A subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.

There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.

Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.

Other partial results and more open problems are presented.

We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over $\mathbb{Q}$ of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use $R=T$ theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from $\text{GSp}(4)$ to $U(4)$. In that case we obtain a corollary for abelian surfaces.

Practical use of Resistive Random Access Memory (ReRAM) depends on thorough understanding of the resistive switching (RS) mechanism in polycrystalline metal oxide films. Based on experimental and theoretical results of NiO based ReRAM, we have proposed a grain surface tiling model, in which grain surfaces (i.e. grain boundaries) are composed by insulating and conductive micro surface structures. This paper reports the adequacy of our model to the NiO based ReRAM and universality of surface electronic properties in metal oxides of NiO, CoO and MgO. Experimental results of RS operating modes suggest that the resistance changes in the grain boundaries, supporting our model. First-principles calculation results suggest that our model can be adopted to other metal oxide materials and the RS from a low resistance to a high resistance can be caused at 1000 K, which agrees with previous experimental reports.

Let $F/\mathbf{Q}$ be a totally real field and $K/F$ a complex multiplication (CM) quadratic extension. Let $f$ be a cuspidal Hilbert modular new form over $F$. Let ${\it\lambda}$ be a Hecke character over $K$ such that the Rankin–Selberg convolution $f$ with the ${\it\theta}$-series associated with ${\it\lambda}$ is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg $L$-values $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$, as ${\it\chi}$ varies over the class group characters of $K$. Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters ${\it\chi}$ such that $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$ increases with the absolute value of the discriminant of $K$. We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg $L$-values. The Waldspurger formula plays an underlying role.

Let $p\geqslant 5$ be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup ${\rm\Gamma}(L)$ of $\text{SL}_{2}(\mathbb{Z}_{p}[[T]])$ for a principal ideal $(L)\neq 0$ of $\mathbb{Z}_{p}[[T]]$ for the canonical ‘weight’ variable $t=1+T$. If $L\notin {\rm\Lambda}^{\times }$, the power series $L$ is proven to be a factor of the Kubota–Leopoldt $p$-adic $L$-function or of the square of the anticyclotomic Katz $p$-adic $L$-function or a power of $(t^{p^{m}}-1)$.

This paper examines two tenth-century woodblock prints from Dunhuang acquired by Sir Aurel Stein. They are registered under the numbers Ch.00151 and Ch.00152 at the British Museum. The first, Ch.00151, is dedicated to Avalokiteśvara and the second, Ch.00152, to Amitābha, both with magical formulas written in Siddham characters in square or circular fashion around the central image of the respective deity. Editions, translations and identifications of the texts are presented along with a study of the Chinese inscriptions on the side of the xylographs. It is shown that these two objects were apparently produced to serve as amulets.

The Antarctic Plateau provides the best terrestrial sites for infrared (IR) and submillimetre (sub-mm) astronomy. In this paper we examine the relative importance of temperature, aerosol content and precipitable water vapour to determine which parameters have the greatest influence on atmospheric transmission and sky brightness. We use the atmospheric modelling program MODTRAN to model the observed sky spectrum at the South Pole from the near-IR to the sub-mm. We find that temperature and aerosol content determine the quality of near-IR observing conditions, aerosol content is the determining factor in the mid-IR up to 20 μm, while at longer wavelengths, including the sub-mm, it is the water vapour content that matters. Finding a location where aerosol levels are minimised is a key constraint in determining the optimum site on the Antarctic Plateau for an IR observatory.

The cold, dry, and stable air above the summits of the Antarctic plateau provides the best ground-based observing conditions from optical to sub-millimetre wavelengths to be found on the Earth. Pathfinder for an International Large Optical Telescope (PILOT) is a proposed 2 m telescope, to be built at Dome C in Antarctica, able to exploit these conditions for conducting astronomy at optical and infrared wavelengths. While PILOT is intended as a pathfinder towards the construction of future grand-design facilities, it will also be able to undertake a range of fundamental science investigations in its own right. This paper provides the performance specifications for PILOT, including its instrumentation. It then describes the kinds of projects that it could best conduct. These range from planetary science to the search for other solar systems, from star formation within the Galaxy to the star formation history of the Universe, and from gravitational lensing caused by exo-planets to that produced by the cosmic web of dark matter. PILOT would be particularly powerful for wide-field imaging at infrared wavelengths, achieving near diffraction-limited performance with simple tip–tilt wavefront correction. PILOT would also be capable of near diffraction-limited performance in the optical wavebands, as well be able to open new wavebands for regular ground-based observation, in the mid-IR from 17 to 40 μm and in the sub-millimetre at 200 μm.

We prove vanishing of the μ-invariant of the p-adic Katz L-function in N. M. Katz [p-adic L-functions for CM fields, Invent. Math. 49 (1978), 199–297].

Embryonic loss occurring in the first week of the incubation is termed early embryonic mortality. Environmental, technological and genetic factors can be responsible forits appearance. The examination of genetic background of embryonic death is justified when the breeding and hatching technological parameters are optimal. Disposition of early embryonic mortality can be inherited and it can accumulate in a stock, which will finally deteriorate its fertility indices. Although the heritability value of reproductiveness is poor, h2=0.1−0.2, even so, during the selection still should be considered. This paper offers a short review on the special features of early embryonic development in poultry and examination methods forpossible genetic background of early embryonic abnormalities and mortality. The application of cytogenetical and embryological methods may be of major relevance in the improvement of the reproduction traits of poultry species.

Hydrothermal lead zirconate titanate (Pb(Zr,Ti)O3: PZT) films with a thick of about 10 μm on both a Ti foil and a Ti layer sputtered stainless steel (SS) foil were obtained. Both PZT films showed same morphology and crystalinity. After repeating the hydrothermal process, the films were formed to cantilever shape. A tip displacement measurement of the cantilevers was carried out in order to estimate transverse piezoelectric constant (d31). The obtained d31 value of the PZT films on Ti sputtered SS foils is -29 pm/V, which is larger than that on Ti foil (-24 pm/V). Amorphous layer consist of Pb and Ti was observed at the interfaces between the both PZT film and the Ti metals. The thickness of the layer for PZT film on Ti foil (350 nm) is thicker than that for PZT film on SS foil (200 nm). The relative permittivity of the layer is estimated to be small compared with that of PZT. Accordingly, the effective electric field for PZT layer is decreased. The deference of d31 values between the obtained PZT films is thought to be attributed to the thickness of amorphous layer.

The Automated Patrol Telescope, operated by the University of New South Wales, has been undertaking a search for extrasolar planets using the transit method. We present lightcurves from two recent promising candidates; spectroscopic follow-up using the ANU 2.3m telescope shows that the companions are probably low mass stars rather than planets, although more data will be needed to be certain. Additionally, we outline future improvements to our transit search: a hardware upgrade scheduled for 2006, and the addition of a robust trend-filtering algorithm to the data reduction software.