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# Papers etc. by Alexander Yong

My papers on the Mathematics ArXiV.

77. Classifying Levi-spherical Schubert varieties (with Yibo Gao and Reuven Hodges), FPSAC 2022 conference proceedings. Version of March 29, 2022

Discusses earlier work on the classification problem of the title together with some new results, conjecture, and evidence for that conjecture.
76. Combinatorics of Newell-Littlewood numbers (with Shiliang Gao, Gidon Orelowitz, and Nicolas Ressayre), FPSAC 2022 conference proceedings. Version of March 29, 2022
Survey of work on the Newell-Littlewood numbers.
75. Minimal equations for Matrix Schubert varieties (with Shiliang Gao), preprint, version of January 17, 2022.
Explicit minimal generators for Fulton's Schubert determinantal ideals are determined along with some implications.
74. Newell-Littlewood numbers III: eigencones and GIT-semigroups (with Shiliang Gao, Gidon Orelowitz, Nicolas Ressayre), preprint, version of July 5, 2021.
We continue our study of Newell-Littlewood numbers.
73. Castelnuovo-Mumford regularity and Schubert geometry, draft, version of June 8, 2021.
Extended notes for my talk at Recent developments in Grobner Geometry'', June 2021.
72. Classification of Spherical Schubert varieties (with Yibo Gao and Reuven Hodges), preprint, version of April 19, 2021.
We classify Levi spherical Schubert varieties in the complete flag variety GL_n/B.
71. The "Grothendieck to Lascoux" conjecture (with Vic Reiner), preprint, version of February 18, 2021.
We formulate a combinatorial conjecture about the expansion of Grothendieck polynomials into Lascoux polynomials.
70. The Kostka semigroup and its Hilbert basis (with Shiliang Gao, Joshua Kiers, and Gidon Orelowitz), preprint, version of February 1, 2021.
We study the Hilbert basis of the Kostka semigroup.
69. Proper permutations, Schubert geometry, and randomness (with David Brewster and Reuven Hodges), preprint, version of December 17, 2020.
We introduce and study proper permutations, in connection to spherical Schubert geometry.
68. Generalized permutahedra and Schubert calculus (with Avery St. Dizier), preprint, version of October 27, 2020. To appear in Arnold Math J.
We use the theory of generalized permutahedra to give a vanishing criterion for Schubert intersection numbers.
67. Newell-Littlewood numbers II: extended Horn inequalities (with Shiliang Gao and Gidon Orelowitz), preprint, version of September 29, 2020. To appear in Algebraic Combinatorics .
We continue our study of Newell-Littlewood numbers.
66. Coxeter combinatorics and spherical Schubert geometry (with Reuven Hodges), to appear in Journal of Lie Theory, version of December 2, 2021.
We introduce spherical elements of Coxeter groups, in connection to Levi spherical Schubert varieties.
65. Multiplicity-free key polynomials (with Reuven Hodges), preprint, version of July 17, 2020.
We classify multiplicity-free key polynomials. This is the companion paper to Coxeter combinatorics and spherical Schubert geometry''.
64. Newell-Littlewood numbers (with Shiliang Gao and Gidon Orelowitz), Trans. AMS (accepted), version of June 13, 2020.
We study Newell-Littlewood numbers by analogy with modern research on their celebrated cousins, the Littlewood-Richardson coefficients.
63. Equivariant cohomology, Schubert calculus, and edge-labeled tableaux (with Colleen Robichaux and Harshit Yadav), to appear in "Facets of Algebraic Geometry: A Volume in Honour of William Fulton's 80th Birthday", Cambridge University Press, London Mathematical Society Lecture Note Series, preprint version of August 29, 2019.
We discuss results concerning the objects of the title. We introduce shifted edge labeled tableau in connection to a ring of Anderson-Fulton.
62. The A.B.C.Ds of Schubert calculus (with Colleen Robichaux and Harshit Yadav), Seminaire Lotharingien de Combinatoire, B85a (2020), 12 pp.
We collect "Atiyah-Bott Combinatorial Dreams" in Schubert calculus.

61. An estimation method for game complexity (with David Yong), preprint, version of January 29, 2019.

We study a method for estimating the game tree size of strategic games.
60. Reduced words, complexity, and randomization (with Cara Monical and Benjamin Pankow), preprint, version of January 30, 2019.
We study algorithmic and complexity aspects of reduced word enumeration.
59. An efficient algorithm for deciding vanishing of Schubert polynomial coefficients (with Anshul Adve and Colleen Robichaux), Advances in Math., to appear, 2021.
This is the main technical part of this preprint whose material was split into this text and the one below.
58. Complexity, Newton polytopes, and Schubert polynomials (with Anshul Adve and Colleen Robichaux), Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana), Sem. Lothar. Combin. 82B (2020), Art. 52, 12 pp.
We discuss an algebraic combinatorics paradigm for computation complexity. Our main results concern Schubert polynomials.

This is an extended abstract of this preprint whose material was split into this text and the one above.

57. Tropicalization, symmetric polynomials, and complexity (with Alexander Woo), accepted to Journal of Symbolic Computation, version of October 9, 2017.
We study the circuit complexity of some tropicalized symmetric polynomials.
56. K-orbits and Barbasch-Evens-Magyar varieties (with Laura Escobar and Benjamin Wyser), preprint, version of August 22, 2017.
We construct and study Barbasch-Evens-Magyar varieties.
55. Vanishing of Littlewood-Richardson polynomials is in P (with Anshul Adve and Colleen Robichaux), to appear in Computational Complexity; version of March 21, 2019.
We give a strongly polynomial time algorithm for deciding the vanishing of Littlewood-Richardson polynomials.
54. Symmetric group representations and Z (with Anshul Adve), C. R. Math. Acad. Sci. Paris 356 (2018), no. 1, 1--4. (Version of June 30, 2017.)
A remark on the representation theory of symmetric groups.
53. Newton polytopes and symmetric Grothendieck polynomials (with Laura Escobar), C. R. Math. Acad. Sci. Paris 355 (2017), no. 8, 831--834. (Version of May 22, 2017.)
We prove symmetric Grothendieck polynomials have saturated Newton polytopes.
52. Newton polytopes in algebraic combinatorics (with Cara Monical and Neriman Tokcan), to appear in Selecta Math, 2019.
We study Newton polytopes of various families of polynomials in algebraic combinatorics.
51. Governing singularities of symmetric orbit closures (with Alexander Woo and Benjamin Wyser), Algebra and Number theory 12 (2018), no. 1, 173-225.
We develop interval pattern avoidance and Mars-Springer ideals to study singularities of symmetric orbit closures in a flag variety. This paper focuses on the case of the Levi subgroup GL(p)xGL(q) acting on the classical flag variety.
50. Partition identities and quiver representations (with Richard Rimanyi and Anna Weigandt), J. Algebraic Comb. 47 (2018), no. 1, 129--169. (Version of June 1, 2017).
We establish a specific connection between classical partition combinatorics and the theory of quiver representations.
49. Rhombic tilings and Bott-Samelson varieties (with Laura Escobar, Oliver Pechenik and Bridget Tenner), Rhombic tilings and Bott-Samelson varieties. Proc. Amer. Math. Soc. 146 (2018), no. 5, 1921--1935. (Version of Jun 9, 2016.)
We observe a connection between Elnitsky's rhombic tilings and Magyar's description of Bott-Samelson varieties.
48. Poset edge densities, nearly reduced words and barely set-valued tableaux (with Victor Reiner and Bridget Tenner), to appear in J. Combinatorial Theory Series A, March 13, 2018.
We study a certain coincidence of expectations of down-degree of posets.
47. Genomic Tableaux (with Oliver Pechenik), J. Algebraic Combinatorics, to appear, version of March 28, 2016.
We describe a theory of genomic tableaux.
46. The prism tableau model for Schubert polynomials (with Anna Weigandt), J. Combin. Theory Ser. A 154 (2018), 551--582. (Version of September 8, 2015.)
We propose a new model for Schubert polynomials.
45. Equivariant K-theory of Grassmannians II: the Knutson-Vakil conjecture (with Oliver Pechenik), to appear in Compositio Math., version of August 3, 2015.
We resolve a conjecture of A. Knutson-R. Vakil.
44. Equivariant K-theory of Grassmannians (with Oliver Pechenik), preprint, June 5, 2015; accepted to Forum of Math, Pi.
We give a combinatorial rule for multiplication of the basis of Schubert structure sheaves in the equivariant K-theory ring of the Grassmannian. (See also the announcement version below.)
43. Genomic Tableaux and Combinatorial K-theory (with Oliver Pechenik) to appear in the Proceedings of Formal Power Series and Algebraic combinatorics 2015 , version of April 8, 2015.
Announcement of results on (equivariant) K-theory of Grassmannians (and maximal orthogonal, Lagrangian Grassmannians).
42. Critique of Hirsch's citation index: a combinatorial Fermi problem preprint (10 pages+appendix), accepted to Notices of the AMS , version of October 13, 2014.
We use the Euler-Gauss partition identity to offer a critique of the bibliometric h-index.
41. Polynomials for symmetric orbit closures in the flag variety (with Benjamin Wyser), preprint (22 pages), version of September 30, 2014.
We continue our study of polynomial representatives of orbit closures in the flag variety for symmetric pairs (G,K) in type A.
40. The Joseph Greenberg problem: combinatorics and comparative linguistics , preprint (6 pages), version of October 10, 2013.

We discuss some combinatorics related to work of linguist J. Greenberg. We give an interpretation of the consensus range of the number of indigenious language families of the Americas. Best read along with the video .
39. Polynomials for GLp x GLq orbit closures in the flag variety (with Benjamin Wyser), to appear in Selecta Math (25 pages), version of March 4, 2014.

We introduce polynomial representatives of GLp x GLq orbit closures in the flag variety.
38. Root-theoretic Young diagrams and Schubert calculus: planarity and the adjoint varieties (with Dominic Searles), J. Algebra to appear (45 pages), version of August 29, 2013.

We study the existence of a root-system uniform and nonnegative combinatorial rule for Schubert calculus. Our main cases are the adjoint varieties of classical type. (An preliminary version appeared as an extended abstract in FPSAC 2013.)
37. Combinatorial rules for three bases of polynomials (with Colleen Ross), Seminaire Lotharingien de Combinatoire (11 pages), 2015.

We present combinatorial rules (one theorem and two conjectures) concerning three bases of Z[x1,x2,...]. Errata note added March 10, 2017: the published version has a misstatement of Conjectures 1.4 and 1.6. The version above has the correction.
36. Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians (with David Anderson and Edward Richmond), accepted to Compositio Math (14 pages), version of April 2, 2013.

We establish a connection between the eigenvalue problem of S. Friedland and equivariant Schubert calculus.
35. Singularities of Richardson varieties (with Allen Knutson and Alexander Woo), Math. Res. Letters (10 pages), version of September 14, 2012.

We prove essentially all singularity questions about Richardson varieties reduce to the case of Schubert varieties.
34. Equivariant Schubert calculus and jeu de taquin (with Hugh Thomas), Annales de l'Institut Fourier (31 pages), accepted pending minor revisions, version of July 11, 2012.

We develop equivariant analogues of jeu de taquin, in relation to Schubert calculus of Grassmannians.
33. Patch ideals and Peterson varieties (with Erik Insko), Transformation Groups (23 pages), version of February 3, 2012.

We study patch ideals of Peterson varieties and some other subvarieties of GLn/B.
32. Kazhdan-Lusztig polynomials and drift configurations (with Li Li), Algebra and Number Theory , (25 pages), June 19, 2010.

We suggest a parallel between Kazhdan-Lusztig polynomials and H-polynomials of local rings of Schubert varieties.
31. Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties (with Li Li), Advances in Math , (31 pages), version of July 1, 2011.

We prove a combinatorial rule for the Hilbert-Samuel multiplicities of covexillary Schubert varieties at singular points. Generalizations of our method to general Schubert varieties in the complete flag variety are suggested.
30. K-theoretic Schubert calculus for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux (with Edward Clifford and Hugh Thomas), to appear in J. Reine Angew Math (Crelle) (13 pages), Feb 15, 2012.

Using recent work of Buch--Ravikumar and Feigenbaum-Sergel we prove the conjectural rule from [Thomas-Yong '09] for $K$-theory Schubert calculus for maximal odd orthogonal Grassmannians.
29. A Grobner basis for Kazhdan-Lusztig ideals (with Alexander Woo), preprint (40 pages), American J. Math , 2012.

We prove a Grobner basis theorem for Kazhdan-Lusztig ideals, and apply this to the study of specializations of Schubert/Grothendieck polynomials, and multiplicities of Schubert varieties.
28. The direct sum map on Grassmannians and Jeu de taquin for increasing tableaux (with Hugh Thomas), International Math. Res. Notices, (20 pages), 2010.

We establish analogues of Schutzenberger's fundamental theorems of jeu de taquin for splitting coefficients defining the direct sum map, as well as products of Schubert boundary ideal sheaves, in K-theory of Grassmannians.
27. Presenting the cohomology of a Schubert variety (with Victor Reiner and Alexander Woo), Transactions of the AMS, (22 pages), 2009.

We extend the short presentation of the cohomology ring of a generalized flag manifold, due to [Borel '53], to a relatively short presentation of the cohomology of any of its Schubert varieties.
26. An approximation algorithm for contingency tables (with Alexander Barvinok, Zur Luria and Alex Samorodnitsky), accepted to Random Structures and Algorithms , 2009 (45 pages).

We present a randomized approximation algorithm for counting contingency tables, m x n non-negative integer matrices with given row sums R=(r_1,...,r_m) and column sums C=(c_1, ..., c_n).

Update (July, 2008): My coauthor A. Barvinok has further studied the typical table in his paper.

25. Longest strictly increasing subsequences, Plancherel measure and the Hecke insertion algorithm (with Hugh Thomas), Advances in Applied Math (Dennis Stanton special issue) (34 pages), 10/02/02. (Contains an appendix with Ofer Zeitouni and myself.)
Companion Maple code HeckeLIS.v0.1.txt.

We define and study the Plancherel-Hecke probability measure on Young diagrams.
24. A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus (with Hugh Thomas), Algebra and Number Theory , 2008 (22 pages).

We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Sch\"{u}tzenberger '77] for standard Young tableaux.

23. An S_3 symmetric Littlewood-Richardson rule (with Hugh Thomas), Mathematical Research Letters , volume 15, no.5-6, 2008; (9 pages).

The classical Littlewood-Richardson coefficients carry a natural $S_3$ symmetry via permutation of indices. Our carton rule'' for computing these numbers transparently and uniformly explains these six symmetries; previous rules manifest at most three of the six.

22. Counting magic squares in quasi-polynomial time (with Alexander Barvinok and Alexander Samorodnitsky), preprint (30 pages), 2007.
Companion Maple code contingency.txt. Here is a supplemental webpage comparing various ways of enumerating contingency tables and magic squares.

We present a randomized algorithm which approximates the number of n x n non-negative integer matrices (magic squares) with the row and column sums equal to t.

21. Cominuscule tableau combinatorics (with Hugh Thomas), (revised July 2013). To appear in MSJ-SI 2012 Proceedings. Here is the Maple code to check Lemma 2.4 for Lie types E6 and E7.

We continue our study of "cominuscule tableau combinatorics" by generalizing constructions of M. Haiman, S. Fomin and M.-P. Schützenberger.

20. What is a Young tableau?
Notices of the AMS , Volume 54, Number 2, February 2007.

A short expository piece on Young tableaux and their basic theorems.

19. A combinatorial rule for (co)minuscule Schubert calculus (with Hugh Thomas), Advances in Math , 2009.
Companion Maple code cominrule.v1.0.txt. Here's an extra note about comparing cominrule.v1.0.txt with Schubert.v0.2.txt, given below.

We prove a root system uniform, concise combinatorial rule for Schubert calculus of _minuscule_ and _cominuscule_ flag manifolds G/P.

18. Stable Grothendieck polynomials and K-theoretic factor sequences (full version) (with A. Buch, A. Kresch, M. Shimozono and H. Tamvakis), Math. Annalen, volume 340, number 2, 2008.

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions.

This is the complete version of the announcement (presented at FPSAC 2005).

17. Tableau complexes (with A. Knutson and E. Miller), Israel Journal of Math , 163 (2008), 317-343.

Let X, Y be finite sets and T a set of functions from X->Y, which we will call tableaux''. We define a simplicial complex whose facets, all the same dimension, correspond to these tableaux, and call it a tableau complex''.

16. Multiplicity-free Schubert calculus (with H. Thomas), Canadian Math. Bulletin, 2007.

We give a short combinatorial classification of which products of Schubert classes in the Grassmannian are multiplicity-free. This answers a question of W. Fulton, and extends work of J. Stembridge.

15. Governing singularities of Schubert varieties (with A. Woo), accepted Journal of Algebra (section on computational algebra), 2007.
Companion Macaulay 2 code Schubsingular.v0.2.m2

We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds.

Update (Nov 11, 2006): 1. My coauthor A. Woo has extended the interval pattern avoidance ideas of the above paper here.

Update (April 6, 2007): N. Perrin has proved our Gorenstein locus conjecture for the case of minuscule G/P flag varieties. See this paper.

Correction (October, 2019): Example 6.18 has a typo; the multiplicity is 3 (as confirmed by the code, and consistent with Proposition 6.21). We thank Jordan Lambert for bringing this to our attention.

14. Grobner geometry of vertex decompositions and of flagged tableaux (with A. Knutson and E. Miller), Journal fur die reine und angewandte Mathematik (Crelle's Journal) , accepted, 2007.

We relate a classic algebro-geometric degeneration technique, dating at least to [Hodge 1941], to the notion of vertex decompositions of simplicial complexes. The good case is when the degeneration is reduced, and we call this a geometric vertex decomposition .

13. When is a Schubert variety Gorenstein? (with A. Woo),
Advances in Math , Vol 207 (2006), Issue 1 205--220. Companion Maple code available.

We determine which Schubert varieties are Gorenstein in terms of a combinatorial characterization using generalized pattern avoidance conditions.
12. Grobner geometry of Schubert transition formulae and Littlewood-Richardson rules (with A. Knutson), preprint.

11. A formula for K-theory truncation Schubert calculus (with A. Knutson),
International Mathematics Research Notices , 70 (2004), 3741-3756.

In certain good cases, the truncation of a Schubert or Grothendieck polynomial may again be a Schubert or Grothendieck polynomial. We use this phenomenon to give subtraction-free formulae for certain Schubert structure constants in K(Flags(C^n)).

10. Lecture notes on the K-theory of the flag variety and the Fomin-Kirillov quadratic algebra (with C. Lenart), 2004.

9. Quiver coefficients are Schubert structure constants (with A. Buch and F. Sottile),
Mathematical Research Letters , Volume 12, Issue 4, 567-574 (2005).

We give a new explanation of the alternation in sign of the K-theory quiver coefficients (via a theorem of Brion) as well as new combinatorial formulas for what they count.

8. Grothendieck polynomials and Quiver formulas (with A. Buch, A. Kresch and H. Tamvakis),
American Journal of Math , 127 (2005), 551-567.

We give a formula that proves that splitting coefficients'' for Grothendieck polynomials alternate in sign according to degree.

7. On Combinatorics of Degeneracy Loci and H*(G/B), a dissertation submitted to the Rackham graduate school, University of Michigan, 2003. Companion Maple code for part 2 available. (Includes an implementation of the classical Monk-Chevalley formula for arbitrary Lie type.)

Contains: 1. the content of the paper 6 below; 2. a Grobner based construction of an alternative (combinatorial) linear basis for H*(G/B) related to the Schubert calculus of a generalized flag manifold.

6. On Combinatorics of Quiver Component Formulas,
Journal of Algebraic Combinatorics , 21, 351-371, 2005.

Buch and Fulton conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver variety. Knutson, Miller and Shimozono proved this conjecture as an immediate consequence of their component formula''. We present an alternative proof of the component formula by substituting combinatorics for Grobner degeneration.

5. Schubert polynomials and Quiver formulas (with A. Buch, A. Kresch and H. Tamvakis),
Duke Math Journal , Volume 122, Issue 1, 125-143 (2004).

Fulton's Universal Schubert polynomials represent general degeneracy loci for maps of vector bundles with rank conditions coming from a permutation. The Buch-Fulton Quiver formula expresses this polynomial as an integer linear combination of products of Schur polynomials in the differences of the bundles. We present a positive combinatorial formula for the coefficients.

4. Degree bounds in quantum Schubert calculus
Proceedings of the AMS , Volume 131, Number 9, 2649-2655 (2003).

We give a combinatorial proof of a conjecture of Fulton (after Fulton-Woodward) for the smallest degree of q that appears in the expansion of the product of two Schubert classes in the (small) quantum cohomology ring of a Grassmannian. We present a combinatorial proof of this result.

3. Tree-like properties of cycle factorizations (with I.P. Goulden)
Journal of Combinatorial Theory Series A, 98 , 106-117 (2002).

We provide a bijection between the set of factorizations, that is, ordered (n-1)-tuples of transpositions in ${\mathcal S}_{n}$ whose product is (12...n), and labelled trees on $n$ vertices. We prove a refinement of a 1959 theorem and question of Dénes that establishes new tree-like properties of factorizations.

2. Dyck paths and a bijection for multisets of hook numbers (with I.P. Goulden),
Discrete Math, 254 , no.1-3, 153-164 (2002).

We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture, by means of a construction involving Dyck paths, a particular type of lattice path.

1. Seeing the factorizations for the trees, M.Math. thesis (1999), University of Waterloo. Available upon request.

Contains 1. the results of paper 2'' above; and 2. a Prufer type code for factorizations providing the first direct bijective proof that n^{n-2} counts the number of factorizations of the long cycle.

# More software

(Code for a particular paper is found with the .ps file above.)

• Maple 7 code to compute Schubert calculus in G/B , website exclusive, 2006.
• At the end of the 2002 CRM Workshop on "Computational Lie Theory", a "wish list" of software as compiled. One of these wishes was for code to compute Schubert calculus for arbitrary G/B. This code implements Allen Knutson's recursion for Schubert calculus (in the classical setting). (Warning: this was written for Maple 7. I do not know if it works for later editions of Maple.) (Code updated June 4, 2006.)

• C++ code for estimating permanents, hafnians and the number of forests in a graph, 2003.


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# Draw a number line and show all numbers that are five units away from 0

###### Question:

Draw a number line and show all numbers that are five units away from 0

### Translate the sentence into an inequality. The sum of a number times 8 and 27 is at most – 29. Use the variable b for the unknown number.

Translate the sentence into an inequality. The sum of a number times 8 and 27 is at most – 29. Use the variable b for the unknown number....

### How did the united states seek to counter the japanese in the pacific and the germans in europe?

How did the united states seek to counter the japanese in the pacific and the germans in europe?...

### Given the following system of equations, what is the value of the y-coordinate of the solution? 3x + 4y = 8 6x - 2y = 11 (Possible Answers} A. -4 B. 1/2 C. -9/4

Given the following system of equations, what is the value of the y-coordinate of the solution? 3x + 4y = 8 6x - 2y = 11 (Possible Answers} A. -4 B. 1/2 C. -9/4...

### Why do you think that Dr.King refers to a lecture made by Ralph Waldo Emerson ?

why do you think that Dr.King refers to a lecture made by Ralph Waldo Emerson ?...

### Fill in the blanks with spanish words missing from the story Sólo el principe habla 11. ___________, ¿verdad? No, él no lo habla. Lo entiende un poco, pero no lo habla. 12. __________ es la princesa que lo haba. 13.____________ ella. No lo habla 14. __________ De hecho, lo habla 15_____________ en la iglesia. Pero la habla casi sin acento. ¿Cómo es que lo habla en la 16. ___________? Aparentemente le gusta hablarlo allá. No le gusta hablarlo en el palacio , parece. ¿Y con quien habla chino en la

Fill in the blanks with spanish words missing from the story Sólo el principe habla 11. ___________, ¿verdad? No, él no lo habla. Lo entiende un poco, pero no lo habla. 12. __________ es la princesa que lo haba. 13.____________ ella. No lo habla 14. __________ De hecho, lo habla 15_____________ e...

### (r= -82) is a example of what correlation

(r= -82) is a example of what correlation...

### What has to happen for the President to get line-item veto power? A) A national referendum on the veto must be held. B) The Supreme Court must give the President the power. C) The Senate must vote to give the President the power. D) The U.S. Constitution must be amended.

What has to happen for the President to get line-item veto power? A) A national referendum on the veto must be held. B) The Supreme Court must give the President the power. C) The Senate must vote to give the President the power. D) The U.S. Constitution must be amended....

### (1.1)3 + 8.6 ÷ 2.15 – 0.086 = Plisss answer the question

(1.1)3 + 8.6 ÷ 2.15 – 0.086 = Plisss answer the question...

### Which statement best evaluates Machiavelli's argument

which statement best evaluates Machiavelli's argument...

### How was religion used to organize or influence society in Ancient Egypt?

How was religion used to organize or influence society in Ancient Egypt?...

### Compute the following 8c8 a. 16 b. 0 c. 64 d. 1

compute the following 8c8 a. 16 b. 0 c. 64 d. 1...

### The Adventures of Tom Sawyer was written by Samuel Clemens who wrote under the name, Mark Twain. When Twain was a boy in school in the 1840s, he always earned the medal for winning the weekly spelling bee. Once, he left out the first "r" in February so that a girl he liked would win. This and other stories became the basis for many of the delightful tales in Twain's writings. He went on to become one of the most famous American writers of all time. How does the author use the story about the spe

The Adventures of Tom Sawyer was written by Samuel Clemens who wrote under the name, Mark Twain. When Twain was a boy in school in the 1840s, he always earned the medal for winning the weekly spelling bee. Once, he left out the first "r" in February so that a girl he liked would win. This and other ...

### There are 56 beads on a necklace. 1/4 of the beads are blue. How many beads are blue?

There are 56 beads on a necklace. 1/4 of the beads are blue. How many beads are blue?...

### Find the surface area of a regular hexagonal prism with a side length of 6 and a height of 12.

find the surface area of a regular hexagonal prism with a side length of 6 and a height of 12....

### I GIVEEEE BRAINLILST

I GIVEEEE BRAINLILST...

### 738.75 rounded to the ones is

738.75 rounded to the ones is...

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Showing top 8 worksheets in the category - Egyptian Fractions. Worksheet 1. The Egyptians insisted on writing all their fractions as sums of fraction with numerators equal to 1. There are over 100 free fraction worksheets in PDFs below to support the many concepts encountered with fractions. Egyptian Fractions. Egyptian Fractions: The Answer Sheet. Articles that describe this calculator. For K-12 kids, teachers and parents. We start you off with the obvious: modeling fractions. Previous. Lesson materials located below the video overview. London WC1R 4HQ. 107, (2000), pages 62-63 of the form $\frac{2}{n}$). Some of the worksheets displayed are Egyptian mathematics, Egyptian fractions work pdf, Egyptian fractions, Egyptian unit fractions, Egyptian fractions what are known and unknown, Egyptian mathematics, Egyptian mathematics, Mega fun fractions. Egyptian Fraction. Stories For Comprehension. This worksheet asks kids to solve fraction word problems about food. Old Egyptian Math cats never repeated the same fraction when adding. This was practically important because many of the Egyptian structures required massive labor work. Includes Grade 2 fraction worksheets. This set features one-step addition and subtraction inequalities such as “5 + x > 7″ and “x – 3″ 21″. Learn fraction in a fun way by coloring or shading the pictures. Old Egyptian Math Cats knew fractions like 1/2 or 1/4 (one piece of a pie). Egyptian fractions; Published: 20/04/2012 KS3 2 pages. First few slides are a fun intro. Basic Reading Comprehension Worksheets Free. The focus is on gaining a conceptual understanding of what a fraction means; this is largely approached by comparing fractions to simple shapes which have been divided into equal parts.Fractions of a group or set are also introduced. There are several NRICH problems based on Egyptian fractions. Some of the worksheets for this concept are Egyptian mathematics, Egyptian mathematics, Egyptian fractions representations as sums of unit fractions, Egyptian mathematics, A complete overview for all those who never, Count like an egyptian, Second edition teachers guide 4, Math mammoth fractions 1 worktext. This is the android version of the Mathcats Old Egyptian Fractions: The app is about: proper fractions and their equivalent egyptian fractions (which in this case, are the addition of 2 or 3 unit fractions). The file includes a student worksheet for Egyptian fractions from 1/2 to 9/10. Introduce the idea of Egyptian Fractions to the class. Scholars of ancient Egypt (ca. Adding Decimals Practice. Math fraction worksheets for children in kindergarten, 1st, 2nd, 3rd, 4th, 5th, 6th and 7th grades. Subtraction Question For Class 1. Worksheet will open in a new window. LaTex does not handle Egyptian hieroglyphs — or at least, I don’t know how to make it do so. In this unit we want to explore that situation. (Be sure to use the words numerator and denominator.) you can see that fractions take part of this. What type of fraction is 1/4? So tables were provided to help with this task. She knew that… Therefore, as soon as she figured out one fraction, she had the answer to all of its equivalents. 1. We look at how fractions can be represented in terms of Egyptian Fractions. Worksheets > Math > Grade 6 > Fractions vs. Decimals > Converting decimals to fractions. The unit fraction is made by writing the number with a “mouth” symbol over the top – $$\dfrac{1}{2}=$$ Egyptians did not like repeating fractions, therefore, each fraction must be unique. Conditions. Pupils can learn about fractions by doing divisions the Egyptian way! 1. Created: Sep 22, 2013| Updated: Feb 22, 2018, Pupils can learn about fractions by doing divisions the Egyptian way! Some of the worksheets displayed are Egyptian mathematics, Egyptian fractions work pdf, Egyptian fractions, Egyptian unit fractions, Egyptian fractions what are known and unknown, Egyptian mathematics, Egyptian mathematics, Mega fun fractions. These fractions worksheets are great for practicing how to add fractional inch measurements that you would find on a tape measure. (intro) 2. Eating fractions is a great way to advance kids' understanding of fractions. The answers are now posted. But to make fractions like 3/4, they had to add pieces of pies like 1/2 + 1/4 = 3/4. Fractions really aren't that difficult to master especially with the support of our wide selection of worksheets. An egyptian number is any number equal which can be expressed as the sum of an integer plus the sum of an Egyptian fraction. Our grade 2 fraction worksheets introduce students to fractions as both parts of a whole and parts of a set.We cover identifying common fractions, comparing common fractions, and reading / writing fractions. This worksheet is designed as a short exercise in drawing human figures in the style of the Ancient Egyptians. After this brief introduction, hand out Egyptian Fractions, and have the students begin working either individually or in small groups to solve the problems. Some of the worksheets for this concept are Egyptian mathematics, Egyptian fractions work pdf, Egyptian fractions, Egyptian unit fractions, Egyptian fractions what are known and unknown, Egyptian mathematics, Egyptian mathematics, Mega fun fractions. You can start by exploring unit fractions at Keep it Simple In this problem we are going to start by considering how the Egyptians might have written fractions with a numerator of 2 (i.e. *(Maybe ¾, too. Egyptians represented fractions differently than we do. Below are six versions of our grade 6 math worksheet on converting decimal numbers to proper fractions. 3000 BCE) were very practical in their approaches to mathematics and always sought answers to problems that would be of most convenience to the people involved. Egyptian fractions are the reciprocals of the positive integers where the numerator is always one. They never wrote: 1/4 + 1/4 + 1/4 = 3/4 Learn how the ancient Egyptians used unit fraction sums to write different fractions… Lesson 5 Homework Practice Solve Multi Step Equations Answers. Egyptian fractions; Egyptian fraction expansion. This problem set is based on this one from Illustrative_Mathematics. 2 Step Subtraction Word Problems Year 4. and 1/4 + 1/28 = our 2/7. Reading Passages For Children. Old Egyptian Math cats never repeated the same fraction when adding. Tes Global Ltd is Ask them if they can think of any reason why the Egyptians were hooked on fractions with a one in the numerator. The answer is 1/20. What Egyptian Fraction is smaller than 0.3 but closest to it? This means that our Egyptian Fraction representation for 4/5 is 4/5 = 1/2 + 1/4 + 1/20; But to make fractions like 3/4, they had to add pieces of pies like 1/2 + 1/4 = 3/4. Reading Comprehension Worksheets For Free. person_outlineAntonschedule 2019-10-29 20:02:56. address one or more of the NCTM fraction standards listed above. Egyptian fractions for 4/n and the Erdös-Straus Conjecture Every fraction of the form 3/n where n is not a multiple of 3 and odd can be written as 1/a + 1/b + 1/c for distinct odd a, b and c. For a proof see A Proof of a Conjecture on Egyptian Fractions T. R. Hagedorn The American Mathematical Monthly, Vol. (Be sure to use the words numerator and denominator.) We would slice each cake into 4 … Fractions worksheets: addition with like denominators. Accept a… The Egyptian fractions were particularly useful when dividing a number of objects equally for more number of people. Learn how the ancient Egyptians used unit fraction sums to write different fractions. Next. registered in England (Company No 02017289) with its registered office at 26 Red Lion All other fractions were written as a sum of unit fractions: 6 ; 5 8 5 6 <. A UNIT fraction is a fraction with a numerator one(1). This bundle contains 12 worksheets on 12 different topics all listed in detail below. But to make fractions like 3/4, they had to add pieces of pies like 1/2 + 1/4 = 3/4. Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets. Welcome to the fractions worksheets page at Math-Drills.com where the cup is half full! D. Russell. Old Egyptian Math Cats knew fractions like 1/2 or 1/4 (one piece of a pie). 4.1 Egyptian Fractions. Yes, it’s a unit fraction. Egyptian Fractions Displaying top 8 worksheets found for - Egyptian Fractions . Found worksheet you are looking for? 3rd grade. Addition And Subtraction For Class 4. An Egyptian fraction is a sum of positive (usually) distinct unit fractions.The famous Rhind papyrus, dated to around 1650 BC contains a table of representations of as Egyptian fractions for odd between 5 and 101. Identify a proper fraction, improper fraction, mixed number, unit fraction, like fractions, unlike fractions like a pro with these printable types of fractions worksheets. For a more exciting powerpoint, please check out cbarclay99’s at this link: ... Fractions of Amounts Differentiated Worksheet … This math worksheet will give your child practice identifying fractions of shapes and filling in the missing numbers in fractions. after each example, there is an exercise consisting of 5 questions. Square EGYPTIAN NUMERATION - FRACTIONS For reasons unknown, the ancient Egyptians worked only with unit fractions, that is, fractions with a numerator of 1. When starting with fractions, begin by focusing on 1/2 and then a 1/4 before moving to equivalent fractions and using the 4 operations with fractions … What type of fraction is 1/4? Your third grader will shade in parts of rectangles and circles in this coloring math worksheet to match a given fraction amount. The number of terms in the Egyptian fraction representation of x/y is the sum of the odd terms after the first in the continued fraction list, which is at most x. These grade 1 worksheets introduce students to fractions. They did this because fractions of the form 1 N have a good intuitive feel to them. Mathematics / Number / Fractions, decimals, percentage, equivalence, 2020 KS2 SATs Revision Ultimate 15-in-1 Maths Organiser, Fractions 2 - Equivalent Fractions & Simplifying (+ worksheet). Old Egyptian Math Cats knew fractions like 1/2 or 1/4 (one piece of a pie). Egyptian fractions A wonderfully different way to practise fraction addition. In this download packet you will find 12 pages of Egyptian Art and one Teacher's Guide. Types of Fractions. Egyptian fraction notation was developed in the Middle Kingdom of Egypt, altering the Old Kingdom's Eye of Horus numeration system. This resource is designed for UK teachers. Old Egyptian Math Cats knew fractions like 1/2 or 1/4 (one piece of a pie). 3 Digit Subtraction With Regrouping Worksheets. 1/16's and there is an option to select 1/32's and 1/64's. You don't have to be an ancient Egyptian to decipher fractions in this activity that focuses on adding fractions with unlike denominators and developing fraction number sense. Fraction worksheets for children from kindergarten to 7th grades. The fact that the sum of the fractions is the original input number is a straightforward but tedious exercise in algebraic manipulation. In this fraction addition worksheet, 5th graders review the term unit fraction and then find the sums for the fraction equations and study how the Egyptians wrote the fractions. For a more exciting powerpoint, please check out cbarclay99âs at this link: http://www.tes.co.uk/teaching-resource/Egyptian-Fractions-6255731/ If you like this resource then please check out my other stuff on here! Reading Comp Worksheet. Students should simplify all fractions. Get Free Access See Review Think of slicing 3 cakes among 4 workers. The representation of rational numbers as sums of unit fractions dates back to the time of ancient Egypt. 1. To download/print, click on pop-out icon or print icon to worksheet to print or download. All Egyptian fractions share this same property: there are always endless ways to write an Egyptian fraction. These worksheets will generate 10 fractional inch problems per worksheet. In this first lesson we have a look at the sum of two Egyptian Fractions to see if we can get another Egyptian Fraction. Worksheets > Math > Grade 1 > Fractions. This calculator allows you to calculate an Egyptian fraction using … As they work, circulate around the room offering encouragement, quietly assessing progress, and answering questions. Reading Comprehension Past Simple Regular Verbs PDF Preposition Worksheet … Some of the worksheets for this concept are Egyptian mathematics, Egyptian mathematics, Egyptian fractions representations as sums of unit fractions, Egyptian mathematics, A complete overview for all those who never, Count like an egyptian, Second edition teachers guide 4, Math mammoth fractions 1 worktext. Students are shown the method of dividing the numerator by the denominator to find the first fraction and then dividing the remainder(s) to find the other fractions. But to make fractions like 3/4, they had to add pieces of pies like 1/2 + 1/4 = 3/4. Egyptian Fractions Ancient Egyptians used unit fractions, such as $\frac{1}{2}$ and $\frac{1}{3}$, to represent all fractions. Ask them if they can think of any reason why the Egyptians were hooked on fractions with a one in the numerator. Cross-curricular activities link fractions to language arts, music, science, art, and social studies. Paragraphs With Comprehension Questions. Guided Reading Handouts. Many of the fractions were easy. Finally we suggest a … Subtraction Problems Year 5. Generally, Egyptian Art spans 5,000 years. Egyptian Art is…. These worksheets will generate 10 fractional inch problems per worksheet. The Secret of Egyptian Fractions. Quickly find that inspire student learning. Simplifying Fraction Common fraction. Subtracting Fractions Worksheets. This bundle contains 12 worksheets on 12 different topics all listed in detail below. Showing top 8 worksheets in the category - Egyptian Fractions. My art worksheets show aspects of Egyptian Art that are unusual, unconventional and relatively unknown. Students draw a 'modern' representation of an Egyptian figure, using the sample and grid This page includes Fractions worksheets for understanding fractions including modeling, comparing, ordering, simplifying and converting fractions and operations with fractions. Egyptian Unit Fractions . Kindergarten Worksheet Examples; Newspaper Worksheet Examples in PDF; When you purchase goods, when you eat, when you clean your house, when you count your money, when you cook or bake, when you see carpenters or architects making a house or a building, when kids play, even in computer games! Egyptian She had the most trouble with the 7ths and 9ths. Egyptian Fractions: The Answer Sheet; To Be Continued… Read all the posts from the January/February 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter. These worksheets are pdf files.. These fractions worksheets are great for practicing how to add fractional inch measurements that you would find on a tape measure. Introduce the idea of Egyptian Fractions to the class. In this unit we want to explore that situation. Unit fractions are written additively: 1/4 1/26 means 1/4 + 1/26. Similar: View US version . All ancient Egyptian fractions, with the exception of 2/3, are unit fractions, that is fractions with numerator 1. These fractions worksheets will use 1/2's, 1/4's, 1/8's. Today this subject survives mainly as a source of mathematical puzzles and problems in abstract number theory, but the subject is also of historical and anthropological In this first lesson we have a look at the sum of two Egyptian Fractions to see if we can get another Egyptian Fraction. Are encouraged to solve this task and teaching resources divisions the Egyptian way cross-curricular link... 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# Determining Electric Potential due to a line of nonuniform charge density [closed]

A rod of length $$L$$ lies along the x axis with its left end at the origin. It has a non uniform charge density $$\lambda = \alpha x$$ where $$\alpha$$ is a positive constant. Calculate the electric potential at $$A(-d,0,0)$$.

First Attempt

$$V = - \int \vec{E}\cdot \mathrm d\vec{l}$$

Calculating The electric field $$E$$:

$$E = \int \frac{\lambda\,\mathrm dl}{4\pi \epsilon r^2 }$$

with $$r^2 = (x+d)^2$$

We then get $$E=\frac{\alpha}{4 \pi \epsilon}\Big(\ln(x+d) + \frac{d}{x+d}\Big)$$

Then, $$V = - \int_0^L E\cdot\mathrm dl= -(L+2d)\ln(l+d)+l+2d\ln(d)$$

Second Attempt

Using the formula $$V = \int_0^L \frac{\lambda\,\mathrm dl}{4\pi \epsilon r}$$

Evaluating this integral leads to the following :

$$V=\frac{\alpha}{4 \pi \epsilon}\Bigg(L-d \ln\Big(\frac{L+d}{d}\Big)\Bigg)$$

Why do both attempts give different solutions?

## closed as off-topic by ZeroTheHero, Jon Custer, Kyle Kanos, Aaron Stevens, Qmechanic♦Jan 30 at 16:32

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ZeroTheHero, Jon Custer, Kyle Kanos, Aaron Stevens, Qmechanic
If this question can be reworded to fit the rules in the help center, please edit the question.

• There are many issues here. In the first method you shouldn't have $r^2=(x+d)^2$. If you wan to go the first attempt route, you are trying to find the potential difference between the point A and the left end of the rod. So you need to find the field at all points on the line from $x=0$ to $x=-d$ (so maybe say $r^2=(x+x')^2$ and do the first integral over $x$?), and then do the line integral over $x'$ from $x=0$ to $x=-d$ – Aaron Stevens Jan 30 at 15:36
• Another issue is that the first method puts $V=0$ at $x=0$, but the second method puts $V=0$ at infinity. I would just stick with the second method. – Aaron Stevens Jan 30 at 15:37


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# NBA Players of the Week - Explanatory Visual Analysis¶

Created: October 27, 2018
Latest Update: November 2, 2018
By: Can Bekleyici - bekleydata.com
In [2]:
# toggle option for slides
from IPython.display import HTML

HTML('''
The raw code for this IPython notebook is by default hidden for easier reading.
To toggle on/off the raw code, click here.''')

Out[2]:
The raw code for this IPython notebook is by default hidden for easier reading. To toggle on/off the raw code, click here.

# Visual Data Analysis¶

For this project, I explored a dataset from kaggle, which contains every Player of the Week awarded between the NBA seasons 1984/85 and 2017/18. I focused heavily on visual data exploration and divided them into three stages of exploratory iteration.

## Age, Height, Weight, and Draft Year Distributions¶

Looking at the modes, the most awarded fictional player would be 25 years old, about 205-210 centimeters tall, and would weigh between 100-105 kilograms. Ideally, he would have been drafted around 1985 or 1998.

In [20]:
# plot the distributions
plt.figure(figsize= [16, 8])
subplots(df2, 1, 'Age', 'Age in years', 20.5, 35.5, 1)
subplots(df2, 2, 'Height', 'Height in cm', 170, 225, 5)
subplots(df2, 3, 'Weight', 'Weight in kg', 70, 140, 5)
subplots(df2, 4, 'Draft Year', 'Draft Year', 1969, 2018, 3);

# Distribution of Positions¶

By looking at the chart below, it is clear that Guards (G) have been awarded 'Player of the Week' the most times from 1985 to 2018, while Guard-forwards (GF) have been awarded the least times.

In [21]:
# Clean inconsistent Position values
df['Position'] = df['Position'].str.replace('F-C', 'FC')
df['Position'] = df['Position'].str.replace('G-F', 'GF')

# position distribution of all
order = df['Position'].value_counts().index.values
plt.figure(figsize= [14, 7])
plt.gca().axes.get_yaxis().set_visible(False)
sb.despine(left=True, top=True, right=True)
plt.title('Number of rewarded NBA Players of the Week by Position',  fontdict={'size':16})
splot = sb.countplot(data=df, x='Position', color='#2b3f48', order=order)
plt.xlabel('Playing Position', size=14)
for p in splot.patches:
splot.annotate(format(p.get_height(), '.0f'), (p.get_x() + p.get_width() / 2., p.get_height()),
ha = 'center', va = 'center', xytext = (0, -20), textcoords = 'offset points', color = 'w', size=13);

# What are the Relations?¶

In this step, I looked deeper into the relations between different variables. An interesting question would be, which position had which individual features (e.g. height, weight, age) to be awarded. Is there a difference in the draft years in terms of the number of positions that have been awarded?

# BMI by Position¶

By setting each players bmi's in relation to their positions, a distinct pattern for each position is revealed. While the median bmi for awarded Shooting-Guards (SG) is about 24, the top performing Forward-Centers (FC) have a higher middle bmi of about 26, which means that the median player has more weight per height. The bmi count is narrower and therefore wider distributed for Centers (C) and Power-Forwards (PF) than for Guards (G) or Guard-Forwards (GF).

In [25]:
# Clean inconsistent Position values
df2['Position'] = df2['Position'].str.replace('F-C', 'FC')
df2['Position'] = df2['Position'].str.replace('G-F', 'GF')

# Creates a violin plot ordered by the median
orders = df2.groupby('Position')['bmi'].median().sort_values(ascending=False).index.values
plt.figure(figsize=[14, 7])
sb.violinplot(data = df2, x = 'Position', y = 'bmi', order=orders, color='#dbbb8d', inner = 'quartile')
sb.despine()
plt.title('Bmi Distribution by Position - NBA Players of the Week', fontdict={'size':16})
plt.xlabel('Playing Position', size=13)
plt.ylabel('Bmi', size=13);

# How does these relations change over time?¶

Are there any changes regarding the height and weight of the PotW throughout the years? What about the bmi distributions for each Position?

# Players Weight on Height by Years¶

From the visualization below, one can see that players that have been drafted more recently had higher weight on height ratios (or higher bmi's), because they tend to be south of the (imaginary) regression line.

In [26]:
# creates a 3 dimensional scatter plot
plt.figure(figsize=[14,8])
cmap = plt.get_cmap('mako', df2['Draft Year'].nunique())
plt.scatter(data = df2, x = 'Weight', y = 'Height', c = 'Draft Year', cmap=cmap, s=100)
sb.despine()
plt.colorbar()
plt.title('Weight vs Height by Draft Year - NBA Players of the Week', fontdict={'size': 16})
plt.xlabel('Weight in kg', size=13)
plt.ylabel('Height in cm', size=13);

# BMI Distribution for each Position over Time¶

The changes of bmi over time for each player of the week's position over time reveals some very interesting insights. Despite the overall trend of higher bmi's for the top performing players, the BMI 10 year average of Forwards (F), Centers (C), and Forward-Centers have actually decreased for players have been drafted more recently.

In [30]:
def mean_poly(x, y, bins = 10, **kwargs):
""" Custom adapted line plot code. """
# set bin edges if none or int specified
if type(bins) == int:
bins = np.linspace(x.min(), x.max(), bins+1)
bin_centers = (bin_edges[1:] + bin_edges[:-1]) / 2

# compute counts
data_bins = pd.cut(x, bins, right = False,
include_lowest = True)
means = y.groupby(data_bins).mean()

# create plot
plt.errorbar(x = bin_centers, y = means, **kwargs)

bin_edges = np.arange(1970, df['Draft Year'].max()+10, 10)
g = sb.FacetGrid(data = df, hue = 'Position', palette='Set2', size = 8, aspect=2)
g.map(mean_poly, "Draft Year", "bmi", linewidth=7, alpha=0.8, bins = bin_edges)
g.set_ylabels('mean(bmi)')
g.set_xlabels('Draft Year', fontsize=15)
g.set_ylabels('Bmi Average', fontsize=15)
plt.title('Bmi 10 Year Averages for each Position by Draft Years - NBA Players of the Week', fontsize=24);

# Conclusion¶

The visual exploration of the dataset revealed, that players in the position of a guard have been chosen the most times as 'Player of the Week' in the NBA.

The body mass index (bmi) of each awarded player have been the highest on average for Forward-Centers and the lowest on average for Shooting-Guards.

Players that have been drafted more recently tend to have a higher bmi on average than players that have been drafted 10-50 years ago.

Despite the overall trend of higher bmi's for the top performing players, the BMI 10 year average of Forwards (F), Centers (C), and Forward-Centers have actually decreased for players have been drafted more recently.


text:
# Appendix without heading, only TOC entry

I'd like to add an appendix to the TOC without showing its headline over the continuous text.

Minimal Example:

\documentclass{scrartcl}
\begin{document}
\tableofcontents

\section{A Section}
Lorem ipsum.

\appendix

This section should not have a headline.
\end{document}

An idea would be using \addcontentsline instead of \section but then I lose the foregoing "B.":

\addcontentsline{toc}{section}{Appendix without Headline}

It's a somewhat strange requirement, but here you go. (Note that this does not take care of the numbering of additional "visible" sections.)

\documentclass{scrartcl}
\begin{document}
\tableofcontents

\section{A Section}
Lorem ipsum.

\appendix


text:
# DAVIDSTUTZ

Check out our latest research on weakly-supervised 3D shape completion.

## The Mathematics of Variational Auto-Encoders

As part of my master thesis, I made heavy use of variational auto-encoders in order to learn latent spaces of shapes — to later perform shape completion. Overall, I invested a big portion of my time in understanding and implementing different variants of variational auto-encoders. This article, a first in a small series, will deal with the mathematics behind variational auto-encoders. The article covers variational inference in general, the concrete case of variational auto-encoder as well as practical considerations.

In general, a variational auto-encoder [] is an implementation of the more general continuous latent variable model. While I used variational auto-encoders to learn a latent space of shapes, they have a wide range of applications — including image, video or shape generation. With this article, I want to start a small series on variational auto-encoders; this article starts discussing the mathematical background of variational inference and variational auto-encoders. However, other parts of the series will cover topics including:

• Denoising variational auto-encoders [];
• Variational auto-encoders with discrete latent space (i.e. categorical/Bernoulli latent variables) [][];
• 2D and 3D Torch implementations of variational auto-encoders including practical tips;

Take a seat and — depending on your background — roughly 30 minutes and I will introduce you to the mathematical framework of variational inference and variational auto-encoders.

Prerequisites. This article is not too heavy on mathematics; however, basic probability theory and linear algebra is necessary for a deeper understanding - including conditional probability, Gaussian distributions, Bernoulli distributions and expectations. Furthermore, the article assumes a basic understanding of (convolutional) neural networks and network training; terms such as cross entropy error or convolutional layers should not be new.

To get a basic understanding of (convolutional) neural networks, have a look at my seminar papers:

### Overview

In general, a continuous latent variable model is intended to learn a latent space $\mathcal{Z} = \mathbb{R}^Q$ using a given set of samples $\{y_m\} \subseteq \mathcal{Y} = \mathbb{R}^R$ where $Q \ll R$, i.e. a dimensionality reduction. The model consists of two components which are summarized using the graphical models shown in Figure 1. In particular, the model consists of the generative model $p(y | z)$ given a fixed prior $p(z)$, and the recognition (inference) model $q(z | y)$. For simple Gaussian models, the recognition model $q(z | y)$ as well as the so-called marginal likelihood

$p(y) = \int p(y, z) dz = \int p(y | z)p(z) dz$(1)

can usually be determined and maximized analytically, e.g. for probabilistic Principal Component Analysis (PCA) []. In the case of variational auto-encoders, both the generative and the recognition model are implemented using neural networks. Then, the recognition model as well as the marginal likelihood can only be approximated -- mainly because the integral in Equation (1) becomes intractable. We are going to follow [] and [] to introduce the framework of variational inference which allows to maximize a lower bound on the likelihood.

### Variational Inference

In general, variational inference is posed as the problem of finding a model distribution $q(z)$ to approximate the true posterior $p(z | y)$

$q(z) = \arg\min_{q} \text{KL}(q(z) | p(z | y))$(2)

where the Kullback-Leibler divergence $\text{KL}$ is a distance measure defined on probability distributions. The Kullback-Leibler divergence can then be rewritten to obtain a lower bound on the intractable marginal likelihood $p(y)$. The Kullback-Leibler divergence is formally defined as:

Definition. The Kullback-Leibler divergence between two probability distributions $q(z)$ and $p(z|y)$ is defined as:

$\text{KL}(q(z) | p(z|y)) = \mathbb{E}_{q(z)}\left[\ln\frac{q(z)}{p(z|y)}\right]$

where $\mathbb{E}_{q(z)}$ denotes the expectation with respect to the distribution $q(z)$.

A close look at the Kullback-Leibler divergence reveals that the optimization problem in Equation (2) involves computing the marginal likelihood:

$\text{KL}(q(z) | p(z|y)) = \mathbb{E}_{q(z)}\left[\ln\frac{q(z)}{p(z|y)}\right]$

$= \mathbb{E}_{q(z)}[\ln q(z)] - \mathbb{E}_{q(z)}[\ln p(z|y)]$

$= \mathbb{E}_{q(z)}[\ln q(z)] - \mathbb{E}_{q(z)}[\ln p(z,y)] + \ln p(y)$.

Re-arranging left- and right-hand side leads to the evidence lower bound which is also referred to as variational lower bound:

$\ln p(y) = \text{KL}(q(z) | p(z|y)) - \mathbb{E}_{q(z)}[\ln q(z)] + \mathbb{E}_{q(z)}[\ln p(z,y)]$

$\geq - \mathbb{E}_{q(z)}[\ln q(z)] + \mathbb{E}_{q(z)}[\ln p(z,y)]$

$= - \mathbb{E}_{q(z)}[\ln q(z)] + \mathbb{E}_{q(z)}[\ln p(z)] + \mathbb{E}_{q(z)}[\ln p(y|z)]$

$= - \text{KL}(q(z) | p(z)) + \mathbb{E}_{q(z)}[\ln p(y|z)]$

The original problem of maximizing the intractable marginal likelihood $p(y)$ in Equation (1) is then approximated by maximizing the evidence lower bound which we formally define as:

Definition. The variational lower bound or evidence lower bound derived from Problem (2) is given by

$-\text{KL}(q(z) | p(z)) + \mathbb{E}_{q(z)}[\ln p(y|z)] = \mathbb{E}_{q(z)}\left[\ln\frac{p(y, z)}{q(z)}\right]$(3)

In this general formulation of the evidence lower bound, the model distribution $q(z)$ can be arbitrary. However, in the context of latent variable models, it makes sense to make the model distribution depend on $y$ explicitly, i.e. $q(z | y)$, as we want to be able to reconstruct any $y$ from the corresponding latent code $z$. Then, the evidence lower bound in Equation (3) takes the form of an auto-encoder where $q(z | y)$ represents the encoder and $p(y | z)$ the decoder. Thus, a variational auto-encoder can be trained by maximizing the right-hand-side of Equation (3) after choosing suitable parameterizations for the distributions $p(z)$ and $q(z | y)$. In the following, we discuss the Gaussian case. I leave the discrete case, e.g. the Bernoulli case for a later article.

### Gaussian Variational Auto-Encoder

In the framework of variational auto-encoders, the model distribution $q(z|y)$ is implemented as neural network; similarly, in the case of Gaussian variational auto-encoders, both $q(z|x)$ and $p(z)$ are assumed to be Gaussian, specifically

$q(z|y) = \mathcal{N}(z; \mu(y;w), \text{diag}(\sigma^2(y;w)))$

$p(z) = \mathcal{N}(z; 0, I_Q)$

where the dependence on the neural network weights $w$ is made explicit; i.e. both the mean $\mu(y;w) \in \mathbb{R}^Q$ and the (diagonal) covariance matrix $\text{diag}(\sigma^2(y;w)) \in\mathbb{R}^{Q \times Q}$ are predicted using a neural network with parameters $w$ that are to be optimized. Here, $I_Q \in \mathbb{R}^{Q \times Q}$ denotes the identity matrix and $\mathcal{N}$ refers to the Gaussian distribution. In the following we will neglect the weights $w$ for brevity.

Considering the evidence lower bound from Equation (3), i.e.

(3)$= - \text{KL}(q(z|y)|p(z)) + \mathbb{E}_{q(z|y)}[\ln p(y|z)]$

the Kullback-Leibler divergence between $q(z|y)$ and $p(z)$ can be computed analytically. However, differentiating the lower bound with respect to the hidden weights in $q(z|y)$ is problematic. A classical Monte-Carlo approximation of the expectation $\mathbb{E}_{q(z|y)}[\ln p(y|z)]$ would require differentiating through the sampling process $z \sim q(z | y)$. Therefore, Kingma and Welling [] proposed the so-called reparameterization trick. In particular, the random variable $z \sim q(z|y)$ is reparameterized using a differentiable transformation $g(z, \epsilon)$ based on an auxiliary variable $\epsilon$ drawn from a unit Gaussian:

$z_i = g_i(y, \epsilon_i) = \mu_i(y) + \epsilon_i \sigma_i^2(y)$(4)

with $\epsilon_i \sim \mathcal{N}(\epsilon; 0, 1)$. Overall, given a sample $y_m \in \mathbb{R}^R$, the objective to be minimized has the form

$\mathcal{L}_{\text{VAE}} (w) = \text{KL}(q(z|y_m) | p(z)) - \frac{1}{L}\sum_{l = 1}^L \ln p(y_m | z_{l,m})$

where $z_{l,m} = g(\epsilon_{l,m}, y)$ and $\epsilon_{l,m} \sim \mathcal{N}(\epsilon ; 0, I_Q)$. Here, $L$ is the number of samples to use for the Monte Carlo estimator of the reconstruction error. In practice, the loss $\mathcal{L}_{\text{VAE}}$ is applied on mini-batches and $L = 1$ is usually sufficient.

Given the neural network weights $w$, the generation process can be summarized as follows: draw $z \sim p(z) = \mathcal{N}(z ; 0,I_Q)$, and draw $y \sim p(y|z)$. Similarly, recognition is performed by drawing $z \sim q(z|y)$, i.e. $\epsilon \sim \mathcal{N}(\epsilon ; 0,I_Q)$ and $z = g(y,\epsilon)$. For evaluation purposes, i.e. for measuring the recognition performance, $z$ is usually set to $z = \mathbb{E}_{q(z|y)}[z]$; this can be accomplished by directly considering $\mu(y)$.

### Practical Considerations

As mentioned above, the Kullback-Leibler divergence of two Gaussian distributions can easily be calculated directly. As we consider diagonal covariance matrices the Kullback-Leibler divergence is separable over $1 \leq i \leq Q$. Then, we have (see [])

$\text{KL}(\mathcal{N}(z_i ; \mu_{1,i}, \sigma_{1,i}^2)|\mathcal{N}(z_i ; \mu_{2,i},\sigma_{2,i}^2)) = \frac{1}{2} \ln\frac{\sigma_{2,i}}{\sigma_{1,i}} + \frac{\sigma_{1,i}^2}{2\sigma_{2,i}^2} + \frac{(\mu_{1,i} - \mu_{2,i})^2}{2 \sigma_{2,i}^2} - \frac{1}{2}.$

and with $\sigma_{2,i} = 1$ and $\mu_{2,i} = 0$ (and for simplicity $\sigma_i^2 := \sigma_i^2(y) = \sigma_{1,i}^2$ and $\mu_i := \mu_i(y) = \mu_{1,i}$) it follows:

$\text{KL}(p(z_i | y) | p(z_i)) = - \frac{1}{2} \ln \sigma_i + \frac{1}{2} \sigma_i^2 + \frac{1}{2} \mu_i^2 - \frac{1}{2}$.

The remaining part of the objective is the reconstruction error, i.e. the negative log-likelihood $- \ln p(y | z_l)$. This depends on how $p(y|z)$ is modeled; for images or volumes, for example, $p(y | z)$ might decompose over pixels/voxels (where we assume the $R$ to be the dimensionality of the vectorized image/volume):

$p(y|z) = \prod_{i = 1}^R p(y_i | z)\quad\Rightarrow\quad -\ln p(y | z) = -\sum_{i = 1}^R \ln p(y_i | z)$

For binary images or volumes, $p(y_i | z)$ can be modeled as Bernoulli distribution, i.e. $p(y_i | z) = \text{Ber}(y_i ; \theta_i(z))$ where $\theta_i(z)$ is the probability of a pixel/voxel being $1$. These probabilities are directly predicted by the decoder. The negative log-likelihood then reduces to the binary cross entropy error. Otherwise, for continuous output, for example color per pixel/voxel, Gaussian distributions can be used, i.e. $p(y_i | z) = \mathcal{N}(y_i; \mu_i(z), \sigma^2)$ where the $\mu_i(z)$'s are predicted by the decoder and the variance $\sigma^2$ is fixed. In this case, the negative log-likelihood leads to the sum-of-squared error.

A possible implementation of a variational auto-encoder is shown in Figure 2. Here, the Kullback-Leibler divergence is implemented as layer denoted $\text{KLD}$ and essentially computes the divergence and adds the corresponding derivatives to the gradient flow in the backward pass. Similarly, the reparameterization layer denoted as $\text{repa}$ implements Equation (4).

In practice, letting the encoder predict $\sigma^2(y)$ directly is problematic as the variance $\sigma^2(y)$ may not be negative. Therefore, we can let the encoder predict log-variances instead, i.e. $l_i (y) := \ln \sigma^2(y)$. This makes sure that the variance $\sigma^2(y) = \exp(l_i (y))$ will always be positive. The Kullback-Leibler divergence and the corresponding derivative with respect to $l_i (y)$ as well as the reparameterization trick are easily adapted.

Training a Gaussian variational auto-encoder in practice means balancing the -- possibly conflicting -- objectives corresponding to reconstruction loss and Kullback-Leibler divergence. While the reconstruction loss is intuitively interpretable, e.g. as binary cross entropy error or scaled sum-of-squared error in the Bernoulli and Gaussian cases, respectively, the Kullback-Leibler divergence is less intuitive. Therefore, it might be beneficial to monitor the latent space statistics over a held-out validation set. Concretely, this means monitoring the first two moments of the predicted means, i.e.

$\overline{\mu} = \frac{1}{QM'} \sum_{i = 1}^Q\sum_{m = 1}^{M'} \mu_i(y_m)$

and

$\text{Var}[\mu] = \frac{1}{QM'}\sum_{i = 1}^Q\sum_{m = 1}^{M'} (\mu_i(y_m) - \overline{\mu})^2$

where $M'$ is the size of the validation set. Additionally, it is worth monitoring the first moment of the predicted log-variances

$\overline{l} = \frac{1}{QM'} \sum_{i = 1}^Q\sum_{m = 1}^{M'} \ln \sigma_i^2(y_m) = \frac{1}{QM'} \sum_{i = 1}^Q\sum_{m = 1}^{M'} l_i(y_m).$

Note that these statistics are computed over all $Q$ dimensions of the latent space -- this is appropriate as the prior $p(z)$ is a unit Gaussian such that all dimensions can be treated equally. During training, these statistics should converge to a unit Gaussian, i.e. $\overline{\mu} \approx 0$ and $\text{Var}[\mu] \approx 1$, and the network should be capable of making certain predictions, i.e. small $\overline{l}$.

### Outlook

The next two articles will cover two interesting extensions of vanilla variational auto-encoders as introduced in []:

• Denoising variational auto-encoders [];
• and variational auto-encoders with categorical/Bernoulli latent variables [][].

Afterwards, we will discuss implementations and practical tips in Torch.

• [] D. M. Blei, A. Kucukelbir, and J. D. McAuliffe. Variational inference: A review for statisticians. CoRR, abs/1601.00670, 2016.
• [] C. Doersch. Tutorial on variational autoencoders. CoRR, abs/1606.05908, 2016.
• [] D. P. Kingma and M. Welling. Auto-encoding variational bayes. CoRR, abs/1312.6114, 2013.
• [] S. Kullback. Information Theory and Statistics. Wiley, New York, 1959.
• [] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Journal of the Royal Statistical Society, Series B, 61:611-622, 1999.
• [] D. J. Im, S. Ahn, R. Memisevic, and Y. Bengio. Denoising criterion for variational auto-encoding framework. In AAAI Conference on Artificial Intelligence, pages 2059-2065, 2017.
• [] E. Jang, S. Gu, and B. Poole. Categorical reparameterization with gumbel-softmax. CoRR, abs/1611.01144, 2016.
• [] C. J. Maddison, A. Mnih, and Y. W. Teh. The concrete distribution: A continuous relaxation of discrete random variables. CoRR, abs/1611.00712, 2016.

What is your opinion on this article? Did you find it interesting or useful? Let me know your thoughts in the comments below or get in touch with me:


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# Learning from small number of sources

I have data from two sources; the two data sets were obtained from different geographical locations and at different historical times. I do not want to pool them into one data set sample and lose information about differing values of parameters that generated those collections, but I also cannot use Bayesian hierarchical model, since there would be just two unobservables from which I could not learn anything about hyperparameters. Is there any possible models that could be used in such a problems?

I know that the question is a bit vague, but I am open to various possibilities.
EDIT
The data are counts, distributed as Poisson with intensities $\lambda _{1}$ and $\lambda _{2}$. Suppose I have two data samples that comes from similar objects and represents number of failures over some period of time. I want to estimate both rates. I don't want to pool them because, for example, the operating conditions (environmental conditions) were different for those two data sources so that intensities were not identical. But I want to be able to borrow some information from each data sample like in hierarchical Bayesian model. But, as I already said, for hierarchical model there is just not enough data sources.

-
 Well, you should at least tell what type of data you have and what to want to do with it... – nico Sep 18 '12 at 17:52 Updated the question. Hope, it will be more informative. – Tomas Sep 18 '12 at 18:03 I did a bunch of editing; I hope it is now what you wanted to ask – Peter Flom Sep 18 '12 at 23:34 Where's the hierarchy? Why Bayesian? If you know that a variable is distributed as Poisson, it is simple to estimate the mean - just find the mean. – Peter Flom Sep 18 '12 at 23:36 Thanks for those few edits. I think it was clear that I do not want to estimate the mean in the simplest way, i.e. by pooling data into one sample. If I would have more data sources, say 10 or 20, then I would just use Bayesian hierarchical model (why Bayesian? Because for me it is the most natural way to do statistical inference. Why hierarchical? Because it is very nice way to account for cour-to-source variability). But hierarchical for two data sources would be of no use. That is way I am looking for another posible way to analyze these two data sets without complete pooling. – Tomas Sep 19 '12 at 7:06


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# String Methods in Regex Find and Replace

227
views
0
6 months ago by
I'm trying to convert between variable naming conventions, from:

exampleVariable

to:

example_variable

I can find the relevant variables with the regex search "([a-z])([A-Z])", and use "\1_\2" to convert to "example_Variable", but to convert to the desired convention using re.sub() would require something like:

re.sub('([a-z])([A-Z])',lambda match: match.group(1)+"_"+match.group(2).lower(),'exampleVariable')

However, simply placing the inline function in Wing's replace line doesn't achieve the desired effect. Is there an alternative way to accomplish what I'm looking to do?


text:
Chord length

• MHB
Gold Member
MHB
View attachment 8904

Ok this should be just an observation solution ..
But isn't the equation for chord length
$$2r\sin{\frac{\theta}{2}}= \textit{chord length}$$

Don't see any of the options
Derived from that..

Attachments

• 20190315_092610.jpg
7.3 KB · Views: 40

Olinguito
Hi karush.

All the information is there. You just need to calculate the angle $\theta$ from the arc length $s=r\theta$.

Gold Member
MHB
Hi karush.

All the information is there. You just need to calculate the angle $\theta$ from the arc length $s=r\theta$.

It is asking for chord lenght!

Gold Member
MHB
I would start with the arc-length formula to find the subtended angle:

$$\displaystyle \theta=\frac{s}{r}=\frac{3.4}{2}=1.7$$

Then, use the law of cosines:

$$\displaystyle \overline{DF}=\sqrt{2^2+2^2-2(2)(2)\cos(1.7)}=2\sqrt{2-2\cos(1.7)}$$

Lastly, a double-angle identity for cosine:

$$\displaystyle \overline{DF}=2\sqrt{4\sin^2(0.85)}=4\sin(0.85)$$

Gold Member
MHB
Ok
So that's where .85 comes from
So then it's D

Olinguito
It is asking for chord lenght!

And to do that, you need to know the angle $\theta$, don’t you? Calculate that first!

Gold Member
MHB
Why of course we do!

However for this SAT question
We should know that 1.7 is not the $\theta$ we need
and we have use sin $\theta$
That was because you were given arc length $s=3.4$ (and radius $r=2$) but not $\theta$. I was therefore instructing you to compute $\theta$ from the formula $s=r\theta$ so you could use it in the formula $2r\sin\dfrac{\theta}2$ for the chord length.


text:
1. ## drawing histogram

Can someone help me with this question?

I have to draw a histogram for the following

Repair cost($1000s)....................Frequency less than 10.....................................3 atleast 10 but less than 20.................5 atleast 20 but less than 30.................7 atleast 30 but less than 40.................5 atleast 40 but less than 60.................3 atleast 60 but less than 100................2 and also estimate the mean and variance. thanks you very much. 2. Originally Posted by shellshock Can someone help me with this question? I have to draw a histogram for the following Repair cost($1000s)....................Frequency
less than 10.....................................3
atleast 10 but less than 20.................5
atleast 20 but less than 30.................7
atleast 30 but less than 40.................5
atleast 40 but less than 60.................3
atleast 60 but less than 100................2

and also estimate the mean and variance.

thanks you very much.
What is the problem? You are given the bins, and the frequencies what is stopping you from drawing the histogram?

CB


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# Fortran Wiki dshiftr

## Description

dshiftr(i, j, shift) combines bits of i and j. The leftmost shift bits of the result are the rightmost shift bits of j, and the remaining bits are the rightmost bits of i.

## Standard

Fortran 2008 and later

## Class

Elemental function

## Syntax

result = dshiftl(i, j, shift)

## Arguments

• i - Shall be of type integer.
• j - Shall be of type integer, and of the same kind as i.
• shift - Shall be of type integer.

## Return value

The return value has same type and kind as i.


text:
IIT JAM Follow
July 3, 2020 5:50 pm 30 pts
1.61. Let A be a real 3 x4 matrix of rank 2. Then the rank of A'A, where A' denotes the eranspose A, is: ) exactly 2 (ii) exactly 3 (ii) exactly 4 (iv) at most 2 but not necessarily 2.
• 0 Likes
• Shares
• Arzoo
see the attachment
• Vishal goswami
ans is options 1 rank(A)=rank(AA')=rank(A')
• Deepak singh
rank(A) = rank (AtA) , so rank (AtA) is less than or equal to 2 . so option 4 is correct.
• Ujjawal vishal
rank (AtA)= rank (A) always


text:
# Related rates - finding hypotenuse of triangle

1. Oct 29, 2016

### cmkluza

1. The problem statement, all variables and given/known data
A plane flying horizontally at an altitude of 3 mi and a speed of 480 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 4 mi away from the station. (Round your answer to the nearest whole number.)

2. Relevant equations
$a^2=b^2+c^2$ where $a$ is the hypotenuse of a triangle.

3. The attempt at a solution
I started by relating the given variables as follows
• Altitude is a constant of 3 mi
• Horizontal distance of the plane will be $x$. We measure when $x=4$ miles.
• Distance from the plane to the radar station will be $y$. We measure when $y^2=4^2+3^2 \longrightarrow y=5$ miles.
• Change in horizontal distance will be $\frac{dx}{dt}$ We are given that this is 480 miles/hour.
We have $y^2=x^2+3^2$. Taking the derivative with respect to time gives $2y\times\frac{dy}{dt}=2x\times\frac{dx}{dt}+0$. Substituting in known values gives: $2(5)\times\frac{dy}{dt}=2(4)(480) \longrightarrow \frac{dy}{dt}=\frac{4\times480}{5}=384$. Yet this is not the answer.

Where am I going wrong? I've actually drawn out the triangle and variables, and I'm fairly stuck as to which part I'm messing up. Any insight will be appreciated!

2. Oct 29, 2016

### phinds

How did "4 mi away from the station" become only the horizontal component of how far the plane is away from the station?

3. Oct 29, 2016

### cmkluza

Looks like I totally misread the question. I thought that the problem statement stated 4 to be the distance traveled in the x direction, not the distance between the plane and the station. That's cleared everything up.

Using the above information, the value for $x$ is $\sqrt{4^2-3^2}=\sqrt{7}$. Plugging the correct values into my above derivative gives the correct answer to this problem: $2(4)\times\frac{dy}{dt}=2(\sqrt{7})(480)\longrightarrow\frac{dy}{dt}=\frac{\sqrt{7}\times480}{4}\approx317$.

Thanks for pointing that out!


text:
Related Tags

algorithms
python
strings

# Longest palindromic substring in O(n) with Manacher's Algorithm

## Problem statement

A palindrome is a string that reads the same backward or forwards. For example, madam or racecar are some famous palindromic strings. We are interested in finding the length of the longest palindromic substring in a string. For example, although the string abaccab has multiple palindromic substrings, like aba and acca, the longest palindromic substring in it is baccab. This is what we are interested in finding.

## Intuition behind Manacher’s algorithm

Although there is a brute force and dynamic programming algorithm that can this problem in O($n^3$) and O($n^2$) respectively, Manacher’s algorithm brings down the time complexity of the algorithm to an astounding O($n$). The reason for this improvement is that Manacher’s algorithm very smartly leverages properties of a palindromic string to avoid needless computation. Let’s look at an example to understand these properties.

Consider a string that contains a palindromic substring of odd length centered around index c. Half of the characters of the substring are to the right of c and the other half are to the left. For example, in the illustration below, we can see a palindromic string with a center around c highlighted in a darker shade of green. The second array shows the number of matching characters if we were to find a palindrome centered around a specific character. For example, the palindrome around the first a is cac and only one character from either side match.

Number of matching characters around c is 2

If we wanted to find the palindrome centered around the second a (pointed by the arrow), it would be the same as the first a. This is because we already know the above string is a palindrome around c, and thus, all the characters in either direction are symmetric. We call these two a's to be the mirror of each other. Similarly, both c's on either end are also each other’s mirrors.

Let’s update our example a little. Now, we have a string that includes the palindromic string we discussed above.

cacac is a palindromic substring centered around c in the above string

Now, we cannot say that the palindrome around the a, pointed by the arrow, will be the same as its mirror a, since mirror a's palindrome exceeds the range covered by the palindromic substring cacac. We can, however, say with certainty that the palindrome around a, pointed out by the arrow, would be valid as long as it is in the range of cacac. This means, although the palindrome around mirror a is bigger and exceeds the range of cacac, the part that lies within the range would still be in the other a's palindrome.

This was the intuition behind Manacher’s algorithm. Instead of finding the palindrome centered around each character from scratch, we can use the previous results.

## Implementation

We had an assumption that the length of the palindrome would be odd, as we discussed the intuition above. This gives us a nice center of the palindrome, so we can apply the above properties.

What about the cases for even length palindromes? We transform our string by placing a # intermediately before and after each character. For example, deed becomes #d#e#e#d# and madam becomes #m#a#d#a#m#. This also has another benefit; remember the array we were using to count the number of matching characters around a center? It will now have the length of the palindrome around each center. We call this array LPS in our implementation below. C is used for the current center and R is used to quantify the range of the palindrome centered around C.

Next, we iterate over the updated string, and at each iteration, we first use the discussed properties of a palindrome to find the minimum definite length of the palindrome around the i$^{th}$ character (lines 15-16). Then, we build on that value and expand outwards to find the actual length of the palindrome around the i$^{th}$ character (lines 18-19). Lastly, if the current palindrome becomes long enough to exceed the range of the palindrome centered around C, we update the C and R to the current palindrome’s center and range (lines 23-25).

In the end, the max value in the LPS list will denote the length of the longest palindromic substring.

Let’s look at a visualization to understand the algorithm.

Manacher("acaac")
1 of 18

## Code

def UpdatedString(string):
newString = ['#']
for char in string:
newString += [char, '#']
return ''.join(newString)

def Manachen(string):
string = UpdatedString(string)
LPS = [0 for _ in range(len(string))]
C = 0
R = 0

for i in range(len(string)):
iMirror = 2*C - i
if R > i:
LPS[i] = min(R-i, LPS[iMirror])
else:
LPS[i] = 0
try:
while string[i + 1 + LPS[i]] == string[i - 1 - LPS[i]]:
LPS[i] += 1
except:
pass

if i + LPS[i] > R:
C = i
R = i + LPS[i]

r, c = max(LPS), LPS.index(max(LPS))
print (string[c - r : c + r].replace("#",""))
return r

print(Manachen("acaac"))

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text:
# commuting the resolution of 1-dim singular locus and 0-dim singularities in a non isolated singularity of a surface

Let $X$ be a surface with a non isolated singularity $C = Sing(X)$ such that the curve $C$ has singularities itself. We can solve $Sing(X)$ by blowing up close points and by normalizing. Indeed, we can first solve the 1-dimensional singularities with only normalizations $\pi_1: X_1 \to X$ . In the surface $X_1$ the preimage of $C$ has at most isolated singularities which we can solve by $\pi_2:X_2 \to X_1$.

Another approach is to solve the 0-dimensional singularities by $\varphi_1:Y_1 \to X$, and then to finish with a one dimensional singular locus on $Y_1$ that we can be solve by normalization $\varphi_2:Y_2 \to Y_1$ such that $Y_2$ is smooth (Maybe some ADE singularities, but it would not matter).

I am wondering if the second approach is always possible, and if "commutes" with the first one in some way. I have this idea that normalization remove the 1-dimensional singularity $C$ without affecting the 0-dimensional ones, even if they are supported on $C$. Is this true? In that fantasy, we can "commute" those processes of solving 0-dimensional singularities, and solving 1-dimensional singularities.

I will appreciate any enlighting

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Normalization can definitely change isolated singularities as we well.

For example, $k[x,y,u,v]/(x,y)\cap(u,v)$ has an isolated singularity (this is not irreducible, but there are irreducible versions with analytically isomorphic singularities). However, it is not normal and the normalization is smooth.

On the other hand, if you first resolve the 0-dimensional singular locus (ie, the singular closed points) by blowing up and then do a normalization, you can have much worse than ADE singularities. In fact, you can have a surface with no isolated singularities whose normalization has any possible isolated singularity.

However, something like your thought should be possible I think, you just might have to do blow-ups of points over the 1-dimensional singular locus as well.

For example, see the notion of semiresolution especially in the works of János Kollár.

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Yes, sorry about that. –  Karl Schwede Mar 22 '12 at 11:28
What exactly do you mean by "solve the $0$-dimensional singularities"? If you mean
Here is an example to contemplate: Let $X=Z(xy^2-z^2)\subset \mathbb A^3$. This is normal crossing away from the origin, but a little worse than that at the origin. It turns out that the normalization of this is smooth, but for now suppose we don't know that. (My point is that you don't necessarily know that the normalization is smooth just by looking at the definition.)