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Topic: Unimodular lattice

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	Unimodular lattice at AllExperts
		In mathematics, a unimodular lattice is a lattice of discriminant 1 or −1.The E
		The second cohomology group of a compact orientable topological 4-manifoldis a unimodular lattice.
		When the manifold is simply connected, Michael Freedman showedthat this lattice almost determines the manifold: there is a unique manifold for each even unimodular lattice, and exactly two for each odd unimodular lattice.
	en.allexperts.com /e/u/un/unimodular_lattice.htm (839 words)

	Lattice (group) Information
		The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial.
		In a rhombic lattice, the shortest distance may either be a diagonal or a side of the rhombus, i.e., the line segment connecting the first two points may or may not be one of the equal sides of the isosceles triangle.
		The rectangular lattices are at the imaginary axis, and the remaining area represents the parallelogrammetic lattices, with the mirror image of a parallelogram represented by the mirror image in the imaginary axis.
	www.bookrags.com /wiki/Lattice_(group) (1439 words)

	Lattice - Biocrawler (Site not responding. Last check: 2007-10-20)
		The elements of a lattice are regularly spaced, reminiscent of the intersection points of a lath lattice.
		This concept is used in materials science, in which a lattice is a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal.
		It also occurs in computational physics, in which a lattice is an n-dimensional geometrical structure of sites, connected by bonds, which represent positions which may be occupied by atoms, molecules, electrons, spins, etc. For an article dealing with the formal representation of such structures see Lattice Geometries (http://www.hermetic.ch/compsci/lattgeom.htm).
	www.biocrawler.com /encyclopedia/Lattice (443 words)

	The Shadow Theory of Modular and Unimodular Lattices - Rains, Sloane (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Quebbemann had extended the bound for even unimodular lattices to strongly N-modular even lattices for N in f1; 2; 3; 5; 6; 7; 11; 14; 15; 23g ; () and analogous bounds are established here for odd lattices satisfying certain technical conditions (which are trivial for N = 1 and 2).
		2 The genus of the Barnes-Wall lattice - Scharlau, Venkov - 1994
		1 The genus of the Coxeter-Todd lattice - Scharlau, Venkov - 1995
	citeseer.ist.psu.edu /318755.html (1033 words)

	Unimodular lattice - Wikipedia, the free encyclopedia
		In mathematics, a unimodular lattice is a lattice of discriminant 1 or −1.
		The signature of a lattice is the signature of the form on the vector space.
		Michael Freedman showed that this lattice almost determines the manifold: there is a unique such manifold for each even unimodular lattice, and exactly two for each odd unimodular lattice.
	en.wikipedia.org /wiki/Unimodular_lattice (800 words)

	Applied Algebra and Cryptology
		Theses problems are naturally connected to the minimum distance between lattice points (or vectors), theta-series that counts the number of lattice points of same distance from the origin, and the number of minimal vectors in a given lattice, and so on.
		For example, the densest sphere packing is achieved via Leech lattice (the even unimodular lattice whose minimal vector is of length 4, the longest among 24 possible even unimodular lattices) in 24-dimensional case and from this lattice can be constructed the famous Golay-code, which amazingly is able to correct upto 3 errors.
		Among even unimodular lattices, one whose minimal length is the largest is called an extremal lattice.
	www.math.snu.ac.kr /~mhkim/bk21-e.html (3967 words)

	DESCRIPTION OF CALC
		The unimodular matrix P of a recent paper of Havas, Majewski and Matthews, is sent to lllgcdmat.out.
		The output unimodular matrix P of a recent paper is sent to lllhermitetrans.out.
		Let L be the lattice spanned by the first m-1 rows of A and let P be the last row of A. Output: All lattice vectors X of L such that X-P
	www.numbertheory.org /calc/krm_calc.html (4480 words)

	Index to Catalogue of Lattices (Site not responding. Last check: 2007-10-20)
		This data-base of lattices is a joint project of Gabriele Nebe, RWTH Aaachen (nebe(AT)math.rwth-aachen.de) and Neil Sloane.
		The E_n lattices and their duals: E6, E6, E7, E7a (a second version of E7), E7, E8, E8 as a Hurwitzian lattice.
		, a 5-modular lattice in the "+" genus.
	www.research.att.com /~njas/lattices (1878 words)

	TJS paper
		Section 3 discusses unimodular loop transformations and introduces loop stretching; in addition, we argue that unimodularity is not sufficient for optimizing communication with tiling for distributed memory machines.
		Since the columns of H and T generate the same lattice, the entries on the diagonal of H are the step sizes of the loops in the nest.
		Barnett and Lengauer [5] conclude that unimodularity is not essential; to derive code for non-unimodular transformations, they generate piecewise loop bounds which are not exact.
	www.ece.lsu.edu /jxr/revised/revised.html (5041 words)

	Self-dual lattices Text - Physics Forums Library
		Self-dual lattices are very special,\nandgt; and they can be identified with the weight lattices of E_8 or\nandgt; spin(16.k) / Z_2 - where the Z_2 subtlety is necessary because only\nandgt; "one half" of the weight lattice should be taken - something in\nandgt; between the weight lattice and the smaller root lattice.
		So I cheated:\n\nhttp://www.research.att.com/~njas/lattices/unimodular.html\n\nIn that table the first unimodular lattice that is not the weight lattice\nof a Lie group is in dimension 12, and the "shorter Leech" (in 23d) is\nthe first one that is not even and not a weight lattice.\n\n[Moderator\'s note: That\'s a cool lattice.
		I have in mind the lattice that a target space might be compactified on (which need not be either even or self-dual, is that right?), rather than the (related) lattice on which closed string momenta live (which I think has to be both even and self dual).
	www.physicsforums.com /archive/index.php/t-41595.html (2433 words)

	Point Lattice -- from Wolfram MathWorld
		Every "point lattice" is a lattice under the ordering inherited from the plane, although a point lattice may not be a sublattice of the plane, since the infimum operation in the plane need not agree with the infimum operation in the point lattice.
		Formally, a lattice is a discrete subgroup of
		That is, a lattice is closed under addition and inverses, and every point has a neighborhood in which it is the only lattice point.
	mathworld.wolfram.com /PointLattice.html (571 words)

	YourArt.com >> Encyclopedia >> lattice (Site not responding. Last check: 2007-10-20)
		Lattice (order), a type of partially ordered set
		Bravais lattice, 14 possible arrangements of repeating points in 3-D
		Lattice model (physics), a model defined not on a continuum, but on a lattice
	www.yourart.com /research/encyclopedia.cgi?subject=/lattice (80 words)

	Integer Lattice -- from Wolfram MathWorld
		Such an array is often called a grid or mesh, and is a special case of a point lattice.
		sufficient condition that a linear transformation transforms a lattice to itself is that it be unimodular.
		M. Ajtai has shown that there is no efficient algorithm for finding any fraction of a set of spanning vectors in a lattice having the shortest lengths unless there is an efficient algorithm for all of them (of which none is known).
	mathworld.wolfram.com /IntegerLattice.html (305 words)

	Citebase - The Shadow Theory of Modular and Unimodular Lattices
		Authors: Rains, E. Sloane, N. It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[n/24] +2, unless n = 23 when the bound must be increased by 1.
		Quebbemann had extended the bound for even unimodular lattices to strongly N-modular even lattices for N in 1,2,3,5,6,7,11,14,15,23...
		For N > 1 in (), lattices* meeting the new bound are constructed that are analogous to the ``shorter'' and ``odd'' Leech lattices.
	www.citebase.org /abstract?id=oai:arXiv.org:math/0207294 (229 words)

	math lessons - Lattice (group)
		The Lenstra-Lenstra-Lovász lattice reduction algorithm (LLL) finds a short lattice basis in polynomial time; it has found numerous applications, particularly in public-key cryptanalysis.
		This concept is used in materials science, in which a lattice is a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal.
		More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that G/Γ is of finite measure, for the measure on it inherited from Haar measure on G (left-invariant, or right-invariant - the definition is independent of that choice).
	www.mathdaily.com /lessons/Lattice_(group) (494 words)

	A Note on Optimal Unimodular Lattices - Conway, Sloane (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Abstract: The highest possible minimal norm of a unimodular lattice is determined in dimensions n 33.
		There are precisely five odd 32-dimensional lattices with the highest possible minimal norm (compared with more than 8:10 20 in dimension 33).
		Unimodular lattices with no roots exist if and only if n 23, n 6= 25.
	citeseer.ist.psu.edu /conway98note.html (574 words)

	TJS paper
		Section 3 discusses unimodular loop transformations and introduces loop stretching; in addition, we argue that unimodularity is not sufficient for optimizing communication with tiling for distributed memory machines.
		Since the columns of H and T generate the same lattice, the entries on the diagonal of H are the step sizes of the loops in the nest.
		Barnett and Lengauer [5] conclude that unimodularity is not essential; to derive code for non-unimodular transformations, they generate piecewise loop bounds which are not exact.
	www.ee.lsu.edu /jxr/revised/revised.html (5041 words)

	Mathematical Utilities
		Resampling matrices that are unimodular can be used to shuffle an underlying signal lattice without introducing upsampling or decimation.
		The function returns both the regular unimodular matrix that performs the pivoting to produce the triangular matrix, and the triangular matrix that results.
		distinct coset vectors for the lattice corresponding to an input resampling matrix.
	documents.wolfram.com /applications/signals/MathematicalUtilities.html (3468 words)

	[No title]
		It is known that their lattice width is bounded by a constant w(d) that only depends on the dimension.
		A unimodular triangulation is constructed for the polytopes that are associated with toric local complete intersections, thus generalizing a result of Dais, Henk and Ziegler.
		Das Existenzproblem unimodularer Triangulierungen von Gitterpolytopen wird untersucht.
	edocs.tu-berlin.de /diss/2000/haase_christian.htm (546 words)

	Mathieu functions
		The main feature of qualitative interest is the absolute value of the eigenvalues, which describes the rate of growth of solutions with respect to the translational lattice.
		The stability chart is a contour plot of the absolute value of the eigenvectors of the period matrix as a function of the two parameters a and q.
		Given that the solution matrix of the equation is unimodular, half its trace is the cosine of the logarithm of the eigenvalue.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node59.html (533 words)

	► » Even Lattices (Site not responding. Last check: 2007-10-20)
		lattice, and the Leech lattice (dim=24) is the smallest example of a
		Are there interesting unimodular lattices with minimum norm = 6 (or
		For a start there's the "odd Leech lattice" of rank 24 and minimum norm 3.
	www.science-chat.org /Even-Lattices-5819286.html (139 words)

	Simplicial signed decompositions
		Theoretically this is bad because the number of lattice points in the parallelepiped is exponentially large already for fixed dimension.
		Therefore once we have a unimodular cone decomposition, the rational generating function of the original cone is a signed sum of ``simplicial'' rational functions.
		be the smallest vector in the lattice spanned by
	www.math.ucdavis.edu /~latte/theory/lattE/node3.html (1296 words)

	Unimodular Lattices (Site not responding. Last check: 2007-10-20)
		Part of the Catalogue of Lattices, which is a joint project of Gabriele Nebe, University of Ulm (nebe@mathematik.uni-ulm.de) and Neil J. Sloane, AT&T Labs-Research (njas@research.att.com).
		A unimodular lattice is an integral lattice which is its own dual.
		The table give the highest possible minimal norm (mu) of an n-dimensional unimodular lattice and the names of lattices meeting the bound (also, whenever possible, links to files containing these lattices).
	www.research.att.com /~njas/lattices/unimodular.html (213 words)

	Abstract of "Conference on Groups and Lie algebras"
		It is known that, in every extremal even unimodular lattice, the set of vectors of a given norm forms a spherical design of certain strength.
		This means that, for example, the root system of type $E_8$ can be used to approximate the integrals of continuous functions defined on the sphere of radius $\sqrt{2}$ quite accurately, in the sense that it is exact up to polynomial approximation of degree less than or equal to 7.
		That is, certain spherical designs necessarily come from an extremal even unimodular lattice.
	www.mm.sophia.ac.jp /cgla/abstract.html (1405 words)

	Bacher, Venkov: Lattices and association schemes: a unimodular example without roots in dimension 28
		Lattices and association schemes: a unimodular example without roots in dimension 28.
		Some interesting lattices can be constructed using association schemes.
		We illustrate this by a unimodular lattice without roots of dimension 28 which admits
	www.numdam.org /numdam-bin/item?id=AIF_1995__45_5_1163_0 (167 words)

	[No title]
		Say you are packing spheres in n dimensions in a checkerboard lattice --- in other words, you color the cubes of an n-dimensional checkerboard alternately red and fl, and you put spheres centered at the center of every red cube, using the biggest spheres that will fit.
		If we have a lattice, we say a vector r in it is a "root" if the reflection through r is a symmetry of the lattice.
		First, he shows that the fundamental roots of the even unimodular Lorentzian lattices in dimensions 10, 18, and 26 are the vectors r with r.r = 2 and r.v = -1, where the "Weyl vector" v is (28,0,1,2,3,4,5,6,7,8) (46,0,1,2,3,......,16) and (70,0,1,2,3,......,70) respectively.
	math.ucr.edu /home/baez/twf_ascii/week95 (3283 words)

	the Jacobi functions sn, cn, and dn
		Beyond that, if the period lattice is square or rhombic, the functions follow suit.
		A periodic lattice would be expected to be invariant under affine transformations -- composites of translations with matrix products -- but Möbius transformations also include inversions, which would produce a different network entirely.
		One is to repeat the derivation of the Schwartz derivative for functions which could be double valued functions of a cubic polynomial, to obtain a differential invariant for this more general class of functions.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node36.html (672 words)

	[No title]
		be a unimodular hermitian O-lattice, M a unimodular sublattice and N its orthogonal complement in ?.
		Oq is the intersection of the corresponding reflection lattices E and E0.
		By the classification of indefinite unimodular O3- sublattices æ is ?(ß)- conjugated to one of the øj.
	www.mathematik.uni-osnabrueck.de /projects/carmen/AP11/test/file132.html (17108 words)

	The magic of 8
		That sounds right: I'm talking about "even unimodular lattices" and you're talking about "invertible symmetric integer matrices with even entries on the diagonal", but they're secretly the same thing as long as your matrix is positive definite...
		A lattice is "unimodular" if the volume of each cell of the lattice is 1.
		To get from an even unimodular lattice to a matrix, pick a basis of vectors in the lattice and form the matrix of their inner products.
	www.lns.cornell.edu /spr/2003-01/msg0047924.html (3017 words)

	The Coset Lattices of E. S. Barnes and G. E. Wall (Site not responding. Last check: 2007-10-20)
		John Conway's analysis in 1968 of the automorphism group of the Leech lattice and his discovery of three sporadic simple groups led to the immediate speculation that other Z-lattices might have interesting automorphism groups which give rise to (possibly new) finite simple groups.
		This family of integral lattices received attention and acclaim in the subsequent literature.
		In particular, it is shown how to associate to each such lattice a continuous, peicewise linear graph in the plane from which all the important properties of the lattice, for example, its minimum, whether it is integral, unimodular, even, or perfect can be read off directly.
	anziamj.austms.org.au /JAMSA/V49/Part3/Hahn.html (302 words)

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