Earth Mover\u27s Distance Between Grade Distribution Data with Fixed Mean

The Earth Mover\u27s Distance (EMD) is examined on all theoretically possible grade distributions with the same grade point average (GPA). The numbers of distributions with the same EMD and GPA are encoded in the coefficients of a generating function. The theoretical mean EMD for grade distributions, that are sampled uniformly and independently at random, is computed from this function, and compared to real world grade data taken from several years. The data is further examined regarding the appearance of clusters that change when varying the distance threshold

Combinatorial problems related to optimal transport and parking functions

In the first part of this work, we provide contributions to optimal transport through work on the discrete Earth Mover\u27s Distance (EMD).We provide a new formula for the mean EMD by computing three different formulas for the sum of width-one matrices: the first two formulas apply the theory of abstract simplicial complexes and result from a shelling of the order complex, whereas the last formula uses Young tableaux. Subsequently, we employ this result to compute the EMD under different cost matrices satisfying the Monge property. Additionally, we use linear programming to compute the EMD under non-Monge cost matrices, giving an interpretation of the EMD as a distance measure on pie charts. Furthermore, we generalize our result to the $n$-dimensional EMD, by providing two different formulas for the sum of width--one tensors: once approaching the problem from the perspective of Young tableaux and once through the theory of abstract simplicial complexes by shelling of the $n$-dimensional order complex. In the second part, we provide contributions to the topic of parking functions.We provide background on the topic and show a connection to the first part of this work through certain statistics on parking functions used in the shuffle conjecture. Furthermore, we provide enumerative formulas for different generalizations of parking functions, allowing cars to have varying lengths. Additionally, we show a surprising connection between certain restricted parking objects and the Quicksort algorithm. At last, we will use the intersection of a subset of parking functions and Fubini rankings to characterize and enumerate Boolean algebras in the weak Bruhat order of $S_n$

The sum of all width-one matrices

A nonnegative integer matrix is said to have width 1 if its nonzero entries lie along a path consisting of steps to the south and to the east. These matrices are important in optimal transport theory: the northwest corner algorithm, for example, takes supply and demand vectors and outputs a width-one matrix. The problem in this paper is to write down an explicit formula for the sum of all width-one matrices (with given dimensions $n \times n$ and given sum $d$ of the entries). We prove two strikingly different formulas. The first, a hypergeometric series with unit argument, is obtained by applying the Robinson-Schensted-Knuth correspondence to the width-one matrices; the second is obtained via Stanley-Reisner theory. Computationally, our two formulas are in a sense dual to each other: the first formula outperforms the second if $d$ is fixed and $n$ increases, while the second outperforms the first if $n$ is fixed and $d$ increases. We also show how our result yields a new non-recursive formula for the mean value of the discrete earth mover's distance (i.e., the solution to the transportation problem), whenever the cost matrix has the Monge property.Comment: 14 pages, 3 figure

The structure and normalized volume of Monge polytopes

A matrix $C$ has the Monge property if $c_{ij} + c_{IJ} \leq c_{Ij} + c_{iJ}$ for all $i < I$ and $j < J$. Monge matrices play an important role in combinatorial optimization; for example, when the transportation problem (resp., the traveling salesman problem) has a cost matrix which is Monge, then the problem can be solved in linear (resp., quadratic) time. For given matrix dimensions, we define the Monge polytope to be the set of nonnegative Monge matrices normalized with respect to the sum of the entries. In this paper, we give an explicit description and enumeration of the vertices, edges, and facets of the Monge polytope; these results are sufficient to construct the face lattice. In the special case of two-row Monge matrices, we also prove a polytope volume formula. For symmetric Monge matrices, we show that the Monge polytope is a simplex and we prove a general formula for its volume

Lucky Cars and the Quicksort Algorithm

Quicksort is a classical divide-and-conquer sorting algorithm. It is a comparison sort that makes an average of $2(n+1)H_n - 4n$ comparisons on an array of size $n$ ordered uniformly at random, where $H_n = \sum_{i=1}^n\frac{1}{i}$ is the $n$th harmonic number. Therefore, it makes $n!\left[2(n+1)H_n - 4n\right]$ comparisons to sort all possible orderings of the array. In this article, we prove that this count also enumerates the parking preference lists of $n$ cars parking on a one-way street with $n$ parking spots resulting in exactly $n-1$ lucky cars (i.e., cars that park in their preferred spot). For $n\geq 2$, both counts satisfy the second order recurrence relation $f_n=2nf_{n-1}-n(n-1)f_{n-2}+2(n-1)!$ with $f_0=f_1=0$.Comment: 8 pages, and 2 figures, to appear in The American Mathematical Monthl

Cost-sharing in Parking Games

In this paper, we study the total displacement statistic of parking functions from the perspective of cooperative game theory. We introduce parking games, which are coalitional cost-sharing games in characteristic function form derived from the total displacement statistic. We show that parking games are supermodular cost-sharing games, indicating that cooperation is difficult (i.e., their core is empty). Next, we study their Shapley value, which formalizes a notion of "fair" cost-sharing and amounts to charging each car for its expected marginal displacement under a random arrival order. Our main contribution is a polynomial-time algorithm to compute the Shapley value of parking games, in contrast with known hardness results on computing the Shapley value of arbitrary games. The algorithm leverages the permutation-invariance of total displacement, combinatorial enumeration, and dynamic programming. We conclude with open questions around alternative solution concepts for supermodular cost-sharing games and connections to other areas in combinatorics.Comment: 12 page

Considering Uncertainties in Research by Probabilistic Modeling

Technical and economical evaluation of solar thermal power plants constantly gains more importance for industry and research. The reliability of the results highly depends on the assumptions made for the applied parameters. Reducing a power plant system to one single, deterministic number for evaluation, like the levelized cost of electricity (LCOE), might end in misleading results. Probabilistic approaches can help to better evaluate systems [1] and scenarios [2]. While industry looks for safety in investment and profitability, research is predominantly interested in the evaluation of concepts and the identification of promising new approaches. Especially for research, dealing with higher and hardly quantifiable uncertainties, it is desirable to get a detailed view of the system and its main influences. However, to get there, also a good knowledge on the stochastic interrelations and its interpretation is required. Therefore, this paper mainly assesses the influences of basic stochastic assumptions and suggests a methodology to consider suitable stochastic input, especially for parameters of systems still under research. As examples, the comparison between a parabolic trough plant with synthetic oil and direct steam generation is used

Guidelines for CSP yield analysis-Definition of elementary terms

With the success of CSP technology in the last years more players are active in the market, inducing the need for harmonization of technical terms and methodologies. The mission of the SolarPACES “guiSmo” project which was started in 2010 is to develop a guideline for CSP yield analysis [1]. Activities carried out so far have shown that people have different understandings of many terms used in daily CSP practice. Especially for the development of guidelines, the essential terms need to be clearly defined in order to avoid inconsistencies within the same project. A first version of a nomenclature has been compiled by the “guiSmo” team and will undergo final discussion. The aim is to come to a harmonized version by Summer 2013 which will then be presented at the ASME Energy Sustainability conference. The compilation so far includes essential definitions of terms like direct normal irradiance, incident angles, heat flows, and efficiencies on a system level. The definitions presented will be discussed together with existing standards like the ISO 80000 (physical quantities and units of measurement), the ISO 9488 (Solar energy-vocabulary) and other relevant sources. Although the list of terms is primarily put together for the work in the “guiSmo” project, it might serve as a basis for standardization in the official councils. An international group of solar experts is involved in the preparation of the document in order to ensure high quality and international support for the results