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Counting Prime Polynomials and Measuring Complexity and Similarity of Information
Counting Prime Polynomials and Measuring Complexity and Similarity of Information by Niko Rebenich B.Eng., University of Victoria, 2007 M.A.Sc., University of Victoria, 2012 A Dissertation Submitted in Partial Fulllment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Electrical and Computer Engineering © Niko Rebenich, 2016 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Counting Prime Polynomials and Measuring Complexity and Similarity of Information by Niko Rebenich B.Eng., University of Victoria, 2007 M.A.Sc., University of Victoria, 2012 Supervisory Committee Dr. Stephen Neville, Co-supervisor (Department of Electrical and Computer Engineering) Dr. T. Aaron Gulliver, Co-supervisor (Department of Electrical and Computer Engineering) Dr. Venkatesh Srinivasan, Outside Member (Department of Computer Science) iii Supervisory Committee Dr. Stephen Neville, Co-supervisor (Department of Electrical and Computer Engineering) Dr. T. Aaron Gulliver, Co-supervisor (Department of Electrical and Computer Engineering) Dr. Venkatesh Srinivasan, Outside Member (Department of Computer Science) ABSTRACT This dissertation explores an analogue of the prime number theorem for polynomi- als over nite elds as well as its connection to the necklace factorization algorithm T-transform and the string complexity measure T-complexity. Specically, a precise asymptotic expansion for the prime polynomial counting function is derived. The approximation given is more accurate than previous results in the literature while requiring very little computational eort. In this context asymptotic series expan- sions for Lerch transcendent, Eulerian polynomials, truncated polylogarithm, and polylogarithms of negative integer order are also provided. -
2 Homework Solutions 18.335 " Fall 2004
2 Homework Solutions 18.335 - Fall 2004 2.1 Count the number of ‡oating point operations required to compute the QR decomposition of an m-by-n matrix using (a) Householder re‡ectors (b) Givens rotations. 2 (a) See Trefethen p. 74-75. Answer: 2mn2 n3 ‡ops. 3 (b) Following the same procedure as in part (a) we get the same ‘volume’, 1 1 namely mn2 n3: The only di¤erence we have here comes from the 2 6 number of ‡opsrequired for calculating the Givens matrix. This operation requires 6 ‡ops (instead of 4 for the Householder re‡ectors) and hence in total we need 3mn2 n3 ‡ops. 2.2 Trefethen 5.4 Let the SVD of A = UV : Denote with vi the columns of V , ui the columns of U and i the singular values of A: We want to …nd x = (x1; x2) and such that: 0 A x x 1 = 1 A 0 x2 x2 This gives Ax2 = x1 and Ax1 = x2: Multiplying the 1st equation with A 2 and substitution of the 2nd equation gives AAx2 = x2: From this we may conclude that x2 is a left singular vector of A: The same can be done to see that x1 is a right singular vector of A: From this the 2m eigenvectors are found to be: 1 v x = i ; i = 1:::m p2 ui corresponding to the eigenvalues = i. Therefore we get the eigenvalue decomposition: 1 0 A 1 VV 0 1 VV = A 0 p2 U U 0 p2 U U 3 2.3 If A = R + uv, where R is upper triangular matrix and u and v are (column) vectors, describe an algorithm to compute the QR decomposition of A in (n2) time. -
Volume of Solids Worksheet Key
Volume Of Solids Worksheet Key chloridizingManky Mika her sometimes quadrireme embarrass smothers any while sequela Ezechiel incubated asphyxiated how. Stockier some friendlies Alonzo spools shillyshally. permanently. Thirtieth and trilingual Esme Find the area of the this distance learning and use appropriate scale factor from the of volume of water is not impossible to calculate the diameter Nothing like volume and pennsylvania department of key geometry information to solid figures show a solid worksheet, a square is ___________ and its endpoints on each pyramid. Students will use of a cube is a segment that children further their volume of solids worksheet key illuminations resources is called regular because they might be sure that all in your idea of. Kindergarten math worksheets like your choices could easily calculate the worksheet contains volume of key download or corner of water in the figure here. One of key which comes as vertical height for volume of solids worksheet key guide database some dimensions. Even calculus formulas of volume solids key types pertain to two decimal places if the. Volume of key concepts the worksheet by application to the following figures worksheets, the top basic safety procedures to it is found by calculation. Cross sections you must also called the nearest tenth of solids with every single content and the surface area of spheres the container without electricity is defined as they get our calculation. How to include the prism have the volume multiple choice digital task cards, which glass can compute the surface area is called the. Bring in solids, volume are parallelograms how do numbers and worksheets? How much time do we have two ways of revolution and gerard kwaitkowski, surface area of a geometric form a half spheres, and a building codes of. -
Rigid Analytic Curves and Their Jacobians
Rigid analytic curves and their Jacobians Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult¨at f¨urMathematik und Wirtschaftswissenschaften der Universit¨atUlm vorgelegt von Sophie Schmieg aus Ebersberg Ulm 2013 Erstgutachter: Prof. Dr. Werner Lutkebohmert¨ Zweitgutachter: Prof. Dr. Stefan Wewers Amtierender Dekan: Prof. Dr. Dieter Rautenbach Tag der Promotion: 19. Juni 2013 Contents Glossary of Notations vii Introduction ix 1. The Jacobian of a curve in the complex case . ix 2. Mumford curves and general rigid analytic curves . ix 3. Outline of the chapters and the results of this work . x 4. Acknowledgements . xi 1. Some background on rigid geometry 1 1.1. Non-Archimedean analysis . 1 1.2. Affinoid varieties . 2 1.3. Admissible coverings and rigid analytic varieties . 3 1.4. The reduction of a rigid analytic variety . 4 1.5. Adic topology and complete rings . 5 1.6. Formal schemes . 9 1.7. Analytification of an algebraic variety . 11 1.8. Proper morphisms . 12 1.9. Etale´ morphisms . 13 1.10. Meromorphic functions . 14 1.11. Examples . 15 2. The structure of a formal analytic curve 17 2.1. Basic definitions . 17 2.2. The formal fiber of a point . 17 2.3. The formal fiber of regular points and double points . 22 2.4. The formal fiber of a general singular point . 23 2.5. Formal blow-ups . 27 2.6. The stable reduction theorem . 29 2.7. Examples . 31 3. Group objects and Jacobians 33 3.1. Some definitions from category theory . 33 3.2. Group objects . 35 3.3. Central extensions of group objects . -
A Fast Algorithm for the Recursive Calculation of Dominant Singular Subspaces N
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 218 (2008) 238–246 www.elsevier.com/locate/cam A fast algorithm for the recursive calculation of dominant singular subspaces N. Mastronardia,1, M. Van Barelb,∗,2, R. Vandebrilb,2 aIstituto per le Applicazioni del Calcolo, CNR, via Amendola122/D, 70126, Bari, Italy bDepartment of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium Received 26 September 2006 Abstract In many engineering applications it is required to compute the dominant subspace of a matrix A of dimension m × n, with m?n. Often the matrix A is produced incrementally, so all the columns are not available simultaneously. This problem arises, e.g., in image processing, where each column of the matrix A represents an image of a given sequence leading to a singular value decomposition-based compression [S. Chandrasekaran, B.S. Manjunath, Y.F. Wang, J. Winkeler, H. Zhang, An eigenspace update algorithm for image analysis, Graphical Models and Image Process. 59 (5) (1997) 321–332]. Furthermore, the so-called proper orthogonal decomposition approximation uses the left dominant subspace of a matrix A where a column consists of a time instance of the solution of an evolution equation, e.g., the flow field from a fluid dynamics simulation. Since these flow fields tend to be very large, only a small number can be stored efficiently during the simulation, and therefore an incremental approach is useful [P. Van Dooren, Gramian based model reduction of large-scale dynamical systems, in: Numerical Analysis 1999, Chapman & Hall, CRC Press, London, Boca Raton, FL, 2000, pp. -
On the Velocity at Wind Turbine and Propeller Actuator Discs Gijs A.M
https://doi.org/10.5194/wes-2020-51 Preprint. Discussion started: 28 February 2020 c Author(s) 2020. CC BY 4.0 License. On the velocity at wind turbine and propeller actuator discs Gijs A.M. van Kuik Duwind, Delft University of Technology, Kluyverweg 1, 2629HS Delft, NL Correspondence: Gijs van Kuik ([email protected]) Abstract. The first version of the actuator disc momentum theory is more than 100 years old. The extension towards very low rotational speeds with high torque for discs with a constant circulation, became available only recently. This theory gives the performance data like the power coefficient and average velocity at the disc. Potential flow calculations have added flow properties like the distribution of this velocity. The present paper addresses the comparison of actuator discs representing 5 propellers and wind turbines, with emphasis on the velocity at the disc. At a low rotational speed, propeller discs have an expanding wake while still energy is put into the wake. The high angular momentum of the wake, due to the high torque, creates a pressure deficit which is supplemented by the pressure added by the disc thrust. This results in a positive energy balance while the wake axial velocity has lowered. In the propeller and wind turbine flow regime the velocity at the disc is 0 for a certain minimum but non-zero rotational speed . 10 At the disc, the distribution of the axial velocity component is non-uniform in all flow states. However, the distribution of the velocity in the plane containing the axis, the meridian plane, is practically uniform (deviation < 0.2 %) for wind turbine disc flows with tip speed ratio λ > 5 , almost uniform (deviation 2 %) for wind turbine disc flows with λ = 1 and propeller ≈ flows with advance ratio J = π, and non-uniform (deviation 5 %) for the propeller disc flow with wake expansion at J = 2π. -
Abelian Varieties
Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties . -
Cgweek Young Researchers Forum 2018
CGWeek Young Researchers Forum 2018 Booklet of Abstracts 2018 This volume contains the abstracts of papers at “Computational Geometry: Young Researchers Forum” (CG:YRF), a satellite event of the 34th International Symposium on Computational Geometry, held in Budapest, Hungary on June 11-14, 2018. The CG:YRF program committee consisted of the following people: Don Sheehy (chair), University of Connecticut Peyman Afshani, Aarhus University Chao Chen, CUNY Queens College Elena Khramtcova, Université Libre de Bruxelles Irina Kostitsyna, TU Eindhoven Jon Lenchner, IBM Research, Africa Nabil Mustafa, ESIEE Paris Amir Nayyeri , Oregon State University Haitao Wang, Utah State University Yusu Wang, The Ohio State University There were 31 papers submitted to CG:YRF. Of these, 28 were accepted with revisions. Copyrights of the articles in these proceedings are maintained by their respective authors. More information about this conference and about previous and future editions is available online at http://www.computational-geometry.org/ Topology-aware Terrain Simplification Ulderico Fugacci Institute of Geometry, Graz University of Technology Kopernikusgasse 24, 8010 Graz, Austria [email protected] https://orcid.org/0000-0003-3062-997X Michael Kerber Institute of Geometry, Graz University of Technology Kopernikusgasse 24, 8010 Graz, Austria [email protected] https://orcid.org/0000-0002-8030-9299 Hugo Manet Département d’Informatique, École Normale Supérieure 45 rue d’Ulm, 75005 Paris, France [email protected] https://orcid.org/0000-0003-4649-2584 Abstract A common issue in terrain visualization is caused by oversampling of flat regions and of areas with constant slope. For a more compact representation, it is desirable to remove vertices in such areas, maintaining both the topological properties of the terrain and a good quality of the underlying mesh. -
In Combinatorics, the Eulerian Number A(N, M), Is the Number of Permutations of the Numbers 1 to N in Which Exactly M Elements A
From Wikipedia, the free encyclopedia In combinatorics, the Eulerian number A(n, m), is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents"). They are the coefficients of the Eulerian polynomials: The Eulerian polynomials are defined by the exponential generating function The Eulerian polynomials can be computed by the recurrence An equivalent way to write this definition is to set the Eulerian polynomials inductively by Other notations for A(n, m) are E(n, m) and . 1History 2 Basic properties 3 Explicit formula 4 Summation properties 5 Identities 6 Eulerian numbers of the second kind 7 References 8 External links In 1755 Leonhard Euler investigated in his book Institutiones calculi differentialis polynomials α1(x) = 1, α2(x) = x + 1, 2 α3(x) = x + 4x + 1, etc. (see the facsimile). These polynomials are a shifted form of what are now called the Eulerian polynomials An(x). For a given value of n > 0, the index m in A(n, m) can take values from 0 to n − 1. For fixed n there is a single permutation which has 0 ascents; this is the falling permutation (n, n − 1, n − 2, ..., 1). There is also a single permutation which has n − 1 ascents; this is the rising permutation (1, 2, 3, ..., n). Therefore A(n, 0) and A(n, n − 1) are 1 for all values of n. Reversing a permutation with m ascents creates another permutation in which there are n − m − 1 ascents. Therefore A(n, m) = A(n, n − m − 1). -
Certification of First-Order Proofs in Classical and Intuitionistic Logics
Certification of First-order proofs in classical and intuitionistic logics 1 Zakaria Chihani August 21, 2015 1This work has been funded by the ERC Advanced Grant ProofCert. Abstract The field of automated reasoning contains a plethora of methods and tools, each with its own language and its community, often evolving separately. These tools express their proofs, or some proof evidence, in different formats such as resolution refutations, proof scripts, natural deductions, expansion trees, equational rewritings and many others. The disparity in formats reduces communication and trust be- tween the different communities. Related efforts were deployed to fill the gaps in communication including libraries and languages bridging two or more tools. This thesis proposes a novel approach at filling this gap for first-order classical and intuitionistic logics. Rather than translating proofs written in various languages to proofs written in one chosen language, this thesis introduces a framework for describing the semantics of a wide range of proof evidence languages through a rela- tional specification, called Foundational Proof Certification (FPC). The description of the semantics of a language can then be appended to any proof evidence written in that language, forming a proof certificate, which allows a small kernel checker to verify them independently from the tools that created them. The use of seman- tics description for one language rather than proof translation from one language to another relieves one from the need to radically change the notion of proof. Proof evidence, unlike complete proof, does not have to contain all details. Us- ing proof reconstruction, a kernel checker can rebuild missing parts of the proof, allowing for compression and gain in storage space. -
Rationale of the Chakravala Process of Jayadeva and Bhaskara Ii
HISTORIA MATHEMATICA 2 (1975) , 167-184 RATIONALE OF THE CHAKRAVALA PROCESS OF JAYADEVA AND BHASKARA II BY CLAS-OLOF SELENIUS UNIVERSITY OF UPPSALA SUMMARIES The old Indian chakravala method for solving the Bhaskara-Pell equation or varga-prakrti x 2- Dy 2 = 1 is investigated and explained in detail. Previous mis- conceptions are corrected, for example that chakravgla, the short cut method bhavana included, corresponds to the quick-method of Fermat. The chakravala process corresponds to a half-regular best approximating algorithm of minimal length which has several deep minimization properties. The periodically appearing quantities (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, etc.) are correctly understood only with the new theory. Den fornindiska metoden cakravala att l~sa Bhaskara- Pell-ekvationen eller varga-prakrti x 2 - Dy 2 = 1 detaljunders~ks och f~rklaras h~r. Tidigare missuppfatt- 0 ningar r~ttas, sasom att cakravala, genv~gsmetoden bhavana inbegripen, motsvarade Fermats snabbmetod. Cakravalaprocessen motsvarar en halvregelbunden b~st- approximerande algoritm av minimal l~ngd med flera djupt liggande minimeringsegenskaper. De periodvis upptr~dande storheterna (jyestha-m~la, kanistha-mula, ksepaka, kuttakara, os~) blir forstaellga0. 0 . f~rst genom den nya teorin. Die alte indische Methode cakrav~la zur Lbsung der Bhaskara-Pell-Gleichung oder varga-prakrti x 2 - Dy 2 = 1 wird hier im einzelnen untersucht und erkl~rt. Fr~here Missverst~ndnisse werden aufgekl~rt, z.B. dass cakrav~la, einschliesslich der Richtwegmethode bhavana, der Fermat- schen Schnellmethode entspreche. Der cakravala-Prozess entspricht einem halbregelm~ssigen bestapproximierenden Algorithmus von minimaler L~nge und mit mehreren tief- liegenden Minimierungseigenschaften. Die periodisch auftretenden Quantit~ten (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, usw.) werden erst durch die neue Theorie verst~ndlich. -
IJR-1, Mathematics for All ... Syed Samsul Alam
January 31, 2015 [IISRR-International Journal of Research ] MATHEMATICS FOR ALL AND FOREVER Prof. Syed Samsul Alam Former Vice-Chancellor Alaih University, Kolkata, India; Former Professor & Head, Department of Mathematics, IIT Kharagpur; Ch. Md Koya chair Professor, Mahatma Gandhi University, Kottayam, Kerala , Dr. S. N. Alam Assistant Professor, Department of Metallurgical and Materials Engineering, National Institute of Technology Rourkela, Rourkela, India This article briefly summarizes the journey of mathematics. The subject is expanding at a fast rate Abstract and it sometimes makes it essential to look back into the history of this marvelous subject. The pillars of this subject and their contributions have been briefly studied here. Since early civilization, mathematics has helped mankind solve very complicated problems. Mathematics has been a common language which has united mankind. Mathematics has been the heart of our education system right from the school level. Creating interest in this subject and making it friendlier to students’ right from early ages is essential. Understanding the subject as well as its history are both equally important. This article briefly discusses the ancient, the medieval, and the present age of mathematics and some notable mathematicians who belonged to these periods. Mathematics is the abstract study of different areas that include, but not limited to, numbers, 1.Introduction quantity, space, structure, and change. In other words, it is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Mathematicians seek out patterns and formulate new conjectures. They resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity.