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Issue No.06 - June (2012 vol.18)
pp: 879-889
Shi-Qing Xin , Nanyang Technological University, Singapore
Ying He , Nanyang Technological University, Singapore
Chi-Wing Fu , Nanyang Technological University, Singapore
ABSTRACT
Closed geodesics, or geodesic loops, are crucial to the study of differential topology and differential geometry. Although the existence and properties of closed geodesics on smooth surfaces have been widely studied in mathematics community, relatively little progress has been made on how to compute them on polygonal surfaces. Most existing algorithms simply consider the mesh as a graph and so the resultant loops are restricted only on mesh edges, which are far from the actual geodesics. This paper is the first to prove the existence and uniqueness of geodesic loop restricted on a closed face sequence; it contributes also with an efficient algorithm to iteratively evolve an initial closed path on a given mesh into an exact geodesic loop within finite steps. Our proposed algorithm takes only an O(k) space complexity and an O(mk) time complexity (experimentally), where m is the number of vertices in the region bounded by the initial loop and the resultant geodesic loop, and k is the average number of edges in the edge sequences that the evolving loop passes through. In contrast to the existing geodesic curvature flow methods which compute an approximate geodesic loop within a predefined threshold, our method is exact and can apply directly to triangular meshes without needing to solve any differential equation with a numerical solver; it can run at interactive speed, e.g., in the order of milliseconds, for a mesh with around 50K vertices, and hence, significantly outperforms existing algorithms. Actually, our algorithm could run at interactive speed even for larger meshes. Besides the complexity of the input mesh, the geometric shape could also affect the number of evolving steps, i.e., the performance. We motivate our algorithm with an interactive shape segmentation example shown later in the paper.
INDEX TERMS
Discrete geodesic, geodesic loop, triangular mesh.
CITATION
Shi-Qing Xin, Ying He, Chi-Wing Fu, "Efficiently Computing Exact Geodesic Loops within Finite Steps", IEEE Transactions on Visualization & Computer Graphics, vol.18, no. 6, pp. 879-889, June 2012, doi:10.1109/TVCG.2011.119
REFERENCES
 [1] H. Lee, Y. Tong, and M. Desbrun, “Geodesics-Based One-to-One Parameterization of 3D Triangle Meshes,” IEEE Multimedia, vol. 12, no. 1, pp. 27-33, Jan. 2005. [2] G. Peyré and L.D. Cohen, “Geodesic Remeshing Using Front Propagation,” Int'l J. Computer Vision, vol. 69, no. 1, pp. 145-156, 2006. [3] Q.-X. Huang, B. Adams, M. Wicke, and L.J. Guibas, “Non-Rigid Registration under Isometric Deformations,” SGP '08: Proc. Sixth Symp. Geometry Processing, pp. 1449-1457, 2008. [4] P.V. Sander, Z.J. Wood, S.J. Gortler, J. Snyder, and H. Hoppe, “Multi-Chart Geometry Images,” SGP '03: Proc. Eurographics/ACM SIGGRAPH Symp. Geometry Processing, pp. 146-155, 2003. [5] K. Zhou, J. Synder, B. Guo, and H.-Y. Shum, “Iso-Charts: Stretch-driven Mesh Parameterization Using Spectral Analysis,” SGP '04: Proc. Eurographics/ACM SIGGRAPH Symp. Geometry Processing, pp. 45-54, 2004. [6] J. Zhang, C. Wu, J. Cai, J. Zheng, and X.-C. Tai, “Mesh Snapping: Robust Interactive Mesh Cutting Using Fast Geodesic Curvature Flow,” Computer Graphics Forum, vol. 29, no. 2, pp. 517-526, 2010. [7] M. Sharir and A. Schorr, “On Shortest Paths in Polyhedral Spaces,” SIAM J. Computing, vol. 15, no. 1, pp. 193-215, 1986. [8] J.S.B. Mitchell, D.M. Mount, and C.H. Papadimitriou, “The Discrete Geodesic Problem,” SIAM J. Computing, vol. 16, no. 4, pp. 647-668, 1987. [9] S.-Q. Xin and G.-J. Wang, “Improving Chen and Han's Algorithm on the Discrete Geodesic Problem,” ACM Trans. Graphics, vol. 28, no. 4, pp. 1-8, 2009. [10] R. Bott, “On the Iteration of Closed Geodesics and the Sturm Intersection Theory,” Comm. Pure and Applied Math., vol. 9, no. 2, pp. 171-206, 1956. [11] M. Freedman, J. Hass, and P. Scott, “Closed Geodesics on Surfaces,” Bull. London Math. Soc., vol. 14, pp. 385-391, 1982. [12] M. Jin, F. Luo, S.-T. Yau, and X. Gu, “Computing Geodesic Spectra of Surfaces,” Proc. ACM Symp. Solid and Physical Modeling, pp. 387-393, 2007. [13] F. Hétroy and D. Attali, “From a Closed Piecewise Geodesic to a Constriction on a Closed Triangulated Surface,” PG '03: Proc. 11th Pacific Conf. Computer Graphics and Applications, p. 394, 2003. [14] D. Reniers and A. Telea, “Part-Type Segmentation of Articulated Voxel-Shapes Using the Junction Rule,” Proc. Pacific Graphics, 2008. [15] M. Dixon, N. Jacobs, and R. Pless, “Finding Minimal Parameterizations of Cylindrical Image Manifolds,” CVPRW '06: Proc. Conf. Computer Vision and Pattern Recognition Workshop, p. 192, 2006. [16] T.K. Dey and S. Guha, “Transforming Curves on Surfaces,” J. Computer and System Sciences, vol. 58, pp. 297-325, 1999. [17] X. Yin, M. Jin, and X. Gu, “Computing Shortest Cycles Using Universal Covering Space,” The Visual Computer, vol. 23, no. 12, pp. 999-1004, 2007. [18] L. Fan, L. Liu, and K. Liu, “Paint Mesh Cutting,” Computer Graphics Forum, vol. 30, no. 2, pp. 603-612, 2011. [19] C. Wu and X. Tai, “A Level Set Formulation of Geodesic Curvature Flow on Simplicial Surfaces,” IEEE Trans. Visualization and Computer Graphics, vol. 16, no. 4, pp. 647-662, July/Aug. 2010. [20] J. Chen and Y. Han, “Shortest Paths on a Polyhedron,” SCG '90: Proc. Sixth Ann. Symp. Computational Geometry, pp. 360-369, 1990. [21] V. Surazhsky, T. Surazhsky, D. Kirsanov, S.J. Gortler, and H. Hoppe, “Fast Exact and Approximate Geodesics on Meshes,” ACM Trans. Graphics, vol. 24, no. 3, pp. 553-560, 2005. [22] K. Polthier and M. Schmies, “Straightest Geodesics on Polyhedral Surfaces,” Mathematical Visualization, p. 391, Springer Verlag, 1998. [23] K. Polthier and M. Schmies, “Geodesic Flow on Polyhedral Surfaces,” Proc. Eurographics-IEEE Symp. Scientific Visualization, pp. 179-188, 1999. [24] S.-Q. Xin and G.-J. Wang, “Efficiently Determining a Locally Exact Shortest Path on Polyhedral Surfaces,” Computer-Aided Design, vol. 39, no. 12, pp. 1081-1090, 2007. [25] R. Kimmel and J.A. Sethian, “Computing Geodesic Paths on Manifolds,” Proc. Nat'l Academy of Sciences of USA, vol. 95, pp. 8431-8435, 1998. [26] S. Har-Peled, “Approximate Shortest paths and Geodesic Diameter on a Convex Polytope in Three Dimensions,” Discrete and Computational Geometry, vol. 21, no. 2, pp. 217-231, 1999. [27] P.K. Agarwal, S. Har-Peled, and M. Karia, “Computing Approximate Shortest Paths on Convex Polytopes,” Proc. Symp. Computational Geometry, pp. 270-279, 2000. [28] A. Spira and R. Kimmel, “Geodesic Curvature Flow on Parametric Surfaces,” Proc. Curve and Surface Design, pp. 365-373, 2002. [29] P. Diaz-Gutierrez, D. Eppstein, and M. Gopi, “Curvature Aware Fundamental Cycles,” Computer Graphics Forum, vol. 28, no. 7, pp. 2015-2024, 2009. [30] W. Zeng, M. Jin, F. Luo, and X. Gu, “Canonical Homotopy Class Representative Using Hyperbolic Structure,” Proc. IEEE Int'l Conf. Shape Modeling and Applications, 2009. [31] W. Zeng, Y. He, J. Xia, X. Gu, and H. Qin, “${\rm C}^\infty$ Smooth Freeform Surfaces over Hyperbolic Domains,” SPM '09: Proc. SIAM/ACM Joint Conf. Geometric and Physical Modeling, pp. 367-372, 2009. [32] J. Zhang, J. Zheng, and J. Cai, “Interactive Mesh Cutting Using Constrained Random Walks,” IEEE Trans. Visualization and Computer Graphics, vol. 17, no. 3, pp. 357-367, Mar. 2011. [33] Y. Zheng and C.-L. Tai, “Mesh Decomposition with Cross-Boundary Brushes,” Computer Graphics Forum, vol. 29, no. 2, pp. 527-535, 2010. [34] A. Golovinskiy and T. Funkhouser, “Randomized Cuts for 3D Mesh Analysis,” Proc. ACM SIGGRAPH ASIA Papers, vol. 27, Dec. 2008. [35] L.J. Guibas and J. Hershberger, “Optimal Shortest Path Queries in a Simple Polygon,” SCG '87: Proc. Third Ann. Symp. Computational Geometry, pp. 50-63, 1987. [36] S. Kakutani, “Ein Beweis des ${\rm s}{\ddot{a}}$ tzes Von Edelheit ${\ddot{u}}$ ber Konvexe Mengen,” Proc. Imp. Acad. Tokyo 13, pp. 93-94, 1937. [37] M. Jin, F. Luo, and X.D. Gu, “Computing General Geometric Structures on Surfaces Using Ricci Flow,” Computer-Aided Design, vol. 39, no. 8, pp. 663-675, 2007. [38] E.C. de Verdière and F. Lazarus, “Optimal System of Loops on an Orientable Surface,” Proc. 43rd Symp. Foundations of Computer Science, pp. 627-636, 2002.


text:
# Re: [isabelle-dev] Towards the Isabelle2017 release - Complete Distributive Lattice

> On 25 Aug 2017, at 20:14, Viorel Preoteasa <viorel.preote...@aalto.fi> wrote:
>
> One possible solution:
>
> Add the new class in Complete_Lattice.thy, replacing the existing class
>
> Prove the instantiations and the complete_linearord subclass later
> in Hilbert_Choice.
>
> On the other hand, it seems inconvenient to have the Hilbert Choice
> to depend on so many other theories.

I’d prefer this provided the instantiations aren’t needed earlier.

The delay in the introduction of the Axiom of Choice is partly historical, but
it’s worth noting how much of HOL can be developed without it.

Larry
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# HSS-MD.A

Calculate expected values and use them to solve problems

## Standards

HSS‑MD.A.1 $(+)$${}^{\huge\star}Define a random variable for a quantity of interest by assigning a... HSS‑MD.A.2 (+)$${}^{\huge\star}$Calculate the expected value of a random variable; interpret it as the mean...
HSS‑MD.A.3 $(+)$${}^{\huge\star}Develop a probability distribution for a random variable defined for a... HSS‑MD.A.4 (+)$${}^{\huge\star}$Develop a probability distribution for a random variable defined for a...


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## G = Dic5⋊5M4(2)  order 320 = 26·5

### 2nd semidirect product of Dic5 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic5⋊5M4(2)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C2×C4×Dic5 — Dic5⋊5M4(2)
 Lower central C5 — C2×C10 — Dic5⋊5M4(2)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for Dic55M4(2)
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c5 >

Subgroups: 334 in 126 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4⋊C8 [×4], C2×C42, C2×M4(2), C2×M4(2), C52C8 [×2], C40 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C4⋊M4(2), C2×C52C8 [×2], C4.Dic5 [×2], C4×Dic5 [×2], C4×Dic5 [×2], C2×C40 [×2], C5×M4(2) [×2], C22×Dic5 [×2], C22×C20, C20.8Q8 [×4], C2×C4.Dic5, C2×C4×Dic5, C10×M4(2), Dic55M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C2×M4(2) [×2], Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C4⋊M4(2), C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, D5×M4(2) [×2], C2×C10.D4, Dic55M4(2)

Smallest permutation representation of Dic55M4(2)
On 160 points
Generators in S160
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 111 6 116)(2 120 7 115)(3 119 8 114)(4 118 9 113)(5 117 10 112)(11 63 16 68)(12 62 17 67)(13 61 18 66)(14 70 19 65)(15 69 20 64)(21 123 26 128)(22 122 27 127)(23 121 28 126)(24 130 29 125)(25 129 30 124)(31 94 36 99)(32 93 37 98)(33 92 38 97)(34 91 39 96)(35 100 40 95)(41 109 46 104)(42 108 47 103)(43 107 48 102)(44 106 49 101)(45 105 50 110)(51 154 56 159)(52 153 57 158)(53 152 58 157)(54 151 59 156)(55 160 60 155)(71 150 76 145)(72 149 77 144)(73 148 78 143)(74 147 79 142)(75 146 80 141)(81 140 86 135)(82 139 87 134)(83 138 88 133)(84 137 89 132)(85 136 90 131)
(1 90 43 54 27 74 34 63)(2 81 44 55 28 75 35 64)(3 82 45 56 29 76 36 65)(4 83 46 57 30 77 37 66)(5 84 47 58 21 78 38 67)(6 85 48 59 22 79 39 68)(7 86 49 60 23 80 40 69)(8 87 50 51 24 71 31 70)(9 88 41 52 25 72 32 61)(10 89 42 53 26 73 33 62)(11 111 136 107 156 127 142 91)(12 112 137 108 157 128 143 92)(13 113 138 109 158 129 144 93)(14 114 139 110 159 130 145 94)(15 115 140 101 160 121 146 95)(16 116 131 102 151 122 147 96)(17 117 132 103 152 123 148 97)(18 118 133 104 153 124 149 98)(19 119 134 105 154 125 150 99)(20 120 135 106 155 126 141 100)
(11 156)(12 157)(13 158)(14 159)(15 160)(16 151)(17 152)(18 153)(19 154)(20 155)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)

G:=sub<Sym(160)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,111,6,116)(2,120,7,115)(3,119,8,114)(4,118,9,113)(5,117,10,112)(11,63,16,68)(12,62,17,67)(13,61,18,66)(14,70,19,65)(15,69,20,64)(21,123,26,128)(22,122,27,127)(23,121,28,126)(24,130,29,125)(25,129,30,124)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,109,46,104)(42,108,47,103)(43,107,48,102)(44,106,49,101)(45,105,50,110)(51,154,56,159)(52,153,57,158)(53,152,58,157)(54,151,59,156)(55,160,60,155)(71,150,76,145)(72,149,77,144)(73,148,78,143)(74,147,79,142)(75,146,80,141)(81,140,86,135)(82,139,87,134)(83,138,88,133)(84,137,89,132)(85,136,90,131), (1,90,43,54,27,74,34,63)(2,81,44,55,28,75,35,64)(3,82,45,56,29,76,36,65)(4,83,46,57,30,77,37,66)(5,84,47,58,21,78,38,67)(6,85,48,59,22,79,39,68)(7,86,49,60,23,80,40,69)(8,87,50,51,24,71,31,70)(9,88,41,52,25,72,32,61)(10,89,42,53,26,73,33,62)(11,111,136,107,156,127,142,91)(12,112,137,108,157,128,143,92)(13,113,138,109,158,129,144,93)(14,114,139,110,159,130,145,94)(15,115,140,101,160,121,146,95)(16,116,131,102,151,122,147,96)(17,117,132,103,152,123,148,97)(18,118,133,104,153,124,149,98)(19,119,134,105,154,125,150,99)(20,120,135,106,155,126,141,100), (11,156)(12,157)(13,158)(14,159)(15,160)(16,151)(17,152)(18,153)(19,154)(20,155)(51,70)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,111,6,116)(2,120,7,115)(3,119,8,114)(4,118,9,113)(5,117,10,112)(11,63,16,68)(12,62,17,67)(13,61,18,66)(14,70,19,65)(15,69,20,64)(21,123,26,128)(22,122,27,127)(23,121,28,126)(24,130,29,125)(25,129,30,124)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,109,46,104)(42,108,47,103)(43,107,48,102)(44,106,49,101)(45,105,50,110)(51,154,56,159)(52,153,57,158)(53,152,58,157)(54,151,59,156)(55,160,60,155)(71,150,76,145)(72,149,77,144)(73,148,78,143)(74,147,79,142)(75,146,80,141)(81,140,86,135)(82,139,87,134)(83,138,88,133)(84,137,89,132)(85,136,90,131), (1,90,43,54,27,74,34,63)(2,81,44,55,28,75,35,64)(3,82,45,56,29,76,36,65)(4,83,46,57,30,77,37,66)(5,84,47,58,21,78,38,67)(6,85,48,59,22,79,39,68)(7,86,49,60,23,80,40,69)(8,87,50,51,24,71,31,70)(9,88,41,52,25,72,32,61)(10,89,42,53,26,73,33,62)(11,111,136,107,156,127,142,91)(12,112,137,108,157,128,143,92)(13,113,138,109,158,129,144,93)(14,114,139,110,159,130,145,94)(15,115,140,101,160,121,146,95)(16,116,131,102,151,122,147,96)(17,117,132,103,152,123,148,97)(18,118,133,104,153,124,149,98)(19,119,134,105,154,125,150,99)(20,120,135,106,155,126,141,100), (11,156)(12,157)(13,158)(14,159)(15,160)(16,151)(17,152)(18,153)(19,154)(20,155)(51,70)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,111,6,116),(2,120,7,115),(3,119,8,114),(4,118,9,113),(5,117,10,112),(11,63,16,68),(12,62,17,67),(13,61,18,66),(14,70,19,65),(15,69,20,64),(21,123,26,128),(22,122,27,127),(23,121,28,126),(24,130,29,125),(25,129,30,124),(31,94,36,99),(32,93,37,98),(33,92,38,97),(34,91,39,96),(35,100,40,95),(41,109,46,104),(42,108,47,103),(43,107,48,102),(44,106,49,101),(45,105,50,110),(51,154,56,159),(52,153,57,158),(53,152,58,157),(54,151,59,156),(55,160,60,155),(71,150,76,145),(72,149,77,144),(73,148,78,143),(74,147,79,142),(75,146,80,141),(81,140,86,135),(82,139,87,134),(83,138,88,133),(84,137,89,132),(85,136,90,131)], [(1,90,43,54,27,74,34,63),(2,81,44,55,28,75,35,64),(3,82,45,56,29,76,36,65),(4,83,46,57,30,77,37,66),(5,84,47,58,21,78,38,67),(6,85,48,59,22,79,39,68),(7,86,49,60,23,80,40,69),(8,87,50,51,24,71,31,70),(9,88,41,52,25,72,32,61),(10,89,42,53,26,73,33,62),(11,111,136,107,156,127,142,91),(12,112,137,108,157,128,143,92),(13,113,138,109,158,129,144,93),(14,114,139,110,159,130,145,94),(15,115,140,101,160,121,146,95),(16,116,131,102,151,122,147,96),(17,117,132,103,152,123,148,97),(18,118,133,104,153,124,149,98),(19,119,134,105,154,125,150,99),(20,120,135,106,155,126,141,100)], [(11,156),(12,157),(13,158),(14,159),(15,160),(16,151),(17,152),(18,153),(19,154),(20,155),(51,70),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(71,87),(72,88),(73,89),(74,90),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146)])

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4N 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 10 ··· 10 2 2 4 4 4 4 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D5 M4(2) D10 D10 Dic10 C4×D5 C5⋊D4 C4×D5 D5×M4(2) kernel Dic5⋊5M4(2) C20.8Q8 C2×C4.Dic5 C2×C4×Dic5 C10×M4(2) C4×Dic5 C22×Dic5 C2×C20 C2×C20 C2×M4(2) Dic5 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 4 4 2 2 2 8 4 2 8 4 8 4 8

Matrix representation of Dic55M4(2) in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 7 40 0 0 1 0
,
 32 16 0 0 0 9 0 0 0 0 27 28 0 0 12 14
,
 9 2 0 0 5 32 0 0 0 0 17 1 0 0 40 24
,
 1 21 0 0 0 40 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,7,1,0,0,40,0],[32,0,0,0,16,9,0,0,0,0,27,12,0,0,28,14],[9,5,0,0,2,32,0,0,0,0,17,40,0,0,1,24],[1,0,0,0,21,40,0,0,0,0,1,0,0,0,0,1] >;

Dic55M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_5M_4(2)
% in TeX

G:=Group("Dic5:5M4(2)");
// GroupNames label

G:=SmallGroup(320,745);
// by ID

G=gap.SmallGroup(320,745);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,387,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

׿
×
𝔽


text:
# chemistry

posted by .

How many grams of oxygen are collected in a reaction where 245 mL of oxygen gas is collected over water at a temperature of 25 ^\circ {\rm C} and a total pressure of 688 torr

• chemistry -

Look up vapor pressure of H2O at 25C and
subtract from 688 torr.
688 torr - vpH2O = pressure dry gas.
Then use PV = nRT and solve for n = number of mols oxygen. Don't forget P must be in atmospheres (torr dry gas/760) and T must be in kelvin. V must be in L.
Then n = grams O2/molar mass O2. Solve for grams.

## Respond to this Question

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text:
# Does air resistance increase the speed of a falling object?

I just saw an experiment where a heavy object (bowling ball) and a light object (feather) are dropped in a vacuum and they both fall at the same speed (almost like a slow motion video) and reach the bottom at the same time. But when the same two objects are dropped in a non vacuum environment, the ball hits the ground immediately while the feather has not even covered half the distance. I know this is somehow related to air resistance and terminal velocity and he air resistance reduces the acceleration of the ball to zero. but what I do not understand is that how come air resistance helps in increasing the speed of the ball ? Isn't it a kind of resistance ? Is it that in the vacuum experiment, the ball reaches the ground very quick and only the video is in slow motion ?

• Either slow motion, or filmed on the moon. Nov 30, 2016 at 3:08
• The experiment in vacuum was probably filmed at slower speed. Drag is always opposite to the direction of motion so it can only slow free falling things down. Nov 30, 2016 at 3:18
• Please comment on the reasons for the downvote ! Nov 30, 2016 at 3:33
• Probably because they had difficulty believing this was a serious question. But indeed, the video does not explicitly say this was slow motion. Would be better if they did.
– user137289
Nov 30, 2016 at 8:41

Well, the experiment was obviously filmed at a slower speed or shown at a slower speed. Both feather and ball should accelerate at around $$9.8~\mathrm{m/s^2}$$ and their velocities will be the same at all times. When there is air, the feather falls at much slower rate compared to the ball. Air resistance will decrease the acceleration of both but the effect of it will be much more on the feather.

• "Fall at 9.8 m/s^2" is terrible wording. Nov 22, 2018 at 15:15
• @Jasper Corrected it. Nov 23, 2018 at 9:48
• It is quite terrible that the educational video does not mention this. It is not at all obvious to lay people. The general population tends to associate vacuum with space and "weightlessness". This expensive video only adds to the confusion.
– user137289
Nov 23, 2018 at 9:57
• @Pieter Yeah. I actually wanted to see the feather falling down as fast as the ball at normal speed but was disappointed by the video. Nov 23, 2018 at 10:12

Actually, to begin with, you were correct, @Crusaderpyro, in your question, that as the speed of a falling object increases, your air resistance increases. The reason this is is that as the object falls faster, it meets up with more and more air molecules per given amount of time. This will manifest as an increase in air resistance. However, we also have to include the factor of mass in the equation. Since the ball has more mass and less surface area, even though it also experiences more air resistance, it will be affected by gravity more and reach the ground before the feather, which has very little mass and more surface area and is therefore less affected by gravity. They both start accelerating at 9.8 m/s2, but the feather gets slowed down by the air resistance much more as you can see in the first part of the video, due its small mass and large surface area. Of course, they slowed down the video in the second part, or the action would have been over in a fraction of a second. ( You can see this in the end of the video as they start to fall quickly together before the film gets slowed). But the main point was to show that, after the air is removed and thereby the air resistance removed, the ball and feather will fall at the same velocity.

• It must be a huge feather if it has more surface area than a typical bowling ball. ;) You should make it clear that you are talking about relative surface area, i.e., the ratio of surface area to volume (or to mass). Nov 22, 2018 at 16:00
• Downvoting because the experiment was in vacuum. Old question, please read previous answers.
– user137289
Nov 23, 2018 at 10:11


text:
# How do you convert 73 mi/h = ____km/s?

Jun 20, 2018

$3.26 \cdot {10}^{-} 2 \setminus \text{km/s}$.

#### Explanation:

Well, we know that

$\text{1 mile = 1.609 km}$

and

$\text{1 hour = 3600 seconds}$

So, we have,

$\text{73 miles/hr" = (73*1.609)/3600 \ "km/s}$

$\setminus = \text{0.0326 km/s}$

$= 3.26 \cdot {10}^{-} 2 \setminus \text{km/s}$


text:
1. ## Weird 2nd Order difeq question. Help Please!!

Here is the question:

y'' = (y')^2 (Assume y and y' are positive)

I looked at the question and didn't know how to start it, never seen a question in this form. Help please!!!

2. Originally Posted by Phyxius117
Here is the question:

y'' = (y')^2 (Assume y and y' are positive)

I looked at the question and didn't know how to start it, never seen a question in this form. Help please!!!
Divide by $y'$ and integrate once

$\frac{y''}{y'} = y'$ so $\ln y' = y + \ln c_1 \; \text{or}\; y' = c_1 e^y.$

Then separate and integrate again.

3. Thanks!!

Now I got

y = C1 e^(y) X + C2

as answer but how do I separate to just y in the equation with e^(y) present?

4. lol I think it's wrong but I plug

y = lny' - C1 into y

and becomes

y= C1 e^(lny' - C1) + C2

which simplifies into

y = C1y' + C2

then I integrate again becomes

yx = C1y + C2x + C3

and finally

y = (C2x + C3) / (X - C1)

which I feel it's wrong...

5. Originally Posted by Phyxius117
lol I think it's wrong but I plug

y = lny' - C1 into y

and becomes

y= C1 e^(lny' - C1) + C2

which simplifies into

y = C1y' + C2

then I integrate again becomes

yx = C1y + C2x + C3

and finally

y = (C2x + C3) / (X - C1)

which I feel it's wrong...
Ya - you can't do that!

Originally Posted by Danny
Divide by $y'$ and integrate once

$\frac{y''}{y'} = y'$ so $\ln y' = y + \ln c_1 \; \text{or}\; y' = c_1 e^y.$

Then separate and integrate again.
From what I have.

$e^{-y} dy = c_1 dx$ so $-e^{-y} = c_1 x + c_2$ (I absorbed the negative into the constants).

Solving for y gives y $= - \ln | c_1 x + c_2|$.

6. Just put
y'=f.

The equation then becomes
$
\frac{df}{dx}=f^2(x)
$

Separate and integrate to get

$
f^{-2}df=dx
$

and
$
-f^{-1}=x+a
$
,
Hence
$
f(x)=-\frac{1}{x+a}.
$

Now,
$
y'=-\frac{1}{x+a}
$

Integrate to get the final answer
$
y(x)=-\ln(|x+a|)+b
$

7. ty so much!!!

8. Note that Danny's answer is also correct:

y
$
y=-\ln|c_1x+c_2|=-\ln(|c_1(x+\frac {c_2}{c_1}|)=-\ln(|c_1|)-\ln(|x+\frac{c_2}{c_1}|)=-\ln(|x+a|)+b
$

where b=-ln(c1) and a=c2/c1


text:
# Fusion Summary?

1. May 1, 2007

### mheslep

Can anyone please suggest a comparative pro/con review of the various fusion approaches for power generation? In those approaches I'd include:

tokamaks,
stellarators,
spheromaks,
pinches (Z, focus, etc)
-ICF
laser, ion beam
-Other
inertial electrostatic
Beam-beam FRC
?

Obviously the available information in each area is deep but Ive not found a good comparison of the physics/engineering issues against one another, or at least an attempt at using some kind of common metrics. I had in mind something like the following, in the few areas where Ive done some background:

tokamaks, etc: thermal magnetic confinement thus limited to DD, DT fuels due to Bremstralung losses at higher temperatures required for aneutronic fuels. Therefore neutron loads a challenging problem for the toroidal geometry ...

IEC - claims of mono energetic operation but analysis to date shows IEC quickly thermalizes and then realizes same problems as above.

etc.

Mark

Last edited: May 1, 2007
2. May 14, 2007

### Astronuc

Staff Emeritus
Is this homework? One can find articles on-line about each concept.

Parameters to compare would be efficiency [(energy out)/(energy in)], confinement times, plasma density, reaction rate, . . .

And what does one mean by IEC quickly thermalizes? Fusion produces fast neutrons, and all neutrons will ultimately be absorbed by a nucleus or decay into a proton, electron and anti-neutrino. Neutrons passing through solid materials will certainly damage the crystal structure/atomic lattice by collisions - that is a given and there is no way around it. Materials degradation in systems using reactions producing neutrons is a significant technical problem/challenge. Even systems based on aneutronic reactions, e.g. D+He3, invariably have to deal with the fact that one can get DD reactions, half of which produce n+He3.

the p+B system is about as purely aneutronic as one can get, but it requires high temperatures with resulting high pressures, which is a major handicap for magnetic confinment. In the inertial confinment systems, it's those darn electrons which get in the way.

3. May 16, 2007

### mheslep

No, though my apologies for the overly broad post. I'm looking into some internal R&D work and trying to obtain a comparative high level view of the trade offs based purely on the soundness of the physics, independent of the $and man hours invested to date, to the degree thats possible. Yes I'm well into the lit. on many of the approaches but then of course the issue in each area is that the researchers can be myopic about their particular approach: Tokamak'ers versus implosion, etc. Lidsky's "Trouble w/ Fusion" points some fingers. . Good observables but I think those metrics illustrate my points above: 1. They show for instance how magnetic confinement schemes completely dominate fusion discussion. Density is, what, 5-6 orders of magnitude different between implosion and mag confinement so thats not useful by itself. Confinement times (Larson), AFAICT, are not relevant to accelerator (driven) schemes like IEC or Rostoker's beam-beam http://www.sciencemag.org/cgi/content/abstract/278/5342/1419?ck=nck" 2.$/time invested. If I want only soundness of idea and not \$, do I look at Eout/Ein for 1970 Tokamaks or 1999 JET?

IEC is basically an accelerator like beam-beam system. I dont refer to the fusion products, but the reactants. That is, the is all reactant ions are accelerated by E fields to, say, 50KeV creating a completely non-maxwellian, mono energetic system. The ideal is they stay that way unto a fussion event, but it seems to be well established that for beam-beam systems that coulomb collisions dominate ~1000:1 and thus the reactants quickly thermalize - either escaping the system on the hot end or becoming unavailable for fusion on the cold end.

Ah, so lets get rid of (most) the electrons ;-) Non-neutral plasma work seems to be on the upswing.

Last edited by a moderator: Apr 22, 2017
4. May 16, 2007

### Astronuc

Staff Emeritus
For IECs this might be of interest - http://fti.neep.wisc.edu/iec/ftisite1.htm

I don't believe non-neutral plasmas are necessarily practical from a stability perspective.

Inertial confinement has the problem of a cyrogenic fuel source (fuel pellet), the repeatability issue, and then net energy production vs input. Energy extraction is an issue - as well as a myriad of materials degradation issues.

Magnetic confinement has much the same problem along with the stabilty of magnetic confinement, and limiations on plasma density, which are a consequence on the limit on the magnetic field strength, which are a consequence of the limits on currents in the superconducting magnets.

Beam systems, including injected beams into magnetic confinement, have the drawback of beam energy efficiency and scattering of the beam by the plasma.

5. May 16, 2007

### mheslep

Yes U. Wisc. seems to be IEC central HQ. The Rider[15] & Nevins [16] papers listed http://fti.neep.wisc.edu/iec/inertial_electrostatic_confineme.htm" [Broken] in the overview section are most frequently cited stakes to heart of IEC, though I dont know that there is any experimental verification of the losses they claim.

Can you say more about that? I see that external mag confinement alone wouldn't work against the resulting space charge but why not w/ electric accelerators or a combination? Penning/Paul Traps? BTW a non-neutral conference:
http://sdphca.ucsd.edu/nnp2001/

This is an area where it appears to me that the basic idea is unsound. The external mag. field doesn't really confine anything, especially not heavy ions (Brillouin limit inversely related to mass). It only retards the inevitable random walk to escape the system. On the other hand an E field based recirculating accelerator or trap is actually confining the ion, theoretically forever.
Yes, no question its going to scatter. The interesting question for me is (in a recirculating system) how fast does the scattering take place (loss power) relevant to the fusion power.

Last edited by a moderator: May 2, 2017
6. May 17, 2007

### Astronuc

Staff Emeritus
Well, in a fusion plasma, one does not want heavy ions, which don't fuse and which cause much greater energy losses as a function of Z (nuclear charge and number electrons in neutral atom). Higher Z nuclei in plasmas lead to greater losses due to bremsstrahlung, recombination, electron-electron collisions (including ionization).

The external magnetic field does confine in Tokamak or other devices, but it is short term due to instabilities, which are exacerbated by non-uniformities in the plasma temperature distribution/ion/electron densities.

Non-neutral plasmas have the disadvantage of Coulomb repulsion in addition to temperature-related pressure. In theory, the dynamic (acceleration of the nuclei) in an IEC overcomes some of the pressure limitations, but by virtue of the continuity equation (rule, law) - what goes in, must come out, otherwise pressure will increase with matter accumulation, particularly in the number of particles does not change, i.e. if one does not get A + A = B, but rather A + A = B + B.

7. May 17, 2007

### mheslep

I meant heavy D or T ions relative to the electron mass; the electron being much easier to confine as a result.

Granted a theoretical non-neutral IEC has Coulomb (and kinetic?) pressure to handle, but it has the advantage of having ~all the plasma mass being at reaction energy (at the bottom of the potential E well); vice a thermal, neutral Tokamak plasma where a) only the ions in the high energy tail of the Maxwellian distribution can practically fuse and most of the lower energy plasma mass is just along for the ride, and b) the electrons radiate significant Bremsstrahlung.

mheslep

Last edited: May 17, 2007
8. Jun 8, 2007

### RogerFox

Are you refering to wire grid IEC devices? I think so. Polywell IEC by Bussard over the last 11 yrs shows virtual cathode & potential well formation with no real electron losses, minimal to near zero maxwellian thermalization of ions. 100,000 times better output in relation to input than any previous Fusor type device.

Dr Bussards former assistant Tom Ligon, hangs here:

A schematic look at electron & ion flow in DR. Bussards WB6 IEC fusion reactor.

http://www.emc2fusion.org

Google tech talk by Dr Bussard

Last edited by a moderator: May 2, 2017
9. Jun 10, 2007

### mheslep

I'm familiar with Bussard and the Polywell design, watched the video, read the papers, (including the ones from the early 90's by Krall), got the T-shirt. None of it documents measuring 'minimal to near zero ...'. What is your source for this claim? All of the physics I know of says that any beam - beam collision of ions will quickly thermalize.

Last edited: Jun 11, 2007
10. Jun 11, 2007

### RogerFox

Since WB6 ran in late 2005, I am not aware of Krall or Rider writing about the topic. I don't understand citing 15 to 20 yr old papers to refute R&D done in late 2005.
If Dr Bussards claim's are accurate.... does that suggest he solved the fuel ion thermalization? Would he have gotten increasing neutron counts if his fuel was thermalizing? Though Bussard does need to duplicate the results outside of the D.O.D. publishing embargo.

Bussard ran WB6 4 times, each time at increased drive conditions, each test yielded higher neutron counts. If Bussards WB6 results are duplicated, then Kral, Rider, etc might want to write anew.

Bussard touches on fuel thermaliztion in the google vid & Tom Ligon covers thermalization in response to Riders paper of 1995, here:

If the ions would live sufficiently long and make enough passes without fusing, they would no doubt thermalize, or at least lose energy. This will also happen if the density is allowed to get too high. The key is achieving a density profile where high-energy collisions occur in the center at fusion energies, very low energy (and thermalized) collisions occur in the high-density ion turn-around zone near the magrid, and very few other collisions occur. A narrow range of optimal ion lifetime versus fusion density exists for any given size of machine and set of operating parameters (too high, thermalize, too low, too few fusions). Larger sizes are required for that optimal density to produce net power.

The thermalized turn-around zone is a key to killing two of Rider's objections. Rider believes A) the plasma will maxwellianize, and B) because it is maxwellian, some of the ions will upscatter in energy by collisions and so be able to exit the potential well complety. The outer collision zone does have maxwellian properties, but at very low energy levels, so it essentially removes the scatter in the energy of the fuel ions on every pass, essentially resetting the population to all near zero kinetic energy. Thus, thermalization keeps the machine from thermalizing.

Rider thinks the machine will have excessive bremsstrahlung losses. Bussard and Krall counter that the central region Rider thinks will cause bremsstrahlung due to mutual repulsion is actually a convergence zone for both electrons and ions. The electrons are making a virtual cathode that attracts ions, the ions make a virtual anode that attracts electrons, and the whole zone is never all that far from a neutral plasma. Control of virtual anode height controls the bremsstrahlung problem.

All of this says you need very good control of the ion population. WB6 and that puff gas system did not offer such control ... I suspect the system put in roughly the amount of gas needed, initially, but the amount continued to rise. Fusion occurred in the short period during which the right density profile existed. A successful machine will need to hold that condition.

It may be that it will occasionally be necessary to stop the reaction and clear the machine of junk gas, which may include fusion products in a power reactor.

Rider's final objection was dead-on right ... for HEPS, PXL-1, and WB-5. Cusp losses of electrons in a box machine are too high. They are irrelevant in a magrid machine. Electrons lost out the cusps come right back in.

Last edited by a moderator: May 2, 2017
11. Jun 11, 2007

### mheslep

No, that won't fly here on PF. I asked for a citation, and you talk 3rd hand about more 'R&D'.

Link from this forum to an enthusiast on another forum? No good here. Citation please.

I don't see any sense in this. The coulomb collision / fusion collision ratio does not change with density. This suggestion is akin to manipulating a light dimmer for effect: "If the light is too hot/drawing too much power Ill turn it down. And, if the light is too dim I'll just turn it up. There! Its bright and no heat."

No, you don't get to merely assert which area will be maxwellian and which will not.

Convergence zones? Bremsstrahlung power in a D-D or D-T plasma is proportional to the electron density and the electron energy, thats it. Its worse w/ the higher Z fuels.
$$P_{Br} [\textrm{Watt/m}^3] = \left[{n_e \over 7.69 \times 10^{18} \textrm{m}^{-3} }\right]^2 T_e[\textrm{eV}]^{1/2}$$

To reduce B. power then you either reduce the density or keep the electrons cold. Period. You don't get to waive this a way buy talking convergence zones.

Last edited by a moderator: May 2, 2017
12. Jun 11, 2007

### Morbius

mheslep,

You got THAT right. Hot electrons are the "death knell" of ANY fusion plasma.

Even in HIGHLY convergent plasmas; like those found in ICF.

Once you couple strongly to the radiation field, which is what electrons do - then
you might as well just blow a hole in the side of your plasma - you have a BIG energy
sink.

Dr. Gregory Greenman
Physicist

13. Jun 11, 2007

### mheslep

,

Agreed for any neutral plasma thats optically thin to Bremsstrahlung. I'm interested in some non-neutral and degenerate plasma ideas for just this reason. For instance:

"Confinement Of Pure Ion Plasma In A Cylindrical Current Sheet"
http://www.osti.gov/bridge/servlets/purl/15113-w7GWk3/webviewable/15113.PDF

"Beyond the Brillouin limit with the Penning Fusion Experiment"
www.physics.udel.edu/~mitchell/journal_articles/bard97.pdf[/URL]

"Aneutronic fusion in a degenerate plasma"
[url]http://w3.pppl.gov/~fisch/fischpapers/2004/Son_PLA_04.pdf[/url]

Last edited by a moderator: Apr 22, 2017
14. Jun 14, 2007

### RogerFox

My apologies for my behavior, being a noobie with a cut & paste post.

By hot electrons, do you mean when electron density is high as well as energy? Ala Bremsstrahlung ? (sp?)

Would electron energy at the center of the potential well (in a device such as WB6), be low energy but high density?

mheslep,

In the google video, From 55:09 to 55:17 RWB comments that they repeated the experiment 4 times. I know, crap as a cite, but it what it is.

EDIT:

http://stinet.dtic.mil/

EMC2-0891-04, and EMC2-1291-02

I was informed that these pertain to the HEPS device, a closed box design with large electron losses, from the DARPA days.

Last edited: Jun 14, 2007
15. Jun 14, 2007

### mheslep

I mean the temperature, or kinetic energy of the electron in eV. Typically Bremmsstrahlung becomes significant > 10KeV as per the above equation. There is X amount of radiation for each electron (on average), so more electrons = more radiation.

.
Ok, what were the results each time? Ive only seen the figure of 3 neutrons quoted for one experiment.

16. Jun 16, 2007

### RogerFox

17. Jun 17, 2007

### mheslep

The electrons would be there w/ the high speed ions and collisions /w the ions will heat up the electrons.

I don't think the paper you list is the relevant reference. The earlier one addresses the howto for 'cold' electrons:
"Bremmstrahlung Radiation Losses in Polywell Systems", Bussard, '92

Ive been through it once and it warrants more time. At 1st look I've got two problems:
1. Everything up front seems to be standard plasma beam physics calculations: running the energy and particle density through the geometry of the system. To simplify the approach he initially disregards the ion collisions. Then finally on page 5 just before equation 11 he gets back to it:

He then proposes that, yes, the electrons will be heated in the core by the ions but that they will be cooled on the edge and that the two areas offset each other yielding a stable electron temperature. Ok, interesting idea. My 1st impression is that this is particle description of a heat transfer engine and as such its governed the thermodyamics (carnot cycle) and hence can't be very efficient. Thats a quick take - I don't really know. In any case this is a point of questionable physics and requires some detail description of why /how it can work. Bussard simply refers to 'Analysis of these processes shows the stable up scattered core energy...'. Come on. Addressing this point is the entire key to answering the question of radiation losses. This is the paper to show that and he says 'Analysis of these processes shows ...'? Furthermore this is problem number one to address in the lab starting back in '92; you don't even need to waste time w/ D or T fuels - just put any low Z ionized gas in there and measure your core electron temperature to verify this cold electron theory, or better yet measure the radiation level vs density if its high enough. Just answer this one question and Bussard et al will have much more serious attention. You don't need a 50 page paper, just a clearly documented note verifying cold electrons via test.

2. The ratio of fusion power to Bremm. power in equation 8 is shown to depend on density. Bremm. power and fusion power are directly dependent on density so that the two would cancel out in the ratio. So I'm not following this; I need another look. [EDIT, scratch this: I was confused by the density symbol ne in the paper and the hand written (dang these old papers ;-) nu e, two different things]

mheslep

Last edited by a moderator: May 2, 2017
18. Jun 21, 2007

### RogerFox

An single ion is introduced into the device. The ion sees the potential well, shoots thru the well. Once past the well it slows (cooling?), turns, again seeing the well.

http://flux.aps.org/meetings/YR03/APR03/baps/abs/S3470011.html

Right, 3 neutrons at the detector. That may be the 12.5kV, mentioned by Tom Ligon here:

Yes, though a nice 50-100 page paper would be OK too, right?

Last edited by a moderator: May 2, 2017
19. Jun 21, 2007

### RogerFox

Edit: I was told table 2 shows Brem-losses as 1/32nd ?

Last edited by a moderator: May 2, 2017
20. Jun 21, 2007

### mheslep

Lerner is the Focus Fusion guy, selling T-shirts on his website. What does Lerner's talk have to do w/ the results of the WB6 tests?

Last edited: Jun 21, 2007


text:
# hantingfootmassage.com

### Categories:

Main / Music / Network-information-theory-by-professor-abbas-el-g

# Network-information-theory-by-professor-abbas-el-g

Köp Network Information Theory av Abbas El Gamal på hantingfootmassage.com g. Antal komponenter: 1; Komponenter: B&W x in or x . Abbas El Gamal is the Hitachi America Chaired Professor in the School of. Abbas El Gamal (born May 30, ) is an Egyptian American electrical engineer , educator and entrepreneur. He is best known for his contributions to network information theory, Field El Gamal was an Assistant Professor in the Department of Electrical Engineering from He has been on the faculty of the. Abbas El Gamal, Young-Han Kim, Network Information Theory, Cambridge University Coordinator, Farshad Lahouti, Associate Professor Robert G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, Visiting Assistant Professor, Stanford University UCSD ECE A Network Information Theory (graduate, 6 quarters) Abbas El Gamal and Young-Han Kim, Network Information Theory, .. Kim, Eren \c{S}a\c{s}o\u{g}lu, and Lele Wang, {\bf A random coding approach to index coding,''} in preparation.


text:
# Source code for pacman.model.graphs.abstract_edge_partition

# Copyright (c) 2017-2019 The University of Manchester
#
# This program is free software: you can redistribute it and/or modify
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.
from spinn_utilities.abstract_base import (
AbstractBase, abstractmethod, abstractproperty)
from spinn_utilities.ordered_set import OrderedSet
from pacman.exceptions import (
PacmanConfigurationException, PacmanInvalidParameterException)
from pacman.model.graphs.common import ConstrainedObject

_REPR_TEMPLATE = "{}(identifier={}, edges={}, constraints={}, label={})"

[docs]class AbstractEdgePartition(ConstrainedObject, metaclass=AbstractBase):
""" A collection of edges which start at a single vertex which have the\
same semantics and so can share a single key or block of SDRAM\
(depending on edge type).
"""

__slots__ = [
# The partition identifier
"_identifier",
# The edges in the partition
"_edges",
# The type of edges to accept
"_allowed_edge_types",
# The weight of traffic going down this partition
"_traffic_weight",
# The label of the graph
"_label",
# class name
"_class_name",
# Safety code generated by the graph when added to that graph
"_graph_code"
]

def __init__(
self, identifier, allowed_edge_types, constraints,
label, traffic_weight, class_name):
"""
:param str identifier: The identifier of the partition
:param allowed_edge_types: The types of edges allowed
:type allowed_edge_types: type or tuple(type, ...)
:param iterable(AbstractConstraint) constraints:
Any initial constraints
:param str label: An optional label of the partition
:param int traffic_weight:
The weight of traffic going down this partition
"""
super().__init__(constraints)
self._label = label
self._identifier = identifier
self._edges = OrderedSet()
self._allowed_edge_types = allowed_edge_types
self._traffic_weight = traffic_weight
self._class_name = class_name
self._graph_code = None

@property
def label(self):
""" The label of the edge partition.

:rtype: str
"""
return self._label

""" Add an edge to the edge partition.

.. note::
This method should only be called by the add_edge method of
the graph that owns the partition. Calling it from anywhere else,
even with the correct graph_code, will lead to unsupported
inconsistency.

:param AbstractEdge edge: the edge to add
:param int graph_code:
A code to check the correct graph is calling this method
:raises PacmanInvalidParameterException:
If the edge does not belong in this edge partition
"""
if graph_code != self._graph_code:
raise PacmanConfigurationException(
"Only one graph should add edges")
if self._graph_code is None:
raise PacmanConfigurationException(
"Only Graphs can add edges to partitions")

# Check for an incompatible edge
if not isinstance(edge, self._allowed_edge_types):
raise PacmanInvalidParameterException(
"edge", str(edge.__class__),
"Edges of this graph must be one of the following types:"
" {}".format(self._allowed_edge_types))

[docs]    def register_graph_code(self, graph_code):
"""
Allows the graph to register its code when the partition is added
"""
if self._graph_code is not None:
raise PacmanConfigurationException(
"Illegal attempt to add partition {} to a second "
"graph".format(self))
self._graph_code = graph_code

@property
def identifier(self):
""" The identifier of this edge partition.

:rtype: str
"""
return self._identifier

@property
def edges(self):
""" The edges in this edge partition.

.. note::
The order in which the edges are added is preserved for when they
are requested later. If not, please talk to the software team.

:rtype: iterable(AbstractEdge)
"""
return self._edges

@property
def n_edges(self):
""" The number of edges in the edge partition.

:rtype: int
"""
return len(self._edges)

@property
def traffic_weight(self):
""" The weight of the traffic in this edge partition compared to\
other partitions.

:rtype: int
"""
return self._traffic_weight

def __repr__(self):
edges = ""
for edge in self._edges:
if edge.label is not None:
edges += edge.label + ","
else:
edges += str(edge) + ","
return _REPR_TEMPLATE.format(
self._class_name, self._identifier, edges, self.constraints,
self.label)

def __str__(self):
return self.__repr__()

def __contains__(self, edge):
""" Check if the edge is contained within this partition

:param AbstractEdge edge: the edge to search for.
:rtype: bool
"""
return edge in self._edges

[docs]    @abstractmethod
def clone_without_edges(self):
""" Make a copy of this edge partition without any of the edges in it

This follows the design pattern that only the graph adds edges to

:return: The copied edge partition but excluding edges
"""

@abstractproperty
def pre_vertices(self):
"""
Provides the vertices associated with this partition

.. note::
Most edge partitions will be
:py:class:AbstractSingleSourcePartition
and therefore provide the pre_vertex method.

:rtype: iter(AbstractVertex)
"""


text:
eigenvalues of 12*12 symbolic matrix

Hi dear friends
I have a 12*12 symbolic matrix in terms of x y z d that I want its eigenvalues
but not mathematica nor matlab can do it for me.My mathematica is "7" so If you have a newer version or even in matlab , would you mind checking my matrix in your software?
this is my matrix in mathematica and matlab:
Attached Files
 matrix.m (1.5 KB, 6 views) matrix.nb (10.6 KB, 5 views)
 PhysOrg.com science news on PhysOrg.com >> Leading 3-D printer firms to merge in $403M deal (Update)>> LA to give every student an iPad;$30M order>> CIA faulted for choosing Amazon over IBM on cloud contract
 In[1]:= LeafCount[Eigenvalues[{{a,b},{c,d}}]] Out[1]= 55 In[2]:= LeafCount[Eigenvalues[{{a,b,c},{d,e,f},{g,h,i}}]] Out[2]= 268 In[3]:= LeafCount[Eigenvalues[{{a,b,c,d},{e,f,g,h}, {i,j,k,l},{m,n,o,p}}]] Out[3]= 1657 In[4]:= LeafCount[Eigenvalues[{{a,b,c,d,e},{f,g,h,i,j}, {k,l,m,n,o},{p,q,r,s,t},{u,v,w,x,y}}]] Out[4]= 11916 In[5]:= LeafCount[Eigenvalues[{{a,b,c,d,e,f}, {g,h,i,j,k,l},{m,n,o,p,q,r},{s,t,u,v,w,x}, {y,z,a1,b1,c1,d1},{e1,f1,g1,h1,i1,j1}}]] Out[5]= 97411 In[6]:= Fit[Map[Log,{55,268,1657,11916}],{1,x},x] Out[6]= 2.1100 + 1.7957*x In[7]:= Show[Plot[2.1100+1.7957 x,{x,1,5}],ListPlot[Map[ Log,{55,268,1657,11916,97411}], Joined->True]] Out[7]= ...plotSnipped... In[8]:= E^(2.1100+1.7957 x)/.x->12 Out[8]= 1.8817*10^10 So for this somewhat simpler approximation of an n by n symbolic matrix the number of elements making up the eigenvalues grows just a bit faster than E^(2.11+1.8x) and thus you would have about 18 billion elements making up the result. Now your matrix is "banded" and has some zeros, which will help, but probably not enough. You could try replacing all your 2d/3 by d and each of your unique -2 Cos functions by other simple fabricated variables. That might reduce the size of your task by perhaps a factor of 3 or 4. But even if your zeros and banding help more than I expect, I doubt your result will have less than a billion elements. Maybe with 64 gigabytes of memory and unlimited time you might get an answer to the simplified problem, but that is just a guess and I would not be surprised if I were wrong.

 Quote by Bill Simpson Bill Simpson;
thanks for your help But I made it more simplify by renaming identical elements and matrix became in the form of below:
Attached Files
 matrix1.nb (3.5 KB, 1 views)

eigenvalues of 12*12 symbolic matrix

Did you try this on a computer with 64 gigabytes of memory?

Did you try solving this problem for 8x8 and 9x9 and 10x10 matricies and use the Mathematica function MaxMemoryUsed[] with a fresh start for each of your smaller problems. See how much memory is needed for the smaller problems and how this is growing and whether you can then estimate how much memory will be needed for the 12x12 matrix. With the banding of your matrix I am not sure how to use smaller matricies and still get memory measurements that will correctly estimate what your 12x12 matrix will require.
 Sorry but what do you mean by computer with 64 gigabytes of memory? and also generally in calculating eigenvalues forexample for 4*4 symbolic matrix, mathematica gives "ROOT[......]" so what should I do for solving these roots and arrive to my eigenvalues?
 If you have an old or inexpensive computer you may only have 256 megabytes of memory. If you have a newer or better computer you may have one gigabyte or two gigabytes of memory. Your eigenvalue calculation will have billions of terms in the result. So I asked if you were able to use a computer that does not have one or two gigabytes of memory, but instead to use one with 64 or 128 gigabytes of memory. Perhaps that will be enough to be able to calculate your result. Now Root[]. Cubic and quartic equations, those with x^3 or x^4 in them and you want to find all the roots, are sometimes very large. Mathematica will sometimes give you a smaller way of displaying the result. It will give Root[equation, 1] for the "first" root of the equation and Root[equation, 2] for the "second" root, etc. These are usually much smaller and sometimes you can see the information you need from this. If you want to see all the details of what each root is you can try ToRadicals[Root[equation,1]] and see if it can display the solution you wish. Here is the documentation on this. http://reference.wolfram.com/mathema...oRadicals.html It is possible that there is no way for ToRadicals to find what you want.
 thanks but my pc is new with core i 7 and 8gig ram and 500gig memory but still it cannot give its eigenvalues! I really dont Know how can I solve my problem anymore !
 In[2]:= aa = a //. {(2*d)/3 -> d, (-2*d)/3 -> -d, -2*Cos[(x + z)/4] -> -e, -2*Cos[(y + z)/4] -> -f, -2*Cos[(x + y)/4] -> -g, -2*Cos[(x - y)/4] -> -h, -2*Cos[(y - z)/4] -> -i, -2*Cos[(x - z)/4] -> -j} Out[2]= { {d, d, d, -e, 0, 0, -f, 0, 0, -g, 0, 0}, {d, d, d, 0, -e, 0, 0, -f, 0, 0, -g, 0}, {d, d, d, 0, 0, -e, 0, 0, -f, 0, 0, -g}, {-e, 0, 0, d, -d, -d, -h, 0, 0, -i, 0, 0}, {0, -e, 0, -d, d, d, 0, -h, 0, 0, -i, 0}, {0, 0, -e, -d, d, d, 0, 0, -h, 0, 0, -i}, {-f, 0, 0, -h, 0, 0, d, -d, d, -j, 0, 0}, {0, -f, 0, 0, -h, 0, -d, d, -d, 0, -j, 0}, {0, 0, -f, 0, 0, -h, d, -d, d, 0, 0, -j}, {-g, 0, 0, -i, 0, 0, -j, 0, 0, d, d, -d}, {0, -g, 0, 0, -i, 0,0, -j, 0, d, d, -d}, {0, 0, -g, 0, 0, -i, 0, 0, -j, -d, -d, d}} In[3]:= aaa=Eigenvalues[aa]; There you have it, the eigenvalues, but not printed out. How big is it? In[4]:= LeafCount[aaa] Out[4]= 424921 In[5]:= aaa[[1]] Out[5] = stunninglyHugeRootObject and there are 11 more of those I am astonished it was able to complete. I did not believe this could be done. Please check all this very carefully to make certain I have made no errors. With your computer you should easily be able to reproduce this. Then you have to understand the twelve roots of this huge polynomial. There may or may not be a solution in radicals for these.
 Really thanks for your complete guidence But how can understand the twelve roots of this huge polynomial? by ToRadicals of aaa[[1]] ? thank you alot
 I do not believe ToRadicals[aaa[[1]]] will be able to find a solution for you, but you can try it and see what happens. Or perhaps there is some other method to at least partly factor the result. Or perhaps there are common subexpressions that appear several times in the large polynomial and you can see some meaning in those. Perhaps there is something about your original problem that will tell something about the polynomial. But it is so large that I believe it will be very difficult for you to make any sense of.
 Recognitions: Science Advisor Assuming Out[2] in post #8 is correct, this has a huge amount of internal structure. You only need to diagonalize the 3x3 diagonal blocks to make it effectively a 4x4 symmetric matrix where each "element" is a 3x3 diagonal matrix. Doing something along those lines will probably give you a lot more understanding than getting "the answer" by brute force.

 Quote by AlephZero Assuming Out[2] in post #8 is correct, this has a huge amount of internal structure. You only need to diagonalize the 3x3 diagonal blocks to make it effectively a 4x4 symmetric matrix where each "element" is a 3x3 diagonal matrix. Doing something along those lines will probably give you a lot more understanding than getting "the answer" by brute force.
yes you are right my matrix is infact a 4*4 matrix where each "element" is a 3x3 diagonal matrix
But I have one question
If I diagonalize each of 16 blocks seperately and then put them in the matrix and after obtaining a symmetric matrix,find its eigenvalues, does this procces preserve the form of first matrix eigenvalues? I mean the finall symmetric matrix will have the same eigenvalues of the first matrix?

thanks

 Similar discussions for: eigenvalues of 12*12 symbolic matrix Thread Forum Replies Math & Science Software 0 Math & Science Software 1 Linear & Abstract Algebra 1 Linear & Abstract Algebra 3 Engineering, Comp Sci, & Technology Homework 1


text:
# Group inclusion- Quotiens Inclusion

Given a group $G$ , and two subgroups $G_1, G_2$ such that $G_1 \subseteq G_2$ , is it true that $G/G_2 \subseteq G/G_1$ ?

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If you know what $G/G_1$ and $G/G_2$ are, then you can use the subgroup test to answer this. – Arkamis Jul 17 '12 at 16:51
But is it always true? – joshua Jul 17 '12 at 17:04
It would be if you can show it generally. Is this a homework problem, by chance? – Arkamis Jul 17 '12 at 17:05

What you probably had in mind is the following: $$G/G_2\cong\left(G/G_1\right)/\left(G_2/G_1\right)$$ by the 2nd or 3d. isomorphism theorem, so what you asked can't generally be as $\,G/G_2\,$ is a quotient of $\,G/G_1\,$ , i.e.: there exists a surjective group homomorphism $$G/G_1\longrightarrow G/G_2\,\,,\,\,xG_1\to xG_2$$

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Grear! Thanks!! – joshua Jul 17 '12 at 18:45

The quotients may not make sense, if $G_1$ and $G_2$ are not normal.

However, it is true that every left-$G_2$ coset is completely contained in a $G_1$ coset. That is, the equivalence relation induced by the larger subgroup is "coarser" than that of the larger group. In other words, $x{}_{G_1}\equiv y$ implies $x{}_{G_2}\equiv y$, but not necessarily conversely.

To see this, simply remember that $x{}_{G_i}\equiv y$ if and only if $y^{-1}x\in G_i$. If $x{}_{G_1}\equiv y$, then $y^{-1}x\in G_1\subseteq G_2$, so $x{}_{G_2}\equiv y$. But if $G_2$ properly contains $G_1$, then let $x\in G_2-G_1$; then $x{}_{G_2}\equiv e$, but $x{}_{G_1}\notequiv e$.

Similar results hold for congruent-on-the-right (equivalently, right cosets).

If both $G_1$ and $G_2$ are normal, so that the set of equivalence classes can be given a group structure, then the coarser equivalence relation induces a smaller group, so that $G/G_2$ is a quotient of $G/G_1$.

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# Is there a notion of field deletions that is like the opposite of field extensions?

I am thinking about my previous question here.

Is there a notion of deleting an element $\alpha$ from a field $F$ that is always well defined? I learned that you cannot always find a proper subfield $E$ such that $E(\alpha)=F$. Take the case of $\mathtt{R}$ and $\alpha=\sqrt{2}$.

However, in the case of $R$, there are many subfields that do not contain the element $\alpha=\sqrt{2}$. Is there a maximal proper subfield? Simply taking unions of subfields would not work as a union of two subfields is not guaranteed to be a field.

I thought if we have two subfields $Q(\beta_1, \dots, \beta_n)$ and $Q(\beta', \dots, \beta_{n'}')$ that do not contain $\sqrt{2}$, we can get a larger subfield $Q(\beta_1, \dots, \beta_n, \beta', \dots, \beta_{n'}')$ that also does not contain $\sqrt{2}$. This was pointed out to be not to be possible in the comments. In some specific cases I think it works, though.

However, my algebra is rusty and I am not sure how to combine subfields that are common extensions of the same underlying field. I know at least one field exists as the algebraic closure of $Q$ exists (take away the imaginary numbers). Certainly a parent subfield exists but in our case will it be proper if they both exclude an element?

My Question: Is there a general notion of field deletion of the element $\alpha$ from the field $F$? I particularly want this notion to undo field extensions, so that would probably be easily gotten.

More specifically, I also want to see the specifics of any such notion in the case of deleting an element of $R$, like $\sqrt{2}$, from $R$.

• Why does the bigger subfield also not contain $\sqrt{2}$? I don't really see that... Anyway, even if it is true it would strongly depend on the element being $\sqrt{2}$. Take for an other example $\alpha = \sqrt{2} + \sqrt{3}$. Then $\alpha \not \in Q(\sqrt{2})$ and $\alpha \not \in Q(\sqrt{3})$ but we have $\alpha \in Q(\sqrt{2},\sqrt{3})$, if I remember correctly we even have $Q(\sqrt{2},\sqrt{3}) = Q(\alpha)$... – Dirk Oct 19 '16 at 12:24
• Ah! Thank you for this correction. – abnry Oct 19 '16 at 12:28
• Zorn's lemma ensures such a field always exists, however I think that uniqueness is hopeless. – Crostul Oct 19 '16 at 12:46

Let $K$ be a field an $\alpha \in K$. Let $M = \{L \leq K \mid \alpha \not \in L\}$ be the set of subfields of $K$ that do not contain $\alpha$. Then using Zorn's Lemma we can show that $M$ contains maximal elements (as long as $M$ is nonempty at least), but we won't get a uniqueness of this element, I'm not even sure we might get uniqueness up to isomorphisms or any structure of these maximal elements at all.
The proof using Zorn is quite simple because if we have an ordered sequence of fields $L_1 \leq L_2 \leq L_3 \leq L_4 \leq \ldots$ then also their union is a field.
And an algorithm to construct a counter example for the uniqueness up to isomorphism: Take $K|Q$ a Galois extension of degree $6$ with primitive element $\alpha$. Then $\alpha$ is not contained in any non trivial subfield of $K$. But $K$ has a maximal subfield of index $2$ and one of index $3$ which then both not contain $\alpha$ but are not isomorphic (as they have different dimensions as a $Q$-vectorspace).
• I am not convinced $M$ is an ordered sequence of fields. Can you give me more? – abnry Oct 19 '16 at 12:30
• Okay, possibly it is unique iff $F=E(\alpha)$? – abnry Oct 19 '16 at 12:32
• If I understand this right, by way of K being a galois extension, $K$ is the smallest field (in $R$) that contains $\alpha$. So as you say any proper subfield cannot contain $\alpha$. It is not clear to me why there are maximal subfields of index $2$ and $3$. Can you give me a particular polynomial $K$ is the splitting field of? Is it something like $p(x)=x^6-8$ with primitive $\alpha=\sqrt{2} e^{i \pi/3}$? Sorry for not quite getting it. – abnry Oct 19 '16 at 20:21
• It occurs to me though that there might be a fix as it seems in this case the "right" field to choose as the resultant of the deletion of $\sqrt{2}e^{i\pi/3}$ from $K$ would be $Q$ itself. But maybe you'd always get $Q$. – abnry Oct 19 '16 at 20:25