Applied Mathematics Commons^™

It Is Important To Take All Available Information Into Account When Making A Decision: Case Of The Two Envelopes Problem, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In situations when we know the probabilities of all possible consequences, traditional decision theory recommends selecting the action that maximizes expected utility. In many practical situations, however, we only have partial information about the corresponding probabilities. In this case, for different possible probability distributions, we get different values of expected utility. In general, possible values of expected utility form an interval. One way to approach this situation is to use the optimism-pessimism approach proposed by Nobelist Leo Hurwicz. Another approach is to select one of the possible probability distributions -- e.g., the one that has the largest possible entropy. Both …

How To Decide Which Cracks Should Be Repaired First: Theoretical Explanation Of Empirical Formulas, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Due to stress, cracks appear in constructions: cracks appear in buildings, bridges, pavements, among other structures, cracks appear in pavements, etc. In the long run, cracks need to be repaired. However, our resources are limited, so we need to decide which cracks are more dangerous. For this, we need to be able to predict how different cracks will grow. There are several empirical formulas describing crack growth. In this paper, we show that by using scale invariance, we can provide a theoretical explanation for these empirical formulas.

Euclidean Distance Between Intervals Is The Only Representation-Invariant One, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

An interval can be represented as a point in a plane, e.g., as a point with its endpoints as coordinates. We can thus define distance between intervals as the Euclidean distance between the corresponding points. Alternatively, we can describe an interval by its center and radius, which leads to a different definition of distance. Interestingly, these two definitions lead, in effect, to the same distance -- to be more precise, these two distances differ by a multiplicative constant. In principle, we can have more general distances on the plane. In this paper, we show that only for Euclidean distance, the …

The Less We Love A Woman, The More She Likes Us: Pushkin's Observation Explained, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Alexander Pushkin, the most famous Russian poet, made this observation in "Eugene Onegin", his novel in verse which is most known to non-Russian readers via Tchaikovsky's opera. This observation may not be an absolute truth -- there are counterexamples -- but the fact that it is still widely cited shows that there is some truth in this statement. In this paper, we recall the usual utility-based explanation for a similar statement, and propose a new explanation, which is even more fundamental -- it is on the biological level.

An Application Of The Unscented Kalman Filter For Spacecraft Attitude Estimation On Real And Simulated Light Curve Data, Kent A. Rush

Master's Theses

In the past, analyses of lightcurve data have been applied to asteroids in order to determine their axis of rotation, rotation rate and other parameters. In recent decades, these analyses have begun to be applied in the domain of Earth orbiting spacecraft. Due to the complex geometry of spacecraft and the wide variety of parameters that can influence the way in which they reflect light, these analyses require more complex assumptions and a greater knowledge about the object being studied. Previous investigations have shown success in extracting attitude parameters from unresolved spacecraft using simulated data. This paper presents a focused …

Optimal Allocation Of Two Resources In Annual Plants, David Mcmorris

Department of Mathematics: Dissertations, Theses, and Student Research

The fitness of an annual plant can be thought of as how much fruit is produced by the end of its growing season. Under the assumption that annual plants grow to maximize fitness, we can use techniques from optimal control theory to understand this process. We introduce two models for resource allocation in annual plants which extend classical work by Iwasa and Roughgarden to a case where both carbohydrates and mineral nutrients are allocated to shoots, roots, and fruits in annual plants. In each case, we use optimal control theory to determine the optimal resource allocation strategy for the plant …

Dynamics Of Discontinuities In Elastic Solids, Arkadi Berezovski, Mihhail Berezovski

Publications

The paper is devoted to evolving discontinuities in elastic solids. A discontinuity is represented as a singular set of material points. Evolution of a discontinuity is driven by the configurational force acting at such a set. The main attention is paid to the determination of the velocity of a propagating discontinuity. Martensitic phase transition fronts and brittle cracks are considered as representative examples.

An A Posteriori Error Analysis Of Stationary Incompressible Magnetohydrodynamics, Ari E. Rappaport

Mathematics & Statistics ETDs

Adjoint based a posteriori error analysis is a technique to produce exact error repre- sentations for quantities of interests that are functions of the solution of systems of partial differential equations (PDE). The tools used in the analysis consist of duality arguments and compatible residuals. In this thesis we apply a posteriori error anal- ysis to the magnetohydrodynamics (MHD) equations . MHD provides a continuum level description of conducting fluids in the presence of electromagnetic fields. The MHD system is therefore a multi-physics system, capturing both fluid and electro- magnetic effects. Mathematically, The equations of MHD are highly nonlinear and …

Numerical Solution Of Ordinary Differential Equations Using Continuous Runge-Kutta Methods (Feldberg Of Order Four And Five), Madeha Yousif

Emirates Journal for Engineering Research

In this paper the continuous Runge-Kutta method (Runge-Kutta Feldberg method of order four and five) have been used to find the numerical solution of ordinary differential equation not only at the mesh points but also the all points between them. the results are computed using matlab program..

Multigrid Methods For Elliptic Optimal Control Problems, Sijing Liu

LSU Doctoral Dissertations

In this dissertation we study multigrid methods for linear-quadratic elliptic distributed optimal control problems.

For optimal control problems constrained by general second order elliptic partial differential equations, we design and analyze a $P_1$ finite element method based on a saddle point formulation. We construct a $W$-cycle algorithm for the discrete problem and show that it is uniformly convergent in the energy norm for convex domains. Moreover, the contraction number decays at the optimal rate of $m^{-1}$, where $m$ is the number of smoothing steps. We also prove that the convergence is robust with respect to a regularization parameter. The robust …

2n-Dimensional Canonical Systems And Applications, Andrei Ludu, Keshav Baj Acharya

Publications

We study the 2N-dimensional canonical systems and discuss some properties of its fundamental solution. We then discuss the Floquet theory of periodic canonical systems and observe the asymptotic behavior of its solution. Some important physical applications of the systems are also discussed: linear stability of periodic Hamiltonian systems, position-dependent effective mass, pseudo-periodic nonlinear water waves, and Dirac systems.

How To Detect Future Einsteins: Towards Systems Approach, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Talents are rare. It is therefore important to detect and nurture future talents as early as possible. In many disciplines, this is already being done -- via gifted and talented programs, Olympiads, and other ways to select kids with unusually high achievements. However, the current approach is not perfect: some of the kids are selected simply because they are early bloomers, they do not grow into unusually successful researchers; on the other hand, many of those who later become very successful are not selected since they are late bloomers. To avoid these problems, we propose to use systems approach: to …

Which Classes Of Bi-Intervals Are Closed Under Addition?, Olga Kosheleva, Vladik Kreinovich, Jonatan Contreras

Departmental Technical Reports (CS)

In many practical situations, uncertainty with which we know each quantity is described by an interval. In processing such data, it is useful to know that the sum of two intervals is always an interval. In some cases, however, the set of all possible value of a quantity is described by a bi-interval -- i.e., by a union of two intervals. It is known that the sum of two bi-intervals is not always a bi-interval. In this paper, we describe all the class of bi-intervals which are closed under addition -- i.e., for which the sum of bi-intervals is a …

Preference For Boys Does Not Necessarily Lead To A Gender Disbalance: A Realistic Example, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Intuitively, it seems that cultural preference for boys should lead to a gender disbalance -- more boys than girls. This disbalance is indeed what is often observed, and this disbalance is what many models predict. However, in this paper, we show, on a realistic example, that preference for boys does not necessarily lead to a gender disbalance: in our simplified example, boys are clearly preferred, but still there are exactly as many girls as there are boys.

Healthy Lifestyle Decreases The Risk Of Alzheimer Disease: A Possible Partial Explanation Of An Empirical Dependence, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent paper showed that for people who follow all five healthy lifestyle recommendations, the risk of Alzheimer disease is only 40% of the risk for those who do not follow any of these recommendations, and that for people two or three of these recommendations, the risk is 63% of the not-followers risk. In this paper, we show that a relation between the two numbers -- namely, that 0.40 is the square of 0.63 -- can be naturally explained by a simple model.

When Can We Be Sure That Measurement Results Are Consistent: 1-D Interval Case And Beyond, Hani Dbouk, Steffen Schön, Ingo Neumann, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, measurements are characterized by interval uncertainty -- namely, based on each measurement result, the only information that we have about the actual value of the measured quantity is that this value belongs to some interval. If several such intervals -- corresponding to measuring the same quantity -- have an empty intersection, this means that at least one of the corresponding measurement results is an outlier, caused by a malfunction of the measuring instrument. From the purely mathematical viewpoint, if the intersection is non-empty, there is no reason to be suspicious, but from the practical viewpoint, if …

What If Not All Interval-Valued Fuzzy Degrees Are Possible?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the applications of intervals is in describing experts' degrees of certainty in their statements. In this application, not all intervals are realistically possible. To describe all realistically possible degrees, we end up with a mathematical question of describing all topologically closed classes of intervals which are closed under the appropriate minimum and maximum operations. In this paper, we provide a full description of all such classes.

Common-Sense-Based Theoretical Explanation For An Empirical Formula Estimating Road Quality, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

The quality of a road is usually gauged by a group of trained raters; the resulting numerical value is known as the Present Serviceability Index (PSI). There are, however, two problems with this approach. First, while it is practical to use trained raters to gauge the quality of major highways, there are also numerous not-so-major roads, and there is not enough trained raters to gauge the quality of all of them. Second, even for skilled raters, their estimates are somewhat subjective: different groups of raters may estimate the quality of the same road segment somewhat differently. Because of these two …

Structure For Regular Inclusions. Ii Cartan Envelopes, Pseudo-Expectations And Twists, David R. Pitts

Department of Mathematics: Faculty Publications

We introduce the notion of a Cartan envelope for a regular inclusion (C,Ɗ). When a Cartan envelope exists, it is the unique, minimal Cartan pair into which (C,Ɗ) regularly embeds. We prove a Cartan envelope exists if and only if (C,Ɗ) has the unique faithful pseudo-expectation property and also give a characterization of the Cartan envelope using the ideal intersection property.

For any covering inclusion, we construct a Hausdorff twisted groupoid using appropriate linear functionals and we give a description of the Cartan envelope for (C,Ɗ) in terms of a twist …

A Study Of The Design Of Adaptive Camber Winglets, Justin J. Rosescu

Master's Theses

A numerical study was conducted to determine the effect of changing the camber of a winglet on the efficiency of a wing in two distinct flight conditions. Camber was altered via a simple plain flap deflection in the winglet, which produced a constant camber change over the winglet span. Hinge points were located at 20%, 50% and 80% of the chord and the trailing edge was deflected between -5° and +5°. Analysis was performed using a combination of three-dimensional vortex lattice method and two-dimensional panel method to obtain aerodynamic forces for the entire wing, based on different winglet camber configurations. …

On The Socles Of Fully Inert Subgroups Of Abelian P-Groups, Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith

Articles

We define the so-called fully inert socle-regular and weakly fully inert socle-regular Abelian p-groups and study them with respect to certain of their numerous interesting properties. For instance, we prove that in the case of groups of length ! these two group classes coincide but that in the case of groups of length ! + 1 they differ. Some structural and characterization results are also obtained. The work generalizes concepts which have been of interest recently in the theory of entropy in algebra and builds on recent investigations by the second and third named authors in Arch. Math. Basel (2009) …

Dynamical Behavior Of A Malaria Relapse Model With Insecticide Treated Nets (Itns) As Protection Measure, Reuben I. Gweryina, Anande R. Kimbir

Applications and Applied Mathematics: An International Journal (AAM)

Malaria is a tropical disease which is mainly spread by plasmodium falciparum which has been the principal enemy to the existence of mankind till date. In this paper a version of a malaria model incorporating the use of treated mosquito nets as a disease control strategy is proposed and then transformed into proportions, so as to assess the global impact of ITNs on the prevalence of malaria. Constructing a Lyapunov function using matrix-theoretic approach, a malaria-free equilibrium state is obtained, which is globally asymptotically stable if the control reproduction number, 𝑅_𝑚<1. This means that malaria can be controlled or eradicated under such a threshold quantity, 𝑅𝑚. On the other hand, a malaria-persistence equilibrium state exists which is globally stable when 𝑅_𝑚>1, using geometric theoretic method with Lozoskii …

Stability Of Regular Thin Shell Wormholes Supported By Vdw Quintessence, A. Eid

Applications and Applied Mathematics: An International Journal (AAM)

The dynamical equations of motion for a thin shell wormhole from regular black holes that are supported by Van der Waals (VDW) quintessence equation of state (EoS) are constructed, through cut and -paste technique. The linearized stability of regular wormhole is derived. The presences of unstable and stable static solutions with different value of some parameters are analyzed.

Variants Of Meir-Keeler Fixed Point Theorem And Applications Of Soft Set-Valued Maps, Akbar Azam, Mohammed Shehu Shagari

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we prove a Meir-Keeler type common fixed point theorem for two mappings for which the range set of the first one is a family of soft sets, called soft set-valued map and the second is a point-to-point mapping. In addition, it is also shown that under some suitable conditions, a soft set-valued map admits a selection having a unique fixed point. In support of the obtained result, nontrivial examples are provided. The novelty of the presented idea herein is that it extends the Meir-Keeler fixed point theorem and the theory of selections for multivalued mappings from the …

Complexity Dynamics Of Gumowski-Mira Map, Sada Nand Prasad, K. R. Meena, Abdullah A. Ansari

Applications and Applied Mathematics: An International Journal (AAM)

In the context of nonlinear dynamics, interesting dynamic behavior of Gumowski-Mira Map has been noted under various feasible circumstances. Evolutionary phenomena are discussed through the study of bifurcation analysis leading to period-doubling and chaos. The appearance of chaos in the method is identified by plotting Lyapunov characteristic exponents (LCE) and Topological Entropy within certain parameter range. Dynamic Lyapunov Indicator (DLI) has been procured for further identification of regular and chaotic motions of the Gumowski-Mira Map. The numerical results through the indicator DLI clearly demonstrate the behavior of our map. The correlation dimension has been calculated numerically for the dimension of …

Stokes Drag On Axially Symmetric Body In Micro Polar Fluid, Deepak K. Srivastava, Nirmal Srivastava

Applications and Applied Mathematics: An International Journal (AAM)

In a recent paper, Srivastava et al. (2016) have tested the proposed formula based on DS-conjecture Datta and Srivastava, (1999) of Stokes drag on axially symmetric bodies placed under micropolar fluid to improve the drag value under Oseen’s limit. In the present work, proof of the proposed drag formula is given for both axial and transverse Stokes flow of micropolar fluid under certain body geometry constraints mainly of continuously turning tangent on body curve in meridional plane as assumed in DS-conjecture. The general expression of drag immediately reduces to the value of drag in classical Newtonian fluid as micro polarity …

Some Summation Theorems For Clausen’S Hypergeometric Functions With Unit Argument, M. I. Qureshi, Mohammad Shadab

Applications and Applied Mathematics: An International Journal (AAM)

Motivated by the work on hypergeometric summation theorems, we establish new summation formula for Clausen’s hypergeometric function with unit argument in terms of pi and natural logarithms of some rational and irrational numbers. For the application purpose, we derive some new and modified summation theorems for Clausen’s hypergeometric functions using our new formula.

Mhd Mixed Convective Flow Of Maxwell Nanofluid Past A Porous Vertical Stretching Sheet In Presence Of Chemical Reaction, Hunegnaw Dessie, Demeke Fissha

Applications and Applied Mathematics: An International Journal (AAM)

In this study, MHD mixed convective flow of Maxwell nanofluid past a porous vertical stretching sheet in the presence of chemical reaction is investigated. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations via Lie group analysis. Numerical solutions of these equations are obtained by Runge-Kutta fourth order method along with shooting technique and the results obtained for different governing flow parameters are drawn graphically and their effects on velocity, temperature and concentration profiles are discussed. The values of skin-friction coefficient, Nusselt number coefficient and Sherwood number coefficient are presented …

Mathematical Modeling For Studying The Sustainability Of Plants Subject To The Stress Of Two Distinct Herbivores, B. Chen-Charpentier, M. C.A. Leite, O. Gaoue, F. B. Agusto

Applications and Applied Mathematics: An International Journal (AAM)

Viability of plants, especially endangered species, are usually affected by multiple stressors, including insects, herbivores, environmental factors and other plant species. We present new mathematical models, based on systems of ordinary differential equations, of two distinct herbivore species feeding (two stressors) on the same plant species. The new feature is the explicit functional form modeling the simultaneous feedback interactions (synergistic or additive or antagonistic) between the three species in the ecosystem. The goal is to investigate whether the coexistence of the plant and both herbivore species is possible (a sustainable system) and under which conditions sustainability is feasible. Our theoretical …

Exploring The Convergence Properties Of A New Modified Newton-Raphson Root Method, Euaggelo E. Zotos, Wei Chen

Applications and Applied Mathematics: An International Journal (AAM)

We examine the convergence properties of a modified Newton-Raphson root method, by using a simple complex polynomial equation, as a test example. In particular, we numerically investigate how a parameter, entering the iterative scheme, affects the efficiency and the speed of the method. Color-coded polynomiographs are deployed for presenting the regions of convergence, as well as the fractality degree of the complex plane. We demonstrate that the behavior of the modified Newton-Raphson method is correlated with the numerical value of the parameter 1. Additionally, there are cases for which the method works flawlessly, while in some other cases we encounter …