Marek R Rychlik

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• Rychlik, M., Tanriover, B., & Han, Y. (2023). Large-scale data extraction from the UNOS organ donor documents. arxiv.
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  In this paper we focus on three major task: 1) discussing our methods: Our method captures a portion of the data in DCD flowsheets, kidney perfusion data, and Flowsheet data captured peri-organ recovery surgery. 2) demonstrating the result: We built a comprehensive, analyzable database from 2022 OPTN data. This dataset is by far larger than any previously available even in this preliminary phase; and 3) proving that our methods can be extended to all the past OPTN data and future data. The scope of our study is all Organ Procurement and Transplantation Network (OPTN) data of the USA organ donors since 2008. The data was not analyzable in a large scale in the past because it was captured in PDF documents known as ``Attachments'', whereby every donor's information was recorded into dozens of PDF documents in heterogeneous formats. To make the data analyzable, one needs to convert the content inside these PDFs to an analyzable data format, such as a standard SQL database. In this paper we will focus on 2022 OPTN data, which consists of $\approx 400,000$ PDF documents spanning millions of pages. The entire OPTN data covers 15 years (2008--20022). This paper assumes that readers are familiar with the content of the OPTN data. [Journal_ref: ]
• , M. M., & , M. R. (2022). Beyond RAID 6 --- an Efficient Systematic Code Protecting Against Multiple Errors, Erasures, and Silent Data Corruption.
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  We describe a replacement for RAID 6, based on a new linear, systematic code,which detects and corrects any combination of $E$ errors (unknown location) and$Z$ erasures (known location) provided that $Z+2E \leq 4$. We investigate somescenarios for error correction beyond the code's minimum distance, using listdecoding. We describe a decoding algorithm with quasi-logarithmic timecomplexity, when parallel processing is used: $\approx O(\log N)$ where $N$ isthe number of disks in the array (similar to RAID 6). By comparison, the error correcting code implemented by RAID 6 allows errordetection and correction only when $(E,Z)=(1,0)$, $(0,1)$, or $(0,2)$. Hence,when in degraded mode (i.e., when $Z \geq 1$), RAID 6 loses its ability fordetecting and correcting random errors (i.e., $E=0$), leading to data lossknown as silent data corruption. In contrast, the proposed code does notexperience silent data corruption unless $Z \geq 3$. The aforementioned properties of our code, the relative simplicity ofimplementation, vastly improved data protection, and low computationalcomplexity of the decoding algorithm, make our code a natural successor to RAID6. As this code is based on the use of quintuple parity, this justifies thename PentaRAID for the RAID technology implementing the ideas of the currentpaper.[Journal_ref: ]
• , M. R., , D. N., , Y. H., & , D. M. (2022). Development of a New Image-to-text Conversion System for Pashto, Farsi and Traditional Chinese.
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  We report upon the results of a research and prototype building project\emph{Worldly~OCR} dedicated to developing new, more accurate image-to-textconversion software for several languages and writing systems. These includethe cursive scripts Farsi and Pashto, and Latin cursive scripts. We alsodescribe approaches geared towards Traditional Chinese, which is non-cursive,but features an extremely large character set of 65,000 characters. Ourmethodology is based on Machine Learning, especially Deep Learning, and DataScience, and is directed towards vast quantities of original documents,exceeding a billion pages. The target audience of this paper is a generalaudience with interest in Digital Humanities or in retrieval of accuratefull-text and metadata from digital images.[Journal_ref: ]
• Rychlik, M. R. (2022). A proof of convergence of multi-class logistic regression network.
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  This paper revisits the special type of a neural network known under twonames. In the statistics and machine learning community it is known as amulti-class logistic regression neural network. In the neural networkcommunity, it is simply the soft-max layer. The importance is underscored byits role in deep learning: as the last layer, whose autput is actually theclassification of the input patterns, such as images. Our exposition focuses onmathematically rigorous derivation of the key equation expressing the gradient.The fringe benefit of our approach is a fully vectorized expression, which is abasis of an efficient implementation. The second result of this paper is thepositivity of the second derivative of the cross-entropy loss function asfunction of the weights. This result proves that optimization methods based onconvexity may be used to train this network. As a corollary, we demonstratethat no $L^2$-regularizer is needed to guarantee convergence of gradientdescent.[Journal_ref: ]
• Rychlik, M. R. (2022). Convergence Rates for Multi-classs Logistic Regression Near Minimum. ArXiv.
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  In the current paper we provide constructive estimation of the convergence rate for training a known class of neural networks: the multi-class logistic regression. Despite several decades of successful use, our rigorous results appear new, reflective of the gap between practice and theory of machine learning. Training a neural network is typically done via variations of the gradient descent method. If a minimum of the loss function exists and gradient descent is used as the training method, we provide an expression that relates learning rate to the rate of convergence to the minimum. The method involves an estimate of the condition number of the Hessian of the loss function. We also discuss the existence of a minimum, as it is not automatic that a minimum exists. One method of ensuring convergence is by assigning positive probabiity to every class in the training dataset. [Journal_ref: ]
• Rychlik, M. R. (2021). Development of a Gold-standard Pashto Dataset and a Segmentation App. Information Technology and Libraries, 40(1). doi:https://doi.org/10.6017/ital.v40i1.12553
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  The article aims to introduce a gold-standard Pashto dataset and a segmentation app. The Pashto dataset consists of 300 line images and corresponding Pashto text from three selected books. A line image is simply an image consisting of one text line from a scanned page. To our knowledge, this is one of the first open access datasets which directly maps line images to their corresponding text in the Pashto language. We also introduce the development of a segmentation app using textbox expanding algorithms, a different approach to OCR segmentation. The authors discuss the steps to build a Pashto dataset and develop our unique approach to segmentation. The article starts with the nature of the Pashto alphabet and its unique diacritics which require special considerations for segmentation. Needs for datasets and a few available Pashto datasets are reviewed. Criteria of selection of data sources are discussed and three books were selected by our language specialist from the Afghan Digital Repository. The authors review previous segmentation methods and introduce a new approach to segmentation for Pashto content. The segmentation app and results are discussed to show readers how to adjust variables for different books. Our unique segmentation approach uses an expanding textbox method which performs very well given the nature of the Pashto scripts. The app can also be used for Persian and other languages using the Arabic writing system. The dataset can be used for OCR training, OCR testing, and machine learning applications related to content in Pashto.
• Rychlik, M. R. (2021). Development of a gold-standard pashto dataset and a segmentation app. Information Technology and Libraries.
• Rychlik, M. R., & Nwaigwe, D. (2021). Convergence Rates for Multi-classs Logistic Regression Near Minimum. ArXiv.
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  In the current paper we provide constructive estimation of the convergence rate for traininga known class of neural networks: the multi-class logistic regression. Despite several decades of successful use, our rigorous results appear new, reflective of the gap between practice and theory of machine learning. Training a neural network is typically done via variations of the gradient descent method. If a minimum of the loss function exists and gradient descent is used as the training method, we provide an expression that relates learning rate to the rate of convergence to the minimum. The method involves an estimate of the condition number of the Hessian of the loss function. We also discuss the existence of a minimum, as it is not automatic that a minimum exists. One method of ensuring convergence is by assigning positive probability to every class in the training dataset.
• Rychlik, M. R. (2020). Deductron—A Recurrent Neural Network. Frontiers in Applied Mathematics and Statistics.
• Rychlik, M., & Rychlik, M. (2020). Deductron—A Recurrent Neural Network. Frontiers in Applied Mathematics and Statistics, 6. doi:10.3389/fams.2020.00029
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  The current paper is a study in Recurrent Neural Networks (RNN), motivated by the lack of examples simple enough so that they can be thoroughly understood theoretically, but complex enough to be realistic. We constructed an example of structured data, motivated by problems from image-to-text conversion (OCR), which requires long-term memory to decode. Our data is a simple writing system, encoding characters 'X' and 'O' as their upper halves, which is possible due to symmetry of the two characters. The characters can be connected, as in some languages using cursive, such as Arabic (abjad). The string 'XOOXXO' may be encoded as '${\vee}{\wedge}\kern-1.5pt{\wedge}{\vee}\kern-1.5pt{\vee}{\wedge}$'. It is clear that seeing a sequence fragment '$|\kern-1.8pt{\wedge}\kern-1.5pt{\wedge}\kern-1.5pt{\wedge}\kern-1.5pt{\wedge}\kern-1.5pt{\wedge}\kern-1.8pt|$' of any length does not allow us to decode the sequence as '\ldots XXX\ldots' or '\ldots OOO \ldots' due to inherent ambiguity, thus requiring long-term memory. Subsequently we constructed an RNN capable of decoding sequences like this example. Rather than by training, we constructed our RNN ``by inspection'', i.e. we guessed its weights. This involved a sequence of steps. We wrote a conventional program which decodes the sequences as the example above. Subsequently, we interpreted the program as a neural network (the only example of this kind known to us). Finally, we generalized this neural network to discover a new RNN architecture whose instance is our handcrafted RNN. It turns out to be a three-layer network, where the middle layer is capable of performing simple logical inferences; thus the name ``deductron''. It is demonstrated that it is possible to train our network by simulated annealing. Also, known variants of stochastic gradient descent (SGD) methods are shown to work.
• Rychlik, M. R. (2019). A proof of convergence of multi-class logistic regression network. ArXiv.
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  This paper revisits the special type of a neural network known under twonames. In the statistics and machine learning community it is known as amulti-class logistic regression neural network. In the neural networkcommunity, it is simply the soft-max layer. The importance is underscored byits role in deep learning: as the last layer, whose autput is actually theclassification of the input patterns, such as images. Our exposition focuses onmathematically rigorous derivation of the key equation expressing the gradient.The fringe benefit of our approach is a fully vectorized expression, which is abasis of an efficient implementation. The second result of this paper is thepositivity of the second derivative of the cross-entropy loss function asfunction of the weights. This result proves that optimization methods based onconvexity may be used to train this network. As a corollary, we demonstratethat no $L^2$-regularizer is needed to guarantee convergence of gradientdescent.[Journal_ref: ]
• Rychlik, M. R. (2019). Deductron -- A Recurrent Neural Network. ArXiv.
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  The current paper is a study in Recurrent Neural Networks (RNN), motivated bythe lack of examples simple enough so that they can be thoroughly understoodtheoretically, but complex enough to be realistic. We constructed an example ofstructured data, motivated by problems from image-to-text conversion (OCR),which requires long-term memory to decode. Our data is a simple writing system,encoding characters 'X' and 'O' as their upper halves, which is possible due tosymmetry of the two characters. The characters can be connected, as in somelanguages using cursive, such as Arabic (abjad). The string 'XOOXXO' may beencoded as'${\vee}{\wedge}\kern-1.5pt{\wedge}{\vee}\kern-1.5pt{\vee}{\wedge}$'. Itfollows that we may need to know arbitrarily long past to decode a currentcharacter, thus requiring long-term memory. Subsequently we constructed an RNNcapable of decoding sequences encoded in this manner. Rather than by training,we constructed our RNN "by inspection", i.e. we guessed its weights. Thisinvolved a sequence of steps. We wrote a conventional program which decodes thesequences as the example above. Subsequently, we interpreted the program as aneural network (the only example of this kind known to us). Finally, wegeneralized this neural network to discover a new RNN architecture whoseinstance is our handcrafted RNN. It turns out to be a 3 layer network, wherethe middle layer is capable of performing simple logical inferences; thus thename "deductron". It is demonstrated that it is possible to train our networkby simulated annealing. Also, known variants of stochastic gradient descent(SGD) methods are shown to work.[Journal_ref: ]
• Rychlik, M. R. (2019). On completion of a linearly independent set to a basis with shifts of a fixed vector. ArXiv.
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  Let $\mathbb{F}$ be an infinite field. Let $n$ be a positive integer and let$1\leq d\leq n$. Let $\vec{f}_1, \vec{f}_2, \ldots, \vec{f}_{d-1} \in\mathbb{F}^{n}$ be $d-1$ linearly independent vectors. Let$\vec{x}=(x_1,x_2,\ldots,x_{d},0,0,\ldots,0)\in\mathbb{F}^{n}$, with $n-d$zeros at the end. Let $\vec{R}: \mathbb{F}^n \to\mathbb{F}^n$ be the cyclicshift operator to the right, e.g. $\vec{R}\,\vec{x} =(0,x_1,x_2,\ldots,x_{d},0,0,\ldots,0)$. Is there a vector $\vec{x} \in\mathbb{F}^n$, such that the $n-d+1$ vectors $\vec{x},\vec{R}\vec{x}, \ldots,\vec{R}^{n-d}\vec{x}$ complete the set $\{\vec{f}_j\}_{j=1}^{d-1}$ to a basisof $\mathbb{F}^n$? The answer is in the affirmative for every linearlyindependent set of $\vec{f}_j$, $j=1,2,\ldots,d-1$. In order to prove thisfact, we prove that the $(n-d+1)\times(n-d+1)$ minors of the$(n-d+1)\times(n-d+1)$ circulant matrix. $\begin{bmatrix} \vec{x}, \vec{R}\vec{x}, \ldots, \vec{R}^{n-d} \vec{x} \end{bmatrix}^\intercal$ form aGr\"obner basis with respect to the graded reverse lexicographic order(grevlex).[Journal_ref: ]
• Rychlik, M. R. (2017). On the Reliability of RAID Systems: An Argument for More Check Drives. ArXiv.
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  In this paper we address issues of reliability of RAID systems. We focus on "big data" systems with a large number of drives and advanced error correction schemes beyond \RAID{6}. Our RAID paradigm is based on Reed-Solomon codes, and thus we assume that the RAID consists of $N$ data drives and $M$ check drives. The RAID fails only if the combined number of failed drives and sector errors exceeds $M$, a property of Reed-Solomon codes. We review a number of models considered in the literature and build upon them to construct models usable for a large number of data and check drives. We attempt to account for a significant number of factors that affect RAID reliability, such as drive replacement or lack thereof, mistakes during service such as replacing the wrong drive, delayed repair, and the finite duration of RAID reconstruction. We evaluate the impact of sector failures that do not result in drive replacement. The reader who needs to consider large $M$ and $N$ will find applicable mathematical techniques concisely summarized here, and should be able to apply them to similar problems. Most methods are based on the theory of continuous time Markov chains, but we move beyond this framework when we consider the fixed time to rebuild broken hard drives, which we model using systems of delay and partial differential equations. One universal statement is applicable across various models: increasing the number of check drives in all cases increases the reliability of the system, and is vastly superior to other approaches of ensuring reliability such as mirroring. [Journal_ref: ]
• Rychlik, M. (2012). Why is Helfenstein's claim about equichordal points false?. New York Journal of Mathematics, 18, 499-521.
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  Abstract: This article explains why a paper by Heinz G. Helfenstein entitled Ovals with equichordal points, J. London Math. Soc. 31 (1956), 54-57, is incorrect. We point out a computational error which renders his conclusions invalid. More importantly, we explain that the method presented there cannot be used to solve the equichordal point problem. Today, there is a solution to the problem: Marek R. Rychlik, A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck, Inventiones Mathematicae 129 (1997), 141-212. However, some mathematicians still point to Helfenstein's paper as a plausible path to a simpler solution. We show that Helfenstein's method cannot be salvaged. The fact that Helfenstein's argument is not correct was known to Wirsing, but he did not explicitly point out the error. This article points out the error and the reasons for the failure of Helfenstein's approach in an accessible, and hopefully enjoyable way.
• Rychlik, M. R. (2012). Why is Helfenstein's claim about equichordal points false?. New York Journal of Mathematics, Volume, 499-521.
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  This article explains why a paper by Heinz G. Helfenstein entitled "Ovalswith equichordal points", published in J.London Math.Soc.31, 54-57, 1956, isincorrect. We point out a computational error which renders his conclusionsinvalid. More importantly, we explain that the method cannot be used to solvethe equichordal point problem with the method presented there. Today, there isa solution to the problem: Marek R. Rychlik, "A complete solution to theequichordal point problem of Fujiwara, Blaschke, Rothe and Weizenb\"ock",Inventiones Mathematicae 129 (1), 141-212, 1997. However, some mathematiciansstill point to Helfenstein's paper as a plausible path to a simpler solution.We show that Helfenstein's method cannot be salvaged. The fact thatHelfenstein's argument is not correct was known to Wirsing, but he did notexplicitly point out the error. This article points out the error and thereasons for the failure of Helfenstein's approach in an accessible, andhopefully enjoyable way.[Journal_ref: New York Journal of Mathematics, Volume 18 (2012), pp. 499-521]
• Rychlik, M. R. (2012). Why is Helfenstein's claim about equichordal points false?. New York Journal of Mathematics.
• Rychlik, M. R. (2003). On the attractivity of a class of homogeneous dynamic economic systems. Nonlinear Analysis, Theory, Methods and Applications.
• Weiye, L. i., Rychlik, M., Szidarovszky, F., & Chiarella, C. (2003). On the attractivity of a class of homogeneous dynamic economic systems. Nonlinear Analysis, Theory, Methods and Applications, 52(6), 1617-1636.
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  Abstract: The attractivity properties of the set of equilibria of a special class of homogeneous dynamic economic systems are examined. The nonlinearity of the models and the presence of eigenvalues with zero real parts make the application of the classical theory impossible. Some principles of the modern theory of dynamical systems and invariant manifolds are applied, and the local attractivity of the set of equilibria is verified under mild conditions. As an application, special labor-managed oligopolies are investigated. © 2003 Elsevier Science Ltd. All rights reserved.
• Rychlik, M. R. (1998). Algebraic non-integrability of the Cohen map. New York Journal of Mathematics.
• Rychlik, M., & Torgerson, M. (1998). Algebraic non-integrability of the Cohen map. New York Journal of Mathematics, 4, 57-74.
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  Abstract: The map φ(x, y) = (√1 + x2 - y, x) of the plane is area preserving and has the remarkable property that in numerical studies it shows exact integrability: The plane is a union of smooth, disjoint, invariant curves of the map φ. However, the integral has not explicitly been known. In the current paper we will show that the map φ does not have an algebraic integral, i.e., there is no non-constant function F(x, y) such that 1. F ○ φ = F; 2. There exists a polynomial G(x, y, z) of three variables with G(x, y, F(x, y)) = 0. Thus, the integral of φ, if it does exist, will have complicated singularities. We also argue that if there is an analytic integral F, then there would be a dense set of its level curves which are algebraic, and an uncountable and dense set of its level curves which are not algebraic.
• Rychlik, M. R. (1997). A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck. Inventiones Mathematicae, 129(1), 141-212.
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  Abstract: The Equichordal Point Problem can be formulated in simple geometric terms. If C is a Jordan curve on the plane and P, Q ∈ C then the segment PQ is called a chord of the curve C. A point inside the curve is called equichordal if every two chords through this point have the same length. The question was whether there exists a curve with two distinct equichordal points O1 and O2. The problem was posed by Fujiwara in 1916 and independently by Blaschke, Rothe and Weizenböck in 1917, and since then it has been attacked by many mathematicians. In the current paper we prove that if O1 and O2 are two distinct points on the plane and C is a Jordan curve such that the bounded region D cut out by C is star-shaped with respect to both O1 and O2 then C is not equichordal. The original question was posed for convex C, and thus we have solved the Equichordal Point Problem completely. Our method is based on the observation that C would be an invariant curve for an algebraic map of the plane. It would also form a heteroclinic connection. We complexify the map and obtain a multivalued algebraic map of ℂ2. We develop criteria for the existence of heteroclinic connections for such maps.
• Rychlik, M. R. (1997). A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck. Inventiones Mathematicae.
• Rychlik, M. (1996). The Equichordal Point Problem. Electronic Research Announcements of The American Mathematical Society, 2(3), 108-123. doi:10.1090/s1079-6762-96-00015-7
• Rychlik, M. (1992). Renormalization of cocycles and linear ODE with almost-periodic coefficients. Inventiones Mathematicae, 110(1), 173-206.
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  Abstract: In the current paper we address the problem of classification of cocycles over an irrational rotation. We use the renormalization group approach. A cocycle means a Cr-mapping u:T →SU (2, ℂ) (r≧2). We fix an irrational rotation τ:T →T. Two cocycles u, v:T →SU(2, ℂ) are considered equivalent (or cohomologous) if there is a continuous map h:T→SU (2, ℂ) such that u·h{o script} τ=h·v. By definition, our problem is to classify cocycles up to this equivalence relation. By introducing a suitable renormalization map we are able to define a notion of a fiber rotation number for a class of cocycles which are in the basin of the attractor of the renormalization map. The attractor itself is built of algebraic Anosov maps on T2. We present a number of results and conjecuters resulting from this approach. We show how this approach sheds some light upon the problem of classifying linear ODE with almost-periodic, skew-hermitian coefficient matrix. © 1992 Springer-Verlag.
• Rychlik, M. R. (1992). Regularity and other properties of absolutely continuous invariant measures for the quadratic family. Communications in Mathematical Physics.
• Rychlik, M. R. (1992). Renormalization of cocycles and linear ODE with almost-periodic coefficients. Inventiones Mathematicae.
• Rychlik, M., & Sorets, E. (1992). Regularity and other properties of absolutely continuous invariant measures for the quadratic family. Communications in Mathematical Physics, 150(2), 217-236.
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  Abstract: In the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of Jakobson's Theorem. In particular, we show that the density of the invariant measure is continuous at Misiurewicz points. From this we deduce that the Lyapunov exponent is also continuous at these points (our considerations apply just to the parameters constructed in the proof of Jakobson's Theorem). Other properties, like the positivity of the Lyapunov exponent, uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system, are also proved. © 1992 Springer-Verlag.
• Rychlik, M. R. (1990). Lorenz attractors through Šil'nikov-type bifurcation. Part I†. Ergodic Theory and Dynamical Systems, 10(4), 793-821. doi:10.1017/s0143385700005915
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  The main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an Ω-explosion. The unperturbed vector field on ℝ3 is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities λ1 > 0, λ2 > 0, λ3 > 0 and |λ2|>|λ1|>|λ3|. Moreover, the unstable manifold of the fixed point is supposed to form a double loop. Under some other natural assumptions a generic two-parameter family containing the unperturbed vector field contains geometric Lorenz attractors.A possible application of this result is a method of proving the existence of geometric Lorenz attractors in concrete families of differential equations. A detailed discussion of the method is in preparation and will be published as Part II.
• Rychlik, M. R. (1990). Lorenz attractors through Šill'nikov-type bifurcation. Part I. Ergodic Theory and Dynamical Systems.
• Rychlik, M. (1989). Regularity of the metric entropy for expanding maps. Transactions of the American Mathematical Society, 315(2), 833-847. doi:10.1090/s0002-9947-1989-0958899-6
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  The main result of the current paper is an estimate of the radius of the nonperipheral part of the spectrum of the Perron-Frobenius operator for expanding mappings. As a consequence, we are able to show that the metric entropy of an expanding map has modulus of continuity x log( 1 /x) on the space of C2-expandings. We also give an explicit estimate of the rate of mixing for Cl-functions in terms of natural constants. It seems that the method we present can be generalized to other classes of dynamical systems, which have a distinguished invariant measure, like Axiom A diffeomorphisms. It also can be adopted to show that the entropy of the quadratic family f,(x) = 1 ,ux2 computed with respect to the absolutely continuous invariant measure found in Jakobson's Theorem varies continuously (the last result is going to appear somewhere else).
• Rychlik, M. R. (1989). Periodic points of the billiard ball map in a convex domain. Journal of Differential Geometry.
• Rychlik, M. R. (1989). Regularity of the metric entropy for expanding maps. Transactions of the American Mathematical Society.
• Rychlik, M. R. (1988). Another proof of Jakobson's Theorem and related results. Ergodic Theory and Dynamical Systems, 8(1), 93-109. doi:10.1017/s014338570000434x
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  The author shows that any family C 2-close to f a (x) = 1 − ax 2 (2 − e ≤ a ≤ 2) satisfies Jakobson’s theorem: For a positive measure set of a the transformation f a has an absolutely continuous invariant measure. He also indicates some generalizations.
• Rychlik, M. R. (1988). Another proof of Jakobson's Theorem and related results. Ergodic Theory and Dynamical Systems.
• Rychlik, M. R. (1988). The Wiener lemma and cocycles. Proceedings of the American Mathematical Society, 104(3), 932-933. doi:10.1090/s0002-9939-1988-0964876-6
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  We give a sufficient and necessary condition for a function with its values in the unit circle to be a multiplicative coboundary. This theorem generalizes the following result of Veech [1]. Let T: T _ T be a rotation of the unit circle T by an irrational angle 0. Let F: T T be a measurable function. Then F is a multiplicative coboundary iff F (x)F(Tx)---F(T'-1x))dp(x) 1, as IlnIf ?0, T where I InOI is the distance of nO from integers and u is the Haar measure. Let T: (X, E,u) -(X, E,u) be a measure preserving automorphism which is ergodic. Let F: X -+ T be a measurable function. We call F a coboundary mod A if there is a function g: X -T and A E T such that F(x) = Ag(x)/g(Tx) a.e. For every integer n > 0, let Fn(x) = F(x)F(Tx) ... F(Tn-lx). For n 0 such that the set {n E Z: IFn (x) f (Tnx) f (x) d (x) > 6} has positive upper density. PROOF. This theorem is an easy corollary of Wiener's Theorem. Let U: L2(X,I ,E) -, LI(XIEl be a unitary operator defined as follows: Uf (x) = F(x)f (Tx). Then the condition of being a coboundary mod A is equivalent to the existence of g E L2(X, , ,u) such that Ug = Ag. In fact, if such g exists then Ig o TI = lgl and by ergodicity the modulus of g is constant a.e., so we can assume that it is equal to 1. Received by the editors November 2, 1987. 1980 Mathematics Subject Classification (1985 Remssion). Primary 58F11.
• Rychlik, M. R. (1988). The wiener lemma and cocycles. Proceedings of the American Mathematical Society.
• Rychlik, M. (1983). Bounded variation and invariant measures. Studia Mathematica, 76(1), 69-80. doi:10.4064/sm-76-1-69-80

Presentations

• Rychlik, M. R. (2023, 01-18-2023).

  Teaching machines to read, and how it will save lives

  . Math 586B Presentation. University of Arizona: Program of Applied Mathematics.
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  Abstract: Teaching machines to read, and how it will save livesOne of the oldest challenges of artificial intelligence is to have machine read documents created for humans, also known as OCR (Optical Character Recognition). It has taken 60 years to solve easy problems, such as accurately transcribing high quality printed text, but new problems crop up in various applications, and the field presents a rich collection of challenges for algorithmic mathematics, machine learning and a variety of related areas.  Currently, I and several math graduate students are engaged in joint research with the College of Medicine that aims at automated reading of medical forms used in organ transplantation, which has the potential to save thousands of lives per year and affect medical policies nationwide. In this talk I will give an overview of this project and the kind of research questions that it brings about.
• Rychlik, M. R. (2023, 01-25-2023). Two technologies in the areas of computer storage and retrieval of data from legacy systems. Delivered Talk to IARPA (Intelligence Advanced Research Projects Activity) Directors in the area Artificial Intelligence. University of Arizona: IARPA (Intelligence Advanced Research Projects Activity).
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  Title: Two technologies in the areas of computer storage and retrieval of data from legacy systems Abstract: In this talk I will describe two outcomes of long-term research on Big Data and AI with potential IC significance. 1) Computer storage, due to rapid growth and cost, is not sustainable. We propose to address it with a new error-correcting technology, based on 5+ redundant blocks. A recent invention at the University provides this technology. The 5-block version received 2 US utility patents and the prototype has been tested. The technology is similar to the dominant RAID 6 in its relative simplicity, but it extends mean time to data loss from an estimated 1 day to one hundred quadrillion years in a realistic application. This property can be leveraged in various ways. For example, substituting RAID 6 algorithm with ours would reduce power consumption by an estimated 50% in today’s data centers. The technology is compatible with DNA-based storage. 2) Legacy information systems use documents, such as scanned images and PDF, in lieu of databases. One such system is the national system for organ transplantation (managed by UNOS). With 90,000 patients awaiting a kidney transplant, only 25,000 transplants per year are performed. There is a disturbing trend: 28% of available kidneys are discarded without finding a compatible recipient, and the discard rate tripled since 1988. By comparison, some countries have discard rates close to 5%. Research to-date demonstrates that conversion to machine readable format is possible, although the content is very challenging by state-of-the-art methods, due to the tabular layout of medical forms and records, perspective distortion due to the use of cell phone cameras, poor lighting, handwritten digits and doctor notes, etc. We propose to satisfactorily address the problem of transplant data, and use the approaches developed on other massive document collections of national importance.
• Rychlik, M. R. (2023, 02-20-2023). Basic parallel programming with MATLAB examples (Part 1). Research Training Group in Data Driven Discovery Seminar (RTG) at UA Math DeptResearch Training Group in Data Driven Discovery Seminar (RTG) at UA Math Dept.
• Rychlik, M. R. (2023, 02-27-2023). Basic parallel programming with MATLAB examples (Part 2). Research Training Group in Data Driven Discovery Seminar (RTG)Research Training Group in Data Driven Discovery Seminar (UA Math Dept).
• Rychlik, M. R. (2023, 03-14-23). Decoding Quintuple Parity in Constant Time, Extending MTTDL to 100 Quadrillion Years. Storage Technology Showcase Conference STS 2023 Engaging the Rising Generation of Storage ProfessionalsStorage Technology Showcase Conference STS.
• Rychlik, M. R. (2023, 07-18-23). Transforming the ailing US organ transplant system and transitioning to AI-driven medical record keeping of the future. ARPA-H Open BAA Virtual WorkshopAdvanced Research Projects Agency for Health (ARPA-H).
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  The US organ transplant system serves millions of Americans whose lives or quality of life are in danger due to organ failure; 100,000 patients are on the waiting list for a kidney transplant; 25,000 transplants are performed per year. The potential beneficiaries of a kidney transplant are the 750,000 Americans who undergo hemodialysis due to kidney failure, usually performed 3 days a week, with each session lasting around 4 hours. A kidney transplant dramatically improves the quality of life of these patients and is often a life-saving treatment. The cost of hemodialysis is $36 billion/year, $100k per patient. Kidney transplant costs $450k, but it lasts ~20 years, resulting in 80% savings per patient.There is an organ shortage with only 1 patient out of 4 receiving a transplant. Yet, about 30% of recovered kidneys are discarded in the US (8,000 kidneys). This alarming statistic is a reflection of the many inadequacies of current legacy systems. The discard ratio was close to 7% in 1988 and it has dramatically increased in the past decades. Furthermore, in many European countries, the discard ratio is close to 5%. The need to overhaul the US organ transplant system is well understood by Congress and the medical community. The results of an extensive study of the system are the subject of public reports. One such report is: https://bloomworks.digital/organdonationreform/Summary/ which cites the number of wasted organs at 28,000 per year.Our project started in 2022 as a collaborative effort between the University of Arizona College of Medicine and Department of Mathematics, with the following goals:•Make the totality of the donor data captured in the UNOS system since 2008 machine readable and analyzable.Currently the data is in the form of unstructured PDF documents (“attachments”), which are known to be very challenging as a source of analyzable data. In addition, many valuable records are images, including images captured with a cell phone camera, with significant perspective distortion. By using and developing cutting edge tools from AI, (Statistical) Machine Learning, Machine Vision and Natural Language Processing we are able to perform the daunting task of converting the large document base (130,000 deceased donors, 20-30 PDF documents per donor) into a database accessible to statistical analysis.•Develop a state-of-the-art organ allocation model.The currently used model developed 30 years ago was based on 10 variables, and considered very poor (C statistic = 0.59). Yet, the model is used as KDPI (Kidney Profile Donor Index) as the main criterion for matching donors with recipients. We will use the developed donor database, combined with the existing recipient database, to build a neural network model that will correctly rank the best matching recipients.•Propose policy and clinical practice changes, supported by compelling evidence based on the obtained data, to be implemented nationwide.
• Rychlik, M. R. (2023, 08-07-2023). Delivered lecture  regarding a professor's expectations of a graduate student. UA Applied Math Integration Workshop Summer 2023UA Applied Math.
• Rychlik, M. R. (2023, 2023). Delivered multiple lectures throughout 2023 (Spring, Summer, Fall) at the weekly Multi-lingual OCR Seminar which I formed.. Multi-lingual OCR SeminarUA Math Dept.
• Tanriover, B. (2023, 01-25-2023). Automation of medical form processing in the context of organ transplantation. Environmental and Occupational Health Seminar at Zuckerman College of Public Health. Zuckerman College of Public Health at the University of Arizona: Zuckerman College of Public Health.
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  Abstract: About 90,000 patients are waiting for kidney transplant in the US, but only 25,000 kidney transplants are performed yearly due to shortage of available organs. Despite the shortage, the percentage of discarded available organs is increasing and has reached 28%, tripling since 1988. We propose to reduce the discard rate by automation of reading and interpreting of the extensive medical records used in deceased donor kidney allocation system. The first part of the project entails developing algorithms and software that will build a database out of 2.5 TB of documents covering all deceased donors in the US between 2015 and 2022. To date, small subsets of the data have been extracted manually, of the order of 10,000 records. Our effort aims at creating a comprehensive data set covering millions of records. The availability of this dataset will affect both research and policy making related to organ transplantation. The major reason for discard has been avoidance by transplant centers of risks associated with accepting less-than-perfect organs. Supplying good data in a prompt fashion will allow a more optimal decision for the patient and appears as the only workable path to reducing the discard rate.
• Tanriover, B. (2023, 05-13-2023). Automation of medical form processing in the context of organ transplantation. Northeastern University Computer Science and Mathematics Graduate SeminarNortheastern University Computer Science and Mathematics Department.
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  Youtube video of talk: https://www.youtube.com/watch?v=sYboqp3u3wcAbstract: About 90,000 patients are waiting for kidney transplant in the US, but only 25,000 kidney transplants are performed yearly due to shortage of available organs. Despite the shortage, the percentage of discarded available organs is increasing and has reached 28%, tripling since 1988. We propose to reduce the discard rate by automation of reading and interpreting of the extensive medical records used in deceased donor kidney allocation system. The first part of the project entails developing algorithms and software that will build a database out of 2.5 TB of documents covering all deceased donors in the US between 2015 and 2022. To date, small subsets of the data have been extracted manually, of the order of 10,000 records. Our effort aims at creating a comprehensive data set covering millions of records. The availability of this dataset will affect both research and policy making related to organ transplantation. The major reason for discard has been avoidance by transplant centers of risks associated with accepting less-than-perfect organs. Supplying good data in a prompt fashion will allow a more optimal decision for the patient and appears as the only workable path to reducing the discard rate.
• Rychlik, M. R. (2021, September). A Quintuple Parity Error Correcting Code - A Game Changer in Data Protection. Storage Developer Conference 2021. Zoom: Storage Networking Industry Association.
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  We describe a replacement for RAID 6, based on a new linear, systematic code, which detects and corrects any combination of E errors (unknown location) and Z erasures (known location) provided that Z+2E≤4. The code is at the core of a RAID technology called PentaRAID, for which the two co-inventors were awarded a US utility patent. The problem that we address is that of weak data protection of RAID 6 systems. The known vulnerability is that RAID 6 may recover from not more than 2 failed disks in a RAID, if we know which disks failed. If we do not know which disks are the source of errors, we are protected from only 1 disk failure. Moreover, if 1 disk fails, the failed data needs to be recovered, and with hard disks reaching 40TB, the recovery process lasts for weeks (degraded mode). While in degraded mode, the second disk failure results in a system with no error detection and correction. In addition, Undetected Disk Errors (UDE) can only be detected but not corrected event with one failed disk. The natural solution is to increase redundancy from 2 to more disks. There is very little payoff from using 3 disks. It turns out that a practical solution is possible with 5 redundant disks, and this solution is employed in PentaRAID. The payoff is immense, as the RAID extends Mean Time to Data Loss (MTDL) from days to far beyond the age of the universe (100 quadrillion years) under typical assumptions in regard to disk error rates. The new RAID can tolerate a loss of 2 drives at unknown locations (thus seemingly operating normally but generating UDE), and up to 4 disks at known locations, e.g. due to power failure (typically detected by the disk controller). In addition, the recovery process involves a fixed, small number of Galois field operations per error, and therefore has virtually fixed computational cost per error, independent of the number of disks in the array. Parity calculation has also constant time per byte of data, if distributed computation is utilized. In short, the computational complexity is on the par with that of RAID 6. Notably, the solution does not require Chien search commonly used in Reed-Solomon coding, which makes it possible to utilize large Galois fields. The application of the new technology will dramatically increase data durability with significant reduction of the number of hard disks necessary to maintain data integrity, and will simplify the logistics of operating RAID in degraded mode and recovery from disk failures.Learning ObjectivesGiving the audience an overview of the new error correcting code and the decoding algorithm aimed as a replacement for RAID 6.Explaining how quantitatively the data protection and durability will be impacted by the new technology.Explaining the technology from the point of view of an IT department.Explaining the technology implementation from the point of view of an electrical or software engineer.
• Rychlik, M. R. (2021, Spring). MATLAB implementation of CTC with Logarithmic Scaling of Probabilities. Multi-ingual OCR Seminar. University of Arizona: University of Arizona.
• Rychlik, M. R. (2021, Spring). Sequence-to-sequence mapping problem and CTC. Seminar given to Applied Math 586b Students. Department of Mathematics: University of Arizona.
• Rychlik, M. R. (2021, Summer). LineBreaker App for Page Segmentation. Multi-ingual OCR Seminar. University of Arizona: University of Arizona.
• Rychlik, M. R. (2021, Summer). Semantic segmentation, medical imaging experiment and potential use in OCR. Multi-ingual OCR Seminar. University of Arizona: University of Arizona.
• Rychlik, M. R. (2020, Fall). Harnessing New Technologies for Learning and Research in the Languages and Cultures of the Middle East. 2020 Middle Eastern Studies Association (MESA) Conference. Zoom: MESA.
• Rychlik, M. R. (2020, January). Image-to-text Conversion as a Playground for Modern AI. TRIPODS Conference, Guanajuato, January 5-8, 2020. Guanajuato, Mexico: UA-TRIPODS and CIMAT.
• Rychlik, M. R. (2019, Fall). CTC - Connectionist Temporal Classification. Multi-ingual OCR Seminar. University of Arizona: University of Arizona.
• Rychlik, M. R. (2019, Fall). Latin and Chinese character outlines as means of extracting features. Multi-ingual OCR Seminar. University of Arizona: University of Arizona.
• Rychlik, M. R. (2019, Fall). Modern OCR and large-scale text retrieval from images. TRIPODS Seminar. University of Arizona: TRIPODS Seminar.
• Rychlik, M. R. (2019, Fall). Multi-page clustering of Traditional Chinese text. Multi-ingual OCR Seminar. University of Arizona: University of Arizona.
• Rychlik, M. R. (2019, October 21). TRIPODS --- Modern OCR and large-scale text retrieval from images. TRIPODS Seminar. University of Arizona: TRIPODS Program.
• Rychlik, M. R. (2019, Spring). Character recognition with multi-class logistic regression. Multi-ingual OCR Seminar. University of Arizona: University of Arizona.
• Rychlik, M. R. (2019, Spring). From Perceptron to Deductron - A tour of Neural Nets and Optical Character Recognition. RTG seminar given to Applied Math Students. Department of Mathematics: University of Arizona.
• Rychlik, M. R. (2019, Spring). Mathematics in the era of Big Data - Error correcting codes and RAID, Optical Character Recognition, Dealing with Policy Data. RTG seminar given to (Pure) Math Students. Department of Mathematics: University of Arizona.
• Rychlik, M. R. (2018, April). An introduction to RNN and Deductron. RWG3 TRIPODS Working Group Meeting. CS Department: TRIPODS.
• Rychlik, M. R. (2018, December). Lightening Talk on Language and Math. Retreat organized by Language, Science and Innovation Collaborative. Hacienda del Sol: University of Arizona.
• Rychlik, M. R. (2018, May). Recurrent Neural Networks for Optical Character Recognition. TRIPODS Southwest Summer Conference. Biosphere 2: NSF.
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  The research presented in the talk is related to the problem ofimage-to-text conversion, also known as optical character recognition(OCR). A special difficulty is posed by languages which use cursive(connected characters), such as Arabic and cursive English.In recent years the most accurate systems for OCR of scripts use arecurrent neural network (RNN) called Long-Short TERM Memory (LSTM).The focus of the research is on providing a simple, mathematicallyrigorous example of data which provably requires long-term memory. Theexample is a writing system using two characters, obtained bytaking the upper halves of the Latin characters 'X' and 'O'. Thecharacters can be connected. Thus a text that looks like 'w' wouldencode the string 'XX', while the 'upside down w' would encode'OO'. The texts in our language are sequences of "waves", thus we callour language the W-language. Subsequently, we construct an RNN forperforming OCR on texts written in the W-language. It is interestingthat the weights of the neural network can be written down byinspection, thus providing an exact solution to the problem ofimage-to-text conversion for the W-language, the only known to usexample of this type. Finally, we generalize the constructed RNN todefine a new 3-layer RNN architecture, which is a general purpose RNN,like LSTM. We call the instances of the architecture deductrons, astheir middle layer indeed performs logic deductions. We demonstratedthat the deductron network can be trained to decode the W-language,using simulated annealing. We will discuss a number of open problemsrelated to deductrons.

Others

• Rychlik, M. R. (2023, 03-15-2023). Invited Panelist on Discussion of Storage Technologies at Storage Technology Showcase Conference STS 2023 Engaging the Rising Generation of Storage Professionals. Storage Technology Showcase Conference STS 2023 Engaging the Rising Generation of Storage Professionals.
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  Invited Panelist on Discussion of Storage Technologies at Storage Technology Showcase Conference STS 2023 Engaging the Rising Generation of Storage Professionals
• Rychlik, M. R. (2023, 05-01-2023). Invited Panelist in discussion of the potential role that generative AI (ChatGPT, etc.) may play in the mathematical sciences for the Research Training Group in Data Driven Discovery (RTG) Seminar at UA Mathematics.. Research Training Group in Data Driven Discovery (RTG) Seminar at UA Mathematics.
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  Invited Panelist in discussion of the potential role that generative AI (ChatGPT, etc.) may play in the mathematical sciences for the Research Training Group in Data Driven Discovery (RTG) Seminar at UA Mathematics.
• Rychlik, M. R. (2020). Convergence rates for multi-classs logistic regression near minimum. arXiv. http://www.scopus.com/inward/record.url?eid=2-s2.0-85101809006&partnerID=MN8TOARS
• Rychlik, M. R. (2020). Development of a new image-to-text conversion system for Pashto, farsi and traditional Chinese. arXiv. http://www.scopus.com/inward/record.url?eid=2-s2.0-85094771149&partnerID=MN8TOARS
• Rychlik, M. R. (2019). A proof of convergence of multi-Class logistic regression network. arXiv. http://www.scopus.com/inward/record.url?eid=2-s2.0-85093139950&partnerID=MN8TOARS
• Rychlik, M. R. (2019). On completion of a linearly independent set to a basis with shifts of a fixed vector. arXiv. http://www.scopus.com/inward/record.url?eid=2-s2.0-85094371146&partnerID=MN8TOARS
• Rychlik, M. R. (2018). Beyond raid 6 - An efficient systematic code protecting against multiple errors, erasures, and silent data corruption. arXiv. http://www.scopus.com/inward/record.url?eid=2-s2.0-85095051481&partnerID=MN8TOARS
• Rychlik, M. R. (2018). Deductron — a recurrent neural network. arXiv. http://www.scopus.com/inward/record.url?eid=2-s2.0-85093172342&partnerID=MN8TOARS