Mathematics Commons^™

From Historically First "Unary" Numbers, Through Egyptian Fractions, Roman Numerals, Leibniz's Binary Numbers And Kepler's Fractions To Modern Ideas Such As Calkin-Wilf Tree: A Unified Approach To Representing Natural Numbers And Fractions, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

In elementary mathematics classes, students are often overwhelmed by different representations of numbers and corresponding operations: usual fractions, decimal representations, binary numbers, etc. What often helps is when students learn the history of these representations, see the limitations of seemingly reasonable representations like Roman numerals, and how other representations overcame these limitations. Still, history was developed somewhat randomly, so the historical sequence is still somewhat chaotic. We believe that providing a unified approach for all these representations would help describe their sequence in a more logical way and thus, help the students even more.

In our analysis, we explore the …

Nessie Notation: A New Tool In Sequential Substitution Systems And Graph Theory For Summarizing Concatenations, Colton Davis

Student Research

While doing research looking for ways to categorize causal networks generated by Sequential Substitution Systems, I created a new notation to compactly summarize concatenations of integers or strings of integers, including infinite sequences of these, in the same way that sums, products, and unions of sets can be summarized. Using my method, any sequence of integers or strings of integers with a closed-form iterative pattern can be compactly summarized in just one line of mathematical notation, including graphs generated by Sequential Substitution Systems, many Primitive Pythagorean Triplets, and various Lucas sequences including the Fibonacci sequence and the sequence of square …

Fair Bankruptcy Solutions Under Interval Uncertainty, Uyen Pham, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

If the overall amount of the company's assets is smaller than its total debts, then a fair solution is to give, to each creditor, the amount proportional to the corresponding debt, e.g., 10 center for each dollar or 50 cents for each dollar. But what if the debt amounts are not known exactly, and for some creditors, we only know the lower and upper bounds on the actual debt amount? What division will be fair in such a situation? In this paper, we show that the only fair solution is to make payments proportional to an appropriate convex combination of …

How Probable Is A Revolution? A Natural Relu-Like Formula That Fits The Historical Data, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In his recent book "Principles for Dealing with the Changing World Order", Ray Dalio considered many historical crisis situations, and came up with several data points showing how the probability of a revolution or a civil war depends on the number of economic red flags. In this paper, we provide a simple empirical formula that is consistent with these data points.

Shape Of An Egg: Towards A Natural Simple Universal Formula, Sofia Holguin, Vladik Kreinovich

Departmental Technical Reports (CS)

Eggs of different bird species have different shapes. There exists formulas for describing the usual egg shapes -- e.g., the shapes of chicken eggs. However, some egg shapes are more complex. A recent paper proposed a general formula for describing all possible egg shapes; however this formula is purely empirical, it does not have any theoretical foundations. In this paper, we use the theoretical analysis of the problem to provide an alternative -- theoretically justified -- general formula. Interestingly, the new general formula is easier to compute than the previously proposed one.

Freedom Of Will, Non-Uniqueness Of Cauchy Problem, Fractal Processes, Renormalization, Phase Transitions, And Stealth Aircraft, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

We all know that we can make different decisions, decisions that change -- at least locally -- the state of the world. This is what is known as freedom of will. On the other hand, according to physics, the future state of the world is uniquely pre-determined by its current state, so there is no room for freedom of will. How can we resolve this contradiction? In this paper, we analyze this problem, and we show that many physical phenomena can help resolve this contradiction: fractal character of equations, renormalization, phase transitions, etc. Usually, these phenomena are viewed as somewhat …

How To Estimate The Present Serviceability Rating Of A Road Segment: Explanation Of An Empirical Formula, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

An accurate estimation of the road quality requires a lot of expertise, and there is not enough experts to provide such estimates for all the road segments. It is therefore desirable to estimate this quality based on easy-to-estimate and easy-to-measure characteristics. Recently, an empirical formula was proposed for such an estimate. In this paper, we provide a theoretical explanation for this empirical formula.

Education In The Era Of Google, Wikipedia, And Deep Learning: Are We Humans Still Needed And If Yes For What?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main purposes of education is to teach skills needed in future life and future jobs. What is important and what is useful changes with time. Before the industrial revolution, routine mechanical work was an important part of human activity – now machines can do it (and do it better). Before printing, copying was an important activity – now copy machines do it. Before computers, humans were needed for computing – now computer do it better. With Wikipedia and Google, there is not much need for scholars being erudite. Even extracting dependencies from data – one of the …

Economy-Related Emotional Attitudes Towards Other People: How Can We Explain Them?, Christopher Reyes, Vladik Kreinovich, Chon Van Le

Departmental Technical Reports (CS)

Research has shown that to properly understand people's economic behavior, it is important to take into account their emotional attitudes towards each other. Behavioral economics shows that different attitudes results in different economy-related behavior. A natural question is: where do these emotional attitudes come from? We show that, in principle, such emotions can be explained by people's objective functions. Specifically, we show it on the example of a person whose main objective is to increase his/her country's GDP: in this case, the corresponding optimization problem leads exactly to natural emotions towards people who contribute a lot or a little towards …

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 …

What About Generic "And"- And "Or"-Operations: From Levels Of Certainty (Philosophical-Physical-Mathematical) To A Natural Interpretation Of Quantum-Like Negative Degrees Of Certainty, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy techniques -- techniques designed to convert imprecise human knowledge into precise computer-understandable terms -- have many successful applications. Traditional applications of fuzzy techniques use only important general features of human reasoning and, to make an implement more efficient, ignore subtle details, details which are not important for the corresponding application. But from the more fundamental viewpoint, it is desirable to understand, in all the detail, how people actually reason. In this paper, we use general ideas of fuzzy approach to answer this question. Interestingly -- and somewhat unexpectedly -- the resulting analysis leads to a natural explanation of existence …

How Can The Opposite To A True Theory Be Also True? A Similar Talmudic Discussion Helps Make This Famous Bohr's Statement Logically Consistent, Miroslav Svitek, Vladik Kreinovich

Departmental Technical Reports (CS)

In his famous saying, the Nobelist physicist Niels Bohr claimed that the sign of a deep theory is that while this theory is true, its opposite is also true. While this statement makes heuristic sense, it does not seem to make sense from a logical viewpoint, since, in logic, the opposite to true is false. In this paper, we show how a similar Talmudic discussion can help come up with an interpretation in which Bohr's statement becomes logically consistent.

In The Absence Of Information, The Only Reasonable Negotiation Scheme Is Offering A Certain Percentage Of The Original Request: A Proof, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the case of complete information, a reasonable solution to a negotiation process is Nash's bargaining solution, in which we maximize the product of all agents' utility gains. This is the only solution that does not depend on the order in which we list the agents, and does not change if we use a different scale for describing each agent's utility. In this paper, we apply similar invariance criteria to a situation when practically all information is absent, and all we know is the smallest and largest possible gains. We show that in this situation, the only invariant negotiation strategy …

Why Some Theoretically Possible Representations Of Natural Numbers Were Historically Used And Some Were Not: An Algorithm-Based Explanation, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Historically, people have used many ways to represent natural numbers: from the original "unary" arithmetic, where each number is represented as a sequence of, e.g., cuts (4 is IIII) to modern decimal and binary systems. However, with all this variety, some seemingly reasonable ways of representing natural numbers were never used. For example, it may seem reasonable to represent numbers as products -- e.g., as products of prime numbers -- such a representation was never used in history. So why some theoretically possible representations of natural numbers were historically used and some were not? In this paper, we propose an …

Performance Of Density Estimators In Additive Measurement Error Models Based On Right Censored Data, Hrishabh Khakurel

Mathematics Dissertations - Archive

In the deconvolution problem for right censored data, one is interested in estimating the density of a contaminated variable X when X satisfies Z= X+ E, where E is a measurement error with a known distribution, and the observable variable Z is right-censored. Zhu, Sun, Khakurel, and Wang (2021) applied the Inverse Probability of Censoring Weighted Average method and derived the estimators of the unknown density of X. In this study, we evaluate the performance of the density estimators both in theory and in simulation. We derive the theoretical upper bounds for Mean Squared Error (MSE) of the estimator and …

3-Uniform 4-Path Decompositions Of Complete 3-Uniform Hypergraphs, Rachel Mccann

Mathematical Sciences Undergraduate Honors Theses

The complete 3-uniform hypergraph of order v is denoted as K_v and consists of vertex set V with size v and edge set E, containing all 3-element subsets of V. We consider a 3-uniform hypergraph P₇, a path with vertex set {v₁, v₂, v₃, v₄, v₅, v₆, v₇} and edge set {{v₁, v₂, v₃}, {v₂, v₃, v₄}, {v₄, v₅, v₆}, {v₅, v_{6 …}

Structure Of Number Theoretic Graphs, Lee Trent

Mathematical Sciences Technical Reports (MSTR)

The tools of graph theory can be used to investigate the structure
imposed on the integers by various relations. Here we investigate two
kinds of graphs. The first, a square product graph, takes for its vertices
the integers 1 through n, and draws edges between numbers whose product
is a square. The second, a square product graph, has the same vertex set,
and draws edges between numbers whose sum is a square.
We investigate the structure of these graphs. For square product
graphs, we provide a rather complete characterization of their structure as
a union of disjoint complete graphs. For …

A Necessary And Sufficient Condition For The Asymptotic Normality Of The Quantile Estimator In The Deconvolution Problem, Jeremy R. Valdez

Mathematics Dissertations - Archive

In this study, we examine the estimation of a quantile function when we have n observations coming from the convolution model contaminated by additive measurement errors. Under certain assumptions, a kernel type deconvolution quantile estimator of the unknown quantile function is proposed. Moreover, we discuss the necessary and sufficient condition on the bandwidth in order to investigate the limiting distribution of the deconvolution kernel quantile estimator when the error terms follow either an ordinary smooth or super smooth distribution. A bootstrap approach is used to select the optimal bandwidth to construct approximate distribution free confidence bands for the quantile function …

Twisting Systems And Some Quantum P³S With Point Scheme A Rank-2 Quadric, Hung Viet Tran

Mathematics Dissertations - Archive

In 1996, J. J. Zhang introduced the concept of twisting a graded algebra by a twisting system, which generalizes the concept of twisting a graded algebra by an automorphism (the latter concept having been introduced in an article by M. Artin, J. Tate and M. Van den Bergh in 1991). Twisting using a twisting system is an equivalence relation and certain important algebraic properties of the original algebra are carried over to the twisted algebra. We call a twisting system nontrivial if it is not given by an automorphism. However, there are very few known examples of nontrivial twisting systems …