Given a filtered ring, we give bounds of its multiplicity in terms of the data of the tangent cone using the technique of the filtered blowing-up. Applying it to each simple K3 singularity of multiplicity two, we find a good coordinate where the Newton boundary of the defining equation contains the point (1, 1, 1, 1) ∈ R⁴. In the course of the proof, we classify simple K3 singularities of multiplicity two into 48 weight types. Furthermore we prove that the weight type of the singularity stays the same under arbitrary one-parameter (FG)-deformations.

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It is shown that for a positive definite continuous function f(x) on \mathbb{R}ⁿ the followings are equivalent:
(i)   f(x) is quasaianalytic in some neighborhood of the origin.
(ii)   f(x) can be expressed as an integral f(x)=∫_{\mathbb{R}ⁿ} e^ixξ dμ (ξ) for some positive Radon measure μ on \mathbb{R}ⁿ such that ∫ exp M (L|ξ|)d μ (ξ) is finite for some L>0 where the function M(t) is a weight function corresponding to the quasaianalyticity.
(iii)   f(x) is quasaianalytic everywhere in \mathbb{R}ⁿ.
Moreover, an analogue for the analyticity is also given as a corollary.

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Ellipticity of (pseudo-differential) operators on a manifold with geometric singularities gives rise to a hierarchy of symbols, associated with the system of lower-dimensional strata of the configuration. Classical examples are boundary value problems with interior and boundary symbols (the latter ones describe Shapiro-Lopatinskij ellipticity of boundary conditions), or operators on manifolds with conical singularities with interior and conormal symbols. Ellipticity on a manifold with smooth edges may be investigated by a suitable combination of ideas from boundary value problems and cone calculus. The present article studies another typical case, namely ellipticity on a manifold that has edges with conical singularities. Locally, we may talk about cones, where the base is a manifold with smooth edges. Parametrices and iterated asymptotics of solutions to elliptic equations are determined by a three-component symbolic hierarchy, with interior, edge and conormal symbols. We construct an operator algebra of 2 × 2-block matrices, where the upper left corners contain the interior operators, together with so-called Green and Mellin operators (caused by analogues of Green's function in boundary value problems as well as by asymptotic phenomena), while the other entries contain trace and potential conditions with respect to the edge and pseudo-differential operators on the edge itself that are of Fuchs type with respect to the conical points. The calculus is organized in an iterative way and can be viewed as a starting point for constructing similar operator algebras with asymptotics for higher polyhedral singularities.

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We construct an integral PBW basis and an integral crystal basis of the quantum affine algebra of type A₂⁽²⁾.

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For developing a theory of *-representations of *-algebras \mathcal{A} on (indefinite) inner product spaces E, (·, ·), the classes of pseudo-Krein and pre-Krein spaces which admit a Hilbert space structure and exactly one maximal Hilbert space structure, respectively, are introduced. Inner characterizations for the topological structure of E, (·, ·) to be a pseudo-Krein and pre-Krein space, respectively, are given. The results are illustrated by some examples.

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