Applied Mathematics Commons^™

How To Gauge A Combination Of Uncertainties Of Different Type: General Foundations, Ingo Neumann, Vladik Kreinovich, Thach N. Nguyen

Departmental Technical Reports (CS)

In many practical situations, for some components of the uncertainty (e.g., of the measurement error) we know the corresponding probability distribution, while for other components, we know only upper bound on the corresponding values. To decide which of the algorithms or techniques leads to less uncertainty, we need to be able to gauge the combined uncertainty by a single numerical value -- so that we can select the algorithm for which this values is the best. There exist several techniques for gauging the combination of interval and probabilistic uncertainty. In this paper, we consider the problem of gauging the combination …

Runge–Kutta–Gegenbauer Explicit Methods For Advection-Diffusion Problems, Stephen O'Sullivan

Articles

In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of arbitrarily high order of accuracy are introduced in closed form. The stability domain of RKG polynomials extends in the the real direction with the square of polynomial degree, and in the imaginary direction as an increasing function of Gegenbauer parameter. Consequently, the polynomials are naturally suited to the construction of high order stabilized Runge-Kutta (SRK) explicit methods for systems of PDEs of mixed hyperbolic-parabolic type.

We present SRK methods composed of L ordered forward Euler stages, with complex-valued stepsizes derived from the roots of RKG stability polynomials of degree $L$. Internal stability …

How To Make A Decision Based On The Minimum Bayes Factor (Mbf): Explanation Of The Jeffreys Scale, Vladik Kreinovich, Olga Kosheleva, Nguyen Duc Trung

Departmental Technical Reports (CS)

In many practical situations, we need to select a model based on the data. It is, at present, practically a consensus that the traditional p-value-based techniques for such selection often do not lead to adequate results. One of the most widely used alternative model selection techniques is the Minimum Bayes Factor (MBF) approach, in which a model is preferred if the corresponding Bayes factor -- the ratio of likelihoods corresponding to this model and to the competing model -- is sufficiently large for all possible prior distributions. Based on the MBF values, we can decide how strong is the evidence …

Why Derivative: Invariance-Based Explanation, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To many students, the notion of a derivative seems unrelated to any previous mathematics -- and is, thus, difficult to study and to understand. In this paper, we show that this notion can be naturally derived from a more intuitive notion of invariance.

How To Reconcile Maximum Entropy Approach With Intuition: E.G., Should Interval Uncertainty Be Represented By A Uniform Distribution, Vladik Kreinovich, Olga Kosheleva, Songsak Sriboonchitta

Departmental Technical Reports (CS)

In many practical situations, we only have partial information about the probabilities; this means that there are several different probability distributions which are consistent with our knowledge. In such cases, if we want to select one of these distributions, it makes sense not to pretend that we have a small amount of uncertainty -- and thus, it makes sense to select a distribution with the largest possible value of uncertainty. A natural measure of uncertainty of a probability distribution is its entropy. So, this means that out of all probability distributions consistent with our knowledge, we select the one whose …

Why The Obvious Necessary Condition Is (Often) Also Sufficient (Toncas): An Explanation Of The Phenomenon, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

In many graph-related problems, an obvious necessary condition is often also sufficient. This phenomenon is so ubiquitous that it was even named TONCAS, after the first letters of the phrase describing this phenomenon. In this paper, we provide a possible explanation for this phenomenon.

Why Lasso, En, And Clot: Invariance-Based Explanation, Hamza Alkhatib, Ingo Neumann, Vladik Kreinovich, Chon Van Le

Departmental Technical Reports (CS)

In many practical situations, observations and measurement results are consistent with many different models -- i.e., the corresponding problem is ill-posed. In such situations, a reasonable idea is to take into account that the values of the corresponding parameters should not be too large; this idea is known as regularization. Several different regularization techniques have been proposed; empirically the most successful are LASSO method, when we bound the sum of absolute values of the parameters, and EN and CLOT methods in which this sum is combined with the sum of the squares. In this paper, we explain the empirical success …

Neutron Lifetime Puzzle And Nuclear Stability: A Possible Relation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that a free neutron decays into a proton, an electron, and an anti-neutrino. Interesting, recent attempts to measure the neutron's lifetime has led to two slightly different estimates: namely, the number of decaying neutrons is somewhat larger than the number of newly created protons. This difference is known as the neutron lifetime puzzle. A natural explanation for this difference is that in some cases, a neutron decays not into a proton, but into some other particle. If this explanation is true, this implies that nuclei with a sufficiently large number of neutrons will be unstable. Based on …

Why Experts Sometimes Do Not Perform Well In Unusual Situations, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

We expect that the quality of experts' decisions increases with their experience. This is indeed true for reasonably routine situations. However, surprisingly, empirical data shows that in unusual situations, novice experts make much better decisions than more experience ones. This phenomenon is especially unexpected for medical emergency situations: it turns out that the mortality rate of patients treated by novice doctors is a third lower than for patients treated by experience doctors. In this paper, we provide a possible explanation for this seemingly counterintuitive phenomenon.

Unexpected Empirical Dependence Of Calf Gender On Insemination Time: System-Based Explanation, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

To improve the efficiency of artificial insemination, farmers equip cows with sensors, based on which a computer system determines the cow's insemination window. Analysis of the resulting calves showed an unexpected dependence of the calf's gender on the insemination time: cows inseminated earlier in their window mostly gave birth to female calves, while cows inseminated later in their window mostly gave birth to males. In this paper, we provide a general system-based explanation for this phenomenon.

Dunning-Kruger Effect: A Simple System-Based Explanation, Griselda Acosta, Eric Smith

Departmental Technical Reports (CS)

In their 1999 paper, psychologists David Dunning and Justin Kruger showed that, in general, experts not only provide better estimates of different situations, but they also provide a better estimates of the accuracy of their estimates. Which this phenomenon has been confirmed by many follow-up experiments, it remains largely unexplained. In this paper, we provide a simple system-based qualitative explanation for the Dunning-Kruger effect.

Extending Statistical Learning For Aneurysm Rupture Assessment To Finnish And Japanese Populations Using Morphology, Hemodynamics, And Patient Characteristics, Felicitas J. Detmer, Sara Hadad, Bong Jae Chung, Fernando Mut, Martin Slawski, Norman Juchler, Vartan Kurtcuoglu, Sven Hirsch, Philippe Bijlenga, Yuya Uchiyama, Soichiro Fujimura, Makoto Yamamoto, Yuichi Murayama, Hiroyuki Takao, Timo Koivisto, Juhana Frösen, Juan R. Cebral

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

OBJECTIVE: Incidental aneurysms pose a challenge for physicians, who need to weigh the rupture risk against the risks associated with treatment and its complications. A statistical model could potentially support such treatment decisions. A recently developed aneurysm rupture probability model performed well in the US data used for model training and in data from two European cohorts for external validation. Because Japanese and Finnish patients are known to have a higher aneurysm rupture risk, the authors' goals in the present study were to evaluate this model using data from Japanese and Finnish patients and to compare it with new models …

Investigation Of Fundamental Principles Of Rigid Body Impact Mechanics, Khalid Alluhydan

Mechanical Engineering Research Theses and Dissertations

In impact mechanics, the collision between two or more bodies is a common, yet a very challenging problem. Producing analytical solutions that can predict the post-collision motion of the colliding bodies require consistent modeling of the dynamics of the colliding bodies. This dissertation presents a new method for solving the two and multibody impact problems that can be used to predict the post-collision motion of the colliding bodies. Also, we solve the rigid body collision problem of planar kinematic chains with multiple contacts with external surfaces.

In the first part of this dissertation, we study planar collisions of Balls and …

Mittag–Leffler Stability Of Systems Of Fractional Nabla Difference Equations, Paul W. Eloe, Jaganmohan Jonnalagadda

Mathematics Faculty Publications

Mittag-Leffler stability of nonlinear fractional nabla difference systems is defined and the Lyapunov direct method is employed to provide sufficient conditions for Mittag-Leffler stability of, and in some cases the stability of, the zero solution of a system nonlinear fractional nabla difference equations. For this purpose, we obtain several properties of the exponential and one parameter Mittag-Leffler functions of fractional nabla calculus. Two examples are provided to illustrate the applicability of established results.

Material Flow Analysis As A Decision Supporting Tool For Faecal Sludge Resource Recovery: Mathematical Formulation And Quantification, Isabela Thomas Mkude

Tanzania Journal of Engineering and Technology (TJET)

In this work, a conceptual framework for faecal sludge (FS) management and resource recovery in Dar es Salaam city was developed. Material Flow Analysis (MFA) approach was used to assess and quantify the current materials and nutrients to support on decisions for nutrient recovery and minimize environmental pollution in three unplanned settlements of Manzese, Keko and Kipawa in Dar es Salaam city. Nitrogen and Phosphorus were chosen as indicators for the evaluation of the process. The results showed that about 75% of the Dar es Salaam city population relies on pit latrines and 15% connected to septic tanks, translating to …

Development Of Improved Characteristic Equations For Lake Rukwa In Tanzania, Patrick Valimba Dr

Tanzania Journal of Engineering and Technology (TJET)

Often Lake Rukwa characteristics have been misreported in literature giving different volumes and surface areas at similar water surface elevations. This study aimed at establishing reliable lake characteristics elevation-area-storage equations for Lake Rukwa by utilising all available data and information to define the bathymetry and derive characteristic equations. A procedure was developed that combines historical lake extents, spot heights from topographical maps and surveyed lake bathymetry to define refined bathymetry to levels it has never reached. It combined spot heights around the lake and selected 13,934 surveyed points (from 107,938 available) within the lake confined by the 820 m land …

Assessment Of Surface Water Resources In Great Ruaha River Sub-Basin Tanzania: Application Of Weap Model For Water Allocation And Utilization Impacts Analysis, Deogratias Mulungu

Tanzania Journal of Engineering and Technology (TJET)

The Great Ruaha River (GRR) sub-basin is one of the most important waterways in Tanzania as it supports important economic activities. The sub-basin is progressively faced with an inevitable situation of increasing water demand among competing users while the quantity and quality of water is diminishing. The focus of this study was to assess allocation of existing (2012) and future (up to 2025) quantities of surface water in the GRR sub-basin with consideration of priorities given in the Tanzania Water Resources Management Act, 2009 in the order: domestic, environment, agriculture, livestock and non-domestic. Simulation of water allocation scenarios of irrigation …

Meshless Modeling Of Flow Dispersion And Progressive Piping In Poroelastic Levees, Anthony Khoury, Eduardo Divo, Alain J. Kassab, Sai Kakuturu, Lakshmi Reddi

Publications

Performance data on earth dams and levees continue to indicate that piping is one of the major causes of failure. Current criteria for prevention of piping in earth dams and levees have remained largely empirical. This paper aims at developing a mechanistic understanding of the conditions necessary to prevent piping and to enhance the likelihood of self-healing of cracks in levees subjected to hydrodynamic loading from astronomical and meteorological (including hurricane storm surge-induced) forces. Systematic experimental investigations are performed to evaluate erosion in finite-length cracks as a result of transient hydrodynamic loading. Here, a novel application of the localized collocation …

A “Rule-Of-Five” Framework For Models And Modeling To Unify Mathematicians And Biologists And Improve Student Learning, C. Diaz Eaton, H. C. Highlander, K. D. Dahlquist, G. Ledder, M.D. Lamar, R.C. Schugart

Department of Mathematics: Faculty Publications

Despite widespread calls for the incorporation of mathematical modeling into the undergraduate biology curriculum, there is lack of a common understanding around the definition of modeling, which inhibits progress. In this paper, we extend the “Rule-of-Four,” initially used in calculus reform efforts, to a “Rule-of-Five” framework for models and modeling that is inclusive of varying disciplinary definitions of each. This unifying framework allows us to both build on strengths that each discipline and its students bring, but also identify gaps in modeling activities practiced by each discipline. We also discuss benefits to student learning and interdisciplinary collaboration.

The Role Of Self-Efficacy And Use Of Teachers’ Feedback On Students’ Mathematics Performance In Tanzanian Secondary Schools, Florence Kyaruzi

Journal of Humanities and Social Sciences

Self-efficacy has been found to have positive relationship with important educational variables such as choice of task, persistence, motivation, and academic performance. This study investigated the role of students’ mathematics self-efficacy and use of teachers’ feedback in improving mathematics performance among secondary schools in Tanzania. The study involved 2767 Form III students sampled from 48 secondary schools in Dar es Salaam and Kilimanjaro regions. A questionnaire was used to measure students’ mathematics self-efficacy and mathematics performance, while focus group discussions were used to explore students’ perceptions of the use teachers’ feedback. Data were analysed using structural equation modelling and content …