CONWAY J. H., SLOANE N. J. A.: Sphere Packings, Lattices, and Groups, 2 ed. Springer-Verlag, New York, 1993. 4  Home/Search   Document Not in Database   Summary   Related Articles   Check

....may contain several range variables. If the variables are all interdependent, then the criterion is referred to as the smallest criterion. Definition 4.3 The smallest criterion partition contains the disjoint subsets of smallest criterion. In terms of the Stirling number of the second kind [5], the smallest criterion partition corresponds to S(n, n) where S(n, k) states the number of partitions of an n set into k blocks. Here n is the number of disjoint smallest criterion. The algorithm in figure 6 is used to generate the smallest criterion partition. C represents the range ....

J. H. Conway and R. K. Guy. The Book of Numbers. Springer-Verlag, New York, 1996.

.... Although the general theory encompasses arbitrary measures in topological spaces [36] well known sample sets such as the Hammersley Halton sequences are designed for low discrepancy over [0; 1] There are exceptions, such as the low dispersion sequence given for a d dimensional torus in [12] (page 115) If the boundary identi cations are taken into account, it should be possible to take advantage of sample sets that have even lower discrepancy. The problem of designing low discrepancy samples for di erent topological spaces is yet to be investigated in the context of motion ....

J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, Berlin, 1988.

....a lattice L ( R s.t. every one of the 2 0 1 combinations of the basis vectors is a lattice point of shortest length, then a parsimonious reduction would become possible. However, the construction of such lattices would be a tremendous breakthrough (see the book of Conway and Sloane [CS93] for much related work that has fallen well short of achieving anything close to this) In fact, the best known lattice construction in this spirit is a construction from the 1950 s, called the Barnes Wall lattice, that has 2 (log ) 2 points of shortest length. 3 USVP is NP hard In this ....

J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 2nd edition,

....r 2 2 Z 2 0 v(R; OE) R 2 Gamma 2Rr cos( Gamma OE) r 2 dOE (3 2) This is well known as the Poisson Integral Formula in the literature. Thus the value of v inside the disk is uniquely determined by the value of points on the disk. 18 3.1. POTENTIAL THEORY 19 It follows from 3 2 [Con] that, if v(R; OE) a 0 P n=1 n=1 (a n cos n b n sin n ) then for r R, v(r; a 0 n=1 X n=1 r R n (a n cos n b n sin n ) 3 3) We note that the coefficients, a 0 = 1 2 R 2 0 v(R; OE)dOE, a n = 1 R 0 v(R; OE) cos nOEdOE n = 1; 2; b n = 1 R 0 v(R; ....

J.Conway Functions of one complex variable. Springer Verlag. 1973.

.... Delta a 6 i 6 j each a t 2 Z=2, and fa t g Z 2 Sg; where S consists of the subsets of fa1 ; a 0 ; a 6 g whose indices are taken from f; 0124; 0235; 0346; 1045; 0156; 1026; 1013g and the complements of these indices in 10123456. Geometrically, O 3 is similar to the E 8 lattice [2]. The problem of finding the factorizations of a given ae 2 O as ae = fffi for ff; fi 2 O and fixed m = ff] and n = fi] has a long history. Factorization results for O in R, C , and H are classical (see [5] However, the methods of associative number theory are not well suited to O ae O since, ....

J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, second edition, 1993.

....in which they have no legal move. A strategy which accomplishes this has essentially the same structure as a proof of wellfoundedness. The idea of developing an arithmetic of competitive advantage for positions in terminating games has been extensively pursued by Conway and others [15] 41] [16]. Conway did not however consider the metamathematical question of how competitive strength is limited by the programming system in which the strategies are written, which seems to be a question more in the province of proof theory. If proofs are programs, it does not follow that programs are ....

J.H. Conway and R.K. Guy. The Book of Numbers. Springer-Verlag, New York, 1996.

....sions [7] We utilize a root lattice, E8, in our scheme. A cube in wavelet coecient domains is illustrated in Fig. 19. This lattice contains (1) all locations in eight dimensions with integer coordinates that are summed as even numbers and (2) a collection of their middle points called the coset [7, 8]. The primary concern in LVQ is the truncation and scaling of the lattice. Although there exist analytical methods [20, 8] based on the number theory) to determine the radius ( p m) of a sphere for truncation and scaling, the adjustment of the radius and a scaling factor (F) is required for ....

....This lattice contains (1) all locations in eight dimensions with integer coordinates that are summed as even numbers and (2) a collection of their middle points called the coset [7, 8] The primary concern in LVQ is the truncation and scaling of the lattice. Although there exist analytical methods [20, 8] (based on the number theory) to determine the radius ( p m) of a sphere for truncation and scaling, the adjustment of the radius and a scaling factor (F) is required for di erent applications [1] The scaling factor F is determined by the following formula: F = p p Maximun Coecient ....

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J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, Springer-Verlag, New York, 1999.

....best apple peeling codes currently realizable is given by replacing (k 1) in the formula for the density in Lemma 3 with the density of the best known sphere packing in R k 2 . This apple peeling code asymptotic density is shown in Figure 2, using a table of the best known sphere packings from [34], along with recent improvements from [35] and [36] The asymptotic density of the wrapped spherical codes is equal or higher in every dimension. 3 Wrapped spherical codes Any spherical code can be described by the projection of its codepoints to the interior of a sphere of one less dimension via ....

....1 O( p d) X 2 f 1 arg min Y 2 kY f(R)k ; which involves only f , f 1 , and the decoding algorithm for . It is known that when points from the packing with minimum distance d are used on an AWGN 22 channel, the probability of symbol error is Q( d 2 ) see, e.g. [34]) where is the average number of codepoints at distance 2d from a codepoint and where Q is the complementary error function de ned by Q(x) 1 p 2 R 1 x e x 2 =2 dx. The following theorem shows that the performance of eciently decoding C is asymptotically close to the performance ....

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer-Verlag, 1993.

....spheres and projecting each codepoint onto S k . We exploit this idea and pack the concentric spheres closely. We place codepoints of one sphere at the radial extension of the holes of codepoints of the next smaller radius sphere, and use a method similar to constructing laminated lattices (e.g. [3]) to construct new spherical codes, which we denote by C L . This is illustrated for k = 2 and k = 3 in Figure 1. Our method is similar to those of [4] and [5] in that a projection from k 1 dimensions to k dimensions is used; the di erence lies in the placement of points prior to the ....

....For some larger code sizes, we obtained codes using a simulating annealing approach which slightly improves upon [5] This method produces good codes, but its computational complexity limits the code size that can be constructed. Spherical codes can also be generated from shells of lattices (e.g. [3, 6]) Figure 6 shows the best codes generated among the rst 1000 point centered shells of the face centered cubic and Z 3 lattices, whose minimum distances were obtained exactly. Figure 6 also shows spherical codes formed from concatenations of MPSK and BPSK codes. For d 0:7, the laminated ....

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J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer-Verlag, 1993.

....laminated spherical code, or another code formed from its deep holes. By nesting the concentric spheres closely, and placing codepoints of one sphere at the radial extension of the deep holes of codepoints of the previous sphere, a method similar to constructing laminated lattices (e.g. [CS93]) is used to construct our spherical codes, which we denote by C . As more of these concentric spheres are stacked up, codepoints start spreading out, and the density lessens. To counteract this, a buffer zone is placed between concentric spheres, and a new, tighter packed (k Gamma ....

J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1993.

....[12] 9 The classi cation of even self dual lattices is extremely restrictive. We will use the notation m;n to refer to an even self dual lattice of signature (m; n) It is known that m and n must satisfy m n = 0 (mod 8) 23) and that if m 0 and n 0 then m;n is unique up to isometries [14, 15]. An isometry is an automorphism of the lattice which preserves the inner product. In our case, one may chose a basis such that the inner product on the basis elements forms the matrix 0 B B B B B B B B B B B B B B B B B B B B B B B B B E 8 E 8 U U U 1 C C C ....

....string are compacti ed on an even self dual lattice of de nite signature. There are two such lattices, which we denote 8 8 and 16 . The former is two copies of the root lattice of E 8 E 8 with which we have become well acquainted in these talks. The second lattice is the Barnes Wall lattice [15]. This may be constructed by supplementing the root lattice of so(32) by the weights of one of its spinors. Such spinor weights are never of length squared 2 are so do not give massless states. Thus, as far as massless states are concerned, the lattice is the root lattice of SO(32) and the string ....

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, Springer-Verlag, 1993.

....As shown in the previous section, there is no direct relation between the magnitude of the inner product between two complex vectors and their Euclidean distance. There is a large body of literature on choosing collections of unit vectors that maximize their pairwise Euclidean distances (see [3] and the many references therein) However, the literature on choosing vectors that minimize their pairwise correlations appears to be smaller [10] 12] 22] Moreover, the constellation design problem in T dimensional complex space does not reduce to a design problem in 2T dimensional real ....

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, 2nd edition, Springer-Verlag, New York, 1993.

....farther these points are apart, measured by the squared Euclidean distances d 2 free , the better they can resist the disturbance of the AWGN. To keep the points as far apart as possible while maintaining a fixed power is the task of code construction. Obviously, this is a sphere packing problem [46]. To calculate the performance of a trellis code, we assume a reference codeword, or the correct sequence and calculate the probability of error when it is disturbed by an infinitive dimensional Gaussian noise. Theoretically we have to calculate the integral in an irregular integration region, ....

J. H. Conway and N. J. A. Sloane, Sphere packings, lattices, and groups. Springer-Verlag, 1988.

....20dimensional lattice with automorphism group isomorphic to 2:M 22 :2. 1 Introduction An even, unimodular lattice can only exist in dimension n which is a multiple of 8, and one knows, from the properties of its theta series as a modular form, that its minimum is bounded by 2( n=24] 1) see [CoS 88] A lattice attaining this bound is said to be extremal. Extremal lattices are known up to dimension 64; the most wanted would be a 72 dimensional lattice of minimum 8, which is not yet proved to exist. In this paper, we construct two extremal lattices in dimension 80. One of them belongs to a ....

....can be described as above: For example the root lattices D 4 = Q(A 2 ; 1; 1; 3) and E 8 = Q(D 4 ; 1; 1; 2) Q(A 4 ; 1; 2; 5) but also the Coxeter Todd lattice K 12 = Q(P 6 ; 1; 2; 7) can be constructed in this way. Examples: Some extremal 2 modular lattices: i) The Barnes Wall lattice (cf. CoS 88] p.129) BW 16 = Q(A 2 Omega D 4 ; 1; 2; 6) ii) The 2 known extremal 2 modular lattices of dimension 20 (cf. PlN 95] p. 49) may be obtained from the Craig lattices (cf. CoS 88] p.222) in dimension 10: 2:M 12 :2] 20 = Q(A (2) 10 ; 1; 3; 11) and [SU 5 (2) 2(2) ffi SL 2 (3) 20 = Q(A (3) ....

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J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer-Verlag 1988

....by arranging spheres in the form of a face centered cubic lattice. Thus, at present, we know that 0:7405 ffi(3) 0:7784, though Rogers [Rogers 58] remarks many mathematicians believe that the correct answer is 0:7405. For the values of ffi(n) n 3, the interested reader can refer to [Conway Sloane 88] 5.1.2 Sphere Packings in Spherical Space A problem that is closely related to the problem of packing spheres in the Euclidean space is that of packing (n Gamma 1) dimensional spheres (spherical caps) of angular diameter OE on the surface of an n dimensional unit sphere. Let us define A(n; ....

....mathematical theory of packing and covering. The interested reader can further study this subject, starting perhaps with the classical book by Rogers [Rogers 64] For more recent results in this field see the survey article by Fejes T oth [Fejes T oth 83] and the book by Conway and Sloane [Conway Sloane 88] 5.2 Solvent Accessible Protein Surfaces In Section 5.2.1, we shall quickly review the terminology of Section 4.2.1 and the concept of neighborhood as described in Section 4.2.2. After that, we shall look at some relevant properties of proteins, the molecules for which the solvent accessible ....

J. H. Conway and N. J. A. Sloane. Sphere Packing, Lattices, and Groups. Springer-Verlag, New York, NY, 1988.

....the symmetry of particle interactions, which may be used to reduce the computation and the communication by a factor of two. It is possible to further improve the distribution by allowing cuts of the domain in any direction. This can be done efficiently by taking a suitable sphere packing lattice [7] and assigning particles to the nearest centre of a sphere. Sphere packing lattices have been used in other areas of scientific computing; for instance, it has been proposed [5] to use them to decrease anisotropy in pseudo spectral PDE solving on multidimensional grids. This method splits the ....

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, Springer-Verlag, New York, 1988.

.... has received much attention as a computationally simpler alternative to fullsearch vector quantization (VQ) The usual approach is to choose an appropriate lattice, then scale and truncate it in such a way that rate and mean square error (MSE) performance is comparable to that of full search VQ [10, 11, 12, 13]. Lattice VQ also has the potential of limiting maximum errors, provided that an untruncated lattice is used [14] The idea is illustrated in Figure 5. For untruncated lattice VQ to perform well for nonuniformly distributed input (the case of interest in image coding) it is essential that the ....

....coding of the quantized scalars. While it is true that a Z N lattice is suboptimal in terms of space filling, the performance penalty for using it instead of a more efficient lattice is known to be quite small. This has been established theoretically in the case of asymptotically high rate [10, 15], and experimentally in the low and medium rate regions for some sources [14] In the example illustrated in Figure 5, the rate MSE performance was examined and found to be comparable to that of standard VQ. If this performance can be shown to be comparable in general, then the proposed ....

J.H. Conway and N.J.A. Sloane. Sphere packings, lattices, and groups. Springer-Verlag, 1988.

.... Gamma a fi fi fi fi 2 fi fi fi fi i 2 Gamma b Gamma n=2 fi fi fi fi 2 1 2 : 6:2) By our condition on a, the first term on the left lies in [0; 1=4] and equals 1=4 just if a = 0. Since Im p is a copy of the D 3 root lattice (it is spanned by i Sigma j and j Sigma k) p. 112 of [6] shows that the covering radius of 1 2 Im p is 1=2. Therefore by choice of n=2 we may suppose that the second term is bounded by 1=4, with equality just if b is a deep hole of i=2 1 2 Im p. All deep holes of Im p are equivalent by translations by elements of p, and 0 is such a deep hole. ....

J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1988.

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CONWAY J. H., SLOANE N. J. A.: Sphere Packings, Lattices, and Groups, 2 ed. Springer-Verlag, New York, 1993. 4

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J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, Berlin, 1999.

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J. H. Conway, and R. K. Guy, The Book of Numbers, Springer-Verlag, New York, 1996.

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J.H. Conway and R.K. Guy. The Book of Numbers. Springer-Verlag, New York, 1996.

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J.B. Conway: Functions of One Complex Variable. Springer-Verlag, Berlin, Heidelberg, 1978.

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J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1988.

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J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1988.

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J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1988.

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