2,280 research outputs found
A product structure on Generating Family Cohomology for Legendrian Submanifolds
One way to obtain invariants of some Legendrian submanifolds in 1-jet spaces
, equipped with the standard contact structure, is through the Morse
theoretic technique of generating families. This paper extends the invariant of
generating family cohomology by giving it a product . To define the
product, moduli spaces of flow trees are constructed and shown to have the
structure of a smooth manifold with corners. These spaces consist of
intersecting half-infinite gradient trajectories of functions whose critical
points correspond to Reeb chords of the Legendrian. This paper lays the
foundation for an algebra which will show, in particular, that
is associative and thus gives generating family cohomology a ring
structure.Comment: 50 pages, 4 figures, minor change
Group-theoretic compactification of Bruhat-Tits buildings
Let GF denote the rational points of a semisimple group G over a
non-archimedean local field F, with Bruhat-Tits building X. This paper contains
five main results. We prove a convergence theorem for sequences of parahoric
subgroups of GF in the Chabauty topology, which enables to compactify the
vertices of X. We obtain a structure theorem showing that the Bruhat-Tits
buildings of the Levi factors all lie in the boundary of the compactification.
Then we obtain an identification theorem with the polyhedral compactification
(previously defined in analogy with the case of symmetric spaces). We finally
prove two parametrization theorems extending the BruhatTits dictionary between
maximal compact subgroups and vertices of X: one is about Zariski connected
amenable subgroups, and the other is about subgroups with distal adjoint
action
On -covering designs
A -covering design , , is a collection
of -spaces of such that every
-space of is contained in at least one element of
. Let denote the minimum number of
-spaces in a -covering design . In this paper
improved upper bounds on , ,
, , and , , are presented. The results are achieved by constructing the related
-covering designs
Pairwise transitive 2-designs
We classify the pairwise transitive 2-designs, that is, 2-designs such that a
group of automorphisms is transitive on the following five sets of ordered
pairs: point-pairs, incident point-block pairs, non-incident point-block pairs,
intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall
into two classes: the symmetric ones and the quasisymmetric ones. The symmetric
examples include the symmetric designs from projective geometry, the 11-point
biplane, the Higman-Sims design, and designs of points and quadratic forms on
symplectic spaces. The quasisymmetric examples arise from affine geometry and
the point-line geometry of projective spaces, as well as several sporadic
examples.Comment: 28 pages, updated after review proces
Equidistant Codes in the Grassmannian
Equidistant codes over vector spaces are considered. For -dimensional
subspaces over a large vector space the largest code is always a sunflower. We
present several simple constructions for such codes which might produce the
largest non-sunflower codes. A novel construction, based on the Pl\"{u}cker
embedding, for 1-intersecting codes of -dimensional subspaces over \F_q^n,
, where the code size is is
presented. Finally, we present a related construction which generates
equidistant constant rank codes with matrices of size
over \F_q, rank , and rank distance .Comment: 16 page
LDPC codes from Singer cycles
The main goal of coding theory is to devise efficient systems to exploit the
full capacity of a communication channel, thus achieving an arbitrarily small
error probability. Low Density Parity Check (LDPC) codes are a family of block
codes--characterised by admitting a sparse parity check matrix--with good
correction capabilities. In the present paper the orbits of subspaces of a
finite projective space under the action of a Singer cycle are investigated.Comment: 11 Page
Nonintersecting Subspaces Based on Finite Alphabets
Two subspaces of a vector space are here called ``nonintersecting'' if they
meet only in the zero vector. The following problem arises in the design of
noncoherent multiple-antenna communications systems. How many pairwise
nonintersecting M_t-dimensional subspaces of an m-dimensional vector space V
over a field F can be found, if the generator matrices for the subspaces may
contain only symbols from a given finite alphabet A subseteq F? The most
important case is when F is the field of complex numbers C; then M_t is the
number of antennas. If A = F = GF(q) it is shown that the number of
nonintersecting subspaces is at most (q^m-1)/(q^{M_t}-1), and that this bound
can be attained if and only if m is divisible by M_t. Furthermore these
subspaces remain nonintersecting when ``lifted'' to the complex field. Thus the
finite field case is essentially completely solved. In the case when F = C only
the case M_t=2 is considered. It is shown that if A is a PSK-configuration,
consisting of the 2^r complex roots of unity, the number of nonintersecting
planes is at least 2^{r(m-2)} and at most 2^{r(m-1)-1} (the lower bound may in
fact be the best that can be achieved).Comment: 14 page
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