Mathematics Commons^™

Co-Designing Elementary-Level Computer Science And Mathematics Lessons: An Expansive Framing Approach, Umar Shehzad, Jody Clarke-Midura, Kimberly Beck, Jessica Shumway, Mimi Recker

Publications

This study examines how a rural-serving school district aimed to provide elementary-level computer science (CS) by offering instruction during students’ computer lab time. As part of a research-practice partnership, cross-context mathematics and CS lessons were co-designed to expansively frame and highlight connections across – as opposed to integration within – the two subjects. Findings indicated that most students who engaged with the lessons across the lab and classroom contexts reported finding the lessons interesting, seeing connections to their mathematics classes, and understanding the programming. In contrast, a three-level logistic regression model showed that students who only learned about mathematics connections …

Machine-Assisted Discovery Of Integrable Symplectic Mappings, T. Zolkin, Y. Kharkov, S. Nagaitsev

Physics Faculty Publications

We present a new automated method for finding integrable symplectic maps of the plane. These dynamical systems possess a hidden symmetry associated with an existence of conserved quantities, i.e., integrals of motion. The core idea of the algorithm is based on the knowledge that the evolution of an integrable system in the phase space is restricted to a lower-dimensional submanifold. Limiting ourselves to polygon invariants of motion, we analyze the shape of individual trajectories thus successfully distinguishing integrable motion from chaotic cases. For example, our method rediscovers some of the famous McMillan-Suris integrable mappings and ultradiscrete Painlevé equations. In total, …

The History Of The Enigma Machine, Jenna Siobhan Parkinson

History Publications

The history of the Enigma machine begins with the invention of the rotor-based cipher machine in 1915. Various models for rotor-based cipher machines were developed somewhat simultaneously in different parts of the world. However, the first documented rotor machine was developed by Dutch naval officers in 1915. Nonetheless, the Enigma machine was officially invented following the end of World War I by Arthur Scherbius in 1918 (Faint, 2016).

Which Interval-Valued Alternatives Are Possibly Optimal If We Use Hurwicz Criterion, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, for each alternative i, we do not know the corresponding gain xi, we only know the interval [li,ui] of possible gains. In such situations, a reasonable way to select an alternative is to choose some value α from the interval [0,1] and select the alternative i for which the Hurwicz combination α*ui + (1 − α)*li is the largest possible. In situations when we do not know the user's α, a reasonable idea is to select all alternatives that are optimal for some α. In this paper, we describe a feasible algorithm for such a selection.

Data Processing Under Fuzzy Uncertainty: Towards More Accurate Algorithms, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Data that we process comes either from measurements or from experts -- or from the results of previous data processing that were also based on measurements and/or expert estimates. In both cases, the data is imprecise. To gauge the accuracy of the results of data processing, we need to take the corresponding data uncertainty into account. In this paper, we describe a new algorithm for taking fuzzy uncertainty into account, an algorithm that, for small number of inputs, leads to the same or even better accuracy than the previously proposed methods.

Non-Localized Physical Processes Can Help Speed Up Computations, Be It Hidden Variables In Quantum Physics Or Non-Localized Energy In General Relativity, Michael Zakharevich, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While most physical processes are localized -- in the sense that each event can only affect events in its close vicinity -- many physicists believe that some processes are non-local. These beliefs range from more heretic -- such as hidden variables in quantum physics -- to more widely accepted, such as the non-local character of energy in General Relativity. In this paper, we attract attention to the fact that non-local processes bring in the possibility of drastically speeding up computations.

Graph Approach To Uncertainty Quantification, Hector A. Reyes, Cliff Joslyn, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional analysis of uncertainty of the result of data processing assumes that all measurement errors are independent. In reality, there may be common factor affecting these errors, so these errors may be dependent. In such cases, the independence assumption may lead to underestimation of uncertainty. In such cases, a guaranteed way to be on the safe side is to make no assumption about independence at all. In practice, however, we may have information that a few pairs of measurement errors are indeed independent -- while we still have no information about all other pairs. Alternatively, we may suspect that for …

Systems Approach Explains Why Low Heart Rate Variability Is Correlated With Depression (And Suicidal Thoughts), Francisco Zapata, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

Depression is a serious medical problem. If diagnosed early, it can usually be cured, but if left undetected, it can lead to suicidal thoughts and behavior. The early stages of depression are difficult to diagnose. Recently, researchers found a promising approach to such diagnosis -- it turns out that depression is correlated with low heart rate variability. In this paper, we show that the general systems approach can explain this empirical relation.

An Argument In Favor Of Piecewise-Constant Membership Functions, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Theoretically, we can have membership functions of arbitrary shape. However, in practice, at any given moment of time, we can only represent finitely many parameters in a computer. As a result, we usually restrict ourselves to finite-parametric families of membership functions. The most widely used families are piecewise linear ones, e.g., triangular and trapezoid membership functions. The problem with these families is that if we know a nonlinear relation y = f(x) between quantities, the corresponding relation between membership functions is only approximate -- since for piecewise linear membership functions for x, the resulting membership function for y is not …

Will Nanotechnology Bring In The Judgement Day?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

There are many current and prospective positive aspects of nanotechnology. However, while we look forward to its future successes, we need to keep our eyes open and be prepared for what will really be a future shock: that quantum computing – an inevitable part of nanotechnology – will enable the future folks to read all our encrypted messages and thus, learn everything that we wanted to keep secret. This will be really the Judgement Day, when all our sins will be open to everyone. How we will react to it? Will this destroy our civilization? Let us hope that the …

Standard Interval Computation Algorithm Is Not Inclusion-Monotonic: Examples, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When we usually process data, we, in effect, implicitly assume that we know the exact values of all the inputs. In practice, these values comes from measurements, and measurements are never absolutely accurate. In many cases, the only information about the actual (unknown) values of each input is that this value belongs to an appropriate interval. Under this interval uncertainty, we need to compute the range of all possible results of applying the data processing algorithm when the inputs are in these intervals. In general, the problem of exactly computing this range is NP-hard, which means that in feasible time, …

How Viscosity Of An Asphalt Binder Depends On Temperature: Theoretical Explanation Of An Empirical Dependence, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

Pavement must be adequate for all the temperatures, ranging from the winter cold to the summer heat. In particular, this means that for all possible temperatures, the viscosity of the asphalt binder must stay within the desired bounds. To predict how the designed pavement will behave under different temperatures, it is desirable to have a general idea of how viscosity changes with temperature. Pavement engineers have come up with an empirical approximate formula describing this change. However, since this formula is purely empirical, with no theoretical justification, practitioners are often somewhat reluctant to depend on this formula. In this paper, …

Why In Mond -- Alternative Gravitation Theory -- A Specific Formula Works The Best: Complexity-Based Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Based on the rotation of the stars around a galaxy center, one can estimate the corresponding gravitational acceleration -- which turns out to be much larger than what Newton's theory predicts based on the masses of all visible objects. The majority of physicists believe that this discrepancy indicates the presence of "dark" matter, but this idea has some unsolved problems. An alternative idea -- known as Modified Newtonian Dynamics (MOND, for short) is that for galaxy-size distances, Newton's gravitation theory needs to be modified. One of the most effective versions of this idea uses so-called simple interpolating function. In this …

Epistemic Vs. Aleatory: Case Of Interval Uncertainty, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Interval computations usually deal with the case of epistemic uncertainty, when the only information that we have about a value of a quantity is that this value is contained in a given interval. However, intervals can also represent aleatory uncertainty -- when we know that each value from this interval is actually attained for some object at some moment of time. In this paper, we analyze how to take such aleatory uncertainty into account when processing data. We show that in case when different quantities are independent, we can use the same formulas for dealing with aleatory uncertainty as we …

Meshfree Methods For Pdes On Surfaces, Andrew Michael Jones

Boise State University Theses and Dissertations

This dissertation focuses on meshfree methods for solving surface partial differential equations (PDEs). These PDEs arise in many areas of science and engineering where they are used to model phenomena ranging from atmospheric dynamics on earth to chemical signaling on cell membranes. Meshfree methods have been shown to be effective for solving surface PDEs and are attractive alternatives to mesh-based methods such as finite differences/elements since they do not require a mesh and can be used for surfaces represented only by a point cloud. The dissertation is subdivided into two papers and software.

In the first paper, we examine the …

How To Get The Most Accurate Measurement-Based Estimates, Salvador Robles, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we want to estimate a quantity y that is difficult -- or even impossible -- to measure directly. In such cases, often, there are easier-to-measure quantities x₁, ..., x_n that are related to y by a known dependence y = f(x₁,...,x_n). So, to estimate y, we can measure these quantities x_i and use the measurement results to estimate y. The two natural questions are: (1) within limited resources, what is the best accuracy with which we can estimate y, and (2) to reach a given accuracy, what amount …

Anomaly Detection In Crowdsourcing: Why Midpoints In Interval-Valued Approach, Alejandra De La Pena, Damian L. Gallegos Espinoza, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations -- e.g., when preparing examples for a machine learning algorithm -- we need to label a large number of images or speech recordings. One way to do it is to pay people around the world to perform this labeling; this is known as crowdsourcing. In many cases, crowd-workers generate not only answers, but also their degrees of confidence that the answer is correct. Some crowd-workers cheat: they produce almost random answers without bothering to spend time analyzing the corresponding image. Algorithms have been developed to detect such cheaters. The problem is that many crowd-workers cannot describe …

Dielectric Barrier Discharge (Dbd) Thrusters -- Aerospace Engines Of The Future: Invariance-Based Analysis, Alexis Lupo, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the most prospective aerospace engines is a Dielectric Barrier Discharge (DBD) thruster -- an effective electric engine without moving parts. Originally designed by NASA for flights over other planets, it has been shown to be very promising for Earth-based flights as well. The efficiency of this engine depends on the proper selection of the corresponding electric field. To make this selection, we need to know, in particular, how its thrust depends on the atmospheric pressure. At present, for this dependence, we only know an approximate semi-empirical formula. In this paper, we use natural invariance requirements to come up …

Hunting Habits Of Predatory Birds: Theoretical Explanation Of An Empirical Formula, Adilene Alaniz, Jiovani Hernandez, Andres D. Munoz, Vladik Kreinovich

Departmental Technical Reports (CS)

Predatory birds play an important role in an ecosystem. It is therefore important to study their hunting behavior, in particular, the distribution of their waiting time. A recent empirical study showed that the waiting time is distributed according to the power law. In this paper, we use natural invariance ideas to come up with a theoretical explanation for this empirical dependence.

Aquatic Ecotoxicology: Theoretical Explanation Of Empirical Formulas, Demetrius R. Hernandez, George M. Molina Holguin, Francisco Parra, Vivian Sanchez, Vladik Kreinovich

Departmental Technical Reports (CS)

To analyze the effect of pollution on marine life, it is important to know how exactly the concentration of toxic substances decreases with time. There are several semi-empirical formulas that describe this decrease. In this paper, we provide a theoretical explanation for these empirical formulas.

Resource Allocation For Multi-Tasking Optimization: Explanation Of An Empirical Formula, Alan Gamez, Antonio Aguirre, Christian Cordova, Alberto Miranda, Vladik Kreinovich

Departmental Technical Reports (CS)

For multi-tasking optimization problems, it has been empirically shown that the most effective resource allocation is attained when we assume that the gain of each task logarithmically depends on the computation time allocated to this task. In this paper, we provide a theoretical explanation for this empirical fact.

Why Color Optical Computing, Victor L. Timchenko, Yury P. Kondratenko, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that requirements that computations be fast and noise-resistant naturally lead to what we call color-based optical computing.

How To Reach A Joint Decision With The Smallest Need For Compromise, Sofia Holguin, Olga Kosheleva

Departmental Technical Reports (CS)

Usually, people's interests do not match perfectly. So when several people need to make a joint decision, they need to compromise. The more people one has to coordinate the decision with, the fewer chances that each person's preferences will be properly taken into account. Therefore, when a large group of people need to make a decision, it is desirable to make sure that this decision can be reached by dividing all the people into small-size groups so that this decision can reach a compromise between the members of each group. In this paper, we use a recent mathematical result to …

Word Representation: Theoretical Explanation Of An Empirical Fact, Leonel Escapita, Diana Licon, Madison Anderson, Diego Pedraza, Vladik Kreinovich

Departmental Technical Reports (CS)

There is a reasonably accurate empirical formula that predicts, for two words i and j, the number Xij of times when the word i will appear in the vicinity of the word j. The parameters of this formula are determined by using the weighted least square approach. Empirically, the predictions are the most accurate if we use the weights proportional to a power of Xij. In this paper, we provide a theoretical explanation for this empirical fact.

Need For Optimal Distributed Measurement Of Cumulative Quantities Explains The Ubiquity Of Absolute And Relative Error Components, Hector A. Reyes, Aaron D. Brown, Jeffrey Escamilla, Ethan D. Kish, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to measure the value of a cumulative quantity, i.e., a quantity that is obtained by adding measurement results corresponding to different spatial locations. How can we select the measuring instruments so that the resulting cumulative quantity can be determined with known accuracy -- and, to avoid unnecessary expenses, not more accurately than needed? It turns out that the only case where such an optimal arrangement is possible is when the required accuracy means selecting the upper bounds on absolute and relative error components. This results provides a possible explanation for the ubiquity of such …

Lstm-Sdm: An Integrated Framework Of Lstm Implementation For Sequential Data Modeling[Formula Presented], Hum Nath Bhandari, Binod Rimal, Nawa Raj Pokhrel, Ramchandra Rimal, Keshab R. Dahal

Arts & Sciences Faculty Publications

LSTM-SDM is a python-based integrated computational framework built on the top of Tensorflow/Keras and written in the Jupyter notebook. It provides several object-oriented functionalities for implementing single layer and multilayer LSTM models for sequential data modeling and time series forecasting. Multiple subroutines are blended to create a conducive user-friendly environment that facilitates data exploration and visualization, normalization and input preparation, hyperparameter tuning, performance evaluations, visualization of results, and statistical analysis. We utilized the LSTM-SDM framework in predicting the stock market index and observed impressive results. The framework can be generalized to solve several other real-world time series problems.

Why 1/(1+D) Is An Effective Distance-Based Similarity Measure: Two Explanations, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Most of our decisions are based on the notion of similarity: we use a decision that helped in similar situations. From this viewpoint, it is important to have, for each pair of situations or objects, a numerical value describing similarity between them. This is called a similarity measure. In some cases, the only information that we can use to estimate the similarity value is some natural distance measure d(a,b). In many such situations, empirical data shows that the similarity measure 1/(1+d) is very effective. In this paper, we provide two explanations for this effectiveness.

How To Describe Variety Of A Probability Distribution: A Possible Answer To Yager's Question, Vladik Kreinovich

Departmental Technical Reports (CS)

Entropy is a natural measure of randomness. It progresses from its smallest possible value 0 -- when we have a deterministic case in which one alternative i occurs with probability 1 (pi = 1), to the largest possible value which is attained at a uniform distribution p1 = ... = pn = 1/n. Intuitively, both in the deterministic case and in the uniform distribution case, there is not much variety in the distribution, while in the intermediate cases, when we have several different values pi, there is a strong variety. Entropy does not seem to capture this notion of variety. …

How The Pavement Strength Changes With Time: Ai Ideas Help To Explain Semi-Empirical Formulas, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we use AI ideas to provide a theoretical explanation for semi-empirical formulas that describe how the pavement strength changes with time, and how we can predict the pavement lifetime.

Machine Learning To Predict Warhead Fragmentation In-Flight Behavior From Static Data, Katharine Larsen

Doctoral Dissertations and Master's Theses

Accurate characterization of fragment fly-out properties from high-speed warhead detonations is essential for estimation of collateral damage and lethality for a given weapon. Real warhead dynamic detonation tests are rare, costly, and often unrealizable with current technology, leaving fragmentation experiments limited to static arena tests and numerical simulations. Stereoscopic imaging techniques can now provide static arena tests with time-dependent tracks of individual fragments, each with characteristics such as fragment IDs and their respective position vector. Simulation methods can account for the dynamic case but can exclude relevant dynamics experienced in real-life warhead detonations. This research leverages machine learning methodologies to …