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# Thomson problem

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The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904[1] after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms.

Related problems include the study of the geometry of the minimum energy configuration and the study of the large N behavior of the minimum energy.

## Mathematical statement

The physical system embodied by the Thomson problem is a special case of one of eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere".[2] The solution of each N-electron problem is obtained when the N-electron configuration constrained to the surface of a sphere of unit radius, ${\displaystyle r=1}$, yields a global electrostatic potential energy minimum, ${\displaystyle U(N)}$.

The electrostatic interaction energy occurring between each pair of electrons of equal charges (${\displaystyle e_{i}=e_{j}=e}$, with ${\displaystyle e}$ the elementary charge of an electron) is given by Coulomb's Law,

${\displaystyle U_{ij}(N)=k_{e}{e_{i}e_{j} \over r_{ij}}.}$

Here, ${\displaystyle k_{e}}$ is Coulomb's constant and ${\displaystyle r_{ij}=|\mathbf {r} _{i}-\mathbf {r} _{j}|}$ is the distance between each pair of electrons located at points on the sphere defined by vectors ${\displaystyle \mathbf {r} _{i}}$ and ${\displaystyle \mathbf {r} _{j}}$, respectively.

Simplified units of ${\displaystyle e=1}$ and ${\displaystyle k_{e}=1}$ are used without loss of generality. Then,

${\displaystyle U_{ij}(N)={1 \over r_{ij}}.}$

The total electrostatic potential energy of each N-electron configuration may then be expressed as the sum of all pair-wise interactions

${\displaystyle U(N)=\sum _{i

The global minimization of ${\displaystyle U(N)}$ over all possible collections of N distinct points is typically found by numerical minimization algorithms.

### Example

The solution of the Thomson problem for two electrons is obtained when both electrons are as far apart as possible on opposite sides of the origin, ${\displaystyle r_{ij}=2r=2}$, or

${\displaystyle U(2)={1 \over 2}.}$

## Known solutions

Schematic geometric solutions of the mathematical Thomson Problem for up to N = 5 electrons.

Minimum energy configurations have been rigorously identified in only a handful of cases.

• For N = 1, the solution is trivial as the electron may reside at any point on the surface of the unit sphere. The total energy of the configuration is defined as zero as the electron is not subject to the electric field due to any other sources of charge.
• For N = 2, the optimal configuration consists of electrons at antipodal points.
• For N = 3, electrons reside at the vertices of an equilateral triangle about a great circle.[3]
• For N = 4, electrons reside at the vertices of a regular tetrahedron.
• For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid.[4]
• For N = 6, electrons reside at vertices of a regular octahedron.[5]
• For N = 12, electrons reside at the vertices of a regular icosahedron.[6]

The geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are Platonic solids whose faces are congruent equilateral triangles. Numerical solutions for N = 8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, whose faces are square and pentagonal, respectively.[citation needed]

## Generalizations

One can also ask for ground states of particles interacting with arbitrary potentials. To be mathematically precise, let f be a decreasing real-valued function, and define the energy functional ${\displaystyle \sum _{i

Traditionally, one considers ${\displaystyle f(x)=x^{-\alpha }}$ also known as Riesz ${\displaystyle \alpha }$-kernels. For integrable Riesz kernels see the 1972 work of Landkof.[7] For non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see the 2004 work of Hardin and Saff.[8] Notable cases include:[9]

• α = ∞, the Tammes problem (packing);
• α = -1, the Thomson problem;
• α = 0, to maximize the product of distances, latterly known as Whyte's problem;
• α = 1 : maximum average distance problem.

One may also consider configurations of N points on a sphere of higher dimension. See spherical design.

## Solution algorithms

Several algorithms have been applied to this problem. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance:[9]

• constrained global optimization (Altschuler et al. 1994),
• steepest descent (Claxton and Benson 1966, Erber and Hockney 1991),
• random walk (Weinrach et al. 1990),
• genetic algorithm (Morris et. al 1996)

## Relations to other scientific problems

The Thomson problem is a natural consequence of Thomson's plum pudding model in the absence of its uniform positive background charge.[10]

"No fact discovered about the atom can be trivial, nor fail to accelerate the progress of physical science, for the greater part of natural philosophy is the outcome of the structure and mechanism of the atom."

—Sir J. J. Thomson[11]

Though experimental evidence led to the abandonment of Thomson's plum pudding model as a complete atomic model, irregularities observed in numerical energy solutions of the Thomson problem have been found to correspond with electron shell-filling in naturally occurring atoms throughout the periodic table of elements.[12]

The Thomson problem also plays a role in the study of other physical models including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps.

The generalized Thomson problem arises, for example, in determining the arrangements of the protein subunits which comprise the shells of spherical viruses. The "particles" in this application are clusters of protein subunits arranged on a shell. Other realizations include regular arrangements of colloid particles in colloidosomes, proposed for encapsulation of active ingredients such as drugs, nutrients or living cells, fullerene patterns of carbon atoms, and VSEPR theory. An example with long-range logarithmic interactions is provided by the Abrikosov vortices which would form at low temperatures in a superconducting metal shell with a large monopole at the center.

## Configurations of smallest known energy

In the following table ${\displaystyle N}$ is the number of points (charges) in a configuration, ${\displaystyle E_{1}}$ is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and ${\displaystyle r_{i}}$ are the positions of the charges. Most symmetry types require the vector sum of the positions (and thus the electric dipole moment) to be zero.

It is customary to also consider the polyhedron formed by the convex hull of the points. Thus, ${\displaystyle v_{i}}$ is the number of vertices where the given number of edges meet, ${\displaystyle e}$ is the total number of edges, ${\displaystyle f_{3}}$ is the number of triangular faces, ${\displaystyle f_{4}}$ is the number of quadrilateral faces, and ${\displaystyle \theta _{1}}$ is the smallest angle subtended by vectors associated with the nearest charge pair. Note that the edge lengths are generally not equal; thus (except in the cases N = 2, 3, 4, 6, 12, and the geodesic polyhedra) the convex hull is only topologically equivalent to the figure listed in the last column.[13]

N ${\displaystyle E_{1}}$ Symmetry ${\displaystyle \left|\sum \mathbf {r} _{i}\right|}$ ${\displaystyle v_{3}}$ ${\displaystyle v_{4}}$ ${\displaystyle v_{5}}$ ${\displaystyle v_{6}}$ ${\displaystyle v_{7}}$ ${\displaystyle v_{8}}$ ${\displaystyle e}$ ${\displaystyle f_{3}}$ ${\displaystyle f_{4}}$ ${\displaystyle \theta _{1}}$ Equivalent polyhedron
2 0.500000000 ${\displaystyle D_{\infty h}}$ 0 2 180.000° digon
3 1.732050808 ${\displaystyle D_{3h}}$ 0 3 2 120.000° triangle
4 3.674234614 ${\displaystyle T_{d}}$ 0 4 0 0 0 0 0 6 4 0 109.471° tetrahedron
5 6.474691495 ${\displaystyle D_{3h}}$ 0 2 3 0 0 0 0 9 6 0 90.000° triangular dipyramid
6 9.985281374 ${\displaystyle O_{h}}$ 0 0 6 0 0 0 0 12 8 0 90.000° octahedron
7 14.452977414 ${\displaystyle D_{5h}}$ 0 0 5 2 0 0 0 15 10 0 72.000° pentagonal dipyramid
8 19.675287861 ${\displaystyle D_{4d}}$ 0 0 8 0 0 0 0 16 8 2 71.694° square antiprism
9 25.759986531 ${\displaystyle D_{3h}}$ 0 0 3 6 0 0 0 21 14 0 69.190° triaugmented triangular prism
10 32.716949460 ${\displaystyle D_{4d}}$ 0 0 2 8 0 0 0 24 16 0 64.996° gyroelongated square dipyramid
11 40.596450510 ${\displaystyle C_{2v}}$ 0.013219635 0 2 8 1 0 0 27 18 0 58.540° edge-contracted icosahedron
12 49.165253058 ${\displaystyle I_{h}}$ 0 0 0 12 0 0 0 30 20 0 63.435° icosahedron
(geodesic sphere {3,5+}1,0)
13 58.853230612 ${\displaystyle C_{2v}}$ 0.008820367 0 1 10 2 0 0 33 22 0 52.317°
14 69.306363297 ${\displaystyle D_{6d}}$ 0 0 0 12 2 0 0 36 24 0 52.866° gyroelongated hexagonal dipyramid
15 80.670244114 ${\displaystyle D_{3}}$ 0 0 0 12 3 0 0 39 26 0 49.225°
16 92.911655302 ${\displaystyle T}$ 0 0 0 12 4 0 0 42 28 0 48.936°
17 106.050404829 ${\displaystyle D_{5h}}$ 0 0 0 12 5 0 0 45 30 0 50.108° double-gyroelongated pentagonal dipyramid
18 120.084467447 ${\displaystyle D_{4d}}$ 0 0 2 8 8 0 0 48 32 0 47.534°
19 135.089467557 ${\displaystyle C_{2v}}$ 0.000135163 0 0 14 5 0 0 50 32 1 44.910°
20 150.881568334 ${\displaystyle D_{3h}}$ 0 0 0 12 8 0 0 54 36 0 46.093°
21 167.641622399 ${\displaystyle C_{2v}}$ 0.001406124 0 1 10 10 0 0 57 38 0 44.321°
22 185.287536149 ${\displaystyle T_{d}}$ 0 0 0 12 10 0 0 60 40 0 43.302°
23 203.930190663 ${\displaystyle D_{3}}$ 0 0 0 12 11 0 0 63 42 0 41.481°
24 223.347074052 ${\displaystyle O}$ 0 0 0 24 0 0 0 60 32 6 42.065° snub cube
25 243.812760299 ${\displaystyle C_{s}}$ 0.001021305 0 0 14 11 0 0 68 44 1 39.610°
26 265.133326317 ${\displaystyle C_{2}}$ 0.001919065 0 0 12 14 0 0 72 48 0 38.842°
27 287.302615033 ${\displaystyle D_{5h}}$ 0 0 0 12 15 0 0 75 50 0 39.940°
28 310.491542358 ${\displaystyle T}$ 0 0 0 12 16 0 0 78 52 0 37.824°
29 334.634439920 ${\displaystyle D_{3}}$ 0 0 0 12 17 0 0 81 54 0 36.391°
30 359.603945904 ${\displaystyle D_{2}}$ 0 0 0 12 18 0 0 84 56 0 36.942°
31 385.530838063 ${\displaystyle C_{3v}}$ 0.003204712 0 0 12 19 0 0 87 58 0 36.373°
32 412.261274651 ${\displaystyle I_{h}}$ 0 0 0 12 20 0 0 90 60 0 37.377° pentakis dodecahedron
(geodesic sphere {3,5+}1,1)
33 440.204057448 ${\displaystyle C_{s}}$ 0.004356481 0 0 15 17 1 0 92 60 1 33.700°
34 468.904853281 ${\displaystyle D_{2}}$ 0 0 0 12 22 0 0 96 64 0 33.273°
35 498.569872491 ${\displaystyle C_{2}}$ 0.000419208 0 0 12 23 0 0 99 66 0 33.100°
36 529.122408375 ${\displaystyle D_{2}}$ 0 0 0 12 24 0 0 102 68 0 33.229°
37 560.618887731 ${\displaystyle D_{5h}}$ 0 0 0 12 25 0 0 105 70 0 32.332°
38 593.038503566 ${\displaystyle D_{6d}}$ 0 0 0 12 26 0 0 108 72 0 33.236°
39 626.389009017 ${\displaystyle D_{3h}}$ 0 0 0 12 27 0 0 111 74 0 32.053°
40 660.675278835 ${\displaystyle T_{d}}$ 0 0 0 12 28 0 0 114 76 0 31.916°
41 695.916744342 ${\displaystyle D_{3h}}$ 0 0 0 12 29 0 0 117 78 0 31.528°
42 732.078107544 ${\displaystyle D_{5h}}$ 0 0 0 12 30 0 0 120 80 0 31.245°
43 769.190846459 ${\displaystyle C_{2v}}$ 0.000399668 0 0 12 31 0 0 123 82 0 30.867°
44 807.174263085 ${\displaystyle O_{h}}$ 0 0 0 24 20 0 0 120 72 6 31.258°
45 846.188401061 ${\displaystyle D_{3}}$ 0 0 0 12 33 0 0 129 86 0 30.207°
46 886.167113639 ${\displaystyle T}$ 0 0 0 12 34 0 0 132 88 0 29.790°
47 927.059270680 ${\displaystyle C_{s}}$ 0.002482914 0 0 14 33 0 0 134 88 1 28.787°
48 968.713455344 ${\displaystyle O}$ 0 0 0 24 24 0 0 132 80 6 29.690°
49 1011.557182654 ${\displaystyle C_{3}}$ 0.001529341 0 0 12 37 0 0 141 94 0 28.387°
50 1055.182314726 ${\displaystyle D_{6d}}$ 0 0 0 12 38 0 0 144 96 0 29.231°
51 1099.819290319 ${\displaystyle D_{3}}$ 0 0 0 12 39 0 0 147 98 0 28.165°
52 1145.418964319 ${\displaystyle C_{3}}$ 0.000457327 0 0 12 40 0 0 150 100 0 27.670°
53 1191.922290416 ${\displaystyle C_{2v}}$ 0.000278469 0 0 18 35 0 0 150 96 3 27.137°
54 1239.361474729 ${\displaystyle C_{2}}$ 0.000137870 0 0 12 42 0 0 156 104 0 27.030°
55 1287.772720783 ${\displaystyle C_{2}}$ 0.000391696 0 0 12 43 0 0 159 106 0 26.615°
56 1337.094945276 ${\displaystyle D_{2}}$ 0 0 0 12 44 0 0 162 108 0 26.683°
57 1387.383229253 ${\displaystyle D_{3}}$ 0 0 0 12 45 0 0 165 110 0 26.702°
58 1438.618250640 ${\displaystyle D_{2}}$ 0 0 0 12 46 0 0 168 112 0 26.155°
59 1490.773335279 ${\displaystyle C_{2}}$ 0.000154286 0 0 14 43 2 0 171 114 0 26.170°
60 1543.830400976 ${\displaystyle D_{3}}$ 0 0 0 12 48 0 0 174 116 0 25.958°
61 1597.941830199 ${\displaystyle C_{1}}$ 0.001091717 0 0 12 49 0 0 177 118 0 25.392°
62 1652.909409898 ${\displaystyle D_{5}}$ 0 0 0 12 50 0 0 180 120 0 25.880°
63 1708.879681503 ${\displaystyle D_{3}}$ 0 0 0 12 51 0 0 183 122 0 25.257°
64 1765.802577927 ${\displaystyle D_{2}}$ 0 0 0 12 52 0 0 186 124 0 24.920°
65 1823.667960264 ${\displaystyle C_{2}}$ 0.000399515 0 0 12 53 0 0 189 126 0 24.527°
66 1882.441525304 ${\displaystyle C_{2}}$ 0.000776245 0 0 12 54 0 0 192 128 0 24.765°
67 1942.122700406 ${\displaystyle D_{5}}$ 0 0 0 12 55 0 0 195 130 0 24.727°
68 2002.874701749 ${\displaystyle D_{2}}$ 0 0 0 12 56 0 0 198 132 0 24.433°
69 2064.533483235 ${\displaystyle D_{3}}$ 0 0 0 12 57 0 0 201 134 0 24.137°
70 2127.100901551 ${\displaystyle D_{2d}}$ 0 0 0 12 50 0 0 200 128 4 24.291°
71 2190.649906425 ${\displaystyle C_{2}}$ 0.001256769 0 0 14 55 2 0 207 138 0 23.803°
72 2255.001190975 ${\displaystyle I}$ 0 0 0 12 60 0 0 210 140 0 24.492° geodesic sphere {3,5+}2,1
73 2320.633883745 ${\displaystyle C_{2}}$ 0.001572959 0 0 12 61 0 0 213 142 0 22.810°
74 2387.072981838 ${\displaystyle C_{2}}$ 0.000641539 0 0 12 62 0 0 216 144 0 22.966°
75 2454.369689040 ${\displaystyle D_{3}}$ 0 0 0 12 63 0 0 219 146 0 22.736°
76 2522.674871841 ${\displaystyle C_{2}}$ 0.000943474 0 0 12 64 0 0 222 148 0 22.886°
77 2591.850152354 ${\displaystyle D_{5}}$ 0 0 0 12 65 0 0 225 150 0 23.286°
78 2662.046474566 ${\displaystyle T_{h}}$ 0 0 0 12 66 0 0 228 152 0 23.426°
79 2733.248357479 ${\displaystyle C_{s}}$ 0.000702921 0 0 12 63 1 0 230 152 1 22.636°
80 2805.355875981 ${\displaystyle D_{4d}}$ 0 0 0 16 64 0 0 232 152 2 22.778°
81 2878.522829664 ${\displaystyle C_{2}}$ 0.000194289 0 0 12 69 0 0 237 158 0 21.892°
82 2952.569675286 ${\displaystyle D_{2}}$ 0 0 0 12 70 0 0 240 160 0 22.206°
83 3027.528488921 ${\displaystyle C_{2}}$ 0.000339815 0 0 14 67 2 0 243 162 0 21.646°
84 3103.465124431 ${\displaystyle C_{2}}$ 0.000401973 0 0 12 72 0 0 246 164 0 21.513°
85 3180.361442939 ${\displaystyle C_{2}}$ 0.000416581 0 0 12 73 0 0 249 166 0 21.498°
86 3258.211605713 ${\displaystyle C_{2}}$ 0.001378932 0 0 12 74 0 0 252 168 0 21.522°
87 3337.000750014 ${\displaystyle C_{2}}$ 0.000754863 0 0 12 75 0 0 255 170 0 21.456°
88 3416.720196758 ${\displaystyle D_{2}}$ 0 0 0 12 76 0 0 258 172 0 21.486°
89 3497.439018625 ${\displaystyle C_{2}}$ 0.000070891 0 0 12 77 0 0 261 174 0 21.182°
90 3579.091222723 ${\displaystyle D_{3}}$ 0 0 0 12 78 0 0 264 176 0 21.230°
91 3661.713699320 ${\displaystyle C_{2}}$ 0.000033221 0 0 12 79 0 0 267 178 0 21.105°
92 3745.291636241 ${\displaystyle D_{2}}$ 0 0 0 12 80 0 0 270 180 0 21.026°
93 3829.844338421 ${\displaystyle C_{2}}$ 0.000213246 0 0 12 81 0 0 273 182 0 20.751°
94 3915.309269620 ${\displaystyle D_{2}}$ 0 0 0 12 82 0 0 276 184 0 20.952°
95 4001.771675565 ${\displaystyle C_{2}}$ 0.000116638 0 0 12 83 0 0 279 186 0 20.711°
96 4089.154010060 ${\displaystyle C_{2}}$ 0.000036310 0 0 12 84 0 0 282 188 0 20.687°
97 4177.533599622 ${\displaystyle C_{2}}$ 0.000096437 0 0 12 85 0 0 285 190 0 20.450°
98 4266.822464156 ${\displaystyle C_{2}}$ 0.000112916 0 0 12 86 0 0 288 192 0 20.422°
99 4357.139163132 ${\displaystyle C_{2}}$ 0.000156508 0 0 12 87 0 0 291 194 0 20.284°
100 4448.350634331 ${\displaystyle T}$ 0 0 0 12 88 0 0 294 196 0 20.297°
101 4540.590051694 ${\displaystyle D_{3}}$ 0 0 0 12 89 0 0 297 198 0 20.011°
102 4633.736565899 ${\displaystyle D_{3}}$ 0 0 0 12 90 0 0 300 200 0 20.040°
103 4727.836616833 ${\displaystyle C_{2}}$ 0.000201245 0 0 12 91 0 0 303 202 0 19.907°
104 4822.876522746 ${\displaystyle D_{6}}$ 0 0 0 12 92 0 0 306 204 0 19.957°
105 4919.000637616 ${\displaystyle D_{3}}$ 0 0 0 12 93 0 0 309 206 0 19.842°
106 5015.984595705 ${\displaystyle D_{2}}$ 0 0 0 12 94 0 0 312 208 0 19.658°
107 5113.953547724 ${\displaystyle C_{2}}$ 0.000064137 0 0 12 95 0 0 315 210 0 19.327°
108 5212.813507831 ${\displaystyle C_{2}}$ 0.000432525 0 0 12 96 0 0 318 212 0 19.327°
109 5312.735079920 ${\displaystyle C_{2}}$ 0.000647299 0 0 14 93 2 0 321 214 0 19.103°
110 5413.549294192 ${\displaystyle D_{6}}$ 0 0 0 12 98 0 0 324 216 0 19.476°
111 5515.293214587 ${\displaystyle D_{3}}$ 0 0 0 12 99 0 0 327 218 0 19.255°
112 5618.044882327 ${\displaystyle D_{5}}$ 0 0 0 12 100 0 0 330 220 0 19.351°
113 5721.824978027 ${\displaystyle D_{3}}$ 0 0 0 12 101 0 0 333 222 0 18.978°
114 5826.521572163 ${\displaystyle C_{2}}$ 0.000149772 0 0 12 102 0 0 336 224 0 18.836°
115 5932.181285777 ${\displaystyle C_{3}}$ 0.000049972 0 0 12 103 0 0 339 226 0 18.458°
116 6038.815593579 ${\displaystyle C_{2}}$ 0.000259726 0 0 12 104 0 0 342 228 0 18.386°
117 6146.342446579 ${\displaystyle C_{2}}$ 0.000127609 0 0 12 105 0 0 345 230 0 18.566°
118 6254.877027790 ${\displaystyle C_{2}}$ 0.000332475 0 0 12 106 0 0 348 232 0 18.455°
119 6364.347317479 ${\displaystyle C_{2}}$ 0.000685590 0 0 12 107 0 0 351 234 0 18.336°
120 6474.756324980 ${\displaystyle C_{s}}$ 0.001373062 0 0 12 108 0 0 354 236 0 18.418°
121 6586.121949584 ${\displaystyle C_{3}}$ 0.000838863 0 0 12 109 0 0 357 238 0 18.199°
122 6698.374499261 ${\displaystyle I_{h}}$ 0 0 0 12 110 0 0 360 240 0 18.612° geodesic sphere {3,5+}2,2
123 6811.827228174 ${\displaystyle C_{2v}}$ 0.001939754 0 0 14 107 2 0 363 242 0 17.840°
124 6926.169974193 ${\displaystyle D_{2}}$ 0 0 0 12 112 0 0 366 244 0 18.111°
125 7041.473264023 ${\displaystyle C_{2}}$ 0.000088274 0 0 12 113 0 0 369 246 0 17.867°
126 7157.669224867 ${\displaystyle D_{4}}$ 0 0 2 16 100 8 0 372 248 0 17.920°
127 7274.819504675 ${\displaystyle D_{5}}$ 0 0 0 12 115 0 0 375 250 0 17.877°
128 7393.007443068 ${\displaystyle C_{2}}$ 0.000054132 0 0 12 116 0 0 378 252 0 17.814°
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130 7632.167378912 ${\displaystyle C_{2}}$ 0.000025622 0 0 12 118 0 0 384 256 0 17.683°
131 7753.205166941 ${\displaystyle C_{2}}$ 0.000305133 0 0 12 119 0 0 387 258 0 17.511°
132 7875.045342797 ${\displaystyle I}$ 0 0 0 12 120 0 0 390 260 0 17.958° geodesic sphere {3,5+}3,1
133 7998.179212898 ${\displaystyle C_{3}}$ 0.000591438 0 0 12 121 0 0 393 262 0 17.133°
134 8122.089721194 ${\displaystyle C_{2}}$ 0.000470268 0 0 12 122 0 0 396 264 0 17.214°
135 8246.909486992 ${\displaystyle D_{3}}$ 0 0 0 12 123 0 0 399 266 0 17.431°
136 8372.743302539 ${\displaystyle T}$ 0 0 0 12 124 0 0 402 268 0 17.485°
137 8499.534494782 ${\displaystyle D_{5}}$ 0 0 0 12 125 0 0 405 270 0 17.560°
138 8627.406389880 ${\displaystyle C_{2}}$ 0.000473576 0 0 12 126 0 0 408 272 0 16.924°
139 8756.227056057 ${\displaystyle C_{2}}$ 0.000404228 0 0 12 127 0 0 411 274 0 16.673°
140 8885.980609041 ${\displaystyle C_{1}}$ 0.000630351 0 0 13 126 1 0 414 276 0 16.773°
141 9016.615349190 ${\displaystyle C_{2v}}$ 0.000376365 0 0 14 126 0 1 417 278 0 16.962°
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149 10096.859907397 ${\displaystyle C_{1}}$ 0.000638186 0 0 14 133 2 0 441 294 0 16.344°
150 10236.196436701 ${\displaystyle T}$ 0 0 0 12 138 0 0 444 296 0 16.405°
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171 13386.355930717 ${\displaystyle D_{3}}$ 0 0 0 12 159 0 0 507 338 0 15.497°
172 13547.018108787 ${\displaystyle C_{2v}}$ 0.000547291 0 0 14 156 2 0 510 340 0 15.292°
173 13708.635243034 ${\displaystyle C_{s}}$ 0.000286544 0 0 12 161 0 0 513 342 0 15.225°
174 13871.187092292 ${\displaystyle D_{2}}$ 0 0 0 12 162 0 0 516 344 0 15.366°
175 14034.781306929 ${\displaystyle C_{2}}$ 0.000026686 0 0 12 163 0 0 519 346 0 15.252°
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182 15206.730610906 ${\displaystyle D_{5}}$ 0 0 0 12 170 0 0 540 360 0 15.155°
183 15378.166571028 ${\displaystyle C_{1}}$ 0.000467899 0 0 12 171 0 0 543 362 0 14.747°
184 15550.421450311 ${\displaystyle T}$ 0 0 0 12 172 0 0 546 364 0 14.932°
185 15723.720074072 ${\displaystyle C_{2}}$ 0.000389762 0 0 12 173 0 0 549 366 0 14.775°
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187 16072.975186320 ${\displaystyle D_{5}}$ 0 0 0 12 175 0 0 555 370 0 14.848°
188 16249.222678879 ${\displaystyle D_{2}}$ 0 0 0 12 176 0 0 558 372 0 14.740°
189 16426.371938862 ${\displaystyle C_{2}}$ 0.000020732 0 0 12 177 0 0 561 374 0 14.671°
190 16604.428338501 ${\displaystyle C_{3}}$ 0.000586804 0 0 12 178 0 0 564 376 0 14.501°
191 16783.452219362 ${\displaystyle C_{1}}$ 0.001129202 0 0 13 177 1 0 567 378 0 14.195°
192 16963.338386460 ${\displaystyle I}$ 0 0 0 12 180 0 0 570 380 0 14.819° geodesic sphere {3,5+}3,2
193 17144.564740880 ${\displaystyle C_{2}}$ 0.000985192 0 0 12 181 0 0 573 382 0 14.144°
194 17326.616136471 ${\displaystyle C_{1}}$ 0.000322358 0 0 12 182 0 0 576 384 0 14.350°
195 17509.489303930 ${\displaystyle D_{3}}$ 0 0 0 12 183 0 0 579 386 0 14.375°
196 17693.460548082 ${\displaystyle C_{2}}$ 0.000315907 0 0 12 184 0 0 582 388 0 14.251°
197 17878.340162571 ${\displaystyle D_{5}}$ 0 0 0 12 185 0 0 585 390 0 14.147°
198 18064.262177195 ${\displaystyle C_{2}}$ 0.000011149 0 0 12 186 0 0 588 392 0 14.237°
199 18251.082495640 ${\displaystyle C_{1}}$ 0.000534779 0 0 12 187 0 0 591 394 0 14.153°
200 18438.842717530 ${\displaystyle D_{2}}$ 0 0 0 12 188 0 0 594 396 0 14.222°
201 18627.591226244 ${\displaystyle C_{1}}$ 0.001048859 0 0 13 187 1 0 597 398 0 13.830°
202 18817.204718262 ${\displaystyle D_{5}}$ 0 0 0 12 190 0 0 600 400 0 14.189°
203 19007.981204580 ${\displaystyle C_{s}}$ 0.000600343 0 0 12 191 0 0 603 402 0 13.977°
204 19199.540775603 ${\displaystyle T_{h}}$ 0 0 0 12 192 0 0 606 404 0 14.291°
212 20768.053085964 ${\displaystyle I}$ 0 0 0 12 200 0 0 630 420 0 14.118° geodesic sphere {3,5+}4,1
214 21169.910410375 ${\displaystyle T}$ 0 0 0 12 202 0 0 636 424 0 13.771°
216 21575.596377869 ${\displaystyle D_{3}}$ 0 0 0 12 204 0 0 642 428 0 13.735°
217 21779.856080418 ${\displaystyle D_{5}}$ 0 0 0 12 205 0 0 645 430 0 13.902°
232 24961.252318934 ${\displaystyle T}$ 0 0 0 12 220 0 0 690 460 0 13.260°
255 30264.424251281 ${\displaystyle D_{3}}$ 0 0 0 12 243 0 0 759 506 0 12.565°
256 30506.687515847 ${\displaystyle T}$ 0 0 0 12 244 0 0 762 508 0 12.572°
257 30749.941417346 ${\displaystyle D_{5}}$ 0 0 0 12 245 0 0 765 510 0 12.672°
272 34515.193292681 ${\displaystyle I_{h}}$ 0 0 0 12 260 0 0 810 540 0 12.335° geodesic sphere {3,5+}3,3
282 37147.294418462 ${\displaystyle I}$ 0 0 0 12 270 0 0 840 560 0 12.166° geodesic sphere {3,5+}4,2
292 39877.008012909 ${\displaystyle D_{5}}$ 0 0 0 12 280 0 0 870 580 0 11.857°
306 43862.569780797 ${\displaystyle T_{h}}$ 0 0 0 12 294 0 0 912 608 0 11.628°
312 45629.313804002 ${\displaystyle C_{2}}$ 0.000306163 0 0 12 300 0 0 930 620 0 11.299°
315 46525.825643432 ${\displaystyle D_{3}}$ 0 0 0 12 303 0 0 939 626 0 11.337°
317 47128.310344520 ${\displaystyle D_{5}}$ 0 0 0 12 305 0 0 945 630 0 11.423°
318 47431.056020043 ${\displaystyle D_{3}}$ 0 0 0 12 306 0 0 948 632 0 11.219°
334 52407.728127822 ${\displaystyle T}$ 0 0 0 12 322 0 0 996 664 0 11.058°
348 56967.472454334 ${\displaystyle T_{h}}$ 0 0 0 12 336 0 0 1038 692 0 10.721°
357 59999.922939598 ${\displaystyle D_{5}}$ 0 0 0 12 345 0 0 1065 710 0 10.728°
358 60341.830924588 ${\displaystyle T}$ 0 0 0 12 346 0 0 1068 712 0 10.647°
372 65230.027122557 ${\displaystyle I}$ 0 0 0 12 360 0 0 1110 740 0 10.531° geodesic sphere {3,5+}4,3
382 68839.426839215 ${\displaystyle D_{5}}$ 0 0 0 12 370 0 0 1140 760 0 10.379°
390 71797.035335953 ${\displaystyle T_{h}}$ 0 0 0 12 378 0 0 1164 776 0 10.222°
392 72546.258370889 ${\displaystyle C_{1}}$ 0 0 0 12 380 0 0 1170 780 0 10.278°
400 75582.448512213 ${\displaystyle T}$ 0 0 0 12 388 0 0 1194 796 0 10.068°
402 76351.192432673 ${\displaystyle D_{5}}$ 0 0 0 12 390 0 0 1200 800 0 10.099°
432 88353.709681956 ${\displaystyle D_{3}}$ 0 0 0 24 396 12 0 1290 860 0 9.556°
448 95115.546986209 ${\displaystyle T}$ 0 0 0 24 412 12 0 1338 892 0 9.322°
460 100351.763108673 ${\displaystyle T}$ 0 0 0 24 424 12 0 1374 916 0 9.297°
468 103920.871715127 ${\displaystyle S_{6}}$ 0 0 0 24 432 12 0 1398 932 0 9.120°
470 104822.886324279 ${\displaystyle S_{6}}$ 0 0 0 24 434 12 0 1404 936 0 9.059°

According to a conjecture, if ${\displaystyle m=n+2}$, p is the polyhedron formed by the convex hull of m points, q is the number of quadrilateral faces of p, then the solution for m electrons is f(m): ${\displaystyle f(m)=0^{n}+3n-q}$.[14][clarification needed]

## References

1. ^ Thomson, Joseph John (March 1904). "On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure" (PDF). Philosophical Magazine. Series 6. 7 (39): 237–265. doi:10.1080/14786440409463107. Archived from the original (PDF) on 13 December 2013.
2. ^ Smale, S. (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer. 20 (2): 7–15. CiteSeerX 10.1.1.35.4101. doi:10.1007/bf03025291. S2CID 1331144.
3. ^ Föppl, L. (1912). "Stabile Anordnungen von Elektronen im Atom". J. Reine Angew. Math. (141): 251–301..
4. ^ Schwartz, Richard (2010). "The 5 electron case of Thomson's Problem". arXiv:1001.3702 [math.MG].
5. ^ Yudin, V.A. (1992). "The minimum of potential energy of a system of point charges". Discretnaya Matematika. 4 (2): 115–121 (in Russian).; Yudin, V. A. (1993). "The minimum of potential energy of a system of point charges". Discrete Math. Appl. 3 (1): 75–81. doi:10.1515/dma.1993.3.1.75. S2CID 117117450.
6. ^ Andreev, N.N. (1996). "An extremal property of the icosahedron". East J. Approximation. 2 (4): 459–462. MR1426716, Zbl 0877.51021
7. ^ Landkof, N. S. Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972. x+424 pp.
8. ^ Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194
9. ^ a b Batagelj, Vladimir; Plestenjak, Bor. "Optimal arrangements of n points on a sphere and in a circle" (PDF). IMFM/TCS. Archived from the original (PDF) on 25 June 2018.
10. ^ Levin, Y.; Arenzon, J. J. (2003). "Why charges go to the Surface: A generalized Thomson Problem". Europhys. Lett. 63 (3): 415. arXiv:cond-mat/0302524. Bibcode:2003EL.....63..415L. doi:10.1209/epl/i2003-00546-1. S2CID 18929981.
11. ^ Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)
12. ^ LaFave Jr, Tim (2013). "Correspondences between the classical electrostatic Thomson problem and atomic electronic structure". Journal of Electrostatics. 71 (6): 1029–1035. arXiv:1403.2591. doi:10.1016/j.elstat.2013.10.001. S2CID 118480104.
13. ^ Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.
14. ^ "Sloane's A008486 (see the comment from Feb 03 2017)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2017-02-08.


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# Tag Info

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BIC and AIC can both be regarded as (asymptotically) Bayesian (under different situations), but they can also be regarded just as penalized (/regularized) maximum likelihood, for example. That is, using BIC to do model selection doesn't mean you're being inherently Bayesian; a frequentist might choose to use it based on the fact that in many situations its ...

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(1) There is an extensive literature on why one should prefer full models to restricted/parsimonious models. My understanding are few reasons to prefer the parsimonious model. However, larger models may not be feasible for many clinical applications. (2) As far as I know, Discrimination/Discrimination indexes aren’t (?should not be) used as a ...

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One option is to use pseudo R-square measures for both models. A strong difference in pseudo R-square would suggest that the model fit strongly decreases by omitting V17. There are different kinds of Pseudo R-squares available. An overview can be found here, for example: http://www.ats.ucla.edu/stat/mult_pkg/faq/general/Psuedo_RSquareds.htm A popular ...

2

I have also another take, based on the book by Anderson on model selection. The authors state as follows: It seems best not to associate the words significant or rejected with results under an information-theoretic paradigm. Questions concerning the strength of evidence for the models in the set are best addressed using the evidence ratio as well ...

2

Just some strayed thoughts. 1) You fit a multiple regression to examine the effect of a particular variable a worker in another department is interested in. The variable comes back insignificant, but your co-worker says that this is impossible as it is known to have an effect. What would you do? Many reasons could have caused this: The study is ...

2

Probably what you will need to use is the Parametric Bootstrap Cross-fitting Method. Here is the basic procedure: Fit each model to the data. Estimate the models' parameters and extract your favorite measure of goodness of fit. We will call the model with the higher value for this GoF measure $A$ and the other model $B$. Calculate the difference $d$ ...

2

You can think of AIC as a providing a more reasonable (i.e., larger) $P$-value cutoff. But model selection based on $P$-values or any other one-variable-at-a-time metric is frought with difficulties, having all the problems of stepwise variable selection. Generally speaking, AIC works best if used to select a unique single parameter (e.g., shrinkage ...

2

I would say it is often appropriate to use AIC in model selection, but rarely right to use it as the sole basis for model selection. We must also use substantive knowledge. In your particular case, you are comparing a model with a 3rd order AR vs. one with a 1st order AR. In addition to AIC (or something similar) I would look at the the autocorrelation and ...

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(1) Do you really need a smaller model? If not, you're set. (2) Can you honestly pre-specify your model? From your knowledge of the field can you choose a subset of predictors your interested in without using your knowledge of this dataset? If so, you're set. (2.5) If all your data valid? Assess this without looking at outcomes. (3) Consider using some form ...

1

Do not perform backward selection. Nor forward. Nor stepwise. Don't do it for Cox or OLS or logistic or any other model. Instead, think about what variables you have, why you have them, what they mean, what theory says about them, how including them affects other variables and so on. If you must use some automated method, LASSO or LAR have nice properties. ...

1

The flat maximum effect is simply that models with very different coefficients can have very similar performance on a particular prediction or classification task. When this happens it makes interpreting model coefficients difficult. One example of where this occurs is in linear regression with multicolinearity of the predictor variables. As another ...

1

You would do better to avoid such a thing because it (very rarely) makes sense. What would make sense in your situation? Well, since you have only three continuous IVs and one categorical one, unless your N is very small indeed, there is little problem with keeping all the variables. Also, you wrote you want to investigate whether there is "any ...

1

1) This is a multiple regression. Presumably there are other variables. It may be the case that the specific variable in question has an effect on the response, but in the presence of the other variables its effect is not significant. I'd use partial least squares pairing the variable in question with the other variables to determine which other variables ...

1

(1) This doesn't quite make sense. "What would you do" is very general. Often variables that are known to be significant/related to the outcome are included in a regression model even if they are found not to be significant (to increase confidence in that you are providing unbiased estimates for the other coefficients). So one answer would be you keep the ...

Only top voted, non community-wiki answers of a minimum length are eligible

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# Find, inside a large circle, the maximum number of small circles placed 60 degrees to each other and

... starts with a small circle in the center of the large circle.

The above picture shows a program I wrote to actually draw the circles out. But you can see that this method does not yield maximum number of blue circles. There are still spaces around the red circle.

The method i used is to draw blue circle "rings" starting from the center outwards. i.e move out in the blue arrow direction for one circle diameter, then go around in the red arrow direction, then repeat next ring in the blue arrow direction.

Anyone can share a smarter method? Thank you all. I need only to calculate the number, but if there is a systematic way to draw will be better.

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How about you just fill a grid of circles and compute an algorithm that removes the wrong ones? –  Patrick Da Silva Nov 3 '11 at 3:46
Some interesting picures are at www2.stetson.edu/~efriedma/cirincir and at hydra.nat.uni-magdeburg.de/packing/cci Close packing is not always optimal –  Ross Millikan Nov 3 '11 at 4:00
@Patrick because when there is large difference in big and small circle diameter, it is not that efficient counting one by one. I was hoping to just have a formula/theory to calculate the max, but I end up drawing (and counting) one by one because it appeared easier to begin solving. –  Jake Nov 3 '11 at 4:11
@Ross I saw those links before I posted. The application for the small circles is to house heating elements. So the packing has to be uniform. –  Jake Nov 3 '11 at 4:13
@Jake: then I think Carl's solution is a good one. The worry would be that there will be slop between the small circles and the big one that might allow them to shift. –  Ross Millikan Nov 3 '11 at 4:40

If your goal is to make a program, then considering you already have that done, it seems like the easiest strategy would just be to: 1. Add on more rings until you know that every circle in a new ring will be outside the main circle. 2. Iterate through the small circles, removing all of the circles who's centers are further than R-r from the main circle, where R and r are the radii of the large and small circles respectively.

That however assumes that you can't fit more circles in by translating your entire set of small circles to the side a bit more. If you want to make sure you have the maximum, you might have to do some more fudging.

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if the radius between the large and small circle is big, I have to draw a lot more rings since the blue hexagon will have sides further to the red circumference. Using (2), I don't know when to stop drawing rings. A method to calculate the perdicular distance from center to edge of blue hexagon could help, but i don't think it will be conclusive. And imho (1) does not work because rings closer to circumference will not be complete rings, cut at the blue hexagon corners. –  Jake Nov 3 '11 at 3:55
in any case, main objective is to calculate the max. The drawing is just to visually verify. –  Jake Nov 3 '11 at 3:56
I will try you suggestion later today. Thanks. –  Jake Nov 3 '11 at 5:19
I figured how the perpendicular distance from center to the side of hexagon and used your suggestion to cull the blue circles that extends outside of the red circumference. So the drawing works now! Thanks. But I still wonder if there is a direct formula to calcualate this. –  Jake Nov 3 '11 at 6:01


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Show that $\cos x$ is the integrating factor of the linear differential equation $\cos x \frac{d y}{d x}=1+y \sin x$ where $\left(-\frac{\pi}{2}<x<\frac{\pi}{2}\right) .$
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# Why is hexadecimal used in binary model files?

This topic is 2230 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Hi,

I am trying to write a Blender exporter for my game engine's data. When I look at other exporters they often use hexadecimal to identify the different chunks and sub-chunks in the binary file. Why is hexadecimal used and what does it represent in the file?

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The final benefit of hexadecimal is that you can spell words in it: 0xDEADC0DE

Funny!

//-----------------------------------------------------------------------------------------------------------------------------------------------

Here are some pages that deal with what you are working on and they use a format that is a little more human readable, I think anyways (decimal).

They aren't entirely complete but they may be the start you need.

http://stackoverflow.com/questions/13327379/how-to-export-per-vertex-uv-coordinates-in-blender-export-script

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Thanks for the great replies. There still is something that I don't get. When hexadecimal is used to describe the chunks of a binary format does it represent the size of each chunk in the memory?

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Binary files don't have "chunks". I'm more than a little confused by that. A binary file is a string of bytes that span the length of the file. The only time I've seen "chunks" was in a human readable format.

Typically a binary file will be formatted to have a header with offsets to the various data and that header will be immediately followed by the file data. What a binary (hexadecimal) string represents to the file loader is entirely defined by the file format.

Here is a binary file. It is a simple 5x5 pixel TGA file. The first 18 bytes are the file header and the image data starts at byte 19 (the 00 immediately following the 08).
00 00 02 00 00 00 00 00 00 00 00 00 05 00 05 00 20 08 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF 00 00 00 FF

The "02" in the header tells the loader that this is an uncompressed, true-color image.
The first "05 00" defines a 16-bit word that tells the loader the width of the image; In this case, 5 pixels.
The second "05 00" defines a 16-bit word that tells the loader the height of the image.
The "20" that follows is 32 in hexadecimal, telling the loader that each pixel in the pixel data is 32-bits (4 bytes).
The "08" is an encoded byte, whose bits tell the loader if the image is flipped as well as the number of alpha channel bits.
The other bytes that I did not explain have special meaning as well, in certain cases, but are unused in this file.
The image data is stored as BGRA (blue, green, red, alpha) and in this case, each pixel has the values "00 00 00 FF".

If you are seeing anything other than hexadecimal in a file, it isn't a binary file. Edited by MarkS

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The final benefit of hexadecimal is that you can spell words in it: 0xDEADC0DE

One of the versions of the BBC Micro OS had as the two bytes starting at location 0xD0D0 the value 0xD1ED.

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So how would I know where the different pieces of the file were to reference in hexadecimal surely that would depend on the size of the data in the file.

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Sounds like your problem has nothing to do with hexadecimal or any other number base, but with the format itself. What actual data to write is determined by the format you want to read or write. You said you're trying to write an exporter to your own game engine, so you should have a good idea what has to be written to the file. Can you explain mode detailed what your problem actually is?

Edited by Brother Bob

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Well, my problem is that I need to be able to load model data into my engine (vertices, uvs, lighting attributes, textures) but I also want skeletal animation and most of the formats I found that supported animation were either too bloated (Collada, FBX) and not really suitable for game use or text based (MD5) which I don't really want. So I resolved to make my own (binary) format and after looking at a few example python exporters (3DS, FBX) I found hex was often used and I needed to know exactly how the exporter worked to be able to produce one myself that wasn't just a copy.

I hope this is enough explanation.

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text: Title Parity-fluctuation induced enlargement of the ratio $\Delta_{E}/k_{B}T_{c}$ in metallic grains Author Croitoru, M.D. Shanenko, A.A. Peeters, F.M. Axt, V.M. Faculty/Department Faculty of Sciences. Physics Publication type article Publication 2011 Lancaster, Pa , 2011 Subject Physics Source (journal) Physical review : B : condensed matter and materials physics. - Lancaster, Pa, 1998 - 2015 Volume/pages 84(2011) :21 , p. 214518,1-214518,12 ISSN 1098-0121 ISI 000298114100003 Article Reference 214518 Carrier E-only publicatie Target language English (eng) Full text (Publishers DOI) Affiliation University of Antwerp Abstract We investigate how the interplay of quantum confinement and particle number-parity fluctuations affects superconducting correlations in ultra-small metallic grains. Using the number-parity projected BCS formalism we calculate the critical temperature and the excitation gap as a function of the grain size for grains with even and odd number of confined carriers. We show that the experimentally observed anomalous increase of the coupling ratio ΔE/kBTc with decreasing superconducting grain size can be attributed to an enhancement of the number-parity fluctuations in ultra-small grains. Full text (open access) https://repository.uantwerpen.be/docman/irua/62a3cd/ff4eddc3.pdf E-info http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000298114100003&DestLinkType=RelatedRecords&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000298114100003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000298114100003&DestLinkType=CitingArticles&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 Handle


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# Showing a function is not continuous in the one-norm

I have the following question:

Consider the normed space $(C([0,1]),||\cdot||_{\infty})$. Let $x_{0}\in [0,1]$ and define $F:C([0,1])\to \mathbb{R}$ by

$F(f)=f(x_{0})$

Show that $F$ is discontinuous with respect to $||\cdot||_{1}$.

I realise that I simply need to find a counterexample where $f_{n}\to f$ with respect to the one-norm, but where $F(f_{n})\not \to F(f)$. However, I see no way in which to do this. I hate being lazy with this so would appreciate a small push in the right direction.

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There are some very simple, familiar functions $f_n$ that are $0$ at $0$, $1$ at $1$, and squeezing towards the $x$-axis as $n$ increases. They give you a nice example when $x_0=1$. For other values of $x_0$ you can adapt the basic idea of that family, most easily by using piecewise linear functions. –  Brian M. Scott Mar 8 '12 at 1:45

## 2 Answers

If I understand your question and you mean $L^1$ norm (you also mention sup norm and $F$ is continuous with respect to that, which is perhaps point of problem), you could let $f_n$ be $0$ on $[0,1/2-1/n^2]$, the line from $0$ to $n$ on $[1/2-1/n^2,1/2]$, the line from $n$ to $0$ on $[1/2,1/2+1/n^2]$, and $0$ on $[1/2+1/n^2]$. Then $f_n \rightarrow 0$ in $L^1$, but $F(f_n)=n$.

Basically any function that is zero except for a "high spike of small area" will work.

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Here is an idea. Squeeze a bump.

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-1, you could keep these ideas that are relatively small enough for an answer as a comment. –  user21436 Mar 9 '12 at 1:08
Actually that is an answer. I am trying to avoid "giving away the shop." –  ncmathsadist Mar 9 '12 at 1:34


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# Can an attacker exploit the nonuniform distribution of letters in a diceware password to improve a brute force search?

I use some diceware pass phrases in some places but I have one question that I can't find the answer to.

Let's say I roll the dice and get $$N$$ words with a total of $$M$$ letters and that I don't put spaces in between the words.

An attacker wishing to brute force my secret could have to try $$7776^N$$ times to break it as a diceware pass phrase OR $$26^M$$ to break it as a "normal" password (assuming I get only words with letters).

But the attacker trying to break it as a normal password could assume that it is made up of English words and then they'd know that some letters and combinations of letters are more common.

So my $$M$$ letters will not have the entropy of letters selected at random but some lower entropy.

My question is, what $$M$$ do I need to make up the same entropy as the $$N$$ words?

But the attacker trying to break it as a normal password could assume that it is made up of English words and then they'd know that some letters and combinations of letters are more common.

They could try this, but they would be stupid to try this, because there are many fewer $$N$$-word diceware passwords than $$M$$-letter strings of English letters of the same length as an $$N$$-word diceware password!

Suppose you use a ten-word diceware password. There are $$7776^{10} \approx 8 \times 10^{38} \approx 2^{129.2}$$ such passwords. This is already a larger haystack than any adversary can find a single needle in. (If passwords are hashed without salt, an extremely powerful adversary may be able to find one of many needles in a haystack that big, but you could pick $$N = 20$$ instead, and the service oughta salt their password hashes.)

Any distribution on sequences of $$M$$ independent English letters that covers more possibilities than these $$7776^{10}$$ diceware passwords is an even larger haystack to search for a needle in.

A smart adversary, who knows all the details of your method and just doesn't know the password you chose by that method, won't give themselves a larger haystack to search in. A stupid adversary who does give themselves a larger haystack to search in will just waste more time searching!

For example, the distribution on $$M$$ independent English letters following the usual letter frequencies in English would give relatively high probability to sequences like aoqothropaaeeaetphtticawglesttgeectoyheldshnfznecr which has a lot of e's and t's and a's but not many z's, x's, q's, etc. This is not a possible diceware password, however, even though it has the same length as a ten-word diceware password with five letters per word. And it would give lower probability to woozydizzybuzzyfuzzyjazzymezzoexxonpizzapyrexvixen, which (with an appropriate word list) is a possible diceware password.

Let's work through a simplified example. We'll use a dictionary of five words: cat, car, act, art, and rat. We'll pick a single word uniformly at random from the dictionary, so that each word has equal probability 1/5, with just over two bits of entropy.

The corpus in this dictionary has letter frequencies a 5/15, c 3/15, r 3/15, t 4/15, under which distribution each letter position has under two bits of entropy. There are $$4^3 = 64$$ possible three-letter words from this alphabet, of which the vast majority, 59, aren't in our dictionary.

1. An adversary trying to guess our password from the dictionary has a $$1/5 = 20\%$$ chance, of getting it right in a single trial.

This is the smartest, most efficient adversary.

2. An adversary trying to guess our password from the letter frequencies independently has a $$\frac{1}{5}\cdot\frac{19}{225} = 19/1125 \approx 1.7\%$$ chance of getting it right in a single trial.

Why? The distribution on three-letter passwords is the product of the letter probabilities in this model. If the adversary guesses a word in our dictionary this way, there's a 1/5 chance that it's the right password, but there's only a $$19/225 \approx 8.5\%$$ chance that the word the adversary guessed with these letter probabilities is in our dictionary. With probability $$206/225 \approx 91.5\%$$, the adversary will be guessing passwords that you would never have chosen: they're wasting most of their time barking up the wrong trees in a much larger haystack.

This is the adversary who takes advantage of the per-letter entropy as you described. Even if they sort by probability rather than choosing randomly, they'll first try aaa, then aac, then aat, aca, etc., wasting time on impossible passwords. Even if every word in the dictionary happened to come first in this ordering, this adversary has no better chance than adversary (1).

3. An adversary trying to guess our password from the letters, ignoring the frequencies, has a $$1/64 \approx 1.5\%$$ chance of getting it right in a single trial.

Why? Under this distribution, every letter has equal probability 1/4 independently, so every three-letter word has equal probability $$(1/4)^3 = 1/64$$, including the one we chose. With probability $$59/64 \approx 92\%$$, the adversary will be guessing passwords that you would never have chosen. They're also rummaging through a much larger haystack, but they're doing it even less efficiently than adversary (2).

This adversary just tries all combinations without regard to letter frequency. As you can see, this adversary is slightly worse at the job than adversary (2), so knowledge of the letter frequencies helps a little bit, but they're both much worse at the job than adversary (1).

Every word you add, if drawn independently, multiplies these probabilities: pick a second word independently from the same dictionary, and adversary (1) now has $$(1/25)^2 = 1/25 = 4\%$$ chance of success, adversary (2) now has $$(19/1125)^2 = 361/1265625 \approx .029\%$$ chance of success, and adversary (3) now has $$(1/64)^2 = 1/4096 \approx .024\%$$ chance of success. If you make the dictionary large enough and the sequence of words long enough that you defeat the smart adversary (1), then you also defeat the stupid adversaries (2) and (3).

My question is, what $$M$$ do I need to make up the same entropy as the $$N$$ words?

Don't worry about $$M$$. Pick $$N = 10$$ for diceware and you're all set.

• But the size of the haystack depends on the entropy of each symbol. DW words at ~13 bits per word vs letters of English words at ? bits per word. I have seen estimates (think it was Schneider) of 0.6-1.3 bits per letter in English text. So 10 words -> ~130 bits but 50 letters could be as low as 30. But what is the entropy per letter in a diceware password? – evading Mar 21 '19 at 15:11
• Ok, let me rephrase that. The difficulty of finding something in the haystack depends on the entropy. Also even if you pick words uniformly at random the letters will not be picked at random. You will have more e's, a's and o's than z's f.ex. – evading Mar 21 '19 at 16:01
• Not really. I see two problems with your explanation. 1) In your example the dictionary is smaller than the alphabet [cat, car]/[a,c,r,t] so 2 vs 4. So of course that will be easier to guess. With diceware the situation is the opposite 7776 vs 26. 2). If your are never helped by knowing the distribution of symbols you chose from you would get the same password strength when choosing from a known distribution as if you were choosing randomly. But that is not the case. – evading Mar 21 '19 at 17:16
• The dictionary is smaller than the alphabet, but that's not the relevant comparison. The relevant comparison is between {cat, car} and {aaa, aac, aar, aat, aca, acc, acr, act, ara, arc, arr, art, ata, atc, atr, att, …}. As you can see by comparing adversary (2) and adversary (3), the distribution of letters does help, but it's still a foolish adversary who would ignore the dictionary you did use and make up a dictionary you didn't use with words like aaa and rac and trt to search through! – Squeamish Ossifrage Mar 21 '19 at 17:25
• @evading I adapted the numbers so that the dictionary is larger than the alphabet. The numbers are more complicated now, but is the point clearer? – Squeamish Ossifrage Mar 21 '19 at 18:49

Diceware passwords are based on guarantees of minimum entropy. the $$7776^N$$ number is the number of possible passwords generated with $$N$$ words from the Diceware list. If you use dice, or an equivalent, that is a guarantee on the strength of the password against brute force attacks.

That paragraph assumes that the fact that you are using Diceware is public knowledge, as is their word list. If you expose everything about your password generation scheme except the key (the actual dice rolls or the actual words), you still have that guarantee.

If your threat model assumes less knowledge of your password generation routine, brute forcing will be more difficult. If your threat model assumes no knowledge of your password generation approach (other than that it is lower case), the difficulty of brute forcing the password will be equivalent to that of a password with $$26^M$$ possibilities. It still only has precisely $$7776^N$$ possibilities, but it appears to have more because your adversary doesn't know everything.

From there, one can discuss the possibility that, since English words do not have a uniform distribution of letters, the entropy of the password is much less than $$26^M$$, which is true. But Diceware's security proofs are not based on that number. They are based on the $$7776^N$$ number, and the random rolls of the dice drawing from that 7776 entry table.

Thus, if I make a 16 character password using Diceware, I should not assume that it is as strong as a 16 character password made via other means (such as random characters). In fact, per character of the password, Diceware provides less entropy and apparent entropy than other approaches. 16 characters may only be 3 words, which is merely $$7776^3$$ possibilities.

The strength of diceware is that it observes that entropy per character is not the best metric for judging a human memorable password. It encourages long passwords (4-8 words) which are easier to memorize because they are words, but still contains a provable amount of entropy in the password to be used in security proofs. The exact number of words you use should be derived directly from how much entropy you require for your password.

... now the challenge of typing said password correctly without typos is another question entirely.

If you're interested in converting between systems, 7776 possibilities per word means your Diceware password generates 12.925 bits of entropy per word. Round it up to 13. Thus a 5 word password has 65 bits of entropy. A 10 word password has 130, meaning it is as hard to brute force as a 128 bit encryption key.

Wikipedia has a great table of how many bits of entropy per random character a password can have, based on the character set. An alphanumeric password, case sensitive, has just shy of 6 bits of entropy per character. So every word in your diceware password is slightly better than 2 random characters of a password. An 8 character random alphanumeric password is comparable in strength to a 4 word Diceware password.

Of course, if your password generation scheme is anything less than a purely random character stream, your actual entropy will fall short of this 6 bits/char line. In such cases, the exchange rate for Diceware words will be even better.


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# Boxing Pythagoras

## Some Unfortunate Choices in Mathematics Terminology

Words can be tricky things. The same word can often carry wholly different meanings depending upon the context in which it is used. Take, for instance, the semantic range of the word “light.” This word can carry very different meanings when used in different contexts, as the following sentences illustrate.

1. That feather is light.
2. That shade of pink is light.
3. That laserbeam is light.

Each one of these sentences is of the form “That <noun phrase> is light,” but the word “light” intends an entirely different thing, in each. In (1), “light” is a description of the weight of the feather. In (2), “light” is a description of the intensity of the shade of pink. In (3), “light” is a description of the physical nature of the laserbeam. There is a well known fallacy of logic called equivocation which involves conflating such definitions in order to arrive at a false conclusion. For example, if I said…

1. Light things weigh less than heavy things
2. This shade of pink is light
3. Therefore, this shade of pink weighs less than heavy things

…my logic would be invalid, because the definitions of “light” used in (1) and (2) are completely different.

Mathematics, unfortunately, contains some terminology which tends to lead to these same sorts of equivocation fallacies, because the common usage of a word very often differs from the mathematical usage of that word. While there are numerous examples from which I could likely choose, today I’m going to focus on a case which I believe to be particularly egregious. Today, I’m going to discuss Real and Imaginary numbers.

If you continue your math education long enough, you will eventually encounter things which mathematicians refer to as Real numbers and Imaginary numbers. Because of these titles, people have come under the unfortunate and entirely incorrect misunderstanding that Real numbers are somehow better than Imaginary numbers, as if Reals correspond to the actual world more accurately than Imaginaries. Put succinctly, people think that Real numbers are real, and Imaginary numbers are just make-believe.

The bare truth of the matter is that Real numbers are no more real than Imaginary numbers are; and Imaginary numbers are no more make-believe than Real numbers are.

This is why there are a significant number of mathematicians (including me) who chafe at the phrase “Imaginary numbers.” We prefer to use the synonymous phrase “Complex numbers.” Personally, even if everyone were to switch to saying “Complex numbers,” I feel that “Real number” is still a rather misleading phrase, but unfortunately that particular bit of terminology is so deeply rooted in mathematical culture that it’d be almost impossible to dig it out and replace it.

It is usually at this point in the discussion when one of my friends says to me, “but Imaginary numbers rely on $\sqrt{-1}$, and that is not a real number!” This is where the confusion tends to set in, because I always answer them by saying, “Actually, $\sqrt{-1}$ is a real number, it’s just not a Real number.” My response is generally met with exasperation and quizzical stares. Unfortunately, that capital-R doesn’t really stand out, in spoken language.

In order to better explain what I mean, let me jump backwards a couple thousand years. There used to be a grand mystery, among mathematicians, regarding the diagonal of a square. Basically, mathematicians wanted to find the proportion of a square’s diagonal to one of its sides. They were convinced that there was an actual number which would satisfy this proportion, but no one had yet been able to discover it. They knew that squaring this proportion (in a unit square) would yield a value of 2, thanks to the Pythagorean Theorem, but no one had been able to calculate the precise number that could satisfy the proposition. Eventually, these mathematicians came to a shocking realization– I have written before about one of the legends which rose up in the wake of this discovery. Regardless of whether these legends actually occurred, it is true that these mathematicians eventually had to deal with an incredible fact: there existed no number which could represent the proportion of a square’s diagonal to one of its sides. To the ancient Greeks, $\sqrt{2}$ was not a number.

So, mathematicians redefined their concept of “number.” They began to incorporate concepts which could not be represented as a ratio of whole numbers, like $\sqrt{2}$, calling them “irrational.” Originally, the word “irrational” simply meant “unable to be expressed as a ratio,” but the extremely counter-intuitive nature of such numbers is the very reason that the word “irrational” has come to be synonymous with “lacking in reason.” Some philosophers and mathematicians stood in very vocal opposition to the idea that a number could be “irrational.” It simply didn’t make any sense to them. However, as time wore on, and the field of mathematics started to become irrevocably entwined with the concept of irrational numbers, the naysayers began to fall away, and eventually it became nigh universally accepted that irrational numbers were every bit as real and legitimate as rational numbers.

However, it would not take long before another curious puzzle emerged. Mathematicians knew that squaring any number, whether positive or negative (and negative numbers had their own fight for acceptance, by the bye), would yield a positive result. So, then, what would happen if we try to find the square root of a negative number? Most mathematicians thought this was a completely preposterous notion, and asserted that it simply could not be done. They proclaimed that, if some formula required that you take the square root of a negative number, you must have done something wrong. However, some mathematicians began to play an interesting game. If we assume that there exists some number $i$ such that $i^2=-1$, then we are suddenly able to resolve any equation which requires us to take the square root of a negative number. This quickly became an incredibly useful and widely utilized tool.

However, just as before, philosophers and mathematicians began to cry out against this usage. They considered such a notion to be a terrible affront to reason, a horrific departure from true understanding; but the power of this new tool was completely undeniable. As a result, some mathematicians really wanted to use it, despite their philosophical aversion to it. One such person was the eminent René Descartes, and it is him we have to blame for the “imaginary” appellation. Descartes advocated the use of this $\sqrt{-1}$, but argued that it should only be considered an imaginary thing, a conceptual tool which did not actually reflect reality. This seemed like a good enough justification to most mathematicians, and so the use of $\sqrt{-1}$ became widespread and popular. Some mathematicians argued it was a true number, others were content to simply call it “imaginary,” but it was not long before all recognized its utility. All the numbers they used which could be represented without $\sqrt{-1}$ became known as the Real numbers, while the rest became known as Imaginary numbers.

As it turns out, $\sqrt{-1}$ actually has some very important applications to the real world. This number actually describes real world quantities just like natural numbers, negative numbers, rational numbers, irrational numbers, and transcendental numbers do. Take, as an example, the Wave Function, which is utilized in physics: $\Psi=e^{i(kx-\omega t)}$. The $i$ in that formula is $\sqrt{-1}$, and it is absolutely necessary to particle physics. It represents an actual quantity describing an actual property of a physical thing which actually exists. This $i$ is no less real than any Real number.

The unfortunate terminology has led to a lot of confusion. Even today, it’s still common to hear mathematics and physics teachers tell their students that “Imaginary numbers” are just make-believe things while the “Real numbers” are truly real. The truth of the matter is that all numbers are things which humanity has invented in order to attempt to describe the patterns which are found in the world around us. The number 2 is not any more real than the number $\frac{7}{4}$ or the number 0 or the number -2 or the number $\sqrt{2}$ or the number $\sqrt{-1}$.

## 6 thoughts on “Some Unfortunate Choices in Mathematics Terminology”

1. Quite !
I used to tell my students that there was nothing real about real numbers either.
However, the well known exp(i t)=cos(t)+i sin(t) still has an air of mystery about it.

While you are on this stuff have you any clue as to how the word derivative arises from the process of differentiation. I haven’t !

Here is another horror which I found recently:

The distributive law of addition: a(b + c) = ab + ac (OK, it’s a definition)

The current school math explanation:
You take the a and distribute it to the b to get ab
and then you distribute the a to the c to get ac
and then you add them together to get ab + ac

I have come across this explanation in several places, and once again real damage is done to the language, and real confusion sown. “Distribute” means “to spread or share out” as in “The Arts Council distributed its funds unevenly, as usual. Opera got the lion’s share.” So it is NOT the a that is distributed. I tried to find a definition of the term in wordy form as it applies to algebra systems but failed. Heavy thinking produced the “answer”. What is being distributed is the second factor on the left.
Example:
Take 3 x 7. We know that the value of this is 21
Distribute, or spread out, the 7 as 2 + 5 . . . . . . . . the b + c
Then 3 x (2 + 5) has the value 21
But so does 3 x 2 + 3 x 5. To check, get out the blocks !
So 3 x (2 + 5) = 3 x 2 + 3 x 5 ……… The Law !

Regarding the “second” version of the distributive property, a(b – c) = ab – ac, this cannot just be stated, and you won’t find it in any abstract algebra texts. Since the students are looking at this before they have encountered the signed number system, a proof must not involve negative numbers, as a, b and c are all natural numbers. It can be done, and it will be put on my next “Language in math” post.

2. I have tried in introducing the different kinds of numbers to say “complex-valued numbers” or “real-valued numbers”, which I like to think emphasizes that these are all numbers and there’s just different kinds of values they have. I’m not sure this changes things much, though, since “complex” particularly does carry a connotation of “too hard to work with except when there’s no way around it”. But, heck, even something as familiar as “negative numbers” has a, well, negative overtone to it.

3. One of the problems of “accepting the meaning” results from the popular idea of extensions of the number system. The only extension is the incorporation of zero into the natural numbers. When fractions are introduced we are creating a new number system, for measurement of quantity, and all the numbers are “fractions”, with 3/1 being the 3 in the new system, and showing the mapping of the natural numbers into the fractional numbers. More trouble is caused by considering the negative numbers as an extension of the (now we have to call them positive) numbers. So bad! In fact the new system can and is (sometimes) called the signed numbers, created for measurements of position and change, and not meaningful as measures of quantity.
The feature of the “complex !!!” number system which is often overlooked is that every point in a plane can be described (measured) with a complex number, and this is where it stops. In 3D we have vectors, but lose one of the features of numbers, the commutative law of multiplication.

4. If the positive integers are seen as the prototypical examples of numbers, then the complex numbers are further from the prototype than the real numbers in quite a significant way, since the field of the complex numbers cannot be ordered, but the real numbers can be. This could be part of the reason people resist thinking of the complex numbers as proper numbers.

5. abyssbrain on said:

That’s a very interesting post 🙂

This made me think of complex analysis and elementary analysis. Complex analysis doesn’t mean that it is “complex”, though it can become very difficult, it just means that complex numbers are used in the analysis. On the other hand, elementary analysis doesn’t mean that it is simple since it just means that it has no complex numbers in it 🙂 It’s quite confusing for people who don’t know about them.

6. kaptinavenger on said:

Using the word Imaginary to describe Complex numbers went out of style in the mid 19th Century until rehashed a few years ago by a Steve, who used the word to describe events that he says defy all the laws of physics. His untestable theory that defies physics is taught as fact in most western schools. Another guy making a bunch of money using a very Pythagorean Platform.


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Vba Int Vs Cint

In my last two articles, I discussed how to read and write to files. If your numbers start getting beyond the standard range for cInt, try using the Int function instead. 型のメンバを動的に呼び出す. How To: Convert a string field to a numeric field Summary. So for example on such a system, intval('1000000000000') will return 2147483647. Excel VBA, Open File Dialog; Although the article was written for excel, the concept can be used in VBA for Word too. INT is similar to the TRUNC function because they both can return the integer part of a number. Además de las funciones de inserción y escritura de secuencias de comandos, puede utilizar etiquetas de formato de ArcGIS en expresiones de etiqueta. The following is an example of a VBA procedure which passes two Variables;. 95 was rounded to the nearest integer by the CInt function. amite V uestra E xcelencia que e ie a ue est mp a s en contacao te pro gamo s solamente esto ano Pardo ast los tenientes-cin de l ione os o r viaj, l os sntime: s a lclam os sa rn -enaor eaeta t -tia do t bieno la en faa c ue a -e o e oO Rr iti cs av o a ,. 5, the CInt function always rounds it to the nearest even number (0. Logical operators come in handy in such cases. To do this, you need to be familiar with VBA functions and data types, and some keyboard shortcuts can make your job even easier. For example, 0. In VBA, the ROUND function will always keep the result as an even number when the first number in the discarded numbers is 5. 67) In this example, the variable called LNumber would now contain the value of 210. Place a command button on your worksheet and add the code lines below. CInt and CLng perform an **actual** rounding operation, up or down, around 0. net (the release executable). This is a quick reference guide to highlight some key syntactical differences between VB. Parse als auch CInt in den meisten meiner täglichen Programmieraufgaben verwendet, aber ich bin ein wenig verwirrt, was der Unterschied zwischen den beiden ist. The Microsoft Excel CINT function converts a value to an integer. Text) This will work so long as txtAge. Logical operators come in handy in such cases. When the fractional part is exactly 0. Private Function RoundOff(ByVal value As Double, ByVal _ digits As Integer) As Double Dim shift As Double shift = 10 ^ digits RoundOff = CInt(value * shift) / shift End Function. ToInt32(txtID. Why differs the results from the function Int and CInt/CLng depending on the value in this example? E. Lesson 9: Creating a Counter in Excel VBA [Back to Excel VBA Sample Code]This is a simple Excel VBA counter that can count the number of passes and the number of failures for a list of marks obtained by the students in an examination. Example, for VBA Variable Sub Macro1() Dim Num As Integer Num = 99 MsgBox " Guru " & Num End Sub When you run this code, you will get the following output in your sheet. So for example on such a system, intval('1000000000000') will return 2147483647. If you do calculation like this, dim a as integer a=12 Debug. If the value of argument is negative, the INT function has the same result as the CEIL function. Integer, which I. If you want to push the complex query processing done by your Access queries to the back end, you'll have to rewrite them in T. Note CInt differs from the Fix and Int functions, which truncate, rather than round, the fractional part of a number. Cint is an insights exchange platform with technology that transforms and accelerates how you gather insights. NET and C# Comparison - This is one of the most accurate and complete charts I've seen comparing VB. Actually, since this is a problem that deals with an algorithmic equation, it's better to create something a bit more universal and tight. The code used in this video: 'I need to convert a string to an integer. Performance: Native VBA vs. For example, if you want to. But then it would getting in the 20% that is used less often in the business world. what i want is that i want to use the datetimepicker and it automatically compute the age in another textbox without clicking the button. A word on Date arithmetic. Back to Excel VBA The example below shows how CInt can be used in Excel VBA to convert to a Integer data type. For example, 0. 5) = 16 and CInt(15. The CInt Function converts a value to an Integer. PK N±tI META-INF/MANIFEST. Hi, Use CINT = Convert to INTeger, i am not sure if it rounds up or not, just try it. VBA provides a universal (or vague) data type you can use for any type of value. CInt Function in VB 6. Posted on August 27, Use. Explanation: until i is higher than 6, Excel VBA places the value 20. age = CInt(v_Age_Calc) End Function. These methods suspend all the activities of Microsoft Excel and sometimes may also prevent you from performing other operations on it until the pause is in effect. Some expressions are created for you automatically. The integer is converted to string. VBA Data Types. Below is an example of using a variable in a message box. There are lots of inbuilt methods you can use on strings of text, and you'll learn about these later in the course. Yes the help file is a little confusing in regard to IsEmpty I think when they say "IsEmpty only returns meaningful information for variants" they are meaning that a variant can be any "expression that can evaluate to numeric, string, or date data, as well as the special values Empty and Null" but not nessecarily a variable dimensioned as variant. In C, it is denoted by short. The Int function and the CInt function are not designed to do the same thing. I am using the CInt() function to convert a string to an integer, e. VBA workers vs. Sorry for the interruption. So if you are ever counting up higher than 32,767 (or down below -32,768) you are going to run into trouble if you have to use cInt on that number. When the fractional part is exactly 0. In the next section, you'll learn about Conditional Logic. NET Framework methods such as ToString(), either on the Convert class or on an individual type structure or class. Print TypeName(CStr(a)) The result is "String" in the immediate window. Built-in Functions of SSRS In Reporting Services, expressions are used to provide dynamic flexibility for controlling the content and appearance of a report. ASP の開発や SQLServer の DTS を作成する際、ど忘れしたVBScript関数を簡単に探せるように構成し直しました。 VBScript関数名は分かっていて、使い方や引数が思い出せない時などは普通のVBScriptヘルプで調べられますが ロール人工芝（芝丈20mm）幅1×長さ10m SST-FME-2010【送料無料 ロールタイプ リアル. CLng( expression. Se aplica para Excel 2013, Excel 2011 para Mac, Excel 2010, Excel 2007, Excel 2003, Excel XP, Excel 2000 y sólo se puede utilizar en código de VBA. The CInt function converts an expression to type Integer. Visual Basic for Applications isn't one of these languages. net Integer. CType for value and reference types. When the fractional part is exactly 0. A 4 part section >> > BUY THE BOOK OF THIS COURSE ; Excel VBA Left and Right functions. ConvertDataTypes. As a VBA function, you can use this function in macro code that is entered through the Microsoft Visual Basic Editor. In VBA, the ROUND function will always keep the result as an even number when the first number in the discarded numbers is 5. [RESOLVED] Val() vs CInt() or CLng() Can someone explain what's the difference between the other two functions? I already know that Val truncates the non-numeric characters, but is there any other difference, for example performance-related, that makes the developer prefer this over the other two??. The VBA Int function works the same as the worksheet Int function. CInt differs from the Fix and Int functions, which truncate, rather than round, the fractional part of a number. Grundsätzlich habe ich sowohl Integer. The CInt function converts an expression to an integer. When the fractional part is exactly 0. So far, we declared variables knowing the types of values we wanted them to hold. VBA CInt Function Examples. TeX has \int as the integral sign. Note that Int truncates to the next smaller number. net: On my laptop 1000 iterations takes 24 seconds - in the compiled vb. net Integer. This banner text can have markup. The have the same syntax but doesn't work the same as if they are called from VBA (simple test, look for the behaviour of the IIf function in VBA and SQL, in VBA always the complete function is resolved, in SQL the first True condition only). Microsoft Access Glossary. Can a variable be used in a VBA message box? Answer: Yes, a variable can be used in a VBA message box. txt) or read online for free. オブジェクトを別の型にキャストするにはDirectCast演算子を使います。TryCast演算子を使うと、キャストできなかった場合でも例外をスローせずキャスト結果をNothingにすることができます。. 5, the CInt function always rounds it to the nearest even number (0. tags for VBA and tags for inline. CInt() converts value to Integer. The included project runs a performance test on each scenario: DirectCast vs. Ver esto desde la ayuda para CInt y otras funciones de conversión de tipo. INT vs TRUNC. CInt() The CInt function converts an expression to type Integer. Cint - Login. 5, CInt and CLng always round it to the nearest even number. Your email address will not be published. And something that’s come up more than a few times now is consumers of the automation interfaces (a. CInt and CLng differ from the Fix and Int functions, which truncate, rather than round, the fractional part of a number. Visual Basic for Applications defines many datatypes Integer vs Long vs Single vs Double CInt(A) – rounds to the nearest integer. The CInt function converts an expression to an integer. How to Name. If you're a VB. The default VBA data type is called Variant, and I explain it below. se x Hernnde. There are no functions in VBA for alternate or random rounding, so far as I'm is aware. Basic, VB (Visual Basic), VB. VBA Arrays And Worksheet Ranges. Excel VBA Learn how to convert string to integer. These functions fall into two distinct groups: simple data conversion and more complex data conversion. Numeric Types Introduction to VBA programming - (c) 2009 Dario Bonino Designed for holding numeric values Can be Integer s Represent signed integer numbers on 16 bits. tags for VBA and tags for inline. String to Int Function. 1, and also uses 15-digit-long "integers"?. I'll wait for V2. The VBA Instr Function checks if a string of text is found in another string of text. The toInt() function allows you to convert a String to an integer number. A short integer can represent a whole number that may take less storage, while having a smaller range, compared with a standard integer on the same machine. Basically, I have been using both Integer. C# allows you to upcast and perform implicit conversions (eg double to int) without needing a cast. In this example, we will add class module to the workbook. As a VBA function, you can use this function in macro code that is entered through the Microsoft Visual Basic Editor. Can a variable be used in a VBA message box? Answer: Yes, a variable can be used in a VBA message box. Excel VBA Convert String to Number: Step-by-Step Guide and 7 Examples to Convert String to Byte, Integer, Long, Single, Double, Currency or Decimal By J. The syntax of this date function looks like this. vb6リファレンス 数値操作 - 型変換、四捨五入など、数値に関する操作。. 32 bit systems have a maximum signed integer range of -2147483648 to 2147483647. Performance: Native VBA vs. Parse Module Module1 Sub Main() ' Your input string. 5” to an integer, the resulting integer will be reported as a “6. VBA Data Types. Private Sub. The following VBA code shows how the CInt function can be used to convert text and numeric values into Integers. ) and strings (a,b,c,. NET with its C# equivalent. Private Function RoundOff(ByVal value As Double, ByVal _ digits As Integer) As Double Dim shift As Double shift = 10 ^ digits RoundOff = CInt(value * shift) / shift End Function. Both Int and Fix remove the fractional part of number and return the resulting integer value. Exit Exit a block of code immediately f FileSystemObject Work with Drives, Folders and Files Filter Produce an array by filtering an existing array Fix(number) Return the integer portion of a number ForNext Repeat a block of statements a given number of times For Each. Converting String to Integer. VBA in Finance. ' Round value with this many digits. 12 part section >> Excel VBA and Charts. pdf), Text File (. Se aplica para Excel 2013, Excel 2011 para Mac, Excel 2010, Excel 2007, Excel 2003, Excel XP, Excel 2000 y sólo se puede utilizar en código de VBA. VBA workers vs. The CAST and CONVERT column functions do not seem to work in MS Access. " y lo que buscaba de String a int MyInteger = CInt(MyString) "o el antiguo parse // que se me habia olvidado que existia ajajajja MyInteger = Integer. 5 rounds to 2. Visual Basic for Applications isn't one of these languages. From The Microsoft Developer Network When the fractional part of a value is exactly 0. O Excel quando se grava uma macro cria um objecto designado por module no workbook onde regista todas as operações gravadas em linguagem Visual Basic for Applications - VBA. Let's make this concept clear using a simple example. By definition: The ceiling and floor functions map a real number to the next largest or next smallest integer. NET GridView Rows On Mouse Over Using VB. The existence of a range of other conversion functions (CInt, CDbl, etc) is ignored. When the fractional part is exactly 0. If you're a VB. Excel VBA Convert String to Number: Step-by-Step Guide and 7 Examples to Convert String to Byte, Integer, Long, Single, Double, Currency or Decimal By J. Depending on the combination and order of those arguments, different code will be executed. et» Named Arguments. 5, the CInt function always rounds it to the nearest even number. 14) とかもできます. x = CInt(Right(my_string, 4)) My issue is that sometimes, jump to content my subreddits. Hope you find this useful! Thank you to Tom Shelton, Fergus Cooney, Steven Swafford, Gjuro Kladaric, and others for your contributions. Private Sub. Search GridView In VB. A leading zero like in 010 means the number is in octal notation, using (int)010 will convert it to the decimal value 8 in base 10. In the case of a negative number argument, the Int function returns the first negative integer less than or equal to the number; the Fix function returns the first negative integer greater than or equal to the number. NET developer learning C# or converting your VB code to C#, here are a few hints, tips and gotchas. The argument is required and needs to represent a value within the range of -32,678 to 32,767. 5 rounds to 2. 5, the CInt function always rounds it to the nearest even number (0. 5 (it actually uses Banker's Rounding so numbers ending exactly in 0. Parseの違いとはなんでしょうか？？CInt(引数の型は Object型）Integer. But that 20% could include some way cool stuff! A link to email me is below. Join GitHub today. The VBA Int function rounds a supplied number down to an integer. 1 2 1 5 1 4 1 2 1 2 20. Printの使い方です。. If you want to push the complex query processing done by your Access queries to the back end, you'll have to rewrite them in T. ColorIndex offers 56 basic colors plus the following special numbers. Cuando la parte fraccionaria es exactamente 0,5, la función CInt siempre se redondea al número par más cercano. VBA Arrays And Worksheet Ranges. Excel VBA: Convert to a Integer Data Type. When the fractional part is exactly 0. Back to Excel VBA The example below shows how CInt can be used in Excel VBA to convert to a Integer data type. Re: [RESOLVED] Val() vs CInt() or CLng() Val() and Str$() do still have an important use though. 09/10/2009 2 Comments Every program must be Well structured Address each sub-problem in an easy to spot and specific program part Well commented All th t il d t d d/ dif th. Boolean variables (and properties) are used frequently in VBA programming to hold data that can be on/off, yes/no, and so on. The way to do that is using the CInt() function. So for example on such a system, intval('1000000000000') will return 2147483647. xls), PDF File (. Basically, I have been using both Integer. O Excel quando grava uma macro cria um objeto designado por module no workbook onde registra todas as operaes gravadas em linguagem Visual Basic for Applications - VBA. C#コードの冒頭から3行目にある「using static System. Note that the Int() function in VBA truncates, whereas the Int() function in Excel rounds - so they are quite different. There are no functions in VBA for alternate or random rounding, so far as I'm is aware. It helps in generating reports, preparing various charts, graphs. The toInt() function allows you to convert a String to an integer number. MyNumber = CInt(Int(99. For example, 0. Converting Decimals to Integers. Cuando la parte fraccionaria es exactamente 0,5, la función CInt siempre se redondea al número par más cercano. IsNumeric() is a function that can be used to determine if a value is numeric or not. Brian, I would use CInt, unless I specifically needed to use Integer. Puede utilizar expresiones de etiqueta para ajustar el formato de las etiquetas. This is Part 3 of a new tutorial series showing you how to convert between VBA data types. Converting string to integer/number. As a VBA function, you can use this function in macro code that is entered through the Microsoft Visual Basic Editor. The VBA Like Operator can be used instead to perform inexact. Today now = now1. 【送料最安値！】東芝LEDベースライト LEKR419523N-LD9 TENQOO 40形 埋込形 下面開放W190 調光タイプ,ショットツヴィーゼル ピュアループ タンブラー 542cc 6脚セット【食器 グラス】 タンブラー グラス トリタン Tritan Pure Loop PURE SCHOTT ZWIESEL ショット ツヴィーゼル グラス ワイン 食器 洋食器 ショット・ツ. This on-line tutorial will show you how. This project is brand new, but I'm going to do my best to post new videos here as often as possible. The difference between Int and Fix is that if number is negative, Int returns the first negative integer less than or equal to number, whereas Fix returns the first negative integer greater than or equal to number. 0 0 1 1 1. VBA Data Types. 5 转换为 2。Cint 和 CLng 函数不同于 Fix 和 Int 函数，Fix 和 Int 函数会将小数部分截断而不是四舍五入。并且 Fix 和 Int 函数总是返回与传入的数据类型相同的值。. If the number is negative, the result is more negative. Likewise the Convert functions, ToString(), etc. Parse(MyString) y no abri otro tema como me lo sugeriste David, xq al buscar en google " de string a int vb. Example, for VBA Variable Sub Macro1() Dim Num As Integer Num = 99 MsgBox " Guru " & Num End Sub When you run this code, you will get the following output in your sheet. CInt() La función CInt convierte una expresión al tipo Integer. Il n'est donc pas nécessaire de s'assurer que toutes les variables soient du même type avant de les utiliser dans une expression. age = CInt(v_Age_Calc) End Function. The most important job of a data type is to ensure the validity of your data. Parse als auch CInt in den meisten meiner täglichen Programmieraufgaben verwendet, aber ich bin ein wenig verwirrt, was der Unterschied zwischen den beiden ist. The whole issue of implicit vs explicit is not mentioned. 5 part section >> An Excel Picture Viewer Project. The CAST and CONVERT column functions do not seem to work in MS Access. That's a relief. C# allows you to upcast and perform implicit conversions (eg double to int) without needing a cast. VBScript does not have native Ceiling and Floor functions. VBA Excel ColorIndex Property is to set color or get color for Objects like Cell color and Shape color. There are restrictions on how certain data types can be modified and manipulated so sometimes it might be more useful. As a result, it's feasible to use R and perform analyses for years without specifying these differences. Back to Excel VBA The example below shows how CInt can be used in Excel VBA to convert to a Integer data type. NET Forums on Bytes. 1 0 0 0 1 5 1 1 1 3 13. 5 rounds to 0, 1. A Variant is a special kind of data type that can contain different kinds of information, depending on how it's used. If you are new VBA (or VB) arrays, but have experience with arrays in other programming languages (e. Introduction. Examples of data types are integers, floating point numbers, and strings. Back To: All Conversion Functions. Hope the tips can save you a couple of hours or days of headaches. Open Referenceは誰でも利用できる、オープンなテクニカルリファレンスです。プログラムに関するテクニックはもちろん、パソコンの基本的な使い方や、裏技的な情報、様々なソフトウェアの操作方法やハードの情報など、およそコンピュータに関するあらゆる情報を誰でも自由に閲覧・投稿. pdf), Text File (. 型のメンバを動的に呼び出す. 0 2 0 14 0 12 0 9 0 3 40. Note that Int truncates to the next smaller number. When the fractional part is exactly 0. Check if a value is an integer using VBA code I know I should be able ot do this but I'm drawing a blank and nothing seems ? cint(5. O Excel quando se grava uma macro cria um objecto designado por module no workbook onde regista todas as operações gravadas em linguagem Visual Basic for Applications - VBA. Note CInt differs from the Fix and Int functions, which truncate, rather than round, the fractional part of a number. So if you are ever counting up higher than 32,767 (or down below -32,768) you are going to run into trouble if you have to use cInt on that number. VBScript uses the method "sand castle" with respect to the security. The VBA CInt function converts an expression (variable) to Integer data type. By default, arguments are of the ByRef type, which means that if a variable is submitted as an argument, its reference will be transmitted. Re: [RESOLVED] Val() vs CInt() or CLng() Val() and Str$() do still have an important use though. Returns the integer portion of a number. Also, Fix and Int always return a value of the same type as is passed in. See this from the help for CInt and other Type Conversion Functions. You can use any valid numeric or string expression, but the string expression must be able to be converted to a number. 1 2 1 5 1 4 1 2 1 2 20. Examples of data types are integers, floating point numbers, and strings. The CAST and CONVERT column functions do not seem to work in MS Access. Integer, which I. If the argument contains a decimal, Access rounds to the next whole number. As a result, it's feasible to use R and perform analyses for years without specifying these differences. Integer is used as a data type to denote an integer number, whereas double is a data type to denote a big floating number. Function Overloading - VBA Function overloading, another wonders of the object oriented world where by the same name functions with different signatures provide the respective functionality based on the invoking arguments passed and their types. VBA provides a full set of functions for converting data from one data type to another. 5” to an integer, the resulting integer will be reported as a “6. 【送料最安値！】東芝LEDベースライト LEKR419523N-LD9 TENQOO 40形 埋込形 下面開放W190 調光タイプ,ショットツヴィーゼル ピュアループ タンブラー 542cc 6脚セット【食器 グラス】 タンブラー グラス トリタン Tritan Pure Loop PURE SCHOTT ZWIESEL ショット ツヴィーゼル グラス ワイン 食器 洋食器 ショット・ツ. When you declare a Boolean variable, it is automatically initialized to False. 5) = 16 and CInt(15. So for example on such a system, intval('1000000000000') will return 2147483647. The included project runs a performance test on each scenario: DirectCast vs. The Integer data type can hold integer values between -32,768 and 32,767. Typeオブジェクトで表された型のインスタンスの作成、メソッドの呼び出し、プロパティ、フィールドの設定と取得の方法について説明します。. VBScript Number Conversion Functions - Number functions help us to convert a given number from one data subtype to another data subtype. In fact, "VBA" functions which are usable in SQL (Access SQL) are not the same as the real VBA functions. com/ and you'll be signed up for the 12 part VBA Crash Course video training!. In this article, we will create a macro to insert a new module in an Excel workbook. 1 Fractional parts can be subject to a special type of rounding called banker's rounding. Numeric Types Introduction to VBA programming - (c) 2009 Dario Bonino Designed for holding numeric values Can be Integer s Represent signed integer numbers on 16 bits. Private Sub Field_BeforeLostFocus(KeepFocus As Boolean, CancelLogic As Boolean) Dim Var1 As String Var1 = VendorID MsgBox "The variable is " + Var1 End Sub. The VBA Int function works the same as the worksheet Int function. The Int function and the CInt function are not designed to do the same thing. If you expect the conversion to succeed most of the time, and fail only in exceptional cases, than using Int32. Information module for a reason - it returns 0 unless the error. The code below will display an open file dialog and ask the user to select the path of the file to open. See "Remarks" for more information. Technically speaking you don't have to use CInt()since "x" has already been declared as an Integer variable an IMPLICIT conversion will be done for you if you remove CInt(). Back To: All Conversion Functions. Back to Excel VBA The example below shows how CLng can be used in Excel VBA to convert to a Long data type. 5, the CInt function always rounds it to the nearest even number. Y la mejor respuesta viene de la MSDN. Puede utilizar expresiones de etiqueta para ajustar el formato de las etiquetas. 5 rounds to 0, and 1. A Boolean variable can only hold one of two values: true or false. TeX has \int as the integral sign. VBA's functions for simple data. , C), you will find that VBA arrays work pretty much the same. I have learned more about VBA than I could have imagined. Parse or Convert. If you want to push the complex query processing done by your Access queries to the back end, you'll have to rewrite them in T. In excel 2003 windows XP, the same code takes only 9 seconds to run. 5 rounds to 2). Diese müssen nicht, wie in C üblich, dem Compiler durch Angabe von Bibliotheken im Kopfabschnitt ( header ) der Code-Datei "angekündigt" werden, sondern sie gehören zu den Grundelementen der Sprache und sind somit ohne weiteres verwendbar. what i want is that i want to use the datetimepicker and it automatically compute the age in another textbox without clicking the button. Here's how to understand and use them. Re: What is difference of Val, Int, CInt etc ? I suppose Val is the safest option, as it will allow a vast number to be converted before enountering an overflow. Note also that CInt rounds to the nearest even integer for integers plus one half. Lesson 9: Creating a Counter in Excel VBA [Back to Excel VBA Sample Code]This is a simple Excel VBA counter that can count the number of passes and the number of failures for a list of marks obtained by the students in an examination. Integers can be described as whole numbers meaning that they do not have any fractional parts. In this article, we will create a macro to insert a new module in an Excel workbook. For example, if you want to. VBA's Fix() function behaves like Int() for positive numbers; but when applied to a negative number it rounds towards zero. Free tutorial to learn with examples. What Are VBScript Data Types? VBScript has only one data type called a Variant. Depending on the combination and order of those arguments, different code will be executed. There are no functions in VBA for alternate or random rounding, so far as I'm is aware. Back to Excel VBA The example below shows how CInt can be used in Excel VBA to convert to a Integer data type. How To: Convert a string field to a numeric field Summary. For example, Int(-17. When working with VBA, a variable is a location in your computer's memory where you can store data. He has authored 12 SQL Server database books, 24 Pluralsight courses and has written over 4900 articles on the database technology on his blog at a https://blog. As a result, it's feasible to use R and perform analyses for years without specifying these differences.