Hard X-rays have exceptional properties that are useful in the chemical, elemental and structure analysis of matter. Although single-nanometre resolutions in various hard-X-ray analytical methods are theoretically possible with a focused hard-X-ray beam, fabrication of the focusing optics remains the main hurdle. Aberrations owing to imperfections in the optical system degrade the quality of the focused beam 1 . Here, we describe an in situ wavefront-correction approach to overcome this and demonstrate an X-ray beam focused in one direction to a width of 7 nm at 20 keV. We achieved focal spot improvement of the X-ray nanobeam produced by a laterally graded multilayer mirror 2 . A grazing-incidence deformable mirror 3 was used to restore the wavefront shape. Using this system, ideal focusing conditions are achievable even if hard-X-ray focusing elements do not achieve sufficient performance. It is believed that this will ultimately lead to single-nanometre spatial resolution in X-ray analytical methods.Synchrotron radiation facilities produce high-quality light with wavelengths ranging from the infrared to hard-X-ray regions. The use of hard X-rays with energies higher than several kiloelectronvolts in conjunction with analysis methods such as X-ray diffraction, X-ray fluorescence, X-ray absorption and X-ray photoelectron spectroscopy offers unique advantages for the investigation of the structure, elemental distribution and chemical bonding state of advanced materials and biological samples. In these analytical methods, the resolution, signal strength and contrast must be as high as possible. In this regard, the development of a hard-X-ray focusing device is important for meeting these demands. To focus light, it is necessary to take advantage of its interactions with matter, such as diffraction, reflection and refraction. There are a variety of hard-X-ray focusing optical systems such as mirrors 4 , zone plates 5 , refractive lenses 6 and multilayer Laue lenses 7 . The minimum achievable spot size has been theoretically investigated by many researchers 8-10 , and it has been concluded that sizes below 10 nm are feasible with kiloelectronvolt X-rays. That is, hard-X-ray analytical techniques have the potential for single-nanometre spatial resolution.However, in such discussions, the imperfections of the focusing elements have not been entirely considered. Rayleigh's quarterwavelength rule 1 states that if the wavefront aberration exceeds a quarter of a wavelength, the quality of the retinal image will be significantly impaired. This rule is also applicable to simple light-focusing optical systems. The wavefront error of the focused beam distorts the shape of the intensity profile on the focal plane and spreads the beam. The short wavelength of X-rays demands unprecedented accuracy in the manufacturing of the LETTERS 1.43 nm (r.m.s.) over a 500-nm-square area, which was directly confirmed by atomic force microscopy. A phase shift occurred at the boundary between the X-rays propagating inside and outside the phase ...

Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called the multiplicative preprojective algebra. In this paper we study the moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there exists a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety (which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is star-shaped, the multiplicative quiver variety parametrizes Simpson's (poly)stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight and associated graded local system around each puncture.

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne-Simpson problem.

Dettweiler-Reiter's additive description of Katz's middle convolution can be interpreted in terms of the Harnad duality. Using this interpretation together with Mumford's geometric invariant theory, we generalize the additive middle convolution to have a multi-parameter and act on systems of linear ordinary differential equations with irregular singularities. We show that the generalized operation holds basic properties of the original one, and additionally, show that Katz's algorithm involving our generalization works well in generic case.