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# Topic: Discrete subgroup

###### In the News (Wed 22 May 19)

 Discrete group - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu)   (Site not responding. Last check: 2007-10-09) A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one. Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. A lattice in a Lie group is a discrete subgroup such that the Haar measure of the quotient space is finite. en.wikipedia.org.cob-web.org:8888 /wiki/Discrete_group   (669 words)

 3. Discrete Subgroups of Euclidean Group The considered groups are discrete, so in every such group we can choose a translation whose vector is minimal. Their translation subgroup is generated by two independent translations, and this class contains 17 non-isomorphic groups. The classification of the discrete subgroups mentioned will be considered in the next three chapters. members.tripod.com /vismath1/ana/ana3.htm   (533 words)

 PlanetMath: idèle is a discrete subgroup of the group of idèles It is, however, possible to define a certain subgroup of the idèles (the subgroup of norm 1 elements) which does have compact quotient under Cross-references: subset, subspace topology, topology, ring, multiplicative, norm, quotient group, embedding, image, diagonal embedding, units, infinite primes, multiplicative groups, restricted direct product, subgroup, open, compact, group, group of units, completion, valuation ring, finite prime, number field planetmath.org /encyclopedia/IdeleGroup.html   (149 words)

 [No title]   (Site not responding. Last check: 2007-10-09) When V is an induced representation from some subgroup K to G, which we realize geometrically as the space of functions on G/K, the corresponding algebra is essentially what is known as the Hecke algebra H(G;K). In this case the Hecke algebra is almost abelian (in fact its large abelian subgroup is itself often called Hecke algebra),and its basis is indexed by a slight extension of the Weyl group.. An element of the principal series corresponds to the induced representation from the upper-triangular subgroup of the character which assigns to an upper diagonal matrix with a, 1/a on the diagonal the number x(a), where x is a character on the real line. www.ma.utexas.edu /~benzvi/math/Langlands2   (1321 words)

 Spectral geometry and group cohomology   (Site not responding. Last check: 2007-10-09) In Small discrete subgroups and the Plancherel theorem we consider discrete subgroups of semisimple Lie groups of higher rank. We define a sort of convex cocompactness of discrete subgroups with respect to a given class of parabolic subgroups. Under additional smallness hypotheses we study the Plancherel decomposition of the Hilbert space of square integrable functions on the quotient of the semisimple group by the discrete subgroup. www.uni-math.gwdg.de /bunke/project1.html   (364 words)

 Dihedral group - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu)   (Site not responding. Last check: 2007-10-09) The subgroup of Dih(H) of elements (h, 0) is a normal subgroup of index 2, isomorphic to H, while the elements (h, 1) are all their own inverse. As subgroups of the isometry group of the set of vertices of a regular n-gon they are different: the reflections in one subgroup all have two fixed points, while none in the other subgroup has (the rotations of both are the same). As subgroups of the isometry group of Z they are different: the reflections in one subgroup all have a fixed point, the mirrors are at the integers, while none in the other subgroup has, the mirrors are in between (the translations of both are the same: by even numbers). en.wikipedia.org.cob-web.org:8888 /wiki/Dihedral_group   (1891 words)

 [No title] Since the subgroup is discrete, the topology is locally the same as SO(3), that is, it's another manifold. Since the subgroup is _not_ normal, the new space is not a group. The subgroup intended is then the image of this representation (then unique, up to conjugacy). www.math.niu.edu /~rusin/known-math/96/pi_1.so3   (758 words)

 Wikinfo | Lattice 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). The supremum is given by the least common multiple and the infimum by the greatest common divisor. The supremum is given by the subgroup generated by the union of the groups and the infimum is given by the intersection. www.wikinfo.org /wiki.php?title=Lattice   (1547 words)

 The GCHART Procedure : HBAR, HBAR3D, VBAR, and VBAR3D Statements SAME specifies that the outline color of a bar or a bar segment or a legend value is the same as the interior pattern color. With subgroups, PERCENT displays the percent contribution of each subgroup to the midpoint value of the bar, based on frequency. The subgroup segments are proportional to the subgroup's contribution to the sum for the bar. www.okstate.edu /sas/v7/sashtml/books/graph/zart-bar.htm   (6171 words)

 Exercises 4 A subgroup of symmetries of a set S is called discrete if there is some distance m such that every element of S is either fixed or moved by at least m by every element of G. If G is a discrete subgroup of I(R), prove that G contains a translation T by a minimum distance and that every translation in G is a power of T. Prove that there are two infinite discrete subgroups of the group I(R), one generated by a translation (which we will call C www-groups.dcs.st-and.ac.uk /~john/geometry/Tutorials/T4.html   (245 words)

 [cryst] 2 Affine crystallographic groups Their infinity is relatively trivial in the sense that they have an abelian normal subgroup of finite index. A latticeequal subgroup has the same translation lattice as the parent, while a classequal subgroup has the same point group as the parent. In the classequal case a maximal subgroup always has prime-power index, whereas in the latticeequal case this is so only in dimensions up to 3. www-groups.dcs.st-and.ac.uk /gap/Manuals/pkg/cryst/htm/CHAP002.htm   (2609 words)

 Spiral Tilings We can summarize this conclusion by saying that the group G of symmetries of a spiral tiling must be a discrete subgroup of multiplication of nonzero complex numbers. If we choose a reference point 1 in some tile containing that point, the images of that point 1*G=G give one reference point in each tile, so the members of G are in one-to-one correspondence with the tiles of the spiral tiling. So if G is the set of symmetries of a spiral tiling, log(G) is a discrete subgroup of the additive complex numbers. www.ics.uci.edu /~eppstein/junkyard/spiraltile   (1762 words)

 Key Policy Letters Signed by the Education Secretary or Deputy Secretary This subgroup is not a discrete demographic subgroup per se, but rather a subgroup described by instructional needs that change as students gain English language proficiency. A State may include in the LEP subgroup a student who had previously been considered an LEP student during the past one or two years, to calculate AYP for schools, districts, and the State. When determining whether the LEP subgroup meets the State-defined minimum group size, these students are not required to be counted as LEP students. www.ed.gov /policy/gen/guid/secletter/040220.html   (1352 words)

 Preprints (via CobWeb/3.1 planetlab2.cs.unc.edu)   (Site not responding. Last check: 2007-10-09) Abstract: Let \$\Gamma\$ be a discrete group acting on a compact manifold \$X\$, let \$V\$ be a \$\Gamma\$-equivariant Hermitian vector bundle over \$X\$, and let \$D\$ be a first-order elliptic self-adjoint \$\Gamma\$-equivariant differential operator acting on sections of \$V\$. In addition, if equality occurs, then the leaf space of the foliation is isometric to the space of orbits of a discrete subgroup of O(q) acting on the standard q-sphere of constant curvature k. We also prove a result about bundle-like metrics: on any Riemannian foliation with bundle-like metric, there exists another bundle-like metric for which the mean curvature is basic and the basic Laplacian for the new metric is the same as that of the original metric. www.math.tcu.edu.cob-web.org:8888 /Preprints/Preprints.html   (725 words)

 Math Seminars: Tim Steger   (Site not responding. Last check: 2007-10-09) In joint work with Yehuda Shalom, we have proved Margulis' Normal Subgroup Theorem for any discrete subgroup \$\Gamma\$ of the automorphism group of a locally finite \$A_2\$-tilde building, \$B\$, provided that the quotient of \$B\$ by \$\Gamma\$ is compact. The conclusion of the Normal Subgroup Theorem for a center-free group like this one is that any normal subgroup is either trivial or of finite index. Curiously, in all these cases the full automorphism group of the building is itself discrete. www.math.ias.edu /abstract.php?event=12250   (369 words)

 Group theory question   (Site not responding. Last check: 2007-10-09) But in 3D, you could also take a discrete subgroup of the\nrotation group, say the octahedron. You could look at this subgroup as a\ndiscrete subgroup of the rotation group in 3D, but you could also write\nit as a matrix acting on the vertices. discrete subgroup of the rotation group in 3D, but you could also write www.physicsforums.com /showthread.php?t=91370   (957 words)

 Discrete Mathematics Several questions arise in considering discrete mathematics in the mathematics curriculum. If discrete mathematics covered in precalculus, then well prepared CS students might have to take precalculus for the discrete mathematics -- even though they are ready for a more rigorous course Grinnell's current curriculum moves discrete mathematics to the second semester of the sophomore year, with a calculus/linear algebra prerequisite. www.math.grin.edu /~walker/curriculum/discrete-math-iowa.html   (1455 words)

 Dave Witte Morris' papers in graph theory with Kevin Keating: On Hamilton cycles in Cayley graphs with cyclic commutator subgroup. This paper finds a hamiltonian cycle in every (undirected) Cayley graph on any group whose commutator subgroup is cyclic of prime-power order. Witte and K. Keating [6] showed that there is a Hamilton cycle in every connected Cayley graph on each group G whose commutator subgroup is cyclic of prime-power order. people.uleth.ca /~dave.morris/GraphTheory.shtml   (1225 words)

 DEPARTMENT OF MATHEMATICS   (Site not responding. Last check: 2007-10-09) Its subgroup SL(2,ZZ) of 2 ×2 invertible integral matrices is the group of ``symmetries'' of the group ZZ of lattice points in IR is naturally of interest in number theory; and properties of SL(2,ZZ) are accessed most easily by using the fact that it is a discrete subgroup of SL(2,IR). Thus the geometry of Riemann surfaces is tied up intimately with discrete subgroups of SL(2,IR). Raghunathan's book ``Discrete Subgroups of Lie Groups'', published by Springer Verlag, Germany, in 1972 is now a classic in the area. www.math.iitb.ac.in /news/raghu.html   (473 words)

 Analysis and homological algebra   (Site not responding. Last check: 2007-10-09) Further given a discrete (arithmetic, or S-arithmetic) subgroup we can consider the cohomology of the subgroup with coefficients in the globalizations of Harish-Chandra modules. If the discrete subgroup is cocompact, then one considers the maximal and minimal globalizations. In the case of finite covolume one is forced to consider the cohomology of the subgroup with coefficients in the distribution vector globalization. www.uni-math.gwdg.de /bunke/project2.html   (253 words)

 The Math Forum - Math Library - Order/Lattices   (Site not responding. Last check: 2007-10-09) A lattice is an infinite arrangement of points spaced with sufficient regularity that one can shift any point onto any other point by some symmetry of the arrangement. More formally, a lattice can be defined as a discrete subgroup of a finite-dimensional vector space (the subgroup is often required not to lie within any subspace of the vector space, which can be expressed formally by saying that the quotient of the space by the lattice is compact). It is a peer-reviewed publication devoted to rapid publication of innovative research which covers discrete mathematics and theoretical computer...more>> mathforum.org /library/topics/lattices   (1129 words)

 The Design of 2-Colour Wallpaper Patterns Using Methods Based on Chaotic Dynamics and Symmetry E(2) is not the identity, then T is either a reflection, or a rotation or a translation or a glide reflection. The user initially specifies the symmetry of the pattern (any discrete subgroup of E(2)), and the type of iteration (deterministic or iterated function system). At this point it is customary to divide the lattice subgroups into five classes (oblique, rectangular, centered rectangular (or rhombic), square and hexagonal) and then classify the wallpaper groups according to the point groups that can occur for each lattice class. arpam.free.fr /mfield.html   (5289 words)

 Lattice (via CobWeb/3.1 planetlab2.cs.unc.edu)   (Site not responding. Last check: 2007-10-09) The Hasse diagrams of these posets look (in some simple cases) like the aforementioned lattices. In another mathematical usage, a lattice is a discrete subgroup of R The elements of a Lattice are regularly spaced, reminiscent of the intersection points of a lath lattice. lattice.iqnaut.net.cob-web.org:8888   (319 words)

 Lattice (group) - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu)   (Site not responding. Last check: 2007-10-09) More generally, lattice models are studied in physics, often by the techniques of computational physics. A lattice is the symmetry group of discrete translational symmetry in n directions. More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that the quotient 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). en.wikipedia.org.cob-web.org:8888 /wiki/Lattice_(group)   (1468 words)

 VII. DISCRETE FOURIER TRANSFORMS If U = U(n), then pi_n is isomorphic to a discrete subgroup U_ pi of U, that acts in U considered as a group of automorphisms of itself, as the group of "basis" permutations at an element of u say PHI. It is important to keep in mind that diagonalization concept is invariant under the discrete group of basis permutations; any diagonalizing transformation may include, an arbitrary permutation of basis. Choosing our particular ordering, amounts to choosing a zero time with discrete time values equidistributed on the unit circle in the complex plane. graham.main.nc.us /~bhammel/FCCR/VII.html   (2706 words)

 Operators Commuting with a Discrete Subgroup of Translations (ResearchIndex) (via CobWeb/3.1 planetlab2.cs.unc.edu)   (Site not responding. Last check: 2007-10-09) Operators Commuting with a Discrete Subgroup of Translations (2005) Abstract: We study the structure of operators from the Schwartz space S (R the tempered distributions S # (R) that commute with a discrete subgroup of translations. The formalism leads to simple derivations of recent results about the frame operator of shift-invariant systems, Gabor and wavelet frames. citeseer.ist.psu.edu.cob-web.org:8888 /726243.html   (176 words)

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