James Spaces Over Partially Ordered Index-Sets

We study the connections between the linear topological structure of James spaces generated by a partially ordered index-set and the set-theoretic properties of that set.     

Scientific discovery with the James Webb Space Telescope

For the past 400 years, astronomers have sought to observe and interpret the Universe by building more powerful telescopes. These incredible instruments extend the capabilities of one of our most important senses, sight, towards new limits such as increased sensitivity and resolution, new dimensions such as exploration of wavelengths across the full electromagnetic spectrum, new information content such as analysis through spectroscopy, and new cadences such as rapid time-series views of the variable sky. The results from these investments, from small to large telescopes on the ground and in space, have completely transformed our understanding of the Universe; including the discovery that Earth is not the centre of the Universe, that the Milky Way is one among many galaxies in the Universe, that relic cosmic background radiation fills all space in the early Universe, that that the expansion rate of the Universe is accelerating, that exoplanets are common around stars, that gravitational waves exist, and much more. For modern astronomical research, the next wave of breakthroughs in fields ranging over planetary, stellar, galactic, and extragalactic science motivate a general-purpose observatory that is optimised at near- and mid-infrared wavelengths, and that has much greater sensitivity, resolution, and spectroscopic multiplexing than all previous telescopes. This scientific vision, from measuring the composition of rocky worlds in the nearby Milky Way galaxy to finding the first sources of light in the Universe to other topics at the forefront of modern astrophysics, motivates the state-of-the-art James Webb Space Telescope (Webb). In this review paper, I summarise the design and technical capabilities of Webb and the scientific opportunities that it enables.     

On a generalized James constant

We introduce a generalized James constant J(a,X) for a Banach space X, and prove that, if J(a,X)<(3+a)/2 for some a∈[0,1], then X has uniform normal structure. The class of spaces X with J(1,X)<2 is proved to contain all u-spaces and their generalizations. For the James constant J(X) itself, we show that X has uniform normal structure provided that , improving the previous known upper bound at 3/2. Finally, we establish the stability of uniform normal structure of Banach spaces.     

The Drawing or Why History Is Not Mathematics

The mystery of whether or not Werner Heisenberg gave Niels Bohr a drawing that Bohr came to believe was that of a German nuclear weapon, is discussed and resolved.     

EPR study of oriented photoexcited triplets of porphyrins and chlorophylls in a liquid crystal

An EPR study of the photoexcited triplet state of porphyrins, Chl a and the covalently linked dimer of Zn pyrochlorophyllide a, in a liquid crystal, is reported The metalloporphyrins interact to form excimers The low intramolecular energy transfer rate in the dimer (? 10⁶s^?1) suggests that the unfolded form predominates.     

Banach空间中的Jordan-von Neumann型常数和正规结构

首先利用James常数J(X)给出了Jordan-von Neumann型常数C_(-∞)(X)的一个估计式,然后由此式说明确实存在C_(-∞)(X)C_Z(X)的例子,最后利用C_(-∞)(X)和Benavides常数R(a,X)的关系,得到了空间具有正规结构的充分条件,并通过一些例子说明我们的结论严格推广了一些文献中的结果.     

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

On the norm attainment set of a bounded linear operator

In this paper we explore the properties of a bounded linear operator defined on a Banach space, in light of operator norm attainment. Using Birkhoff–James orthogonality techniques, we give a necessary condition for a nonzero bounded linear operator to attain norm at a particular point of the unit sphere. We prove four corollaries to establish the importance of our study. As part of our exploration, we also obtain a characterization of smooth Banach spaces in terms of operator norm attainment and Birkhoff–James orthogonality. Restricting our attention to $l_p^2(p\in \mathbb{N}?\{1\})$ spaces, we obtain an upper bound for the number of points at which any linear operator, which is not a scalar multiple of an isometry, may attain norm.