Geometry of Lattice Angles, Polygons, and Cones
... [0, 1], while the values of integer sines are non-negative integers. Further if we consider angles defined by the rays {y = 0, x ≥ 0}, and {y = αx, x ≥ 0} for some α ≥ 1 then their tangents and lattice tangents coincide, nevertheless the arithmetics of angles is different: one can add two angles in ...
... [0, 1], while the values of integer sines are non-negative integers. Further if we consider angles defined by the rays {y = 0, x ≥ 0}, and {y = αx, x ≥ 0} for some α ≥ 1 then their tangents and lattice tangents coincide, nevertheless the arithmetics of angles is different: one can add two angles in ...
Locally normal subgroups of totally disconnected groups. Part II
... said to be h.j.i. if every non-trivial closed locally normal subgroup is open. • atomic type: |LN (G)| > 2 but LC(G) = {0, ∞}, there is a unique least element of LN (G) r {0}, the action of G on LN (G) is trivial and G is not abstractly simple. • non-principal filter type (abbreviated by NPF type): ...
... said to be h.j.i. if every non-trivial closed locally normal subgroup is open. • atomic type: |LN (G)| > 2 but LC(G) = {0, ∞}, there is a unique least element of LN (G) r {0}, the action of G on LN (G) is trivial and G is not abstractly simple. • non-principal filter type (abbreviated by NPF type): ...
Lecture 5
... τP and the spectral topology τS . How are these two topologies related to each other? We recall that τP is generated by B = {φ(a) − φ(b) : a, b ∈ L} and τS is generated by S = {φ(a) : a ∈ L}. It follows at once that τS is a subtopology of τP , that is τS ⊆ τP . But more is true. In fact, each elemen ...
... τP and the spectral topology τS . How are these two topologies related to each other? We recall that τP is generated by B = {φ(a) − φ(b) : a, b ∈ L} and τS is generated by S = {φ(a) : a ∈ L}. It follows at once that τS is a subtopology of τP , that is τS ⊆ τP . But more is true. In fact, each elemen ...
Scott - Modifying Minkowski`s theorem
... At the turn of the century, Minkowski published his famous “convex body” theorem which became the basis for the geometry of numbers. Suppose that A is a lattice in Euclidean n-space, E”, having determinant d(A). Now Minkowski’s theorem states that if K is a convex body which is symmetric about the o ...
... At the turn of the century, Minkowski published his famous “convex body” theorem which became the basis for the geometry of numbers. Suppose that A is a lattice in Euclidean n-space, E”, having determinant d(A). Now Minkowski’s theorem states that if K is a convex body which is symmetric about the o ...
Lattice Topologies with Interval Bases
... Let us first recall some basic notions from topology. A topological space is said to be zero-dimensional if it has a base of clopen sets, totally disconnected if its connected components are singletons, and totally separated if for any two distinct points there exists a clopen set containing one of ...
... Let us first recall some basic notions from topology. A topological space is said to be zero-dimensional if it has a base of clopen sets, totally disconnected if its connected components are singletons, and totally separated if for any two distinct points there exists a clopen set containing one of ...
A Categorical View on Algebraic Lattices in Formal Concept
... Definition 3.3. Consider a partially ordered set P. A subset I ⊆ P is an ideal if it is a directed lower set. The ideal completion Idl(P) is the collection of all ideals of P partially ordered via subset inclusion. Note that lower sets ↓x are always ideals—the principle ideals generated by the eleme ...
... Definition 3.3. Consider a partially ordered set P. A subset I ⊆ P is an ideal if it is a directed lower set. The ideal completion Idl(P) is the collection of all ideals of P partially ordered via subset inclusion. Note that lower sets ↓x are always ideals—the principle ideals generated by the eleme ...
A Categorical View on Algebraic Lattices in Formal
... Definition 3.4 Consider a partially ordered set P. A subset I ⊆ P is an ideal if it is a directed lower set. The ideal completion Idl(P) is the collection of all ideals of P partially ordered via subset inclusion. Note that lower sets ↓x are always ideals — the principle ideals generated by the elem ...
... Definition 3.4 Consider a partially ordered set P. A subset I ⊆ P is an ideal if it is a directed lower set. The ideal completion Idl(P) is the collection of all ideals of P partially ordered via subset inclusion. Note that lower sets ↓x are always ideals — the principle ideals generated by the elem ...
Part VI - TTU Physics
... such as <ψk(r)|O|ψk(r)>, used in calculating probabilities for transitions from one band to another when discussing optical & other properties (later in the course), can be shown by symmetry to vanish: So, some transitions are forbidden. This gives ...
... such as <ψk(r)|O|ψk(r)>, used in calculating probabilities for transitions from one band to another when discussing optical & other properties (later in the course), can be shown by symmetry to vanish: So, some transitions are forbidden. This gives ...
A TOPOLOGICAL CONSTRUCTION OF CANONICAL EXTENSIONS
... Is there a subcategory of Top that is dually equivalent to Lat? Here, Top is the category of topological spaces and continuous maps and Lat is the category of bounded lattices and lattice homomorphisms. To date, the question has been answered positively either by specializing Lat or by generalizing ...
... Is there a subcategory of Top that is dually equivalent to Lat? Here, Top is the category of topological spaces and continuous maps and Lat is the category of bounded lattices and lattice homomorphisms. To date, the question has been answered positively either by specializing Lat or by generalizing ...
ON MINIMAL, STRONGLY PROXIMAL ACTIONS OF LOCALLY
... curvature, but does not admit a locally symmetric one. Let H be a locally compact group which admits an embedding j : Γ → H with j(Γ) being a cocompact lattice in H. Then Hilbert-Smith conjecture implies that Γ is contained in a closed subgroup H0 ⊆ H of finite index in H, so that H0 is isomorphic t ...
... curvature, but does not admit a locally symmetric one. Let H be a locally compact group which admits an embedding j : Γ → H with j(Γ) being a cocompact lattice in H. Then Hilbert-Smith conjecture implies that Γ is contained in a closed subgroup H0 ⊆ H of finite index in H, so that H0 is isomorphic t ...
as a PDF
... 2. Wilcox lattices and A"6.A projective space is an incidence space such that there are at least three points on each line, and any two coplanar lines intersect. As is well known, the concepts of projective and affine space are coextensive; in particular, one can obtain an affine space by deleting a ...
... 2. Wilcox lattices and A"6.A projective space is an incidence space such that there are at least three points on each line, and any two coplanar lines intersect. As is well known, the concepts of projective and affine space are coextensive; in particular, one can obtain an affine space by deleting a ...
Group actions in symplectic geometry
... semisimple Lie groups of higher rank do not admit eective Hamiltonian actions on symplectically hyperbolic manifolds. An alternative proof of the Polterovich theorem has been given by Gal and K¦dra in [2]. The original proof by Polterovich is published in [6]. Examples of hyperbolic symplectic form ...
... semisimple Lie groups of higher rank do not admit eective Hamiltonian actions on symplectically hyperbolic manifolds. An alternative proof of the Polterovich theorem has been given by Gal and K¦dra in [2]. The original proof by Polterovich is published in [6]. Examples of hyperbolic symplectic form ...
Convexity conditions for non-locally convex lattices
... It turns out that most naturally arising function spaces are L-convex lattices (e.g. the Lp-spaces, Orlicz spaces, Lorentz spaces including the spaces L(p,c°) introduced above). However we shall give examples of non L-convex lattices. We shall show that X is L-convex if and only if X is lattice p-co ...
... It turns out that most naturally arising function spaces are L-convex lattices (e.g. the Lp-spaces, Orlicz spaces, Lorentz spaces including the spaces L(p,c°) introduced above). However we shall give examples of non L-convex lattices. We shall show that X is L-convex if and only if X is lattice p-co ...
ON TAMAGAWA NUMBERS 1. Adele geometry Let X be an
... becomes a locally compact space and is called the adele space of X. We identify XQ as a subset of XA by the diagonal imbedding. If X is quasi-affine, XQ is discrete in XA. The adele geometry is the study of the pair (XA, XQ), together with the imbedding above. Therefore, one has to define all concei ...
... becomes a locally compact space and is called the adele space of X. We identify XQ as a subset of XA by the diagonal imbedding. If X is quasi-affine, XQ is discrete in XA. The adele geometry is the study of the pair (XA, XQ), together with the imbedding above. Therefore, one has to define all concei ...
PDF
... 3. Rings. The lattice L(R) of ideals of a ring R is also complete, the join of a set of ideals of R is the ideal generated by elements in each of the ideals in the set. Any ideal I is the join of cyclic ideals generated by elements r ∈ I. So L(R) is algebraic. 4. Modules. The above two examples can ...
... 3. Rings. The lattice L(R) of ideals of a ring R is also complete, the join of a set of ideals of R is the ideal generated by elements in each of the ideals in the set. Any ideal I is the join of cyclic ideals generated by elements r ∈ I. So L(R) is algebraic. 4. Modules. The above two examples can ...
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results of Borel, Harish-Chandra, Mostow, Tamagawa, M. S. Raghunathan, Margulis, Zimmer obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of nilpotent Lie groups and semisimple algebraic groups over a local field. In the 1990s, Bass and Lubotzky initiated the study of tree lattices, which remains an active research area.