Mathematics Commons^™

On The Uniqueness Of Solutions To Martingale Problems For Diffusion Operators With Progressively Measurable Random Coefficients, Masaaki Tsuchiya

Journal of Stochastic Analysis

No abstract provided.

Particle Representation For The Solution Of The Filtering Problem. Application To The Error Expansion Of Filtering Discretizations, Dan Crisan, Thomas G. Kurtz, Salvador Ortiz-Latorre

Journal of Stochastic Analysis

No abstract provided.

An Anti-Symmetric Version Of Malliavin Calculus, Jirô Akahori, Tomo Matsusita, Yasufumi Nitta

Journal of Stochastic Analysis

No abstract provided.

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 …

Tech & Toys To Teach Linear Algebra, Kelly W. Remijan

Publications & Research

6th - 12 Grade Math teachers will Discover activities involving tech and/or toys:

1. CNC Machines- graphing points & line segments
2. Desmos/GeoGebra & flags- graphing linear equations/inequalities w/ domain/range
3. CBR & movement- graphing/analyzing lines
4. Push toys- systems of equations
5. Water growing toys- growth rate
6. Ziplines, Balloons, Barbie, & Lego People - slope & linear equations

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.

Completing Partial Transversals Of Cayley Tables Of Abelian Groups, Jaromy Kuhl, Donald Mcginn, Michael W. Schroeder

Mathematics Faculty Research

In 2003 Grüttmüller proved that if n ⩾ 3 is odd, then a partial transversal of the Cayley table of ℤₙ with length 2 is completable to a transversal. Additionally, he conjectured that a partial transversal of the Cayley table of ℤₙ with length k is completable to a transversal if and only if n is odd and either n ∈ {k, k + 1} or n ⩾ 3k - 1. Cavenagh, Hämäläinen, and Nelson (in 2009) showed the conjecture is true when k = 3 and n is prime. In this paper, we prove Grüttmüller's conjecture for k = …

Control Strategies To Contain Sars-Cov-2 In A Data-Driven Sir Model For The State Of Michigan, Norma Ortiz-Robinson, Cyrus Foster-Bey

Open Access Publishing Support Funded Articles

In this paper we use a data driven SIR model to capture the dynamics of the spread of the SARS-CoV-2 pandemic in the state of Michigan. The model is then used to formulate an optimal control problem in which we perform sensitivity analysis involving vaccine efficacy and capacity, and vaccination willingness by the public. We obtain numerical approximations for best strategies for vaccination, treatment, and social distancing measures and their effect on the spread of the virus.

Linking Numbers In Three-Manifolds, Patricia Cahn, Alexandra Kjuchukova

Mathematics Sciences: Faculty Publications

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of S3 branched along a knot α⊂ S3. Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot α can be derived from dihedral covers of α. The linking numbers we compute are necessary for evaluating one …

Clifford Harmonics, Samuel L. Hosmer

Dissertations, Theses, and Capstone Projects

In 1980 Michelsohn defined a differential operator on sections of the complex Clifford bundle over a compact Kähler manifold M. This operator is a differential and its Laplacian agrees with the Laplacian of the Dolbeault operator on forms through a natural identification of differential forms with sections of the Clifford bundle. Relaxing the condition that M be Kähler, we introduce two differential operators on sections of the complex Clifford bundle over a compact almost Hermitian manifold which naturally generalize the one introduced by Michelsohn. We show surprising Kähler- like symmetries of the kernel of the Laplacians of these operators in …

A Geometric Model For Real And Complex Differential K-Theory, Matthew T. Cushman

Dissertations, Theses, and Capstone Projects

We construct a differential-geometric model for real and complex differential K-theory based on a smooth manifold model for the K-theory spectra defined by Behrens using spaces of Clifford module extensions. After writing representative differential forms for the universal Pontryagin and Chern characters we transgress these forms to all the spaces of the spectra and use them to define an abelian group structure on maps up to an equivalence relation that refines homotopy. Finally we define the differential K-theory functors and verify the axioms of Bunke-Schick for a differential cohomology theory.

Representing The Derivative Of Trace Of Holonomy, Jeffrey Peter Kroll

Dissertations, Theses, and Capstone Projects

Trace of holonomy around a fixed loop defines a function on the space of unitary connections on a hermitian vector bundle over a Riemannian manifold. Using the derivative of trace of holonomy, the loop, and a flat unitary connection, a functional is defined on the vector space of twisted degree 1 cohomology classes with coefficients in skew-hermitian bundle endomorphisms. It is shown that this functional is obtained by pairing elements of cohomology with a degree 1 homology class built directly from the loop and equipped with a flat section obtained from the variation of holonomy around the loop. When the …

Analytical Properties For The Fifth Order Camassa-Holm (Foch) Model, Mingxuan Zhu, Lu Cao, Zaihong Jiang, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

This paper devotes to present analysiswork on the fifth order Camassa-Holm (FOCH) modelwhich recently proposed by Liu and Qiao. Firstly, we establish the local and global existence of the solution to the FOCH model. Secondly, we study the property of the infinite propagation speed. Finally, we discuss the long time behavior of the support of momentum density with a compactly supported initial data.

Differentiability Of The Liouville Map Via Geodesic Currents, Xinlong Dong

Dissertations, Theses, and Capstone Projects

For a conformally hyperbolic Riemann surface, the Teichmüller space is the space of quasiconformal maps factored by an equivalence relation, and it is a complex Banach manifold. The space of geodesic currents endowed with the uniform weak* topology is a subset of a Fréchet space of Hölder distributions. We introduce an appropriate topology on the space of Hölder distributions and this new topology coincides with the uniform weak* topology on the space of geodesic currents. The Liouville map of the Teichmüller space becomes differentiable in the Fréchet sense. In particular, the derivative of Liouville currents exists and belongs to the …

Classroom And Computational Investigations Of Camel Up, Thomas J. Clark

Faculty Work Comprehensive List

Camel Up is a popular board game in which players score points by betting on camels which move randomly via a dice mechanic. The game is available both as a board game [1], as well as an IOS App [2]. Because of the random nature of the camels it is generally difficult to play optimally, but one can nevertheless develop various strategies. Probabilistic knowledge proves helpful in assigning relative value to potential game choices. We discuss how this game can be used to motivate and provide context for learning about the concepts of conditional probability and expected value. Also we …

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. …

Eulerian Walk And Dna Sequence Assembly, Yifan Zhu

Symposium of Student Scholars

Children like jigsaw puzzles, and the way to assemble the puzzle is by putting together pieces that match, one by one, till the puzzle is complete. DNA sequence assembly could be thought of as something similar: when a strand of DNA is passed into a particular “sequencing machine”, it gives a large number of short reads of the DNA sequence. This type of technology is called shotgun sequencing. These reads form the jigsaw pieces in the puzzle and one must put together these pieces in an intelligent way to obtain the original sequence. There is, however, a catch here; apriori, …

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 …

Modeling Dewetting, Demixing, And Thermal Effects In Nanoscale Metal Films, Ryan Howard Allaire

Dissertations

Thin film dynamics, particularly on the nanoscale, is a topic of extensive interest. The process by which thin liquids evolve is far from trivial and can lead to dewetting and drop formation. Understanding this process involves not only resolving the fluid mechanical aspects of the problem, but also requires the coupling of other physical processes, including liquid-solid interactions, thermal transport, and dependence of material parameters on temperature and material composition. The focus of this dissertation is on the mathematical modeling and simulation of nanoscale liquid metal films, which are deposited on thermally conductive substrates, liquefied by laser heating, and subsequently …

Modeling And Design Optimization For Membrane Filters, Yixuan Sun

Dissertations

Membrane filtration is widely used in many applications, ranging from industrial processes to everyday living activities. With growing interest from both industrial and academic sectors in understanding the various types of filtration processes in use, and in improving filter performance, the past few decades have seen significant research activity in this area. Experimental studies can be very valuable, but are expensive and time-consuming, therefore theoretical studies offer potential as a cost-effective and predictive way to improve on current filter designs. In this work, mathematical models, derived from first principles and simplified using asymptotic analysis, are proposed for: (1) pleated membrane …

In Vivo Antidiabetic Activities Of Polyherbal Extracts Against Streptozotocin-Nicotinamide Induced Type 2 Diabetic Mice Model, Pradnya Jagtap, Yashashri Nimbalkar, Vidya Patil, Ravindra Y. Patil

Tanzania Journal of Science

Diabetes mellitus is a chronic metabolic disorder and rapidly increasing cases of diabetes throughout the world are major concerns in both developed and developing countries. Currently available oral antidiabetic medicines have limitations in efficacy and safety, which in conjunction with the emergence of diabetes mellitus as a global epidemic have aided the popularity of alternative therapies to manage the disease safely and more efficiently. Herbal medicines are accepted as alternative therapies for blood sugar control. Gymnema sylvestre, Boerhavia diffusa, Tinospora cordifolia, and Argyreia nervosa are some potent herbs used for the management of diabetes. Mixing of these plants in different …

An Age-Structured Model For Transmission Dynamics Of Malaria With Infected Immigrants And Asymptomatic Carriers, Asha S Kalula, Eunice Mureithi, Theresia Marijani, Isambi Mbalawata

Tanzania Journal of Science

Abstract An age-structured (children and adults) model for the transmission dynamics of malaria with asymptomatic carriers and infected immigrants has been analyzed. We first analyze a model without infected immigrants. It shows that the disease-free equilibrium exists and is stable when and unstable for . Also, we compute the sensitivity indices of the basic reproduction number. The basic reproduction number is most sensitive to the mosquito biting rate. Besides, the sensitivity of the basic reproduction number shows that the children's class parameters are more sensitive than those of adults. In the presence of infected immigrants, the model does not admit …

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 …

Spatial Modelling For Cholera Incidence Rates In Morogoro Municipality, Tanzania, Caroline Ngereza, Rosalia S. Katapa, Ali R. Mniachi

Tanzania Journal of Science

Spatial modelling was conducted to examine community factors associated with cholera incidence rates in Morogoro Municipality. The study employed both secondary (cholera cases) and primary (geographic coordinates of community risk factors) data. Spatial lag model was applied in examining association between the variables. All wards of Morogoro Municipality were considered in the study to capture their variations because cholera cases have a tendency to be clustered. Results indicated that market density, distance to the market and distance to the dumpster are significant factors associated with cholera incidence rates in the wards (p < 0.05). Geographically weighted Poisson model was used to show the variations of those factors between the wards in Municipality. A statistically significant positive association of cholera incidence rates; and market density was only found in Mazimbu ward (p < 0.05) and distance from the community to the dumpster was found in Kihonda, Kingolwira, Bigwa, Kichangani, Kilakala and Boma wards (p < 0.001) and some wards at the centre of the municipality which are Mji Mkuu and Kingo (p < 0.05). A statistically significant negative association of cholera incidence rates and distance from the centre of the community to the market was found in Kihonda, Kingolwira and Kichangani (p < 0.001) and Bigwa wards (p < 0.05). Therefore, measures taken to control and prevent cholera disease should base on the variations of the risk factors found in the Municipal wards.  Keywords:    Cholera incidence rate, Spatial lag regression model, Community risk factor, Geographically weighted Poisson model