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We study the regularity of the higher secant varieties of
${{\mathbb{P}}^{1}}\times {{\mathbb{P}}^{n}}$
, embedded with divisors of type
$\text{(}d\text{,}\,\text{2)}$
and
$(d,3)$
. We produce, for the highest defective cases, a “determinantal” equation of the secant variety. As a corollary, we prove that the Veronese triple embedding of
${{\mathbb{P}}^{n}}$
is not Grassmann defective.
We associate with the Farey tessellation of the upper half-plane an
$\text{AF}$
algebra
$\mathfrak{A}$
encoding the “cutting sequences” that define vertical geodesics. The Effros–Shen
$\text{AF}$
algebras arise as quotients of
$\mathfrak{A}$
. Using the path algebra model for
$\text{AF}$
algebras we construct, for each
$\tau \,\,\in \,\,\left( 0 \right.,\left. \frac{1}{4} \right]$
, projections
$({{E}_{n}})$
in
$\mathfrak{A}$
such that
${{E}_{n}}{{E}_{n\pm 1}}E\le \tau {{E}_{n}}$
.
Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.
Let
$T$
be a sectorial operator. It is known that the existence of a bounded (suitably scaled)
${{H}^{\infty }}$
calculus for
$T$
, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra
$\Lambda _{\infty ,1}^{\alpha }({{\mathbb{R}}^{+}})$
. Such an algebra includes functions defined byMikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for
$T$
. In this paper, we use fractional derivation to analyse in detail the relationship between
$\Lambda _{\infty ,1}^{\alpha }$
and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.
We investigate the problem of deforming
$n$
-dimensional mod
$p$
Galois representations to characteristic zero. The existence of 2-dimensional deformations has been proven under certain conditions by allowing ramification at additional primes in order to annihilate a dual Selmer group. We use the same general methods to prove the existence of
$n$
-dimensional deformations.
We then examine under which conditions we may place restrictions on the shape of our deformations at
$p$
, with the goal of showing that under the correct conditions, the deformations may have locally geometric shape. We also use the existence of these deformations to prove the existence as Galois groups over
$\mathbb{Q}$
of certain infinite subgroups of
$p$
-adic general linear groups.
Hua’s fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of all
$n\times n$
hermitianmatrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization formaps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.
Let
$F$
be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field
${{F}_{0}}$
, and let
$G$
be a symplectic group over
$F$
or an unramified unitary group over
${{F}_{0}}$
. Following the methods of Bushnell–Kutzko for
$\text{GL}(N,F)$
, we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in
$G$
. In particular, we obtain an irreducible supercuspidal representation of
$G$
like
$\text{GL}(N,F)$
.
We give a complete classification of mixed Tsirelson spaces
$T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$
for finitely many pairs of given compact and hereditary families
${{F}_{i}}$
of finite sets of integers and
$0<{{\theta }_{i}}<1$
in terms of the Cantor–Bendixson indices of the families
${{F}_{i}}$
, and
${{\theta }_{i}}(1\le i\le r)$
. We prove that there are unique countable ordinal
$\alpha $
and
$0<\theta <1$
such that every block sequence of
$T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$
has a subsequence equivalent to a subsequence of the natural basis of the
$T({{S}_{{{\omega }^{\alpha }}}},\theta )$
. Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces.
We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree
$N$
is naturally associated to a subset of
${{\mathbb{R}}^{N-1}}$
. We calculate the volume of this set, prove the set is homeomorphic to the
$N-1$
ball and that its isometry group is isomorphic to the dihedral group of order
$2N$
.
We examine the fine structure of the short time behavior of solutions to various linear and nonlinear Schrödinger equations
${{u}_{t}}=i\Delta u+q(u)$
on
$I\times {{\mathbb{R}}^{n}}$
, with initial data
$u(0,x)=f(x)$
. Particular attention is paid to cases where
$f$
is piecewise smooth, with jump across an
$(n-1)$
-dimensional surface. We give detailed analyses of Gibbs-like phenomena and also focusing effects, including analogues of the Pinsky phenomenon. We give results for general
$n$
in the linear case. We also have detailed analyses for a broad class of nonlinear equations when
$n=1$
and 2, with emphasis on the analysis of the first order correction to the solution of the corresponding linear equation. This work complements estimates on the error in this approximation.