On the Polytope Escape Problem for Continuous Linear Dynamical Systems

The Polyhedral Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ and a convex polyhedron $\mathcal{P} \subseteq \mathbb{R}^{d}$, whether, for some initial point $\boldsymbol{x}_{0}$ in $\mathcal{P}$, the trajectory of the unique solution to the differential equation $\dot{\boldsymbol{x}}(t)=f(\boldsymbol{x}(t))$, $\boldsymbol{x}(0)=\boldsymbol{x}_{0}$, is entirely contained in $\mathcal{P}$. We show that this problem is decidable, by reducing it in polynomial time to the decision version of linear programming with real algebraic coefficients, thus placing it in $\exists \mathbb{R}$, which lies between NP and PSPACE. Our algorithm makes use of spectral techniques and relies among others on tools from Diophantine approximation.Comment: Accepted to HSCC 201

On Termination of Integer Linear Loops

A fundamental problem in program verification concerns the termination of simple linear loops of the form x := u ; while Bx >= b do {x := Ax + a} where x is a vector of variables, u, a, and c are integer vectors, and A and B are integer matrices. Assuming the matrix A is diagonalisable, we give a decision procedure for the problem of whether, for all initial integer vectors u, such a loop terminates. The correctness of our algorithm relies on sophisticated tools from algebraic and analytic number theory, Diophantine geometry, and real algebraic geometry. To the best of our knowledge, this is the first substantial advance on a 10-year-old open problem of Tiwari (2004) and Braverman (2006).Comment: Accepted to SODA1

English economists: Descriptive accounts of location theory: Working paper series--97-03

The spatial aspects of economic theory have been developed through time in spatial price theory and location theory. This paper will concentrate on the interdisciplinary topic: location theory. Most of the major contributions to location theory have come from economists, engineers, and geographers. Location theory has at least four historical roots. If one uses the classification system of agricultural location, market area analysis, industrial location, and central places; then, one would point to Thünen, Launhardt, Weber, and Christaller, respectively. Johann Heinrich von Thünen published his isolated State in 1826. Wilhelm Launhardt's Mathematische der Volkswirtschaftslihre der Volkswirtschaftslihre was published in 1885. Alfred Weber contributed his Theory of the Location of Industries in 1909. Walter Christaller's Central Places in Southern Germany was released in 1933. Most of the theoretical contributions found in modern location theory literature can be traced to these four men. After summarizing the major contributions of these recognized founders of the field, a detailed analyses of "English" economists will be provided. The presentation will be limited to the works of Petty, Smith, Marshall, and Hobson. These authors did not give major emphasis to problems of location in their writings, but the extent of their discussions on the subject go beyond what one might expect. All four realized that space could make a difference in basic economic reasoning. Locational rent can be found in Petty, Smith, and Marshall. Some form of market area analysis can be found in Marshall and Hobson. Transportation cost or transportation system improvements were important to all four writers. In conclusion, this author feels that these discussions are too important to ignore

Embedding of global attractors and their dynamics

Using shape theory and the concept of cellularity, we show that if $A$ is the global attractor associated with a dissipative partial differential equation in a real Hilbert space $H$ and the set $A-A$ has finite Assouad dimension $d$, then there is an ordinary differential equation in ${\mathbb R}^{m+1}$, with $m >d$, that has unique solutions and reproduces the dynamics on $A$. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor $X$ arbitrarily close to $LA$, where $L$ is a homeomorphism from $A$ into ${\mathbb R}^{m+1}$

Reducing high attrition rate in a business statistics course using an interpretive approach encompassing diverse teaching and learning styles: Working paper series--03-11

An interpretive-learner-centered approach is implemented to improve students learning and address the issue of high attrition rate for a business statistics course. It focuses on the interpretation and implications of statistical results through real business problems and data while the mechanical steps of computation via formulae are relegated to the background. The philosophy that "students are responsible for their education" is embraced; thus, a mastery approach to learning was adopted utilizing pre-lecture, post-lecture and lab web quizzes all with multiple attempts allowed. Cooperative learning serves as a common thread in the design through the use of teams in lectures, labs and on two team projects. The projects result in business reports where all statistical jargon is translated into everyday language. Results from data collected on students learning style, pre and post assessments indicate a significant reduction in attrition rate and improved student learning

Impacts of supply chain management on competition: Working paper series--02-10

The American food supply system provides a clear illustration of recent institutional evolution and some likely future outcomes of "supply chain management" (SCM), a required component of the curriculum in many business schools. In the extreme, recommended SCM practices seek to maximize profits for the entire chain of firms involved in bringing a product to consumers, even if the profits of individual firms within the group are reduced. This requires obliteration of many traditional competitive relationships among businesses. Much of the thrust of SCM is accomplished through contractual arrangements that leave intact the independent status of the firms involved. However, several models exist in which firms bend others to their wills and lessen competition through various SCM practices. The antitrust bar has taken note of these situations, but textbooks on SCM and Principles of Economics ignore the issue, to the detriment of understanding

What do regressions estimate? Working paper series--08-04

Along with the readily available computer software that performs least squares regressions, comes the proliferation of misuse and misinterpretation of the regression results. We illustrate in the paper that the solutions to the intercept and slope coefficients in a simple linear regression model are dependent on the definition of the loss function used in the optimization problem. Likewise, there is not a universal population linear regression line. If a squared error loss function is used, the population regression line turns out to be the conditional mean function while an absolute loss function yields the conditional median. When the asymmetric loss function defined in Koenker and Bassett (1978) is used, the population regression line becomes the conditional quantile function. Therefore, what the sample regression is estimating and how the estimated intercept and slope coefficients should be interpreted are dependent upon what loss function is being used in the optimization problem. We illustrated a few mistakes commonly made in interpreting the least squares regression results and discuss some potential misuses of the least square regressions. By pointing out these specific mistakes, we hope that practitioners and educators will have a better understanding of what exactly we are estimating when we perform a least squares regression or any regression for that matter

Limitations of cost-effectiveness analysis: Defective data and significant digits: Working paper series--97-01

Some applications of Cost-Effectiveness Analysis in the process of allocating resources for health care use hospital "charges" as cost data. These are not appropriate for decision making within the framework of standard economic analysis. Also disputable is the accuracy of some calculations in these analyses, caused by failure to consider the limitations imposed by significant digits in the data. A Cost-Effectiveness Analysis of a proposed program for mass vaccinations for chicken pox provides a case study of the general problem, with additional specific evaluation of the problems encountered in using aggregated "charge" data from a statewide data base, compounding the initial error of using "charges" to depict economic costs. Institutionalist economic theory, based on the insights of John Dewey, is the basis for the evaluation. The article concludes that Cost-Effectiveness Analysis is misused in the reviewed article, and that the misuse is economically and politically detrimental