties of the $p$-adics by using examples. In particular, we will show that $\sqrt{6},i \in \mathbb{Q}_5$ and $\sqrt{2} \in \mathbb{Q}_7 $. $p$-adic numbers have a wide range of applicationsnin fields such as string theory, quantum mechanics, and transportation in porous disordered media in geology.

Winding Numbers And Full Extendability In Holomorphic Motions, Yunping Jiang

Publications and Research

We construct an example of a holomorphic motion of a five-point subset of the Riemann sphere over an annulus such that it satisfies the zero winding number condition but is not fully extendable.

Estimating Population Immunity Without Serological Testing, Andrew Lesniewski

Publications and Research

We propose an approximate methodology for estimating the overall level of immunity against COVID-19 in a population that has been affected by the recent epidemic. The methodology relies on the currently available mortality data and utilizes the properties of the SIR model. We illustrate the application of the method by estimating the recent levels of immunity in 10 US states with highest case numbers of COVID-19.

The Tsukano Conjectures On Exponential Sums, Brad Isaacson

Publications and Research

We prove three conjectures of Tsukano about exponential sums stated in his Master’s thesis written at Osaka University. These conjectures are variations of earlier conjectures made by Lee and Weintraub which were first proved by Ibukiyama and Saito.

Graded Quivers, Generalized Dimer Models And Toric Geometry, Sebastián Franco, Azeem Hasan

Publications and Research

The open string sector of the topological B-model on CY (m+2)-folds is described by m-graded quivers with superpotentials. This correspondence extends to general m the well known connection between CY (m+2)-folds and gauge theories on the world-volume of D(5-2m)-branes for m = 0, ..., 3. We introduce m-dimers, which fully encode the m-graded quivers and their superpotentials, in the case in which the CY (m+2)-folds are toric. Generalizing the well known m = 1,2 cases, m-dimers significantly simplify the connection between geometry and m-graded quivers. A key …