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# zbMATH — the first resource for mathematics

## Magaard, Kay

Compute Distance To:
 Author ID: magaard.kay Published as: Magaard, Kay; Magaard, K. External Links: MGP · dblp · GND
 Documents Indexed: 69 Publications since 1988, including 1 Book Biographic References: 1 Publication
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#### Co-Authors

 4 single-authored 11 Le, Tung 7 Frohardt, Daniel E. 7 Gluck, David 7 Völklein, Helmut 6 Guralnick, Robert Michael 5 Shpectorov, Sergey Viktorovich 4 Tiep Pham Huu 3 Paolini, Alessandro 3 Parker, Christopher W. 3 Savin, Gordan 3 Waldecker, Rebecca 2 Aguiar, Marcelo 2 André, Carlos Alberto Martins 2 Benedetti, Carolina 2 Bergeron, Nantel 2 Chen, Zhi 2 Diaconis, Persi Warren 2 Fairbairn, Ben Thomas 2 Goodwin, Simon M. 2 Hendrickson, Anders O. F. 2 Himstedt, Frank 2 Hiss, Gerhard 2 Hsiao, Samuel K. 2 Isaacs, I. Martin 2 Jedwab, Andrea 2 Kantor, William M. 2 Karaali, Gizem 2 Lauve, Aaron 2 Li, Huilan 2 Malle, Gunter 2 Marberg, Eric 2 Novelli, Jean-Christophe 2 O’Brien, Eamonn A. 2 Pang, Amy 2 Saliola, Franco V. 2 Stroth, Gernot 2 Tevlin, Lenny 2 Thibon, Jean-Yves 2 Thiem, Nathaniel 2 Venkateswaran, Vidya 2 Vinroot, C. Ryan 2 Wilson, Robert Arnott 2 Yan, Ning 2 Zabrocki, Mike 1 Baumeister, Barbara 1 Breuer, Thomas 1 Brunat, Olivier 1 Cohen, Arjeh Marcel 1 Frohardt, Dan 1 Huang, Jing-Song 1 Husen, William J. 1 James, Adam 1 Lübeck, Frank 1 Marin, Ivan 1 Moori, Jamshid 1 Riese, Udo 1 Röhrle, Gerhard E. 1 Saxl, Jan 1 Schmid, Peter P. 1 Seress, Ákos 1 Shaska, Tanush 1 Strambach, Karl 1 Testerman, Donna M. 1 Tong-Viet, Hung P. 1 Wang, Gehao 1 Wiesend, Götz
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#### Serials

 13 Journal of Algebra 6 Communications in Algebra 4 Israel Journal of Mathematics 3 Proceedings of the London Mathematical Society. Third Series 3 Transactions of the American Mathematical Society 2 Journal of Pure and Applied Algebra 2 Journal für die Reine und Angewandte Mathematik 2 Forum Mathematicum 2 Journal of Group Theory 2 LMS Journal of Computation and Mathematics 2 Albanian Journal of Mathematics 1 Mathematics of Computation 1 Advances in Mathematics 1 Archiv der Mathematik 1 Compositio Mathematica 1 Geometriae Dedicata 1 Journal of Combinatorial Theory. Series A 1 Journal of the London Mathematical Society. Second Series 1 Manuscripta Mathematica 1 Memoirs of the American Mathematical Society 1 Pacific Journal of Mathematics 1 Proceedings of the American Mathematical Society 1 European Journal of Combinatorics 1 Journal of Symbolic Computation 1 Indagationes Mathematicae. New Series 1 Experimental Mathematics 1 Annals of Mathematics. Second Series 1 Journal of the Australian Mathematical Society 1 Bulletin of the Institute of Mathematics. Academia Sinica. New Series
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#### Fields

 58 Group theory and generalizations (20-XX) 16 Algebraic geometry (14-XX) 9 Functions of a complex variable (30-XX) 5 Combinatorics (05-XX) 5 Field theory and polynomials (12-XX) 4 Computer science (68-XX) 3 Number theory (11-XX) 2 Associative rings and algebras (16-XX) 2 Topological groups, Lie groups (22-XX) 1 Nonassociative rings and algebras (17-XX) 1 Geometry (51-XX) 1 Manifolds and cell complexes (57-XX)

#### Citations contained in zbMATH Open

57 Publications have been cited 436 times in 306 Documents Cited by Year
Cross characteristic representations of symplectic and unitary groups. Zbl 1025.20002
Guralnick, Robert M.; Magaard, Kay; Saxl, Jan; Pham Huu Tiep
2002
Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras. Zbl 1237.05208
Aguiar, Marcelo; André, Carlos; Benedetti, Carolina; Bergeron, Nantel; Chen, Zhi; Diaconis, Persi; Hendrickson, Anders; Hsiao, Samuel; Isaacs, I. Martin; Jedwab, Andrea; Johnson, Kenneth; Karaali, Gizem; Lauve, Aaron; Le, Tung; Lewis, Stephen; Li, Huilan; Magaard, Kay; Marberg, Eric; Novelli, Jean-Christophe; Pang, Amy; Saliola, Franco; Tevlin, Lenny; Thibon, Jean-Yves; Thiem, Nathaniel; Venkateswaran, Vidya; Vinroot, C. Ryan; Yan, Ning; Zabrocki, Mike
2012
On the minimal degree of a primitive permutation group. Zbl 0911.20003
Guralnick, Robert; Magaard, Kay
1998
The solution of the $$k(GV)$$-problem. Zbl 1058.20010
Gluck, David; Magaard, Kay; Riese, Udo; Schmid, Peter
2004
Irreducibility of tensor squares, symmetric squares and alternating squares. Zbl 1072.20013
Magaard, Kay; Malle, Gunter; Tiep, Pham Huu
2002
Base sizes and regular orbits for coprime affine permutation groups. Zbl 0951.20003
Gluck, David; Magaard, Kay
1999
Irreducible tensor products of representations of finite quasi-simple groups of Lie type. Zbl 0992.20009
Magaard, Kay; Pham Huu Tiep
2001
Irreducibility of alternating and symmetric squares. Zbl 0919.20009
Magaard, Kay; Malle, Gunter
1998
Exceptional $$\Theta$$-correspondence. I. Zbl 0878.22011
Magaard, K.; Savin, G.
1997
Composition factors of monodromy groups. Zbl 1004.20001
Frohardt, Daniel; Magaard, Kay
2001
Character degree sums in finite nonsolvable groups. Zbl 1242.20012
Magaard, Kay; Tong-Viet, Hung P.
2011
Character and fixed point ratios in finite classical groups. Zbl 0865.20012
Gluck, David; Magaard, Kay
1995
Imprimitive irreducible modules for finite quasisimple groups. Zbl 1378.20002
Hiss, Gerhard; Husen, William J.; Magaard, Kay
2015
The monodromy group of a function on a general curve. Zbl 1075.14027
Magaard, Kay; Völklein, Helmut
2004
Characters of the Sylow $$p$$-subgroups of the Chevalley groups $$D_4(p^n)$$. Zbl 1242.20010
Himstedt, Frank; Le, Tung; Magaard, Kay
2011
A GAP package for braid orbit computation and applications. Zbl 1068.12002
Magaard, Kay; Shpectorov, Sergey; Völklein, Helmut
2003
Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces. Zbl 1286.20013
Fairbairn, Ben; Magaard, Kay; Parker, Christopher
2013
Constructive recognition of $$\text{SL}_3(q)$$. Zbl 1155.20048
Lübeck, F.; Magaard, K.; O’Brien, E. A.
2007
On the characters of the Sylow $$p$$-subgroups of untwisted Chevalley groups $$Y_{n}(p^{a})$$. Zbl 1360.20008
Himstedt, Frank; Le, Tung; Magaard, Kay
2016
On the character degrees of Sylow $$p$$-subgroups of Chevalley groups $$G(p^f)$$ of type $$E$$. Zbl 1323.20013
Le, Tung; Magaard, Kay
2015
Constructing characters of Sylow $$p$$-subgroups of finite Chevalley groups. Zbl 1370.20006
Goodwin, Simon M.; Le, Tung; Magaard, Kay; Paolini, Alessandro
2016
The extraspecial case of the $$k(GV)$$ problem. Zbl 0984.20010
Gluck, David; Magaard, Kay
2002
Monodromy and sporadic groups. Zbl 0809.20008
Magaard, Kay
1993
Grassmannian fixed point ratios. Zbl 0970.20003
Frohardt, Daniel; Magaard, Kay
2000
Black box exceptional groups of Lie type. Zbl 1285.20012
Kantor, W. M.; Magaard, K.
2013
Recognition of small dimensional representations of general linear groups. Zbl 1216.20005
Magaard, Kay; O’Brien, E. A.; Seress, Ákos
2008
Black box exceptional groups of Lie type. II. Zbl 1304.20020
Kantor, William M.; Magaard, Kay
2015
Symmetric and alternating powers of Weil representations of finite symplectic groups. Zbl 07069218
Guralnick, Robert M.; Magaard, Kay; Tiep, Pham Huu
2018
Genus 0 actions of groups of Lie rank 1. Zbl 1065.20001
Frohardt, Dan; Guralnick, Robert; Magaard, Kay
2002
Unipotent representations of $$G_2$$ arising from the minimal representation of $$D^E_4$$. Zbl 1065.22012
Huang, Jing-Song; Magaard, Kay; Savin, Gordan
1998
The generic character table of a Sylow $$p$$-subgroup of a finite Chevalley group of type $$D_4$$. Zbl 06812802
Goodwin, Simon M.; Le, Tung; Magaard, Kay
2017
Transitive permutation groups where nontrivial elements have at most two fixed points. Zbl 1312.20001
Magaard, Kay; Waldecker, Rebecca
2015
Fixed point ratios in exceptional groups of rank at most two. Zbl 1019.20011
Frohardt, Daniel; Magaard, Kay
2002
The $$k(GV)$$ conjecture for modules in characteristic 31. Zbl 1007.20009
Gluck, David; Magaard, Kay
2002
Affine distance-transitive graphs: The cross characteristic case. Zbl 0941.05038
Cohen, Arjeh M.; Magaard, Kay; Shpectorov, Sergey
1999
Generating sets of affine groups of low genus. Zbl 1317.14066
Magaard, Kay; Shpectorov, Sergey; Wang, Gehao
2012
Corrigendum: Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces. Zbl 1287.20019
Fairbairn, Ben; Magaard, Kay; Parker, Christopher
2013
On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups. Zbl 1287.20017
Magaard, Kay; Röhrle, Gerhard; Testerman, Donna M.
2013
Cross-characteristic character and fixed point space ratios for groups of Lie type. Zbl 0912.20009
Gluck, David; Magaard, Kay
1998
On Weierstrass points of Hurwitz curves. Zbl 1099.14022
Magaard, K.; Völklein, H.
2006
Incidence matrices, permutation characters, and the minimal genus of a permutation group. Zbl 1008.20002
Frohardt, Daniel; Guralnick, Robert; Magaard, Kay
2002
Some remarks on maximal subgroups of finite classical groups. Zbl 1448.20022
Magaard, Kay
2017
Genus 2 curves that admit a degree 5 map to an elliptic curve. Zbl 1174.14025
Magaard, Kay; Shaska, Tanush; Völklein, Helmut
2009
Bi-elliptic Weierstrass points on curves of genus 5. Zbl 1259.14038
Kato, T.; Magaard, K.; Völklein, H.
2011
The irreducible characters of the Sylow $$p$$-subgroups of the Chevalley groups $$\mathrm{D}_{6}(p^f)$$ and $$\mathrm{E}_{6}(p^f)$$. Zbl 1453.20025
Le, Tung; Magaard, Kay; Paolini, Alessandro
2019
Monodromy composition factors among exceptional groups of Lie type. Zbl 0828.20014
Frohardt, Daniel; Magaard, Kay
1993
On the irreducibility of alternating powers and symmetric squares. Zbl 0824.20012
Magaard, K.
1994
The combinatorics of degenerate covers and an application for general curves of genus 3. Zbl 1175.14017
Magaard, Kay; Völklein, Helmut; Wiesend, Götz
2008
Flag-stabilizers in the symmetric and alternating groups and their overgroup lattices. Zbl 0662.20005
Magaard, Kay
1988
Finite quotients of the pure symplectic braid group. Zbl 0926.12002
Magaard, Kay; Strambach, Karl; Völklein, Helmut
1998
Absolute fourth moments and finiteness of linear groups. Zbl 1106.20001
Gluck, David; Magaard, Kay
2006
Algorithmic construction of Chevalley bases. Zbl 1344.17001
Magaard, K.; Wilson, R. A.
2012
Groups of even type which are not of even characteristic. I. Zbl 1355.20011
Magaard, Kay; Stroth, Gernot
2016
Imprimitive irreducible modules for finite quasisimple groups. II. Zbl 06993252
Hiss, Gerhard; Magaard, Kay
2019
The lift invariant distinguishes components of Hurwitz spaces for $$A_{5}$$. Zbl 1373.14028
James, Adam; Magaard, Kay; Shpectorov, Sergey
2015
Representations of unitriangular groups. Zbl 1268.20015
Le, Tung; Magaard, Kay
2012
Transitive permutation groups with trivial four point stabilizers. Zbl 1336.20001
Magaard, Kay; Waldecker, Rebecca
2015
The irreducible characters of the Sylow $$p$$-subgroups of the Chevalley groups $$\mathrm{D}_{6}(p^f)$$ and $$\mathrm{E}_{6}(p^f)$$. Zbl 1453.20025
Le, Tung; Magaard, Kay; Paolini, Alessandro
2019
Imprimitive irreducible modules for finite quasisimple groups. II. Zbl 06993252
Hiss, Gerhard; Magaard, Kay
2019
Symmetric and alternating powers of Weil representations of finite symplectic groups. Zbl 07069218
Guralnick, Robert M.; Magaard, Kay; Tiep, Pham Huu
2018
The generic character table of a Sylow $$p$$-subgroup of a finite Chevalley group of type $$D_4$$. Zbl 06812802
Goodwin, Simon M.; Le, Tung; Magaard, Kay
2017
Some remarks on maximal subgroups of finite classical groups. Zbl 1448.20022
Magaard, Kay
2017
On the characters of the Sylow $$p$$-subgroups of untwisted Chevalley groups $$Y_{n}(p^{a})$$. Zbl 1360.20008
Himstedt, Frank; Le, Tung; Magaard, Kay
2016
Constructing characters of Sylow $$p$$-subgroups of finite Chevalley groups. Zbl 1370.20006
Goodwin, Simon M.; Le, Tung; Magaard, Kay; Paolini, Alessandro
2016
Groups of even type which are not of even characteristic. I. Zbl 1355.20011
Magaard, Kay; Stroth, Gernot
2016
Imprimitive irreducible modules for finite quasisimple groups. Zbl 1378.20002
Hiss, Gerhard; Husen, William J.; Magaard, Kay
2015
On the character degrees of Sylow $$p$$-subgroups of Chevalley groups $$G(p^f)$$ of type $$E$$. Zbl 1323.20013
Le, Tung; Magaard, Kay
2015
Black box exceptional groups of Lie type. II. Zbl 1304.20020
Kantor, William M.; Magaard, Kay
2015
Transitive permutation groups where nontrivial elements have at most two fixed points. Zbl 1312.20001
Magaard, Kay; Waldecker, Rebecca
2015
The lift invariant distinguishes components of Hurwitz spaces for $$A_{5}$$. Zbl 1373.14028
James, Adam; Magaard, Kay; Shpectorov, Sergey
2015
Transitive permutation groups with trivial four point stabilizers. Zbl 1336.20001
Magaard, Kay; Waldecker, Rebecca
2015
Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces. Zbl 1286.20013
Fairbairn, Ben; Magaard, Kay; Parker, Christopher
2013
Black box exceptional groups of Lie type. Zbl 1285.20012
Kantor, W. M.; Magaard, K.
2013
Corrigendum: Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces. Zbl 1287.20019
Fairbairn, Ben; Magaard, Kay; Parker, Christopher
2013
On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups. Zbl 1287.20017
Magaard, Kay; Röhrle, Gerhard; Testerman, Donna M.
2013
Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras. Zbl 1237.05208
Aguiar, Marcelo; André, Carlos; Benedetti, Carolina; Bergeron, Nantel; Chen, Zhi; Diaconis, Persi; Hendrickson, Anders; Hsiao, Samuel; Isaacs, I. Martin; Jedwab, Andrea; Johnson, Kenneth; Karaali, Gizem; Lauve, Aaron; Le, Tung; Lewis, Stephen; Li, Huilan; Magaard, Kay; Marberg, Eric; Novelli, Jean-Christophe; Pang, Amy; Saliola, Franco; Tevlin, Lenny; Thibon, Jean-Yves; Thiem, Nathaniel; Venkateswaran, Vidya; Vinroot, C. Ryan; Yan, Ning; Zabrocki, Mike
2012
Generating sets of affine groups of low genus. Zbl 1317.14066
Magaard, Kay; Shpectorov, Sergey; Wang, Gehao
2012
Algorithmic construction of Chevalley bases. Zbl 1344.17001
Magaard, K.; Wilson, R. A.
2012
Representations of unitriangular groups. Zbl 1268.20015
Le, Tung; Magaard, Kay
2012
Character degree sums in finite nonsolvable groups. Zbl 1242.20012
Magaard, Kay; Tong-Viet, Hung P.
2011
Characters of the Sylow $$p$$-subgroups of the Chevalley groups $$D_4(p^n)$$. Zbl 1242.20010
Himstedt, Frank; Le, Tung; Magaard, Kay
2011
Bi-elliptic Weierstrass points on curves of genus 5. Zbl 1259.14038
Kato, T.; Magaard, K.; Völklein, H.
2011
Genus 2 curves that admit a degree 5 map to an elliptic curve. Zbl 1174.14025
Magaard, Kay; Shaska, Tanush; Völklein, Helmut
2009
Recognition of small dimensional representations of general linear groups. Zbl 1216.20005
Magaard, Kay; O’Brien, E. A.; Seress, Ákos
2008
The combinatorics of degenerate covers and an application for general curves of genus 3. Zbl 1175.14017
Magaard, Kay; Völklein, Helmut; Wiesend, Götz
2008
Constructive recognition of $$\text{SL}_3(q)$$. Zbl 1155.20048
Lübeck, F.; Magaard, K.; O’Brien, E. A.
2007
On Weierstrass points of Hurwitz curves. Zbl 1099.14022
Magaard, K.; Völklein, H.
2006
Absolute fourth moments and finiteness of linear groups. Zbl 1106.20001
Gluck, David; Magaard, Kay
2006
The solution of the $$k(GV)$$-problem. Zbl 1058.20010
Gluck, David; Magaard, Kay; Riese, Udo; Schmid, Peter
2004
The monodromy group of a function on a general curve. Zbl 1075.14027
Magaard, Kay; Völklein, Helmut
2004
A GAP package for braid orbit computation and applications. Zbl 1068.12002
Magaard, Kay; Shpectorov, Sergey; Völklein, Helmut
2003
Cross characteristic representations of symplectic and unitary groups. Zbl 1025.20002
Guralnick, Robert M.; Magaard, Kay; Saxl, Jan; Pham Huu Tiep
2002
Irreducibility of tensor squares, symmetric squares and alternating squares. Zbl 1072.20013
Magaard, Kay; Malle, Gunter; Tiep, Pham Huu
2002
The extraspecial case of the $$k(GV)$$ problem. Zbl 0984.20010
Gluck, David; Magaard, Kay
2002
Genus 0 actions of groups of Lie rank 1. Zbl 1065.20001
Frohardt, Dan; Guralnick, Robert; Magaard, Kay
2002
Fixed point ratios in exceptional groups of rank at most two. Zbl 1019.20011
Frohardt, Daniel; Magaard, Kay
2002
The $$k(GV)$$ conjecture for modules in characteristic 31. Zbl 1007.20009
Gluck, David; Magaard, Kay
2002
Incidence matrices, permutation characters, and the minimal genus of a permutation group. Zbl 1008.20002
Frohardt, Daniel; Guralnick, Robert; Magaard, Kay
2002
Irreducible tensor products of representations of finite quasi-simple groups of Lie type. Zbl 0992.20009
Magaard, Kay; Pham Huu Tiep
2001
Composition factors of monodromy groups. Zbl 1004.20001
Frohardt, Daniel; Magaard, Kay
2001
Grassmannian fixed point ratios. Zbl 0970.20003
Frohardt, Daniel; Magaard, Kay
2000
Base sizes and regular orbits for coprime affine permutation groups. Zbl 0951.20003
Gluck, David; Magaard, Kay
1999
Affine distance-transitive graphs: The cross characteristic case. Zbl 0941.05038
Cohen, Arjeh M.; Magaard, Kay; Shpectorov, Sergey
1999
On the minimal degree of a primitive permutation group. Zbl 0911.20003
Guralnick, Robert; Magaard, Kay
1998
Irreducibility of alternating and symmetric squares. Zbl 0919.20009
Magaard, Kay; Malle, Gunter
1998
Unipotent representations of $$G_2$$ arising from the minimal representation of $$D^E_4$$. Zbl 1065.22012
Huang, Jing-Song; Magaard, Kay; Savin, Gordan
1998
Cross-characteristic character and fixed point space ratios for groups of Lie type. Zbl 0912.20009
Gluck, David; Magaard, Kay
1998
Finite quotients of the pure symplectic braid group. Zbl 0926.12002
Magaard, Kay; Strambach, Karl; Völklein, Helmut
1998
Exceptional $$\Theta$$-correspondence. I. Zbl 0878.22011
Magaard, K.; Savin, G.
1997
Character and fixed point ratios in finite classical groups. Zbl 0865.20012
Gluck, David; Magaard, Kay
1995
On the irreducibility of alternating powers and symmetric squares. Zbl 0824.20012
Magaard, K.
1994
Monodromy and sporadic groups. Zbl 0809.20008
Magaard, Kay
1993
Monodromy composition factors among exceptional groups of Lie type. Zbl 0828.20014
Frohardt, Daniel; Magaard, Kay
1993
Flag-stabilizers in the symmetric and alternating groups and their overgroup lattices. Zbl 0662.20005
Magaard, Kay
1988
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#### Cited by 299 Authors

 31 Tiep Pham Huu 26 Magaard, Kay 20 Guralnick, Robert Michael 10 Burness, Timothy C. 10 Zalesski, Alexandre Efimovich 8 Thiem, Nathaniel 7 O’Brien, Eamonn A. 7 Panov, Aleksandr Nikolaevich 7 Praeger, Cheryl Elisabeth 6 Kantor, William M. 6 Leedham-Green, Charles R. 6 Liebeck, Martin Walter 6 Nguyen, Hung Ngoc 6 Savin, Gordan 6 Seress, Ákos 6 Shalev, Aner 5 Di Martino, Lino 5 Gluck, David 5 Halasi, Zoltán 5 Keller, Thomas Michael 5 Malle, Gunter 5 Navarro, Gabriel 5 Spiga, Pablo 4 Aliniaeifard, Farid 4 Gan, Wee Teck 4 Goodwin, Simon M. 4 Le, Tung 4 Madanha, Sesuai Yash 4 Maróti, Attila 4 Paolini, Alessandro 4 Pirola, Gian Pietro 4 Riese, Udo 4 Robinson, Geoffrey Raymond 4 Saxl, Jan 4 Schmid, Peter P. 4 Waldecker, Rebecca 3 Carocca, Angel 3 Frohardt, Daniel E. 3 Giudici, Michael 3 Gül, Şükran 3 Himstedt, Frank 3 Hiss, Gerhard 3 Katz, Nicholas Michael 3 Larsen, Michael Jeffrey 3 Morotti, Lucia 3 Pellegrini, Marco Antonio 3 Qian, Guohua 3 Röhrle, Gerhard E. 3 Sun, Yujiao 3 Tong-Viet, Hung P. 3 Völklein, Helmut 2 Adrianov, Nikolai M. 2 André, Carlos Alberto Martins 2 Andrews, Scott 2 Aschbacher, Michael George 2 Baumeister, Barbara 2 Bergeron, Nantel 2 Cohen, Arjeh Marcel 2 Craven, David A. 2 De Stavola, Dario 2 Diaconis, Persi Warren 2 Dietrich, Heiko 2 Duyan, Hülya 2 Fowler, Christopher F. 2 Fried, Michael David 2 Garcia, Stephan Ramon 2 Gurevich, Nadya 2 Harper, Scott 2 Hendrickson, Anders O. F. 2 Hidalgo, Rubén Antonio 2 Holt, Derek F. 2 Husen, William J. 2 Jeon, Daeyeol 2 Kleshchev, Alexander S. 2 König, Joachim 2 Lee, Melissa 2 Lempken, Wolfgang 2 Lewis, Mark L. 2 Lübeck, Frank 2 Marberg, Eric 2 Moretó, Alexander 2 Mosch, Peter 2 Moschetti, Riccardo 2 Parker, Chris W. 2 Podoski, Károly 2 Pyber, László 2 Rodriguez, Rubi Elena 2 Rowley, Peter J. 2 Schlesinger, Enrico 2 Segal, Avner 2 Semeraro, Jason 2 Shaska, Tanush 2 Shpectorov, Sergey Viktorovich 2 Sin, Peter K. W. 2 Szechtman, Fernando 2 van Bon, John 2 Woodbury, Michael 1 Accola, Robert D. M. 1 Allender, Eric W. 1 Andrica, Dorin ...and 199 more Authors
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#### Cited in 89 Serials

 79 Journal of Algebra 20 Communications in Algebra 17 Transactions of the American Mathematical Society 16 Journal of Group Theory 11 Israel Journal of Mathematics 10 Journal of Pure and Applied Algebra 6 Journal of Algebraic Combinatorics 5 Advances in Mathematics 5 Archiv der Mathematik 4 Duke Mathematical Journal 4 Journal of Combinatorial Theory. Series A 4 Journal of Symbolic Computation 4 Journal of Mathematical Sciences (New York) 4 Séminaire Lotharingien de Combinatoire 3 Mathematics of Computation 3 Mathematische Zeitschrift 3 Monatshefte für Mathematik 3 Proceedings of the American Mathematical Society 3 European Journal of Combinatorics 3 Advances in Applied Mathematics 3 Representation Theory 3 Algebras and Representation Theory 3 Journal of the European Mathematical Society (JEMS) 2 Mathematical Proceedings of the Cambridge Philosophical Society 2 Compositio Mathematica 2 Journal of Number Theory 2 Proceedings of the Edinburgh Mathematical Society. Series II 2 International Journal of Algebra and Computation 2 Indagationes Mathematicae. New Series 2 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 2 Algebra Colloquium 2 LMS Journal of Computation and Mathematics 2 Journal of the Australian Mathematical Society 2 Groups, Geometry, and Dynamics 2 Albanian Journal of Mathematics 2 Forum of Mathematics, Pi 2 Research in the Mathematical Sciences 1 Bulletin of the Australian Mathematical Society 1 Discrete Mathematics 1 Letters in Mathematical Physics 1 Linear and Multilinear Algebra 1 Acta Mathematica Vietnamica 1 Bulletin of the London Mathematical Society 1 Canadian Journal of Mathematics 1 Geometriae Dedicata 1 Inventiones Mathematicae 1 Journal of Combinatorial Theory. Series B 1 Journal of Functional Analysis 1 Journal of the Mathematical Society of Japan 1 Journal für die Reine und Angewandte Mathematik 1 Manuscripta Mathematica 1 Le Matematiche 1 Mathematische Annalen 1 Memoirs of the American Mathematical Society 1 Nagoya Mathematical Journal 1 Osaka Journal of Mathematics 1 Pacific Journal of Mathematics 1 Rendiconti del Seminario Matematico della Università di Padova 1 Ricerche di Matematica 1 SIAM Journal on Computing 1 Acta Applicandae Mathematicae 1 Order 1 Revista Matemática Iberoamericana 1 Journal of the American Mathematical Society 1 Forum Mathematicum 1 Applicable Algebra in Engineering, Communication and Computing 1 Vestnik St. Petersburg University. Mathematics 1 Experimental Mathematics 1 Journal of Algebraic Geometry 1 Combinatorics, Probability and Computing 1 St. Petersburg Mathematical Journal 1 Turkish Journal of Mathematics 1 Finite Fields and their Applications 1 The Electronic Journal of Combinatorics 1 Transformation Groups 1 Doklady Mathematics 1 The Ramanujan Journal 1 Vietnam Journal of Mathematics 1 Journal of Algebra and its Applications 1 Chebyshevskiĭ Sbornik 1 Algebra & Number Theory 1 Asian-European Journal of Mathematics 1 Japanese Journal of Mathematics. 3rd Series 1 Groups, Complexity, Cryptology 1 Science China. Mathematics 1 International Journal of Group Theory 1 Forum of Mathematics, Sigma 1 European Journal of Mathematics 1 Algebraic Combinatorics
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#### Cited in 33 Fields

 243 Group theory and generalizations (20-XX) 45 Combinatorics (05-XX) 44 Algebraic geometry (14-XX) 35 Number theory (11-XX) 18 Associative rings and algebras (16-XX) 17 Computer science (68-XX) 15 Topological groups, Lie groups (22-XX) 10 Field theory and polynomials (12-XX) 7 Functions of a complex variable (30-XX) 6 Nonassociative rings and algebras (17-XX) 5 Manifolds and cell complexes (57-XX) 4 Linear and multilinear algebra; matrix theory (15-XX) 3 Commutative algebra (13-XX) 3 Special functions (33-XX) 3 Geometry (51-XX) 3 Probability theory and stochastic processes (60-XX) 2 Order, lattices, ordered algebraic structures (06-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Ordinary differential equations (34-XX) 2 Abstract harmonic analysis (43-XX) 2 Quantum theory (81-XX) 1 History and biography (01-XX) 1 General algebraic systems (08-XX) 1 Category theory; homological algebra (18-XX) 1 Real functions (26-XX) 1 Partial differential equations (35-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Approximations and expansions (41-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Functional analysis (46-XX) 1 Operator theory (47-XX) 1 Convex and discrete geometry (52-XX) 1 Numerical analysis (65-XX)


text:
# How can we do LDU decomposition modulo $p$?

If we have an $n \times n$ matrix, with all entries taken modulo $p$, how can we output the three matrixes in LDU decomposition, with all entries again modulo $p$? We can assume the input matrix is invertible.

That is, $LDU=A$, with $A$ given. $L$ is a lower-triangular matrix, $D$ is a diagonal matrix, and $U$ is an upper-triangular matrix. The entries of the results are modulo $p$.

• Do LUDecomposition[mat,Modulus->p] and separate out the diagonal from the upper part. – Daniel Lichtblau Jun 4 '16 at 22:46
• @DanielLichtblau: Thanks! That Modulus trick makes it easy! – Matt Groff Jun 4 '16 at 22:59
• Note that LUDecomposition[] pivots, so you have an extra permutation matrix to contend with. If you need the version without pivoting, you'll have to write your own. – J. M.'s ennui Jun 5 '16 at 2:57
• @J.M. I'm not sure it pivots in the modular case, except when it must. But I don't recall for certain (I think I added that option, and it would have been around 20 years ago). – Daniel Lichtblau Jun 5 '16 at 15:44
• @Daniel, yep, here's a random example where it pivots: LUDecomposition[{{4, 4, 4, 3}, {0, 0, 4, 1}, {3, 3, 4, 2}, {4, 1, 0, 4}}, Modulus -> 5] But that would be because the leading $2\times 2$ block is singular, which is exactly when pivoting is necessary. – J. M.'s ennui Jun 5 '16 at 15:57

For reference, here is how to generate a modular $\mathbf L\mathbf D\mathbf U$ decomposition, as suggested by Daniel in the comments:
mat = {{1, 0, 2}, {2, 1, 3}, {2, 1, 2}}; m = 5;

You can check that Mod[l.d.u, m] gives a result that is congruent to mat. If the matrix has a singular leading submatrix, one should then account for an additional permutation matrix constructed from the vector piv; how to do this is left as an exercise for the interested reader.

format: plain


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# Tree Iteration Traversal

Remember how we learned to iterate through lists? There was a way to iterate through lists that felt natural. Just start at the beginning... and keep going.

Or maybe we did some things that were a little strange, like iterate through the reverse of the list. Recall we also wrote iterators in discussion to skip over students who didn't write descriptions on the Office Hours queue.

Now how do you iterate over a tree? What's the correct 'order'?

Before we answer that question, we must not use the word iteration. Instead, we'll call it 'traversing through a tree' or a 'tree traversal'. Why? No real reason, except that everyone calls iteration through trees 'traversals'. Maybe it's because the world likes alliterations.

So what are some natural ways to 'traverse' through a tree? As it turns out, there are a few –– unlike a list which basically has one natural way to iterate through it:

1. Level order traversal.
2. Depth-First traversals –– of which there are three: pre-order, in-order and post-order.

Let's test out these traversals mentioned above on the tree below.

## Level Order Traversal

We'll iterate by levels, left to right. Level 0? D. Level 1? B and then F. Level 2? A, C, E, and G.

This gives us D B F A C E G.

Imagine each level was a sentence in english, and we just read it off line by line.

Exercise 17.2.1. Write the code for performing a level order traversal (warning, this is more difficult than the writing the other traversals). Hint: You will want to keep track of what level you are at.

## Pre-order Traversal

Here's the idea behind pre-order traversal. Start at the root. Visit the root (aka, do the action you want to do.) The action here is "print".

So, we'll print the root. D. There we go.

Now, go left, and recurse. Then, go right and recurse.

So now we've gone left. We're at the B node. Print it. B. We'll go left after printing. (Remember, after we're done with B's left branch, we'll come back up and visit B's right.)

Keep following this logic, and you get D B A C F E G.

preOrder(BSTNode x) {
if (x == null) return;
print(x.key)
preOrder(x.left)
preOrder(x.right)
}

## In-order Traversal

Slightly different, but same big-picture idea. Here, instead of visiting (aka printing) first, we'll first visit the left branch. Then we'll print. Then we'll visit the right branch.

So we start at D. We don't print. We go left.

We start at B. We don't print. We go left.

We start at A. We don't print. We go left. We find nothing. We come back up, and print A.

Then go to A's right. Find nothing. Go back up. Now we're at B. Remember, we print after visiting left and before visiting right, so now we'll print B, then we'll visit right.

Keep following this and you get A B C D E F G.

First, we're at D. We know we'll print out the items from left, then D, then items from right.

[items from left] D [items from right].

Now what's [items from left] equal to? We'll start at B, print out left, then print B, then print stuff from right of B.

[items from left] = [items from B's left] B [items from B's right] = A B C.

A B C D [stuff from right] = A B C D E F G.

inOrder(BSTNode x) {
if (x == null) return;
inOrder(x.left)
print(x.key)
inOrder(x.right)
}

## Post-order Traversal

Again, same big-picture idea, but now we'll print left branch, then right branch, then ourselves.

Using the method we devised earlier, the result looks like:

[items from left] [items from right] D.

What's [items from left]? It's the output from the B subtree.

If we're at B, we'd get [items from left of B] [items from right of B] B, which is equal to A C B.

Following this through, we get: A C B E G F D

postOrder(BSTNode x) {
if (x == null) return;
postOrder(x.left)
postOrder(x.right)
print(x.key)
}


text:
Find Paper, Faster
Example：10.1021/acsami.1c06204 or Chem. Rev., 2007, 107, 2411-2502
Hadron spectroscopy using the light-front holographic Schrödinger equation and the ’t Hooft equation
Physical Review D  (IF5.296),  Pub Date : 2021-10-18, DOI: 10.1103/physrevd.104.074013
Light-front holographic QCD provides a successful first approximation to hadron spectroscopy in the chiral limit of ($3+1$)-dim light-front QCD, where a holographic Schrödinger-like equation, with an emerging confining scale, $\kappa$, governs confinement in the transverse direction. In its supersymmetric formulation, light-front holography predicts that each baryon has two superpartners: a meson and a tetraquark, with their degenerate masses being generated by the same scale, $\kappa$. In nature, this mass degeneracy is lifted by chiral symmetry breaking and longitudinal confinement. In this paper, we show that the latter can be successfully captured by the ’t Hooft equation of ($1+1$)-dim, large ${N}_{c}$, QCD. Together, the holographic Schrödinger equation and the ’t Hooft equation provide a good global description of the data across the full hadron spectrum with a universal $\kappa$.


text:
## question

given a list of numbers, return the order index of each element;

for example, if input is:

input = [ 6, 4, 9, 3 ]

we should return:

output = [ 2, 1, 3, 0 ]

there is an elegant solution with numpy.argsort:

output = numpy.argsort(numpy.argsort(input))

to understand how it works, first of all, we know argsort returns indices that would sort an array; that means, argsort acts as if it first sorts elements by their values, then label elements by their original indices;

to visualize what is happening, we can label the elements with their indices, turning them into tuples:

      label     argsort     label       argsort
6 4 9 3 => 6 4 9 3 => 3 4 6 9    3 4 6 9    6 4 9 3
-> 0 1 2 3    3 1 0 2 => 3 1 0 2 => 0 1 2 3
-> 0 1 2 3    2 1 3 0

note that the 2nd argsort places elements into their original order, but keeps their indices in the sorted list: those are the order indices; now this explains how this solution works;

finally, argsort has an interesting property: repeatedly doing argsort will ping-pong between two lists; in the above example, they are:

3 1 0 2 <=> 2 1 3 0


text:
Select Page

# Tumor growth inhibition model exploration

### Introduction

The purpose of this case study is the exploration of a tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy. The model has been published in Ribba et al. in Clinical Cancer Research (18(18); 5071–80). The research goal is to describe the tumor size evolution in patients treated with chemotherapy or radiotherapy, and to explore the impact of treatments on the tumor growth, along with the impact of inter-individual variability.

### Tumor growth inhibition model

The tumor is composed of proliferative (P) and nonproliferative quiescent tissue (Q), expressed in millimeters. The transition of proliferative tissue into quiescence is governed by a rate constant denoted $k_{PQ}$. The treatment directly eliminates proliferative cells by inducing lethal DNA damage while cells progress through the cell cycle. The quiescent cells are also affected by the treatment and become damaged quiescent cells ($Q_{P}$). Damaged quiescent cells, when re-entering the cell cycle, can repair their DNA and become proliferative once again (transition from $Q_{P}$ to P) or can die because of unrepaired damages.
The pharmacokinetics of the PCV chemotherapy is modeled using a PK/PD approach, in which drug concentration is assumed to decay according to an exponential function. In this model, the three administrated drugs were not considered separately. Rather, the authors assumed the treatment to be represented as a whole by a unique variable (C), which represents the concentration of a virtual drug encompassing the 3 chemotherapeutic components of the PCV regimen. Following Ribba et al., the model writes

$\left\{\begin{array}{l}\frac{dC}{dt} = - KDE \times C \\\frac{dP}{dt} = \lambda_{p}P(1-\frac{P^{\star}}{K})+k_{Q_{P}P}\times Q_{P}-k_{PQ}\times Q - \gamma_{P} KDE \times C \times P\\\frac{dQ}{dt} = k_{PQ}P-\gamma_Q \times C \times KDE \times Q \\\frac{dQ_{P}}{dt} = \gamma_{Q}\times C \times KDE \times Q - k_{Q_{P}P}Q_{P}-\delta_{QP}\times Q_{P}\\P^{\star} = P+Q+Q_{P}\end{array}\right.$

Notice that, in the publication, $\gamma_P=\gamma_Q=\gamma$ was assumed for identifiability reasons. The initial condition writes

$\left\{\begin{array}{l}C(t=0) = 0 \\P(t=0) = P_{0}\\Q(t=0) = Q_{0}\\Q_{P}(t=0) = 0\end{array}\right.$

A schematic view of the model proposed in Ribba et al. is represented here:

### How to model it in Mlxtran

The purpose here is to define the model in Mlxtran language. The model writes as 4 ODEs. The resulting set of parameters is $(\lambda_P, k_{PQ}, k_{Q_{P}P}, \delta_{Q_{P}}, \gamma, {\rm KDE})$, with 2 initial conditions, $(P_{0}, Q_{0})$. Therefore, the [LONGITUDINAL] subsection starts with

[LONGITUDINAL]
input={lambdaP, kPQ, kQPP, gamma, deltaQP, KDE, P0, Q0}

Then, we start the EQUATION: block with the initial conditions:

EQUATION:
; definition of the initial time
t_0 = 0
; initialization of the ODEs
P_0 = P0
Q_0 = Q0
C_0 = 0
QP_0 = 0

and continue it with the 4 ODEs and the definition of $P^{\star}$

K = 100
PSTAR = P+Q+QP
ddt_C = -KDE*C
ddt_P = lambdaP*P*(1-PSTAR/K) + kQPP*QP - kPQ*P - gamma*P*KDE*C
ddt_Q = kPQ*P - gamma*Q*KDE*C
ddt_QP = gamma*Q*KDE*C - kQPP*QP - deltaQP*QP

The individual parameters corresponding to the 8 population parameters (6 parameters and 2 initial conditions) were assumed to be log normally distributed across individuals. Therefore, we have to define an [INDIVIDUAL] subsection as follows:

[INDIVIDUAL]
input = {lambdaP_pop, kPQ_pop, kQPP_pop, gamma_pop, deltaQP_pop, KDE_pop, P0_pop, Q0_pop,
omega_lambdaP, omega_kPQ, omega_kQPP, omega_deltaQP, omega_KDE, omega_P0, omega_Q0}

DEFINITION:
lambdaP = {distribution = lognormal, typical = lambdaP_pop, sd = omega_lambdaP}
kPQ     = {distribution = lognormal, typical = kPQ_pop,     sd = omega_kPQ}
kQPP    = {distribution = lognormal, typical = kQPP_pop,    sd = omega_kQPP}
gamma   = {distribution = lognormal, typical = gamma_pop,   sd = omega_gamma}
deltaQP = {distribution = lognormal, typical = deltaQP_pop, sd = omega_deltaQP}
KDE     = {distribution = lognormal, typical = KDE_pop,     sd = omega_KDE}
P0  = {distribution = lognormal, typical = P0_pop,          sd = omega_P0}
Q0  = {distribution = lognormal, typical = Q0_pop,          sd = omega_Q0}

The model is then the sum of all these codes and is implemented in TumorGrowthInhibitionModel_mlxt.txt

### Model and treatment exploration: project definition

To define the project, we must define the model (in the <MODEL> section, as done in the previous paragraph), all parameter values (for the *_pop and omega_*, in the <PARAMETER> section) and the output (name and time grid, in the <OUTPUT> section). To explore the model, we define in the section  <DESIGN> three administrations where we vary the inter dose timing and keep the same annual amount.

• “admYear”, where we administrate an amount of 2 each 12 months starting after 12 months
• “admMiYear”, where we administrate an amount of 1 each 6 months starting after 12 months
• “admTri”, where we administrate an amount of .5 each 3 months starting after 12 months

Finally, a section is added to define the graphic we want to look at. Then, the project is implemented in TumorGrowthModel_mlxplore.txt as follows:

<MODEL>
file = './TumorGrowthInhibitionModel_mlxt.txt'

<DESIGN>

<PARAMETER>
; Parameters for the PCV treatment
lambdaP_pop = 0.121
kPQ_pop = 0.0295
kQPP_pop = 0.0031
gamma_pop = 0.729
deltaQP_pop = 0.00867
KDE_pop = 0.24
P0_pop = 7.13
Q0_pop = 41.2

omega_lambdaP = .03
omega_kPQ = .03
omega_kQPP = .03
omega_deltaQP = .03
omega_KDE = .03
omega_P0 = .03
omega_Q0 = .03

<OUTPUT>
list = {PSTAR}
grid = -10:1:100

<RESULTS>
[GRAPHICS]
pstar = {y={PSTAR},  ylabel = 'Pstar', xlabel = 'Time (month)'}

### Model exploration: graphical results

Two aspects can be analyzed: the predictions and the inter-individual variability. First we can explore the predictions associated with the three kinds of administrations:

Then, if one wants to explore the impact on the inter-individual variability, one can click on iiv leading to the following figure:

### Conclusion

In this example, a relatively complex ODE model has been implemented in Mlxtran, and its behavior (predictions following different treatments, as well as inter-individual variability) as been explored with Mlxplore.


text:
# Pi

ππ\pi3.14ratio of a circle's circumference to its diameter (pi)3.1416circumference-to-diameter ratio3.14159
The number is a mathematical constant.wikipedia
889 Related Articles

### Mathematical constant

constantconstantsmathematical constants
The number is a mathematical constant.
Constants arise in many areas of mathematics, with constants such as [[e (mathematical constant)|]] and [[pi|]] occurring in such diverse contexts as geometry, number theory, and calculus.

### Approximations of π

Approximations of Approximating approximation
Still, fractions such as 22/7 and other rational numbers are commonly used to approximate.
Approximations for the mathematical constant pi in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era (Archimedes).

### Irrational number

irrationalirrational numbersirrationality
Being an irrational number, cannot be expressed as a common fraction (equivalently, its decimal representation never ends and never settles into a permanently repeating pattern).
Among irrational numbers are the ratio [[Pi|]] of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational.

### Squaring the circle

square the circlequadrature of the circlesquaring of the circle
This transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.

### Archimedes

Archimedes of SyracuseArchimedeanArchimedes Heat Ray
Around 250 BC the Greek mathematician Archimedes created an algorithm for calculating it. In the 5th century AD Chinese mathematics approximated to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques.
Other mathematical achievements include deriving an accurate approximation of pi, defining and investigating the spiral bearing his name, and creating a system using exponentiation for expressing very large numbers.

### Transcendental number

transcendentaltranscendencetranscendental irrational
Also, is a transcendental number; that is, it is not the root of any polynomial having rational coefficients.
The best-known transcendental numbers are e''.

### Piphilology

digit span memorizationmemorize the digitsmemorize the value of
Attempts to memorize the value of with increasing precision have led to records of over 70,000 digits.
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant [[pi|]].

### Real number

realrealsreal-valued
In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry.
Included within the irrationals are the transcendental numbers, such as [[pi|]] (3.14159265...).

### Normal number

normalabsolutely normal numberequidistributed
In particular, the digit sequence of is conjectured to satisfy a specific kind of statistical randomness, but to date, no proof of this has been discovered. The digits of have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.
It is widely believed that the (computable) numbers √2, e'' are normal, but a proof remains elusive.

### Circle

circularcircles360 degrees
Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics.
The result corresponds to 256⁄81 (3.16049...) as an approximate value of [[Pi|]].

### Rational number

rationalrational numbersrationals
Also, is a transcendental number; that is, it is not the root of any polynomial having rational coefficients.
]], [[Pi|]], [[E (mathematical constant)|

### Proof that π is irrational

Proof that is irrational is irrationalbelow
There are several proofs that is irrational; they generally require calculus and rely on the reductio ad absurdum technique.
In the 1760's, Johann Heinrich Lambert proved that the number irrational: that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer.

### History of mathematics

historian of mathematicsmathematicshistory
Ancient civilizations required fairly accurate computed values to approximate for practical reasons, including the Egyptians and Babylonians.
He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, 310⁄71 < π < 310⁄70.

### Repeating decimal

recurring decimalrepeatrepeating fraction
Being an irrational number, cannot be expressed as a common fraction (equivalently, its decimal representation never ends and never settles into a permanently repeating pattern).
Examples of such irrational numbers are the square root of 2 and [[pi|]].

### Statistical randomness

The digits of have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.
A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll or the digits of π exhibit statistical randomness.

### Circumference

circumferentialgirthcircumferential line
Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics.
This constant, pi, is represented by the Greek letter [[Pi (letter)|]].

### Pi (letter)

πpiϖ
It has been represented by the Greek letter "[[Pi (letter)|]]" since the mid-18th century, though it is also sometimes spelled out as "pi". It is also called Archimedes' constant.
The mathematical real transcendental (and thus irrational) constant π = 3.14159..., the ratio of a circle's circumference to its diameter in Euclidean geometry. The letter "π" is the first letter of the Greek words "περιφέρεια" 'periphery' and "περίμετρος" 'perimeter', i.e. the circumference.

### Chinese mathematics

Chinese mathematicianmathematicsChinese mathematical
Around 250 BC the Greek mathematician Archimedes created an algorithm for calculating it. In the 5th century AD Chinese mathematics approximated to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques.
However, the mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used.

### Euler's identity

Euler's identity, e^{i\pi}=-1
= in Euler's formula results in Euler's identity, celebrated by mathematicians because it contains the five most important mathematical constants:
: is pi, the ratio of the circumference of a circle to its diameter.

Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.
Yasumasa Kanada is a Japanese mathematician most known for his numerous world records over the past three decades for calculating digits of [[pi|]].

### Algebraic number

algebraicalgebraic numbersAlgebraic form
Because is known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational.
They include [[Pi|]] and [[e (mathematical constant)|]].

### Geometry

geometricgeometricalgeometries
Because its most elementary definition relates to the circle, is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres.
Archimedes (c. 287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.

### Six nines in pi

999999sequence of six consecutive 9s
Thus, because the sequence of 's digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of.
A sequence of six 9s occurs in the decimal representation of the number pi, starting at the 762nd decimal place.

### Complex number

complexreal partimaginary part
, of a complex variable
As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of π.

### Ratio

ratiosproportionratio analysis
Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics.
The earliest discovered example, found by the Pythagoreans, is the ratio of the length of the diagonal to the length of a side of a square, which is the square root of 2, formally Another example is the ratio of a circle's circumference to its diameter, which is called algebraically irrational number, but a transcendental irrational.


text:
Kattis

# Foosball Dynasty

Balaji and his coworkers like to play Foosball on their lunch break. Foosball is a game typically played by $2$ players (individual matchup) or $4$ players (team play). However, due to the increasing interest and availability of coworkers, Balaji has created a new variation that supports $5$ or more players. Each individual point is played by four players: two on the White team and two on the Black team.

The remaining players wait in line for their turn to play. On each point, the players on the scoring team switch positions (e.g., if White scores the point, then the White Offense player becomes the White Defense player, and vice versa). The Offense player of the team that lost the point becomes the Defense player of the same team, while the Defense player of the team that lost the point goes to the back of the line and waits for their next chance to play. The person at the front of the line becomes the new Offense player of the team that lost the point.

Based on these rules, a team that continues scoring consecutive points gets to keep playing together (swapping positions with each other after each point) until the other team breaks the streak. Each such streak of points creates a dynasty for the winning team. Dynasties can be short-lived (a single point) or long-lived, but they are always broken when the opposing team scores a point. The “winners” of this variation of foosball are the players that can create the longest dynasty.

## Input

Input begins with a line containing an integer $n$ representing the number of players ($5 \leq n \leq 10$). The next line contains the $n$ names of all participating players. The first four are the names of the four players who initially arrive at the table (in that order): the first person to arrive starts at White Offense, the second starts at Black Offense, the third starts at White Defense, and the fourth starts at Black Defense. The remaining players get in line to wait for their turn. The final input line contains a non-empty string indicating which side scored each point, with a White team point represented by ‘W’ and a Black team point represented by ‘B’. The maximum length of this string will be $1\, 000$.

## Output

Display the team that has achieved the longest dynasty. If more than one team ties for the record, then display each of these teams in chronological order of when the record was set, one team per line. When displaying a team, names should be displayed in the order in which the players arrived at the table for that team. Note that it is possible for the same team to appear more than once in the output, potentially with the player names in a different order.

Sample Input 1 Sample Output 1
6
Balaji David Alex Scott Andrew Ravi
WWBWBBWBW

Balaji Alex
Andrew David

Sample Input 2 Sample Output 2
6
Amy Jinu Kasey Sarah Sheetal Julia
BBBBB

Jinu Sarah


text:
25-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
03-02-2017, 10:00 PM
Post: #41
 min21 Junior Member Posts: 2 Joined: Mar 2017
RE: 25-FEB-2017 SolveSys - Equation Library and Multiple Equations Solver
Hello,
When I hit the plot, there is an error saying "Check exactly one equation."
I installed the equation library app through merged version.
HP connectivity toolkit Build: 2016 12.8 Rev:11226
Thanks.
03-02-2017, 11:36 PM
Post: #42
 Han Senior Member Posts: 1,775 Joined: Dec 2013
RE: 25-FEB-2017 SolveSys - Equation Library and Multiple Equations Solver
(03-02-2017 10:00 PM)min21 Wrote:  Hello,
When I hit the plot, there is an error saying "Check exactly one equation."
I installed the equation library app through merged version.
HP connectivity toolkit Build: 2016 12.8 Rev:11226
Thanks.

That error message is never used in the app. That error messages is reported by the built-in app. So seeing that message suggests that the app was not properly installed. Can you give some details as to how you installed the merged version? Go into the app, press [Shift] [1] to see the Program Browser menu. At the very top should be an entry that ends with (App) -- it should be called Equation Library (App) if you installed using the hpappdir format. Open it and it should show a version number toward the top.

Graph 3D | QPI | SolveSys
03-03-2017, 06:00 AM
Post: #43
 min21 Junior Member Posts: 2 Joined: Mar 2017
RE: 25-FEB-2017 SolveSys - Equation Library and Multiple Equations Solver
(03-02-2017 11:36 PM)Han Wrote:  That error message is never used in the app. That error messages is reported by the built-in app. So seeing that message suggests that the app was not properly installed. Can you give some details as to how you installed the merged version? Go into the app, press [Shift] [1] to see the Program Browser menu. At the very top should be an entry that ends with (App) -- it should be called Equation Library (App) if you installed using the hpappdir format. Open it and it should show a version number toward the top.
Hello, Han.
I just updated my calculator again using the toolkit. And reinstalled the Equation library. It worked this time. Thank you for your help and for this amazing app.
03-05-2017, 10:19 PM
Post: #44
 akmon Member Posts: 185 Joined: Jun 2014
RE: 05-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
A work of art this reference manual, could be part of the HP Prime´s users´ guide.
03-06-2017, 04:57 PM
Post: #45
 EdDereDdE Junior Member Posts: 27 Joined: Jan 2017
RE: 05-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
Where can I get the HP Prime Keys font please?
03-06-2017, 05:09 PM
Post: #46
 Han Senior Member Posts: 1,775 Joined: Dec 2013
RE: 05-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
(03-06-2017 04:57 PM)EdDereDdE Wrote:  Where can I get the HP Prime Keys font please?

Graph 3D | QPI | SolveSys
03-06-2017, 05:11 PM
Post: #47
 EdDereDdE Junior Member Posts: 27 Joined: Jan 2017
RE: 05-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
Thanks Han
03-06-2017, 05:45 PM
Post: #48
 compsystems Senior Member Posts: 1,083 Joined: Dec 2013
RE: 05-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
IDEA: For the creation of the equation library, we can contact the author of formlua PRO to generate a common database..

forum FormulaPro
https://tiplanet.org/forum/viewtopic.php?f=50&t=9576

Formula ProGenerator
https://tiplanet.org/scripts/FormulaProGenerator/

03-06-2017, 07:07 PM (This post was last modified: 03-08-2017 09:35 PM by Han.)
Post: #49
 Han Senior Member Posts: 1,775 Joined: Dec 2013
RE: 05-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
(03-06-2017 05:45 PM)compsystems Wrote:  IDEA: For the creation of the equation library, we can contact the author of formlua PRO to generate a common database..

forum FormulaPro
https://tiplanet.org/forum/viewtopic.php?f=50&t=9576

Formula ProGenerator
https://tiplanet.org/scripts/FormulaProGenerator/

I think that it would be great to have formulas and accompanying documentation of the formulas. To that end, users are more than welcome to submit their own *.lib files and I will be happy to maintain them (i.e. update them so that any format changes will be applied to the library files) and include them with the app. Just putting together the list of equations in from the original Equation Library was a tedious task, and I suspect that there might still be typos yet to be discovered. It is not something I want to spend much time on. (I ended up having to learn about black body radiation, fluid dynamics, and a number of other topics just to ensure that the equations from the original Equation Library were fully compatible in the Prime version. Learning is fun, but I prefer choosing my own topics rather being forced into any particular one.) Any future changes to the format of the *lib files will have an accompanying "update" utility/program so that anyone can manage their own library as well. However, that is the extent of my interest in terms of maintenance of the equations themselves.

Right now, all the equations and related data are stored in "Equation Library.lib" -- the next update will allow users to have multiple *.lib files (e.g. one for Chemistry only, another for Physics only, etc.) to help organize their formulas. The next update will also feature the images that were used in the old Equation Library on the HP48 series in addition to allowing users to "switch" libraries. Those images are already in the most recent reference manual (see first post).

Graph 3D | QPI | SolveSys
03-09-2017, 04:56 AM
Post: #50
 math7 Member Posts: 162 Joined: Feb 2017
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
Thanks to Han for this important program !!! I was testing some equations and in the electricity section when I try to solve resistors in parallel, enter the values for r1 and r2 and when I try to solve Rp and Rs, the program shows me the message "Error evaluating F!".

Attached File(s) Thumbnail(s)

03-09-2017, 04:57 AM
Post: #51
 math7 Member Posts: 162 Joined: Feb 2017
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
Thanks to Han for this important program !!! I was testing some equations and in the electricity section when I try to solve resistors in parallel, enter the values for r1 and r2 and when I try to solve Rp and Rs, the program shows me the message "Error evaluating F!".
03-09-2017, 05:04 AM
Post: #52
 Han Senior Member Posts: 1,775 Joined: Dec 2013
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
(03-09-2017 04:56 AM)math7 Wrote:  Thanks to Han for this important program !!! I was testing some equations and in the electricity section when I try to solve resistors in parallel, enter the values for r1 and r2 and when I try to solve Rp and Rs, the program shows me the message "Error evaluating F!".

$$\mathbf{F}$$ here refers to the vector
$\mathbf{F} = \begin{bmatrix} f_1(x_1, x_2, \dotsm, x_m) \\ f_2(x_1, x_2, \dotsm, x_m) \\ \vdots \\ f_n(x_1, x_2, \dotsm, x_m) \\ \end{bmatrix}$

where each $$f_i(x_1,x_2,\dotsm , x_m)$$ corresponds to the $$i$$-th equation $$f_i(x_1,x_2, \dotsm, x_m) = 0$$. In other words, if your equation was $$x_1+x_2 = 5$$ then $$f(x_1,x_2) = x_1+x_2 - 5$$ is the function equivalent. So what does the message mean when there is an error evaluating $$\mathbf{F}$$? Look at your initial guess for $$R_p$$ -- it's 0. This results in division by 0 in the equation involving $$R_p$$. This just means you picked a poor initial guess -- try a value that does not lead to division by 0.

Graph 3D | QPI | SolveSys
03-10-2017, 05:56 AM
Post: #53
 math7 Member Posts: 162 Joined: Feb 2017
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
Yes it is correct Han, when giving other values to Rs and Rp there is no problem, the results are correct !!! Thank you very much...
03-11-2017, 09:23 PM
Post: #54
 akmon Member Posts: 185 Joined: Jun 2014
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
Fantastic upgrade. This app is now a must have for Hp Prime. As Eric Reching says in his web: "Get it".

As a suggestion, a shorcut to the picture will improve the information. I remember to use it often even inserting values, it helps a lot. So if there were a quick way to watch the picture (for example, long press on the screen, or a virtual button on the screen or physical button) both in the equation menu and variable menu will be useful.
03-13-2017, 10:16 PM
Post: #55
 StephenG1CMZ Senior Member Posts: 695 Joined: May 2015
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
Certainly a great app.
Just for information: Microsoft's PDF Reader on Windows 8.1 renders manual.pdf as 63 blank pages (or else rendering is very slow)....
Perhaps it would be useful to keep the installation instructions separate.

Stephen Lewkowicz (G1CMZ)
03-13-2017, 10:57 PM
Post: #56
 Han Senior Member Posts: 1,775 Joined: Dec 2013
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
(03-13-2017 10:16 PM)StephenG1CMZ Wrote:  Certainly a great app.
Just for information: Microsoft's PDF Reader on Windows 8.1 renders manual.pdf as 63 blank pages (or else rendering is very slow)....
Perhaps it would be useful to keep the installation instructions separate.

There should have been an install.txt (or did I name it readme.txt?) file separate from the manual. I will make sure that it is included in the next release.

Graph 3D | QPI | SolveSys
03-13-2017, 11:13 PM
Post: #57
 StephenG1CMZ Senior Member Posts: 695 Joined: May 2015
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
(03-13-2017 10:57 PM)Han Wrote:
(03-13-2017 10:16 PM)StephenG1CMZ Wrote:  Certainly a great app.
Just for information: Microsoft's PDF Reader on Windows 8.1 renders manual.pdf as 63 blank pages (or else rendering is very slow)....
Perhaps it would be useful to keep the installation instructions separate.

There should have been an install.txt (or did I name it readme.txt?) file separate from the manual. I will make sure that it is included in the next release.

Sorry, yes there is a readme- I should have read it first.

Stephen Lewkowicz (G1CMZ)
03-14-2017, 02:48 AM
Post: #58
 Han Senior Member Posts: 1,775 Joined: Dec 2013
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
KNOWN ISSUES (v0.900)

1. Deleting systems fails when there are more than 70 systems of equations. This is due to a limitation of the input() command, which only allows up to 7 rows per page, and 10 pages max. The current equation library has over 100 systems. There will be a workaround in an upcoming update.

2. Not really an issue, but the library file can be made slightly smaller by removing image width and height parameters (can use grobh and grobw to find the heights and widths). There will be a slight modification to the *.lib file format, and animations will be supported in the next update.

Graph 3D | QPI | SolveSys
03-15-2017, 06:58 PM (This post was last modified: 03-15-2017 07:10 PM by Han.)
Post: #59
 Han Senior Member Posts: 1,775 Joined: Dec 2013
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
So it appears the HP Prime's built-in Solve app has the same issue as the HP48's multiple equation solver (MES): they do not properly analyze inconsistent systems, and return "zeros" that are actually not solutions to the system. Here's an example for the HP48:

The same problem occurs on the HP Prime. Ask it to solve X+Y=5 and X+Y=6. Set Y to 3 and solve for X to get a "zero" of X=2 and Y=3. I think the problem could be easily fixed by having the Solve app actually check to see if it is indeed a zero of the system, and not just one of the equations. My plan was to include a the Solve app's solver engine as option (among three: quasi-Newton engine in SolveSys, Solve app solver, and a hybrid of the two). It's hard to include it as an option when it fails to check if the system is inconsistent.

Graph 3D | QPI | SolveSys
03-21-2017, 11:28 AM
Post: #60
 jorge_rui Junior Member Posts: 1 Joined: Mar 2017
RE: 08-MAR-2017 SolveSys - Equation Library and Nonlinear Systems Solver
I've registered,
Just to say thank you!
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text:
## Pages

### Equivalent fractions

Equivalent fractions are obtained by dividing or multiplying the numerator and denominator of a fraction by the same number. If you divide the numerator and denominator of a fraction by the same number, it should be a common factor of the numerator and denominator. If you multiply the numerator and denominator of a fraction by the same number, then it can be any integer.

Dividing both numerator and denominator of 15/20 by 5, we get 3/4.
Multiplying both numerator and denominator of 15/20 by 2, we get 30/40.
Multiplying both numerator and denominator of 15/20 by 3, we get 45/60.
Therefore 3/4, 30/40 and 45/50 are equivalent fractions to 15/20.


text:
# zbMATH — the first resource for mathematics

Bipartite distance-regular graphs: the $$Q$$-polynomial property and pseudo primitive idempotents. (English) Zbl 1297.05264
Summary: Let $$\Gamma$$ denote a bipartite distance-regular graph with diameter at least 4 and valency at least 3. Fix a vertex of $$\Gamma$$ and let $$T$$ denote the corresponding Terwilliger algebra. Suppose that $$\Gamma$$ is $$Q$$-polynomial and there are two non-isomorphic irreducible $$T$$-modules with endpoint 2. We show that, unless the intersection numbers of $$\Gamma$$ fit one exceptional case (which is not known to correspond to an actual graph), the entry-wise product of pseudo primitive idempotents associated with these modules is a linear combination of two pseudo primitive idempotents. This result relates to a conjecture of MacLean and Terwilliger.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
Full Text:
##### References:
 [1] Bannai, Eiichi; Ito, Tatsuro, Algebraic combinatorics. I. association schemes, (1984), The Benjamin/Cummings Publishing Co. Inc. Menlo Park, CA · Zbl 0555.05019 [2] Biggs, Norman, Algebraic graph theory, Cambridge Mathematical Library, (1993), Cambridge University Press Cambridge · Zbl 0797.05032 [3] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., (Distance-Regular Graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18, (1989), Springer-Verlag Berlin) · Zbl 0747.05073 [4] Caughman, John S., The Terwilliger algebras of bipartite $$P$$- and $$Q$$-polynomial schemes, Discrete Math., 196, 1-3, 65-95, (1999) · Zbl 0924.05067 [5] Curtin, Brian, Almost 2-homogeneous bipartite distance-regular graphs, European J. Combin., 21, 7, 865-876, (2000) · Zbl 1002.05069 [6] Lang, Michael S., Tails of bipartite distance-regular graphs, European J. Combin., 23, 8, 1015-1023, (2002) · Zbl 1012.05159 [7] Lang, Michael S., A new inequality for bipartite distance-regular graphs, J. Combin. Theory Ser. B, 90, 1, 55-91, (2004) · Zbl 1051.05082 [8] MacLean, Mark S.; Terwilliger, Paul, Taut distance-regular graphs and the subconstituent algebra, Discrete Math., 306, 15, 1694-1721, (2006) · Zbl 1100.05104 [9] Nomura, K., Homogeneous graphs and regular near polygons, J. Combin. Theory Ser. B, 60, 1, 63-71, (1994) · Zbl 0793.05130 [10] Nomura, K., Spin models on bipartite distance-regular graphs, J. Combin. Theory Ser. B, 64, 2, 300-313, (1995) · Zbl 0827.05060 [11] Pascasio, Arlene A.; Terwilliger, Paul, The pseudo-cosine sequences of a distance-regular graph, Linear Algebra Appl., 419, 2-3, 532-555, (2006) · Zbl 1110.05105 [12] Terwilliger, Paul, The subconstituent algebra of an association scheme. I, J. Algebraic Combin., 1, 4, 363-388, (1992) · Zbl 0785.05089 [13] Terwilliger, Paul, The subconstituent algebra of an association scheme. II, J. Algebraic Combin., 2, 1, 73-103, (1993) · Zbl 0785.05090 [14] Terwilliger, Paul; Weng, Chih-Wen, Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra, European J. Combin., 25, 2, 287-298, (2004) · Zbl 1035.05104 [15] Janoš Vidali, There is no distance-regular graph with intersection array 55,54,50,35,10;1,5,20,45,55. Preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Isosceles trapezoid

Calculate the circumference and the contents of the isosceles trapezoid if you know the size of the bases is 8 and 12 cm and the size of the arms is 5 cm.

Result

o =  30 cm
S =  45.826 cm2

#### Solution:

$o=8+12+2 \cdot \ 5=30 \ \text{cm}$
$S=\sqrt{ 5^{ 2 }-(4/2)^{ 2 } } \cdot \ (8+12)/2=45.826 \ \text{cm}^2$

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# How the calculate $\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x$?

Just as the title say, consider the integral: $$I=\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x=\frac{1}{2}\int_{-\infty}^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x,$$ how to apply the residue theorem to get the answer?

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Why do you have to apply the residue theorem? Because $\dfrac{\sin^2 z}{z^2}$ is entire, it may suffice to use Cauchy's theorem with some contour. Here's a related question, but the answers there probably don't use methods you're currently looking for: math.stackexchange.com/questions/13344/… –  Jonas Meyer Dec 17 '12 at 8:18

Hint:

\begin{align} \int_0^\infty\frac{\sin^2(x)}{x^2}\,\mathrm{d}x &=\frac12\int_{-\infty}^\infty\frac{\sin^2(x)}{x^2}\,\mathrm{d}x\\ &=-\frac18\int_{-i-\infty}^{-i+\infty}\frac{e^{2ix}+e^{-2ix}-2}{x^2}\,\mathrm{d}x\\ &=-\frac18\left(\int_{\gamma_+}\frac{e^{2ix}-1}{x^2}\,\mathrm{d}x +\int_{\gamma_-}\frac{e^{-2ix}-1}{x^2}\,\mathrm{d}x\right)\tag{1} \end{align} where $\gamma_+$ goes from $-i-R$ to $-i+R$ then circles back counterclockwise on $-i+Re^{i\theta}$ for $\theta\in[0,\pi]$ as $R\to\infty$, and where $\gamma_-$ goes from $-i-R$ to $-i+R$ then circles back clockwise on $-i+Re^{-i\theta}$ for $\theta\in[0,\pi]$ as $R\to\infty$.

Each of the integrals in $(1)$ can easily be handled with the residue theorem.

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$$\int_{0}^{\infty}\frac{(\sin x)^2}{x^2}dx=-\int_{0}^{\infty}(\sin x)^2{d\frac{1}{x}}=-\frac{(\sin x)^2}{x}\Bigg|^{\infty}_{0}+\int_{0}^{\infty}\frac{(2\sin x\cos x)}{x}dx=\int_{0}^{\infty}\frac{(\sin2x)}{2x}d2x=\ldots$$

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Really? How does the first equality follow? In any event, you should give at least some verbal explanation... –  tomasz Dec 17 '12 at 9:14
@tomasz Integration by parts. –  Ragib Zaman Dec 17 '12 at 9:29
How to calculate $\int_0^\infty \sin x/x d x$ then? –  van abel Dec 17 '12 at 12:30
OK, I saw it in the older post:math.stackexchange.com/questions/5248/… –  van abel Dec 17 '12 at 12:32