## Technical Reports

This page serves as a publicly available repository for pre-prints written by faculty and/or students in the Mathematics Department. Readers should be aware that papers appearing here may undergo significant modification before appearing in a professional journal. Readers should also keep in mind that some papers appearing here may not appear in a professional journal. For information about published versions of technical reports appearing on this page, please contact the NEIU-affiliated faculty author directly.

**"Marked CS Movies," Prof Matthew Graham (Northeastern Illinois University). **We present a marked analogue of Carter and Saito’s movie theorem. Our definition of marking was chosen to coincide with the markings that arise in link Floer homology. In order to deal with complications arising from certain isotopies, we define three equivalences for marked surfaces and work over an equivalence class of marked surfaces when proving our marked CS movie theorem. Read the complete report.

**"Temperature-dependent Transport and Thermal-diffusion Effects on Diffusion Flames," Prof Joseph Hibdon (Northeastern Illinois University) and Moshe Matalon (UIUC). **In this paper we examine the effects of temperature-dependent transport and thermal diffusion on the structure and characteristics of a diffusion flame. The configuration adopted is the planar unstrained flame with a bulk flow directed toward the reaction zone from either the fuel or the oxidizer sides. Included in this discussion is the noflow case, where the reactants reach the reaction zone purely by diffusion. The model allows for non-unity and distinct Lewis numbers, for the fuel and oxidizer. Results show that the variations of the thermal conductivity and the diffusion coefficients with temperature affect the flame standoff distance and flame temperature and the profiles of temperature and concentration, in accord with experimental data. The predicted extinctions conditions are exhibited by a critical Damkohler number ¨ Dc below which the flame extinguishes. This Dc is significantly smaller for the temperature dependent case when compared with previous analysis without the temperature dependence. Thermal diffusion, also known as the Soret effect, also affects the flame standoff distance by shifting it towards the fuel/oxidizer and affects the flame temperature by making it smaller/larger for heavy/light fuels respectively. Predicted extinctions Dc are minimally affected by the Soret effects, except when having very heavy fuels. The amount of leakage across the reaction sheet that causes extinction is more/less for light/heavy fuels, respectively. Read the complete report.

**"Mathematical Modeling of Non-Small Cell Lung Cancer Response to Therapy," Russell Injerd, Prof. Emma Turian (Northeastern Illinois University)** Non-small cell lung cancer (NSCLC) represents the largest proportion of lung cancers in the United States. Image guided radiotherapy allows tumor volume dynamics to be measured at certain intervals during treatment. This has improved our ability to study the evolution of tumors such as NSCLC during treatment using time series approach models. The main goal of our research study is to identify the model that best describes the existing radio-therapeutic treatment options: Stereotactic body radiation therapy (SBRT), also known as stereotactic ablative radiotherapy (SABR), and standard therapy (ST). Our mathematical structure builds on the linear quadratic model from the radiation oncological field and therefore, introduces parameters related to tumor's radio-sensitivity. Previous such one and two population ODE models of tumor volume dynamics, treating NSCLC, were designed using exponential and logistic growth functions. These studies indicate that a two population exponential model provided the best balance between fit and mathematical complexity and may serve a functional role in clinical practice. Our study reevaluates previous findings for treating NSCLC using both, the standard and SABR regimens, and tests the suitability of the hyper-Gompertz, hyper-logistic, Richards, Von-Bertalanffy, and a non-linear model derived using fluid mechanics laws by assessing their goodness of fit versus their mathematical complexity. These models are calibrated using data from eleven patients treated using SABR regimen, and four patients treated using standard therapy, extracted from a previous study. Models pertaining both treatment regimens are evaluated using statistical approaches, such as the Akaike Information Criterion. Model comparison indicates that the models fitting patient data perform differently based on the treatment regimen. Our study suggests that for the SABR patients the non-linear model derived from fluid mechanics laws overall outperforms the rest of the studied models, and in the case of the standard treatment the logistic model seems to better represent patient data. Our hope is that our findings will benefit research regarding NSCLC, as well as other cancer field types. Read the complete report. (Updated April 12, 2018)

**"Local Isometric Immersions of Pseudo-spherical Surfaces and k-th Order Evolution Equations," by Prof. Nabil Kahouadji (Northeastern Illinois University), Niky Kamran (McGill University), Keti Tenenblat (Universidade de Brasilia) **We consider the class of evolution equations that describe pseudo-spherical surfaces of a form classified by Chern-Tenenblat. This class of equations is characterized by the property that to each solution of a differential equation within this class, there corresponds a 2-dimensional Riemannian metric of curvature -1. We investigate the following problem: given such a metric, is there a local isometric immersion in real 3-dimensional space such that the coefficients of the second fundamental form of the surface depend on a jet of finite order of u? By extending our previous result for the second-order evolution equation to kth order equations, we prove that there is only one type of equation that admits such an isometric immersion. We prove that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions u are universal, i.e., they are independent of u. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion. Keywords: evolution equations; pseudo-spherical surfaces; isometric immersions. MSC 2010: 35L60, 37K25, 47J35, 53B10, 53B25

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**"Dade's Ordinary Conjecture for the Finite Special Unitary Groups: Part III," by Prof. Katherine Bird (Northeastern Illinois University) **Let G be a finite group. An ordinary character of G is the character of a representation of G over a field of characteristic 0. In the p-modular representation theory of G,where p is a prime dividing the order of G, the ordinary irreducible characters of G are divided into disjoint sets called p-blocks which reflect the decomposition of the group algebra of G over a field of characteristic p into indecomposable two-sided ideals. An important problem is to classify the p-blocks, and a first step is to count the number of ordinary characters in a block. The aim of Dade’s Ordinary Conjecture (DOC) is to prove an alternating sum, which counts the number of characters in B in terms of corresponding numbers in subgroups of G which are normalizers of chains of certain p-subgroups of G. This has been shown for p-blocks, p dividing q, for GLn(q) , SLn(q) and Un(q). Moreover, we have proved DOC for SUn(q). The main difficulties involved arose because the structure of the unitary groups is more complicated than that of the linear groups. In particular, the cancellations in the alternating sum in the unitary case are very different from the cancellations that occur in the general linear case. A key result utilized is that a version of the parametrization of characters used by Ku for Un(q) survives restriction to SUn(q). This report is devoted to presenting an example which aims to elucidate cancellation that occurs in the previously described sub-sums. Read the complete report.

**"Dade's Ordinary Conjecture for the Finite Special Unitary Groups: Part II," by Prof. Katherine Bird (Northeastern Illinois University) **This report is a continuation of the proof of Dade's Ordinary Conjecture (DOC) for the nite special unitary groups. Several reductions of the main alternating sum were completed in Part I [1] resulting in an important reformulation of the main theorem. The alternating sum in this theorem was immediately decomposed into two sub alternating sums. The aim of this paper is to prove the second of these sub alternating sums, completing the proof of DOC for the finite special unitary groups in the defining characteristics. Read the complete report.

**"Dade's Ordinary Conjecture for the Finite Special Unitary Groups: Part I," by Prof. Katherine Bird (Northeastern Illinois University) **This report is devoted firstly to some background and context for DOC for the finite special unitary groups. Then several reductions of the main alternating sum are completed, resulting in an important reformulation of the main theorem. The alternating sum in this theorem is then immediately decomposed into two sub-alternating sums. The final aim of this paper is to prove the rest of these sub-alternating sums, the so-called Levi sum. Read the complete report.

**"Some Solutions for Baseball Manager's Problems: Choosing a Set of Starters...," by Prof. Marina Polyashuk (Northeastern Illinois University)** If you, by chance, decided to use the key-word “baseball” in your search of a scientific database, you would be buried under thousands upon thousands of titles. These titles cover all possible aspects of this exhilarating game, from the absorption of moisture by baseball jerseys to the trajectories of fly balls, to the comparative importance of pitching and hitting in winning a baseball game. And then, of course, there is sabermetrics (derived from SABR, Society for American Baseball Research), which became known to the general audience through the movie “Moneyball”. Most mathematicians, particularly statisticians, are aware that baseball is an ultimate breeding ground for statistical studies and analyses. In short, baseball is not just “America’s favorite pastime”, but also an inspiration for numerous research and business ideas. These ideas allow many people to have fun not only by watching the game, but by analyzing, exploring, and predicting its outcomes. Read the complete report.

### Technical Reports from Department Visitors

**"Precious metals as alternative investments," by Prof. Izabela Pruchnicka-Grabias (Warsaw School of Economics) **The paper is devoted to one of the kinds of alternative investments. Precious metals futures and their risk-return profile in comparison to such stock index futures as Dow Jones and S&P are analyzed. Usually alternative investments are thought to be long-term ones. The author checks if they may be also helpful in a shorter period of time. The following research question is verified: do investments in precious metals let us realize more attractive risk return profiles than in stock indexes in the medium period of time (two years)? The study was conducted with data from January 2015 to December 2016 and daily rates of return were applied. Read the complete report.

**"Lower Partial Moments and Maximum Drawdown Measures in Hedge Fund Risk - Return Profile Analysis," by Prof. Izabela Pruchnicka-Grabias (Warsaw School of Economics) **This paper shows hedge fund rankings made both with traditional effectiveness measures and alternative ones using lower partial moments and maximum drawdown. Its aim is to show whether there are any differences between hedge fund efficiency when different measures are applied. The effectiveness is understood as the relation between rate of return and risk. The author used data from the Hedge Fund Research and the analyzed period is from January 2005 to April 2011. The results show that the traditional risk measure, the Sharpe Ratio, leads to conclusions on hedge fund effectiveness similar to mentioned alternative risk measures. The strong side of the paper is that the research takes into consideration the period of 2008–2009 when the American mortgage crisis appeared and the majority of hedge funds realized substantial losses along with the dramatic decrease of their assets under management. Read the complete report.