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arXiv 2020年

作者： Ovchinnikov, Alexey Pillay, Anand Pogudin, Gleb Scanlon, Thomas CUNY Queens College Department of Mathematics 65-30 Kissena Blvd QueensNY11367 United States CUNY Graduate Center Ph.D. Programs and Mathematics and Computer Science 365 Fifth Avenue New YorkNY10016 United States University of Notre Dame Department of Mathematics Notre DameIN46556 United States LIX CNRS École Polytechnique Institute Polytechnique de Paris Palaiseau France University of California Berkeley Department of Mathematics BerkeleyCA94720-3840 United States

Structural identifiability is a property of an OdE model with parameters that allows for the parameters to be determined from continuous noise-free data. This is a natural prerequisite for practical identifiability. Conducting multiple independent experiments could make more parameters or functions of parameters identifiable, which is a desirable property to have. How many experiments are sufficient? In the present paper, we provide an algorithm to determine the exact number of experiments for multi-experiment local identifiability and obtain an upper bound that is off at most by one for the number of experiments for multi-experiment global identifiability. Interestingly, the main theoretical ingredient of the algorithm has been discovered and proved using model theory (in the sense of mathematical logic). We hope that this unexpected connection will stimulate interactions between applied algebra and model theory, and we provide a short introduction to model theory in the context of parameter identifiability. As another related application of model theory in this area, we construct a nonlinear OdE system with one output such that single-experiment and multiple-experiment identifiability are different for the system. This contrasts with recent results about single-output linear systems. We also present a Monte Carlo randomized version of the algorithm with a polynomial arithmetic complexity. Implementation of the algorithm is provided and its performance is demonstrated on several examples. The source code is available at https://***/pogudingleb/ExperimentsBound. Copyright © 2020, The Authors. All rights reserved.

关键词： Ordinary differential equations

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Iterative Refinement and Oversampling for Low Rank Approximation

arXiv

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arXiv 2019年

作者： Pan, Victor Y. Luan, Qi Go, Soo Ph.D. Programs in Computer Science and Mathematics The Graduate Center of the City University of New York New YorkNY10036 United States Department of Computer Science Lehman College of the City University of New York BronxNY10468 United States

Iterative refinement is particularly popular for numerical solution of linear systems of equations. We extend it to Low Rank Approximation of a matrix (LRA) and observe close link of the resulting algorithm to oversampling techniques, commonly used in randomized LRA algorithms. We elaborate upon this link and revisit oversampling and some efficient randomized LRA algorithms. Applied with sparse sketch matrices they run significantly faster and in particular yield Very Low Rank Approximation (VLRA) at sublinear cost, using much fewer scalars and flops than the input matrix has entries. This is achieved at the price of deterioration of output accuracy, but according to our formal and empirical study subsequent oversampling improves accuracy to near-optimal level under the spectral norm for a large sub-class of matrices with fast decaying spectra of singular values. Copyright © 2019, The Authors. All rights reserved.

关键词： Linear systems

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CUR Low Rank Approximation of a Matrix at Sublinear Cost

arXiv

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arXiv 2019年

作者： Pan, Victor Y. Luan, Qi Svadlenka, John Zhao, Liang Department of Computer Science Lehman College of the City University of New York BronxNY10468 United States Ph.D. Programs in Computer Science and Mathematics The Graduate Center of the City University of New York New YorkNY10036 United States

Low rank approximation of a matrix (hereafter LRA) is a highly important area of Numerical Linear and Multilinear Algebra and data Mining and Analysis. One can operate with LRA at sublinear cost, that is, by using much fewer memory cells and flops than an input matrix has entries,1 but no sublinear cost algorithm can compute accurate LRA of the worst case input matrices or even of the matrices of small families in our Appendix A. Nevertheless we prove that Cross-Approximation (C-A) celebrated iterations and even more primitive sublinear cost algorithms output quite accurate LRA for a large subclass of the class of all matrices that admit sufficiently close LRA and in a sense for most of such matrices. Moreover, we accentuate the power of sublinear cost LRA by means of multiplicative pre-processing of an input matrix, and also compare C-A iterations with AdI-based variant of Randomized and Sketching LRA algorithms. Our tests are in good accordance with our formal *** Codes *** (Numerical Analysis) Copyright © 2019, The Authors. All rights reserved.

关键词： Matrix algebra

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AdI/ACA-like Superfast Iterative Refinement of Low Rank Approximation of a Matrix

arXiv

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arXiv 2019年

作者： Luan, Qi Pan, Victor Y. Ph.D. Programs in Computer Science and Mathematics The Graduate Center of the City University of New York New YorkNY10036 United States Department of Computer Science Lehman College of the City University of New York BronxNY10468 United States

We call a matrix algorithm superfast (aka running at sublinear cost) if it involves much fewer flops and memory cells than an input matrix has entries. Computations with Big data frequently involve matrices of immense size and can be reduced to computations with low rank approximations (LRA) of these matrices, which can be performed superfast. Unfortunately, such LRA algorithms fail for worst case input matrices admitting their LRAs, but Adaptive Cross-Approximation (ACA) method, which can be viewed as LRA version of Alternative directions Implicit (AdI) method, is highly efficient empirically for LRA of a large and important class of input matrices. Adequate formal support for that empirical behavior, however, is still a challenge. This motivated us to propose a novel AdI/ACA-like superfast randomized iterative refinement of a crude LRA. Like ACA iterations we output CUR LRA, which is a particularly memory efficient form of LRA. Unlike AdI/ACA method we use sampling probability to reduce our task to recursive solution of generalized Linear Least Squares Problem (generalized LLSP) and to prove monotone convergence to a close LRA with a high probability (whp) under mild assumptions on an initial LRA. For crude initial LRA lying reasonably far from optimal we have consistently observed significant improvement in two or three iterations in our numerical tests . Copyright © 2019, The Authors. All rights reserved.

关键词： Matrix algebra

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Counting Roots of a Polynomial in a Convex Compact Region by Means of Winding Number Calculation via Sampling

arXiv

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arXiv 2019年

作者： Zaderman, Vitaly Zhao, Liang Ph.D. Program in Mathematics Graduate Center of the City University of New York New YorkNY10016 United States Department of Computer Science

In this paper, we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and exclusion tests in a subdivision algorithms for polynomial root-finding, and would be especially useful in application scenarios where high-precision polynomial coefficients are hard to obtain but we succeed with counting already by using polynomial evaluation with lower precision. We provide the pseudo code of the algorithm, proof of its correctness as well as estimation of its complexity. Copyright © 2019, The Authors. All rights reserved.

关键词： Winding

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A TOPOLOGICAL APPROACH TO UNdEFINABILITY IN ALGEBRAIC EXTENSIONS OF Q

arXiv

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arXiv 2020年

作者： Eisenträger, Kirsten Miller, Russell Springer, Caleb Westrick, Linda Department of Mathematics The Pennsylvania State University University ParkPA16802 United States Department of Mathematics Queens College – City University of New York 65-30 Kissena Blvd. FlushingNY11367 United States Ph.D. Programs in Mathematics and Computer Science Graduate Center - City University of New York 365 Fifth Avenue New YorkNY10016 United States Department of Mathematics University College London LondonWC1H 0AY United Kingdom The Heilbronn Institute for Mathematical Research Bristol United Kingdom

For any subset Z ⊆ Q, consider the set SZ of subfields L ⊆ Q which contain a co-infinite subset C ⊆ L that is universally definable in L such that C ∩ Q = Z. Placing a natural topology on the set Sub(Q) of subfields of Q, we show that if Z is not thin in Q, then SZ is meager in Sub(Q). Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers OL is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every ∃-definable subset of an algebraic extension of Q is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible *** Codes 11U05, 12L05, 11U09, 03d45 © 2020, CC BY.

关键词： Topology

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FORCING AS A COMPUTATIONAL PROCESS

arXiv

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arXiv 2020年

作者： Hamkins, Joel david Miller, Russell Williams, Kameryn J. University of Notre Dame 100 Malloy Hall Notre DameIN46556 United States University of Oxford United Kingdom The Graduate Center of CUNY Ph.D. Programs in Mathematics & Computer Science 365 Fifth Avenue New YorkNY10016 United States Queens College of CUNY Mathematics Dept. 65-30 Kissena Blvd. FlushingNY11367 United States Bard College at Simon's Rock 84 Alford Rd Great BarringtonMA01230 United States

We investigate how set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for information about a model of set theory hM, ∈M i, we explain senses in which one may compute M -generic filters G ⊆ P ∈ M and the corresponding forcing extensions M [G]. Specifically, from the atomic diagram of M one may compute a generic filter G, from the ∆0-diagram of M one may compute a presentation of the corresponding forcing extension M [G] and its ∆0-diagram, and from the elementary diagram of M one may compute the elementary diagram of M [G]. We also examine the information necessary to make the process functorial, and conclude that in the general case, no such computational process will be functorial-for any candidate process there will be different isomorphic presentations of a model of set theory M that lead to non-isomorphic forcing extensions M [G]. Indeed, there is no Borel function providing generic filters that is functorial in this sense. Copyright © 2020, The Authors. All rights reserved.

关键词： Set theory

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The Reverse construction of complete (k, n)- arcs in PG(2,q) where q=2,4,8 related with linear codes

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Journal of physics: Conference Series 2020年第1期1591卷

作者： Aidan Essa Mustafa Sulaimaan Nada Yassen Kasm Yahya Ph.D. Student Department of Mathematics College of Computer Sciences and Mathematics University of Mosul Mosul-Iraq Department of Mathematics College of Education for Pure Science University of Mosul Mosul-Iraq

The aim of this work is to study The reverse construction of complete (k, n)- arcs in PG(2,q) where q=2,4,8 is related to linear codes, and n = q, q-1, ..., 2. And n = q, q-1, ... By removing points from the complete ...

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The Reverse construction of complete (k, n)- arcs in three-dimensional projective space PG(3,4)

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Journal of physics: Conference Series 2020年第1期1591卷

In this work, the complete (k, n) arcs in PG(3,4) over Galois field GF(4) can be created by removing some points from the complete arcs of degree m, where m = n + 1, 3 n q2 + q is used. In addition, where k ≤ 85, we geometrically prove that the minimum complete (k, n)–arc in PG(3,4) is (5,3)-arc. A(k, n)–arcs is a set of k points no n+1 of which collinear. A(k, n)–arcs is complete unless it is embedded in an arc (k+l,n).

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Input-output equations and identifiability of linear OdE models

arXiv

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arXiv 2019年

作者： Ovchinnikov, Alexey Pogudin, Gleb Thompson, Peter CUNY Queens College Department of Mathematics 65-30 Kissena Blvd QueensNY11367 United States CUNY Graduate Center Ph.D. Programs in Mathematics and Computer Science 365 Fifth Avenue New YorkNY10016 United States Courant Institute of Mathematical Sciences New York University New YorkNY10012 United States Higher School of Economics Faculty of Computer Science Moscow109028 Russia LIX CNRS École Polytechnique Institute Polytechnique de Paris 1 rue Honoré d'Estienne d'Orves Palaiseau91120 France CUNY Graduate Center Ph.D. Program in Mathematics 365 Fifth Avenue New YorkNY10016 United States

Structural identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. The method of input-output equations is one method for verifying structural identifiability. This method stands out in its importance because the additional insights it provides can be used to analyze and improve models. However, its complete theoretical grounds and applicability are still to be established. A subtlety and key for this method to work correctly is knowing whether the coefficients of these equations are identifiable. In this paper, to address this, we prove identifiability of the coefficients of input-output equations for types of differential models that often appear in practice, such as linear models with one output and linear compartment models in which, from each compartment, one can reach either a leak or an input. This shows that checking identifiability via input-output equations for these models is legitimate and, as we prove, that the field of identifiable functions is generated by the coefficients of the input-output equations. Finally, we exploit a connection between input-output equations and the transfer function matrix to show that, for a linear compartment model with an input and strongly connected graph, the field of all identifiable functions is generated by the coefficients of the transfer function matrix even if the initial conditions are *** Codes 12H05, 34A55, 92B05, 93C15, 93B25, 93B30 Copyright © 2019, The Authors. All rights reserved.

关键词： Transfer functions

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