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This book has a long prehistory. It began in 1989 when the author undertook her first excavations at Baitai cemetery. The analysis and publication of the material goes some way to make up for the damage done by the archaeologist who destroys the burials (s)he investigates. Studying the grave sets of an unknown deceased person is a way of restoring their individuality and revealing some of the characteristics of the local community, in order to compensate for the anonymity by which burials are numbered. The main aim of this book is to show the place of the Baitai community in the Late Roman Period against the background of the culture of the entire West Balt area and the cultural heritage of the various Baltic Sea regions. If the numbers of the Baitai graves are remembered and cited in the future as examples of particular units of grave goods that reflect the local traditions in the Lithuanian coastland and illustrate the impact of inter-regional communication, then perhaps the inevitable damage done by the archaeologist may be compensated for by her efforts to revive the people of Baitai almost 1,600 years later.

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In this article, we re-examine some of the classical pointwise multiplication theorems in Sobolev–Slobodeckij spaces, in part motivated by a simple counter-example that illustrates how certain multiplication theorems fail in Sobolev–Slobodeckij spaces when a bounded domain is replaced by Rn. We identify the source of the failure, and examine why the same failure is not encountered in Bessel potential spaces. To analyze the situation, we begin with a survey of the classical multiplication results stated and proved in the 1977 article of Zolesio, and carefully distinguish between the case of spaces defined on the all of Rn and spaces defined on a bounded domain (with e.g. a Lipschitz boundary). However, the survey we give has a few new wrinkles; the proofs we include are based almost exclusively on interpolation theory rather than Littlewood–Paley theory and Besov spaces, and some of the results we give and their proofs, including the results for negative exponents, do not appear in the literature in this form. We also include a particularly important variation of one of the multiplication theorems that is relevant to the study of nonlinear PDE systems arising in general relativity and other areas. The conditions for multiplication to be continuous in the case of Sobolev–Slobodeckij spaces are somewhat subtle and intertwined, and as a result, the multiplication theorems of Zolesio in 1977 have been cited (more than once) in the standard literature in slightly more generality than what is actually proved by Zolesio, and in cases that allow for construction of counter-examples such as the one included here.

Let X⊂Pr be an integral and non-degenerate variety. Set n:=dim(X). We prove that if the (k+n−1)-secant variety of X has (the expected) dimension (k+n−1)(n+1)−1<r and X is not uniruled by lines, then X is not k-weakly defective and hence the k-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique S⊂X with ♯(S)=k. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures G1,d. If X is the Segre embedding of a multiprojective space we prove identifiability for the k-secant variety (assuming that the (k+n−1)-secant variety has dimension (k+n−1)(n+1)−1<r, this is a known result in many cases), beating several bounds on the identifiability of tensors.

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant between the classical LS cocategory and the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck’s Torus Theorem, and obtain corresponding results for p-compact groups and p-Noetherian groups.

We discuss the problem of deciding when a metrisable topological group G has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on G, that we characterise intrinsically in terms of a linear growth condition on powers of group elements.

We study analytic integrable deformations of the germ of a holomorphic foliation given by df=0 at the origin 0∈Cn,n≥3. We consider the case where f is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, outside of a dimension ≤n−3 analytic subset Y⊂X, the analytic hypersurface Xf:(f=0) has only normal crossings singularities. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of integrable germs writing as ω=df+fη where f is quasi-homogeneous. Under the same hypotheses for Xf:(f=0) we prove that ω also admits a holomorphic first integral. Finally, we conclude that an integrable germ ω=adf+fη admits a holomorphic first integral provided that: (i) Xf:(f=0) is irreducible with an isolated singularity at the origin 0∈Cn,n≥3; (ii) the algebraic multiplicities of ω and f at the origin satisfy ν(ω)=ν(df). In the case of an isolated singularity for (f=0) the writing ω=adf+fη is always assured so that we conclude the existence of a holomorphic first integral. Some questions related to Relative Cohomology are naturally considered and not all of them answered.

We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion T such that the regular module has a special T-preenvelope. In particular, every torsion-enveloping class in Mod-R are of the form Gen(T) for a minimal silting module T. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T), where T is a cosilting module.

It is known that if the special automorphism group SAut(X) of a quasiaffine variety X of dimension at least 2 acts transitively on X, then this action is infinitely transitive. In this paper we question whether this is the only possibility for the automorphism group Aut(X) to act infinitely transitively on X. We show that this is the case, provided X admits a nontrivial Ga or Gm-action. Moreover, 2-transitivity of the automorphism group implies infinite transitivity.

Hölder Regularity for Degenerate Parabolic Obstacle Problems