Mathematics Commons^™

Curvature: A Geometric Villain That Ruins Our Instinctive Perception Of Nature, Vehbi Emrah Paksoy

Mathematics Colloquium Series

Our perception of nature is based on evolutionary wiring of our brain and observations we make via our senses. But, in reality, many scientific and technological advancements are based on non-intuitive rules and principles that can only be explained by the ultimate abstraction that is embedded in mathematics. In this talk, I will discuss the concept of curvature and argue how it explains the “unexplainable”. We will see how the curvature proves that the earth is rotating, how good the soap bubbles are at proving profound mathematical results, and if the two dimensional residents can determine the shape of their …

Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of PSL₂(q). A quasi-platonic action of a group G on a closed Riemann S surface is a conformal action for which S/G is a sphere and S->S/G is branched over {0, 1,infinity}. The unit interval in S/G may be lifted to a dessin d'enfant D, an embedded bipartite graph in S. The dessin forms the edges and vertices of a tiling on S by dihedrally symmetric polygons, generalizing the idea of a …

Canonoid And Poissonoid Transformations, Symmetries And Bihamiltonian Structures, Giovanni Rastelli, Manuele Santoprete

Mathematics Faculty Publications

We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler’s equations of the rigid body (on so*(3) and so*(4)) and for an integrable …

Discrete Grüss Type Inequality On Fractional Calculus, Elvan Akin, Serkan Asliyuce, Ayse Feza Guvenilir, Billur Kaymakcalan

Mathematics and Statistics Faculty Research & Creative Works

We give a discrete Grüss type inequality on fractional calculus.

Solving Equations (And Systems Of Equations) Under Uncertainty: How Different Practical Problems Lead To Different Mathematical And Computational Formulations, Vladik Kreinovich

Departmental Technical Reports (CS)

Many practical problems are naturally reduced to solving systems of equations. There are many efficient techniques for solving well-defined systems of equations, i.e., systems in which we know the exact values of all the parameters and coefficients. In practice, we usually know these parameters and coefficients with some uncertainty -- uncertainty usually described by an appropriate granule: interval, fuzzy set, rough set, etc. Many techniques have been developed for solving systems of equations under such granular uncertainty. Sometimes, however, practitioners use previously successful techniques and get inadequate results. In this -- mostly pedagogical -- paper, we explain that to obtain …

From Tertullian's Credo Quia Absurdum To Bohr's Crazy Theories: A Rational Explanation Of A Seemingly Irrational Idea, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, Tertullian's idea -- that absurdity of a statement makes it more believable -- sounds irrational, maybe appropriate for theology but definitely not for science. However, somewhat surprisingly, a similar idea was successfully used by the Nobelist Niels Bohr in theoretical physics -- an epitome of rationality in science. In this paper, we show that this Tertullian-Bohr idea actually has a simple rational explanation. Specifically, if previous attempts to construct a theory which is consistent with what is perceived as common sense were unsuccessful, this implies that a true theory much contradict common sense -- and thus, the …

Towards Selecting The Best Abstraction For A Patrolling Game, Anjon Basak, Chris Kiekintveld, Vladik Kreinovich

Departmental Technical Reports (CS)

When the number of possible strategies is large, it is not computationally feasible to compute the optimal strategy for the original game. Instead, we select our strategy based on an approximate approximate description of the original game. The quality of the resulting strategy depends on which approximation we select. In this paper, on an example of a simple game, we show how to find the optimal approximation, the approximation whose use results in the best strategy.

Lower Bound For Ranks Of Invariant Forms, Harm Derksen, Zach Teitler

Mathematics Faculty Publications and Presentations

We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad-Schreyer-Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic symmetric matrices.

Differentialgeometry In Brno, Ian M. Anderson

Presentations

This page will provide files supporting Ian Anderson's presentations in Brno, December 2015. The files can be found and downloaded from "Additional Files", below.

The files include:

(1) DifferentialGeometryUSU.mla: This is the Maple Library Archive file which provides all the DifferentialGeometry functionality. Here are Installation Instructions.

(2) DifferentialGeometry.help : this is the latest version of the DifferentialGeometry documentation. Copy this file to the same directory used for DifferentialGeometryUSU.mla (from step (1)).

Review: A Short Introduction To De Branges-Rovnyak Spaces, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

How To Predict Nesting Sites?, Stephen Escarzaga, Craig Tweedie, Vladik Kreinovich

Departmental Technical Reports (CS)

How to predict nesting sites? Usually, all we know is the past nesting sites, and the fact that the birds select a site which is optimal for them (in some reasonable sense), but we do not know the exact objective function describing this optimality. In this paper, we propose a way to make predictions in such a situation.

Why It Is Healthy To Regularly Challenge Authority: An Algorithmic Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One way to make group decisions is to select the best decision maker(s) in the group as the authority, and to follow his or her decisions. At first glance, it seems that if the selected authority is indeed the best decision maker, it is beneficial for everyone to obey his or her authority. However, history shows that in many cases, challenges to the authority (even to the authority of the best decision maker) were beneficial to the group. In this paper, we provide an algorithmic explanation for this phenomenon. The main idea behind this explanation is that most practical general …

Interval Methods For Data Fitting Under Uncertainty: A Probabilistic Treatment, Vladik Kreinovich, Sergey P. Shary

Departmental Technical Reports (CS)

How to estimate parameters from observations subject to errors and uncertainty? Very often, the measurement errors are random quantities that can be adequately described by the probability theory. When we know that the measurement errors are normally distributed with zero mean, then the (asymptotically optimal) Maximum Likelihood Method leads to the popular least squares estimates. In many situations, however, we do not know the shape of the error distribution, we only know that the measurement errors are located on a certain interval. Then the maximum entropy approach leads to a uniform distribution on this interval, and the Maximum Likelihood Method …

Simplest Innovations Are, Empirically, The Most Promising: An Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many examples show that the simplest innovation are the most promising. In this paper, we provide a theoretical explanation for this empirical observation.

Two-Dimensional 1-Meixner Random Vectors And Their Semi-Quantum Operators, Gabriela Popa, Aurel I Stan

Communications on Stochastic Analysis

No abstract provided.

A Poisson Shot-Noise Process Of Pulses And Its Scaling Limits, Mine Çaglar

Communications on Stochastic Analysis

No abstract provided.

Solving A Class Of Linear Skorokhod Stochastic Differential Equations, Karl-Heinz Fichtner, Steffen Klaere, Volkmar Liebscher

Communications on Stochastic Analysis

No abstract provided.

Large Deviations For Stochastic Tidal Dynamics Equation, Murugan Suvinthra, Sivaguru S Sritharan, Krishnan Balachandran

Communications on Stochastic Analysis

No abstract provided.

On The Lévy-Khinchin Decomposition Of Generating Functionals, Uwe Franz, Malte Gerhold, Andreas Thom

Communications on Stochastic Analysis

No abstract provided.

White Noise Analysis For The Canonical Lévy Process, Rolando D Navarro Jr., Frederi G Viens

Communications on Stochastic Analysis

No abstract provided.