93 research outputs found
A new metric invariant for Banach spaces
We show that if the Szlenk index of a Banach space is larger than the
first infinite ordinal or if the Szlenk index of its dual is larger
than , then the tree of all finite sequences of integers equipped with
the hyperbolic distance metrically embeds into . We show that the converse
is true when is assumed to be reflexive. As an application, we exhibit new
classes of Banach spaces that are stable under coarse-Lipschitz embeddings and
therefore under uniform homeomorphisms.Comment: 22 page
Asymptotic geometry of Banach spaces and uniform quotient maps
Recently, Lima and Randrianarivony pointed out the role of the property
of Rolewicz in nonlinear quotient problems, and answered a
ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman.
In the present paper, we prove that the modulus of asymptotic uniform
smoothness of the range space of a uniform quotient map can be compared with
the modulus of of the domain space. We also provide conditions under
which this comparison can be improved
ON THE COARSE GEOMETRY OF JAMES SPACES
International audienceIn this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space J nor into its dual J *. It is a particular case of a more general result on the non equi-coarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic stucture. This allows us to exhibit a coarse invariant for Banach spaces, namely the non equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property Q of Kalton. We conclude with a remark on the coarse geometry of the James tree space J T and of its predual
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II
We continue the development, by reduction to a first order system for the
conormal gradient, of \textit{a priori} estimates and solvability for
boundary value problems of Dirichlet, regularity, Neumann type for divergence
form second order, complex, elliptic systems. We work here on the unit ball and
more generally its bi-Lipschitz images, assuming a Carleson condition as
introduced by Dahlberg which measures the discrepancy of the coefficients to
their boundary trace near the boundary. We sharpen our estimates by proving a
general result concerning \textit{a priori} almost everywhere non-tangential
convergence at the boundary. Also, compactness of the boundary yields more
solvability results using Fredholm theory. Comparison between classes of
solutions and uniqueness issues are discussed. As a consequence, we are able to
solve a long standing regularity problem for real equations, which may not be
true on the upper half-space, justifying \textit{a posteriori} a separate work
on bounded domains.Comment: 76 pages, new abstract and few typos corrected. The second author has
changed nam
Shift invariant preduals of ℓ<sub>1</sub>(ℤ)
The Banach space ℓ<sub>1</sub>(ℤ) admits many non-isomorphic preduals, for
example, C(K) for any compact countable space K, along with many more
exotic Banach spaces. In this paper, we impose an extra condition: the predual
must make the bilateral shift on ℓ<sub>1</sub>(ℤ) weak<sup>*</sup>-continuous. This is
equivalent to making the natural convolution multiplication on ℓ<sub>1</sub>(ℤ)
separately weak*-continuous and so turning ℓ<sub>1</sub>(ℤ) into a dual Banach
algebra. We call such preduals <i>shift-invariant</i>. It is known that the
only shift-invariant predual arising from the standard duality between C<sub>0</sub>(K)
(for countable locally compact K) and ℓ<sub>1</sub>(ℤ) is c<sub>0</sub>(ℤ). We provide
an explicit construction of an uncountable family of distinct preduals which do
make the bilateral shift weak<sup>*</sup>-continuous. Using Szlenk index arguments, we
show that merely as Banach spaces, these are all isomorphic to c<sub>0</sub>. We then
build some theory to study such preduals, showing that they arise from certain
semigroup compactifications of ℤ. This allows us to produce a large number
of other examples, including non-isometric preduals, and preduals which are not
Banach space isomorphic to c<sub>0</sub>
Profiling of spatial metabolite distributions in wheat leaves under normal and nitrate limiting conditions
The control and interaction between nitrogen and carbon assimilatory pathways is essential in both photosynthetic and non-photosynthetic tissue in order to support metabolic processes without compromising growth. Physiological differences between the basal and mature region of wheat (Triticum aestivum) primary leaves confirmed that there was a change from heterotrophic to autotrophic metabolism. Fourier Transform Infrared (FT-IR) spectroscopy confirmed the suitability and phenotypic reproducibility of the leaf growth conditions. Principal Component–Discriminant Function Analysis (PC–DFA) revealed distinct clustering between base, and tip sections of the developing wheat leaf, and from plants grown in the presence or absence of nitrate. Gas Chromatography-Time of Flight/Mass Spectrometry (GC-TOF/MS) combined with multivariate and univariate analyses, and Bayesian network (BN) analysis, distinguished different tissues and confirmed the physiological switch from high rates of respiration to photosynthesis along the leaf. The operation of nitrogen metabolism impacted on the levels and distribution of amino acids, organic acids and carbohydrates within the wheat leaf. In plants grown in the presence of nitrate there was reduced levels of a number of sugar metabolites in the leaf base and an increase in maltose levels, possibly reflecting an increase in starch turnover. The value of using this combined metabolomics analysis for further functional investigations in the future are discussed
- …