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The structure and geometry of 4×4 pandiagonal matrices

1981
*
Discrete Mathematics
*

*The*sum

*of*these columns would be 2 + l l l + 16 = 72, ; '

*the*equivalence class [ 1, 1, 0, O]invariant gives

*the*fine permutation

*structure*

*of*these

*pandiagonal*

*matrices*. ... Furthermore,

*the*elements

*of*

*the*

*4*~

*4*

*pandiagonal*

*matrices*in this equivalence class are obtained by

*the*action

*of*

*the*group gen:erated by A

*and*B on

*the*single

*4*-tuple, (j, k, r, s) = (8, invariance property ...

##
###
Page 779 of Mathematical Reviews Vol. , Issue 97B
[page]

1997
*
Mathematical Reviews
*

It is useful for

*the*solution*of*some problems*of*distribution (modulo some prime number p)*of*combinatorial coefficients*and*numbers in some arithmetic*structures*. V. A. ... Normalized p-Latin squares*and*algorithms for their application to problems*of**the*elements modulo p in arithmetic*structures*. (Russian. Russian summary) Voprosy Vychisl.i Prikl. Mat. ...##
###
Page 2447 of Mathematical Reviews Vol. , Issue 2000d
[page]

2000
*
Mathematical Reviews
*

*The*following are necessary conditions for

*the*existence

*of*a simple DTS(v,

*4*): Av(v — 1) =0 mod 3,

*4*< 3(v —2). Let (X,A) be a DTS(v, 2)

*and*(¥ B) be a DTS(u,

*4*). ... (English summary)

*Geometry*, combinatorial designs

*and*related

*structures*(Spetses, 1996), 149-159, London Math. Soc. Lecture Note Ser., 245, Cambridge Univ. Press, Cambridge, 1997. ...

##
###
Magic Squares and Sudoku

2012
*
The American mathematical monthly
*

We introduce a family

doi:10.4169/amer.math.monthly.119.09.759
fatcat:y3yddh5wx5dcjkgmtc37wi5gba
*of*magic squares, called linear magic squares,*and*show that any parallel linear sudoku solution*of*sufficiently large order can be relabeled so that all*of*its subsquares are linear ... I thank*the*referees*and**the*Monthly editor for leaving this article far better than they found it. ... We show that if q > 3 then we lose nothing by replacing*the*word "sudoku" with*the*words "magic sudoku" in*the*previous sentence.*The*strategy*and**structure**of**the*paper are as follows. ...##
###
Page 2395 of Mathematical Reviews Vol. , Issue 82f
[page]

1982
*
Mathematical Reviews
*

Praeger (Perth)
Turner, Barbara; Warner, Ken 82f:20012

*The**structure**and**geometry**of**4*<*4**pandiagonal**matrices*. Discrete Math. 34 (1981), no. 3, 301-310. ... In this paper*the*authors find some generators*and*relations for G*and*investigate*the**structure**of*some*of*its subgroups. C. A. Rowley (Milton Keynes) VySenskii, V. A. [ViSins’kii, V. A.}; Klin, M. ...##
###
Magic Mathematics Based on New Matrix Transformations (2D and 3D) for Interdisciplinary Physics, Mathematics, Engineering and Energy Management
[chapter]

2010
*
Products and Services; from R&D to Final Solutions
*

, ...) by Laplace Rule directly from Leibniz, too % Both rules

doi:10.5772/10400
fatcat:tmv77jmnqffjjeeo3corz6ynia
*of*Sarrus*and*Laplace are visualised*structures**of**the*% Leibniz Formula detA_Leibniz (above shown for 4x4-*Matrices*) % This Leibniz Formula ... this point, W.Stanek provides an algorithm to obtain any 3D magic*structure*, especially 64 cells cube, primarily*the*visualised*structure**of**the*algorithm. ... step 2. 11 541*4*12 6 9 7*4*13 3 16 2 81 051 1 NOTES: •*The**matrices*M1 until M4 are magic-*matrices*with*the*cells content*of*number 1 to 16 ...##
###
Superstochastic matrices and magic Markov chains

2009
*
Linear Algebra and its Applications
*

A brief account

doi:10.1016/j.laa.2008.12.010
fatcat:jet2zrtcbbhnhnip4xsoybx7mu
*of**the*conceptual formulation*of**the*two entities in this paper's title, plus an initial preliminary investigation*of*some*of*their mathematical properties, is given. ... Acknowledgments We mentioned in our proof*of*Theorem 2.1 that we need only doubly stochastic*matrices*there. ... In that connection, one*of**the*referees pointed out three important books [19, 20, 21] where we may find ...##
###
Polyhedral Cones of Magic Cubes and Squares
[chapter]

2003
*
Algorithms and Combinatorics
*

Using computational algebraic

doi:10.1007/978-3-642-55566-4_2
fatcat:k2lw36kccnef5bgwpuopg27t4q
*geometry*techniques*and*Hilbert bases*of*polyhedral cones we derive explicit formulas*and*generating functions for*the*number*of*magic squares*and*magic cubes. ...*The*work*of*Ehrhart*and*Stanley [14, 15, 25, 26] when applied to*the*study*of*semi-magic squares showed that many enumerative*and**structural*properties*of*magic arrays can actually be formulated in terms ... From now on, when referring to any*of*these*structures*, we will use*the*terminology magic arrays. ...##
###
Polyhedral Cones of Magic Cubes and Squares
[article]

2002
*
arXiv
*
pre-print

Using computational algebraic

arXiv:math/0201108v2
fatcat:vaxg5gltyzd7nlohktfdlhrupe
*geometry*techniques*and*Hilbert bases*of*polyhedral cones we derive explicit formulas*and*generating functions for*the*number*of*magic squares*and*magic cubes. ...*The*work*of*Ehrhart*and*Stanley [14, 15, 25, 26] when applied to*the*study*of*semi-magic squares showed that many enumerative*and**structural*properties*of*magic arrays can actually be formulated in terms ... For example,*the*stochastic semi-magic squares are*the*well-known bistochastic*matrices*(n × n*matrices*whose row*and*column sums are one)*and*P is*the*famous Birkhoff-von Neumann polytope [9, 26] . ...##
###
Page 749 of Mathematical Reviews Vol. 17, Issue Index
[page]

*
Mathematical Reviews
*

(with Proskurowski, Andrzej) Construction

*of**pandiagonal*magic squares from circulant*pandiagonal**matrices*. Ars Combin. 15 (1983), 153-162. (J. ... Construction*of*magic squares from circulant*pandiagonal**matrices*. Ars Combin. 15 (1983), 153-162. (J. Dénes) 85b:05041 05B15 (with Systo, Maciej M.) Edge-coloring*of*trees*and*unicyclic graphs. ...##
###
Page 141 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

(English summary) 98¢:05031
Pott, Alexander * Finite

*geometry**and*character theory. 98j:05032 Qiu, Weisheng On*the*multiplier conjecture. ...*The*use*of*combinatorial*structures*in communication signal design 98m:94005 Gysin, Mare New D-optimal designs via cyclotomy*and*generalised cyclotomy. (English summary) 98a:05039 Hamilton, G. M. ...##
###
Page 195 of Mathematical Reviews Vol. 14, Issue Index
[page]

*
Mathematical Reviews
*

Fundamental regions in S, for

*the*simple quaternary Gj... 82b:20007 Turner, Barbara (with Warner, Ken)*The**structure**and**geometry**of*44*pandiagonal**matrices*. 82f:20012 Visins’kii, V. A. ... (Russian) (82g:05049) Wefelscheid, Heinrich Verallgemeinerte Minkowski-*Geometrie*. (82b:51014) Chelwood Gate * Finite*geometries**and*designs. (82g:05003) Conference: Isle*of*Thorns, Finite*Geometries**and*...##
###
Page 535 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

Foregger) 87e:15012 15A15
Minimum permanent on faces

*of*staircase type*of**the*polytope*of*doubly stochastic*matrices*. Linear*and*Multilinear Algebra 18 (1985), no.*4*, 271-306. (Thomas H. ... Banchoff) 87¢:53011 53A10 (51M20, 52A25) (with Andersson, Sten) Differential*geometry**of*crystal*structure*descriptions, relationships*and*phase transformation. Z. ...##
###
Subject Index Volumes 1–200

2001
*
Discrete Mathematics
*

Erdös

doi:10.1016/s0012-365x(00)00471-4
fatcat:znxg54q6ifgutnhzj3urcw455y
*and*Sós, 4925 Erdös*and*Szekeres, 4879 Erdös et al., 5529, 5888 Erdös Rubin Taylor, 3815 Erdös Sós Turán, 506 Everett, 2901 Faigle*and*Sands, 2122 Faudree et al., 4399 Faudree Gould Jacobson ... Slater, 5299 Hertz*and*de Werra, 3748 Hibi, 4839 Hoàng, 3748, 4811 Holton, 1660 Holyer*and*Cockayne, 1118 Howorka, 2636 ... 5284, see also three-way regular 3-way regular tripartite graph, 5284*of*degree four, 5284 3-wise, see also three-wise balanced design, 3912*4*×*4**pandiagonal*matrix, 1160*4*array, 2059*4*colour conjecture ...##
###
Page 456 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

On

*pandiagonal*symmetrical magic squares*of*order 6. J. Recreational Math. 16 (1983/84), no.*4*, 263-267. (Robert Girse) 86a:05024 05B15 Johnson, Bruce McK. ... Linear*and*Muitilinear Algebra 16 (1984), no. 1-*4*, 29-38. (Robert Beauwens) 86b:15022. 15A48 (15A15) (with Rodman, L.) Inertia possibilities for completions*of*partial Hermitian*matrices*. ...
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