# Technical Reports (AU-CAS-MathStats)

## Pages

- Technical Report No. 2000-1, 15 pages, The unified Bayesian-Frequentist measure of Berger, Brown and Wolpert.(1994) is considered here in the context of repeated significance tests and change point problems. It was shown that the new evidential measure is quite different from the conventional P-value as a measure of evidence provided by the data against the null hypothesis., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T19:22:10Z No. of bitstreams: 1 SKundu 2000-1.pdf: 3087667 bytes, checksum: caa1abc9041498c4a73f64c9af5f9df7 (MD5), Made available in DSpace on 2014-04-03T19:22:10Z (GMT). No. of bitstreams: 1 SKundu 2000-1.pdf: 3087667 bytes, checksum: caa1abc9041498c4a73f64c9af5f9df7 (MD5)

- Technical Report No. 2000-2, 28 pages, In a phase I clinical trial, we are interested in finding a dose L that will produce toxicity at an acceptable probability level r in the target population. In this paper, we investigate different estimators of the target dose L to be used with the up-and-down Biased Coin Design (BCD) introduced by Durham and Flournoy (1994). These estimators of L are derived using isotonic regression, maximum likelihood, weighted least squares and the simple empirical mean. Given a vector of probability of toxicity at the different doses,we show how to derive the exact distribution of these (and many other) estimators in the BCD setting. However, due to computational limitations, for modest samples (n > 15) the exact method becomes infeasible and bootstrap methods are used. A modified isotonic regression estimate is shown to perform very well, in terms of mean square error (MSE) and average time to converge, in all the scenarios we have studied., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T18:46:34Z No. of bitstreams: 1 NFlournoy 2000-2.pdf: 8162978 bytes, checksum: dbd39c5ad34026013e95d38531be2118 (MD5), Made available in DSpace on 2014-04-03T18:46:34Z (GMT). No. of bitstreams: 1 NFlournoy 2000-2.pdf: 8162978 bytes, checksum: dbd39c5ad34026013e95d38531be2118 (MD5)

- Technical Report No. 2000-3, 15 pages, Assume that the probability of success is unimodal as function of dose. Take the response to be binary and the possible treatment space to be a lattice. The Optimizing Up-and-Down Design allocates treatments to pairs of subjects in a way that causes the treatment distribution to cluster around the treatment with maximum success probability. This procedure is constructed to use accruing information to limit the number of patients that are exposed to doses with high probabilities of failure. The Optimizing Up-and-Down Design is motivated by Kiefer-Wolfowitz's stochastic approximation procedure. In this paper, we compare its performance to stochastic approximation. As an estimator of the best dose, simulation studies demonstrate that the mode of the empirical treatment distribution using the Optimizing Up-and-Down Design converges faster than does the usual estimator using stochastic approximation., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T18:55:50Z No. of bitstreams: 1 NFlournoy 2000-3.pdf: 5256754 bytes, checksum: 7f660d83fc8574b4b064fa3c0513d242 (MD5), Made available in DSpace on 2014-04-03T18:55:50Z (GMT). No. of bitstreams: 1 NFlournoy 2000-3.pdf: 5256754 bytes, checksum: 7f660d83fc8574b4b064fa3c0513d242 (MD5)

- Technical Report No. 2000-5, 21 pages, Closed form expressions for multilocus probabilities are given for the crossover process when it is a renewal process with the distance between crossovers modeled by a Erlang distribution. Closed form expressions are also given for the multilocus probabilities for the chiasma process on the four strand bundle under the same model of recombination for single gamete and for tetrad data. These expressions yield explicit formulas for the map functions, coincidence functions and distributions of the identity- by-descent process for a class of models that incorporate interference. The alternating renewal models used may be of interest in other fields, e.g. telecommunication networks and queues, where they can be used to model the busy/non-busy state of a system with buffers., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T19:31:02Z No. of bitstreams: 1 JNolan erlang 2000-5.pdf: 441285 bytes, checksum: 8e510beb49f58f59504943f457cd6710 (MD5), Made available in DSpace on 2014-04-03T19:31:02Z (GMT). No. of bitstreams: 1 JNolan erlang 2000-5.pdf: 441285 bytes, checksum: 8e510beb49f58f59504943f457cd6710 (MD5)

- Technical Report No. 2012-1, 28 pages, Stable distributions with elliptical contours are a class of distributions that are useful for modeling heavy tailed multivariate data. This paper describes the theory of such distributions, presents formulas for calculating their densities, and methods for fitting the data and assessing the fit. Efficient numerical routines are implemented and evaluated in simulations. Applications to data sets of a financial portfolio with 30 assets and to a bivariate radar clutter data set are presented., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T19:51:58Z No. of bitstreams: 1 JNolan EllipticalStableTechReport 2012-1.pdf: 368532 bytes, checksum: 06483ca9a6a9c47f65cb6fc2e8de4df5 (MD5), Made available in DSpace on 2014-04-03T19:51:58Z (GMT). No. of bitstreams: 1 JNolan EllipticalStableTechReport 2012-1.pdf: 368532 bytes, checksum: 06483ca9a6a9c47f65cb6fc2e8de4df5 (MD5)

- Technical Report No. 2013-1, 46 pages, A weighted social network data type is analyzed using two different methods. The first method used is the Exponential Random Graph Models (ERGM), which is a model used on social network analysis (SNA) that includes as parameters the structural characteristics of the network and the network’s nodes attributes. However, ERGM does not take into account the weights associated with the network’s edges. The second method used is the Cumulative Logistic Regression, which incorporates the weights associated to the network’s edges, but it doesn’t take into account the network’s structural characteristics. Both methods are illustrated using a weighted one-mode data., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-04T15:02:11Z No. of bitstreams: 1 ARebatta Project 2013-1.pdf: 899919 bytes, checksum: 70e3bb6256fba7b46bb10101f5abddcc (MD5), Made available in DSpace on 2014-04-04T15:02:11Z (GMT). No. of bitstreams: 1 ARebatta Project 2013-1.pdf: 899919 bytes, checksum: 70e3bb6256fba7b46bb10101f5abddcc (MD5)

- Technical Report No. 2013-2, 53 pages, In the study of Sampling Theory, the Nyquist-Shannon sampling theorem, proved in the first half of the 20th century, showed that a bandwidth limited signal could be exactly reconstructed when sampled at an appropriate rate. This is the basis for digital representation of audio files (CDs), since human hearing is bandwidth limited. Dr. Stephen Casey proved in his recent research that such a signal, sampled at multiple specifically chosen incommensurate rates, can also be exactly reconstructed. This research takes his theorem and applies it to a regularly sampled audio signal to construct new data sets, sampled at incommensurate rates. The signal can be reconstructed from one or more of these sets, with quality increasing as more data sets are included. Each data set can buffer individually when streamed over a broadband connection. Thus, when bandwidth changes the audio player can add or drop a data set without having to completely re-buffer., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T20:10:35Z No. of bitstreams: 1 SCasey EriksThesisFinal 2013-2.pdf: 517912 bytes, checksum: d74e11f482fc4911448c809b77a8ae3b (MD5), Made available in DSpace on 2014-04-03T20:10:35Z (GMT). No. of bitstreams: 1 SCasey EriksThesisFinal 2013-2.pdf: 517912 bytes, checksum: d74e11f482fc4911448c809b77a8ae3b (MD5)

- Technical Report No. 2013-3, 17 pages, Every college dorm displays at least one print by M.C. Escher. Many people admire Escher's work, but do not know its mathematical roots. Escher's interest in divisions of planes goes back to his early work, but the mathematical in uence in his work did not fully appear until he journeyed through the Mediterranean around 1936. Particularly, when visiting La Alhambra in Spain, he became fascinated with the order and symmetry of the tiling. He then studied mathematical papers on topics such as symmetry groups, non-Euclidean geometries, and impossible shapes, later incorporating them into his artwork. When one looks at an Escher print, e.g. Angels and Demons, one immediately sees the complicated tiling within the circle. However, what is not necessarily realized is that the work is a representation of a hyperbolic geometric space. Though inspired by the at tiling of the Alhambra, Escher strayed away from Euclidean geometry in many of his works, creating tiling in spherical and hyperbolic geometries. These three types of geometries make up the world we live in. On a local scale, we live on a at surface, i.e. in Euclidean geometry, but calculating distances on Earth's surface requires spherical geometry. On a larger scale, the universe acts under the laws of hyperbolic geometry, the same as in Escher's Angels and Demons. In a similar manner, much of Escher's work is the visualization of key mathematical concepts from Complex Analysis and Group Theory., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T20:16:27Z No. of bitstreams: 1 SCasey From Complex Analysis and Group Theory to Geometry and Art 2013-3.pdf: 3938436 bytes, checksum: 6f5d77d457aa6232da59571fcc467cf7 (MD5), Made available in DSpace on 2014-04-03T20:16:27Z (GMT). No. of bitstreams: 1 SCasey From Complex Analysis and Group Theory to Geometry and Art 2013-3.pdf: 3938436 bytes, checksum: 6f5d77d457aa6232da59571fcc467cf7 (MD5)

- Technical Report No. 2013-4, 45 pages, The goal of this thesis is to exposit the key results of the subject, with an eye towards presenting two open problems related to the structure of the groups of automorphisms of a model of arithmetic. Section 2 will present preliminary results and definitions, discussing the axioms of Peano Arithmetic (PA), and deriving key theorems in the model theory of arithmetic. Section 3 will discuss recursive saturation, which is the key to building automorphisms of models of arithmetic. Essentially, recursive saturation is all about building models “just rich enough” to have many elements that are free to move under an automorphism. Finally, Section 4 will cover results at the cutting edge1 of the subject, and present two open problems., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T20:32:03Z No. of bitstreams: 1 SCasey MCasselMAThesis 2013-4.pdf: 433324 bytes, checksum: c88b2a2da3a098625e74dcc2546423d4 (MD5), Made available in DSpace on 2014-04-03T20:32:03Z (GMT). No. of bitstreams: 1 SCasey MCasselMAThesis 2013-4.pdf: 433324 bytes, checksum: c88b2a2da3a098625e74dcc2546423d4 (MD5)

- Technical Report No. 2013-5, 43 pages, The Uniformization Theorem from the study of Riemann surfaces gives a fundamental understanding of the geometry of our world. The theorem tells us that there are three different types of geometries for orientable surfaces: Euclidean (flat), spherical, and hyperbolic. We live in all three at once; it all depends on scale. We travel through our communities as if we were on a flat surface, since the scale is small enough not to notice the curvature of the earth. At this level, we experience Euclidean (or “flat”) geometry. If we increase the scale further, as we do when we fly planes or track satellites, we experience spherical geometry. Furthermore, due to Einstein’s theory of relativity, space itself exhibits a negative curvature, which means at the largest scales we experience hyperbolic geometry. On these larger scales, Euclidean geometry does not accurately measure angles and lengths. While Euclidean geometry is easily visualized by students, spherical and hyperbolic geometries prove to be more challenging. This paper provides new ways to visualize these geometries. Felix Klein and colleagues created a research program that studied geometry in a new way, known as the Erlangen Program. The Erlangen Program used projective geometry as the unifying frame of all other geometries and group theory to abstract and organize our geometric knowledge. The groups studied are the groups of functions called isometries – bijective maps that preserve length. For example, in Euclidean geometry the isometries are rotations, reflections, and translations. Of course, spherical and hyperbolic functions isometries are different from the ones we are familiar with in flat geometry. Consequently, distances and angles are not the same. My project addresses this, with computer programs and graphics that represent spherical and hyperbolic worlds. I have created visual tools for understanding the different geometries. My goal was to give any interested person, from an elementary school student to an academic, the ability to understand Euclidean, spherical, and hyperbolic worlds., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T20:38:45Z No. of bitstreams: 1 SCasey RLaRochelleMAPaper 2013-5.pdf: 331205 bytes, checksum: 54a005dbaa7bad5d44bbd45fea2dec2c (MD5), Made available in DSpace on 2014-04-03T20:38:45Z (GMT). No. of bitstreams: 1 SCasey RLaRochelleMAPaper 2013-5.pdf: 331205 bytes, checksum: 54a005dbaa7bad5d44bbd45fea2dec2c (MD5)

- Technical Report No. 2013-6, 8 pages, When one represents a physical system of harmonic oscillators, it is possible to represent the system in many ways. For a system with three degrees of freedom, one could represent a single particle in three dimensions which would be equivalent to three particles in one dimension. The main idea here is that in each case, the total degrees of freedom must be equal for different representations of a given system. There are many other ways that someone can represent a physical system in three dimensions. The system can be represented in Cartesian coordinates, cylindrical coordinates, or spherical coordinates, just to name a few. Then within each of these representations, one could imagine rotating the coordinate systems or scaling them differently. As a result, one specific state of the multidimensional quantum harmonic oscillator can be represented in many different ways. The purpose of this research project is to calculate different coefficients for translating from one representation to another., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T20:44:47Z No. of bitstreams: 1 Travis_Volz_QM_Research_Results 2013-6.pdf: 4598959 bytes, checksum: b028826168cef80197243be3be3ef417 (MD5), Made available in DSpace on 2014-04-03T20:44:47Z (GMT). No. of bitstreams: 1 Travis_Volz_QM_Research_Results 2013-6.pdf: 4598959 bytes, checksum: b028826168cef80197243be3be3ef417 (MD5)

- Technical Report No. 2014-5, 23 pages, Because the traditional method for evaluating integrals over higher dimensional simplices can be computationally challenging, Lasserre and Avrachenkov established an equation for evaluating integrals of symmetric multilinear forms over simplices. Before an integral can be evaluated in this manner the starting homogeneous polynomial must be expressed as a symmetric multilinear form, by way of a polarization identity. In this paper, the Lasserre-Avrachenkov method for evaluating integrals over simplices is explained and explored, beginning with a homogenous polynomial and a simplex, and ending with an exact value. This method can be used in computer programs that provide an efficient method for precisely evaluating integrals over simplices in higher dimensions.

- Technical Report No. 2014-1, 34 pages, The goal of this project is to develop and validate image processing algorithms for measuring wind direction over the ocean. We plan to use wind truth data from (1) oceanographic buoys and other anchored sensors and (2) wind measurements from other satellite sensors to validate our SAR-derived wind direction estimates. Both of these sources of truth data are of necessarily lower resolution than what is available from TerraSAR-X. It is worth noting that buoy validation is already routinely done against the ASCAT sensor [14]. Our collection campaign centers on several buoys described in the NOAA’s database [20]. We are using the resulting concurrent data stream for validation of spectral shape, but have not completely validated the recovery of wind direction from the image., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T20:50:34Z No. of bitstreams: 1 MRobinson 201311_DLR_Report_small 2014-1.pdf: 3038831 bytes, checksum: 5b705ea86a4451385a8d3c0f714e03c9 (MD5), Made available in DSpace on 2014-04-03T20:50:34Z (GMT). No. of bitstreams: 1 MRobinson 201311_DLR_Report_small 2014-1.pdf: 3038831 bytes, checksum: 5b705ea86a4451385a8d3c0f714e03c9 (MD5)

- Technical Report No. 2014-2, 37 pages, The primary objective of this project was to construct, classify, and exploit invariants for discriminating information systems that are based on abstracted structural descriptions. This main objective was split into three smaller objectives: (1) Construct invariants for information systems that exploit coarse and multiscale structural specifications about their underlying network or communication topology, (2) Classify the semantic and dynamic features of the systems that these in- variants consider, and (3) Exploit the classification results to provide actionable design and analysis rules that can be incorporated into experimental and simulation workflows. All of these objectives were met. Several interesting (and potentially important) discoveries were made as a result of the project. These discoveries have been reported to the scientifi c community, and they are being written as articles for archival journals. In addition, the Principal Investigator (PI), Prof. Michael Robinson, completed a draft of a manuscript entitled Topological Signal Processing that can be used to teach the techniques discovered on this project to beginning graduate student researchers. Finally, Prof. Robinson's research group grew from one student at the start of the project to ve students (partially funded by this project), partially as a means to apply the new algorithmic techniques discovered on this program., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-03T20:58:33Z No. of bitstreams: 1 MRobinson_2012_Robinson_AFOSR_FinalReport 2014-2.pdf: 628436 bytes, checksum: de338421ead40d7f065ce73c16cb94eb (MD5), Made available in DSpace on 2014-04-03T20:58:33Z (GMT). No. of bitstreams: 1 MRobinson_2012_Robinson_AFOSR_FinalReport 2014-2.pdf: 628436 bytes, checksum: de338421ead40d7f065ce73c16cb94eb (MD5)

- The main result of this thesis deals with continuous functions on metric spaces. Specifically, we show that given a continuous function f from a metric space (M1; d1) to a metric space (M2; d2), there is a function such that is continuous. The above result was established by Enayat in [Ena00] using the machinery of partitions of unity. Our expository account of Enayat’s paper contains a substantial body of results in general topology, including a thorough discussion of paracompact spaces and partitions of unity., Submitted by Michele Mazzocchetti (mazzocch@american.edu) on 2014-04-14T14:30:33Z No. of bitstreams: 1 Moskey MA-thesis.pdf: 417413 bytes, checksum: 64e32f38f8a1cc7f49610a4a0b1c7057 (MD5), Made available in DSpace on 2014-04-14T14:30:33Z (GMT). No. of bitstreams: 1 Moskey MA-thesis.pdf: 417413 bytes, checksum: 64e32f38f8a1cc7f49610a4a0b1c7057 (MD5)

- Technical Report No. 2015-2, 6 pages., This report shows that pseudosections are not theoretically any more or less powerful than sections. Indeed, the pseudosections of any consistency structure can be recovered as sections of a different sheaf over an enriched base space. It remains to be seen if pseudosections offer greater economy of thought or are easier to compute as pseudosections than as sections.

- Technical Report No. 2015-3, 8 pages., Given a connected reductive group G̃ over a finite field k, and a semisimple k-automorphism ε of G̃ of finite order, let G denote the connected part of the group of ε-fixed points. Then two of the authors have previously shown that there exists a natural lifting from series of representations of G(k) to series for G̃ (k). In the case of Deligne-Lusztig representations, we show that this lifting satisfies a character relation analogous to that of Shintani.

- Technical Report No. 2015-3, 17 pages., This project enables target classification from unprocessed synthetic aperture sonar (SAS) collections in various clutter contexts by providing principled, foundational analysis of target echo structure through the lens of topological signal processing. It intends to provide systematic validation of target/environment simulations by detecting hidden symmetries in data. Spurious symmetries in simulated data can pose a risk to classification algorithms trained on that simulated data, because they may overfit according to those symmetries. On the other hand, experimental data may contain unleveraged symmetries that are not adequately captured by simulation. By combining both sets of symmetries, this project will enable the extraction of actionable, physics-aware features for classification from simulated and experimental data.