A Hausdorff topological space X is called superconnected (resp. coregular) if for any nonempty open sets U1,…Un⊆X, the intersection of their closures U‾1∩…∩U‾n is not empty (resp. the complement ...X∖(U‾1∩…∩U‾n) is a regular topological space). A canonical example of a coregular superconnected space is the projective space QP∞ of the topological vector space Q<ω={(xn)n∈ω∈Qω:|{n∈ω:xn≠0}|<ω} over the field of rationals Q. The space QP∞ is the quotient space of Q<ω∖{0}ω by the equivalence relation x∼y iff Q⋅x=Q⋅y.
We prove that every countable second-countable coregular space is homeomorphic to a subspace of QP∞, and a topological space X is homeomorphic to QP∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets (Xn)n∈ω such that (i) X0=X, ⋂n∈ωXn=∅, (ii) for every n∈ω and a nonempty relatively open set U⊆Xn the closure U‾ contains some set Xm, and (iii) for every n∈ω the complement X∖Xn is a regular topological space. Using this topological characterization of QP∞ we find topological copies of the space QP∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.
The present paper is devoted to the study of biharmonic submanifolds in quaternionic space forms. After giving the biharmonicity conditions for submanifolds in these spaces, we study different ...particular cases for which we obtain curvature estimates. We study biharmonic submanifolds with parallel mean curvature and biharmonic submanifolds which are pseudo-umbilical in the quaternionic projective space. We find the relation between the bitension field of the inclusion of a submanifold in the n-dimensional quaternionic projective space and the bitension field of the inclusion of the corresponding Hopf-tube in the unit sphere of dimension 4n+3.
Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than 1 are derived. An application in coding theory is ...illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of Fqn) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed n and q.
We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations of C2, using ordinary cohomology graded on representations of the fundamental ...groupoid, with coefficients in the Burnside ring Mackey functor. This extension of the RO(C2)-graded theory allows for the definition of Euler classes, which are used as generators of the cohomology of the projective spaces. As an application, we give an equivariant version of Bézout's theorem.
We construct an explicit map from a generic minimal δ(2)-ideal Lagrangian submanifold of Cn to the quaternionic projective space HPn−1, whose image is either a point or a minimal totally complex ...surface. A stronger result is obtained for n=3, since the above mentioned map then provides a one-to-one correspondence between minimal δ(2)-ideal Lagrangian submanifolds of C3 and minimal totally complex surfaces in HP2 which are moreover anti-symmetric. Finally, we also show that there is a one-to-one correspondence between such surfaces in HP2 and minimal Lagrangian surfaces in CP2.
We calculate the ordinary C2-cohomology, with Burnside ring coefficients, of CPC2∞=BC2U(1), the complex projective space, a model for the classifying space for C2-equivariant complex line bundles. ...The RO(C2)-graded Bredon ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators. These generators include the Euler class of the tautological bundle, which lies outside of the RO(C2)-graded theory.
In classical projective algebraic geometry, ℙ
n
was seen mainly as a linear subspace. The modern setting has produced in the last 40 years several remarkable abstract characterizations of projective ...space. We survey some interaction between these two points of view.
We denote by
$ \Delta _\nu $
Δ
ν
the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective ...n-space, this operator has a discrete spectrum consisting on eigenvalues
$ \beta _m, \ m\in \mathbb {Z}_+ $
β
m
,
m
∈
Z
+
. For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of
$ \Delta _\nu $
Δ
ν
. Using a suitable polynomial decomposition of the multiplicity of each
$ \beta _m $
β
m
, we write down a trace formula for the heat operator associated with
$ \Delta _\nu $
Δ
ν
in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as
$ t\searrow 0^+ $
t
↘
0
+
by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with
$ \Delta _\nu $
Δ
ν
.
There are two known families of maximum scattered Fq-linear sets in PG(1,qt): the linear sets of pseudoregulus type and for t≥4 the scattered linear sets found by Lunardon and Polverino. For t=4 we ...show that these are the only maximum scattered Fq-linear sets and we describe the orbits of these linear sets under the groups PGL(2,q4) and PΓL(2,q4).
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