225 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    GG, Γ\Gamma, G/ΓG/\Gamma

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    The purpose of this book is to provide an introduction to one of the fundamental tools of abstract harmonic analysis, namely the Selberg trace formula

    Classifying ^*-homomorphisms I: Unital simple nuclear CC^*-algebras

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    We classify the unital embeddings of a unital separable nuclear CC^*-algebra satisfying the universal coefficient theorem into a unital simple separable nuclear CC^*-algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear CC^*-algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of KK-theory and traces.Comment: 130 page

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Quasifinite fields of prescribed characteristic and Diophantine dimension

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    Let P\mathbb{P} be the set of prime numbers, P\overline {\mathbb{P}} the union P{0}\mathbb{P} \cup \{0\}, and for any field EE, let char(E)(E) be its characteristic, ddim(E)(E) the Diophantine dimension of EE, GE\mathcal{G}_{E} the absolute Galois group of EE, and cd(GE)(\mathcal{G}_{E}) the Galois cohomological dimension GE\mathcal{G}_{E}. The research presented in this paper is motivated by the open problem of whether cd(GE)ddim(E)(\mathcal{G}_{E}) \le {\rm ddim}(E). It proves the existence of quasifinite fields Φq ⁣:qP\Phi _{q}\colon q \in \mathbb{P}, with ddim(Φq)(\Phi _{q}) infinity and char(Φq)=q(\Phi _{q}) = q, for each qq. It shows that for any integer m>0m > 0 and qPq \in \overline {\mathbb{P}}, there is a quasifinite field Φm,q\Phi _{m,q} such that char(Φm,q)=q(\Phi _{m,q}) = q and ddim(Φm,q)=m(\Phi _{m,q}) = m. This is used for proving that for any qPq \in \overline {\mathbb{P}} and each pair kk, (N{0,})\ell \in (\mathbb{N} \cup \{0, \infty \}) satisfying kk \le \ell , there exists a field Ek,;qE _{k, \ell ; q} with char(Ek,;q)=q(E _{k, \ell ; q}) = q, ddim(Ek,;q)=(E _{k, \ell ; q}) = \ell and cd(GEk,;q)=k(\mathcal{G}_{E_{k, \ell ; q}}) = k. Finally, we show that the field Ek,;qE _{k, \ell ; q} can be chosen to be perfect unless k=0k = 0 \neq \ell .Comment: 19 pages, LaTeX: Incorporates referee's suggestions; references updated; to appear in Analele Sci. ale Univ. Ovidius, Constanta, Seria Matematic

    Models for spin-dependent transport in helical molecules

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    Chiral molecules act as strong spin filters for transmitted electrons (chiral-induced spin selectivity). The interplay of geometry and spin mediated by spin-orbit coupling is commonly assumed as the cause of the effect, but the theoretical description remains incomplete. In this thesis, two models for electron transport through helical molecules were investigated: an atomistic tight binding model for the molecule helicene and a simple continuum model for an electron in a helix-shaped potential. In an attempt to cover the middle ground between phenomenological tight binding approaches and detailed first principle simulations, the helicene model starts with a lattice of carbon atoms represented by a minimal basis of local atomic s- and p-orbitals including electronic nearest-neighbor and spin-orbit interactions. Löwdin partitioning is used to reduce the model to a p-orbital tight binding representation, providing numeric values for all the couplings dependent on geometry. Transport calculations showed helicity dependent spin polarization several orders of magnitude smaller than experimentally observed. To understand the effect on a more fundamental level, an electron moving through a helix-shaped confinement potential in 3D space with spin-orbit coupling was considered. By taking the limit of strong confinement, an approximate model with one-dimensional configuration space (the helix) was obtained. Novel onsite spin-orbit coupling terms appear in the effective Hamiltonian, leading to sizeable spin polarization in transport calculations. These new terms are thoroughly justified by the adiabatic limiting procedure which was adapted to include spin-orbit coupling and might thus provide one of the missing pieces for the theory of chiral-induced spin selectivity

    A Geometric Approach to the Projective Tensor Norm

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    The main focus of this thesis is on the projective norm on finite-dimensional real or complex tensor products. There are various mathematical subjects with relations to the projective norm. For instance, it appears in the context of operator algebras or in quantum physics. The projective norm on multipartite tensor products is considered to be less accessible. So we use a method from convex algebraic geometry to approximate the projective unit ball by convex supersets, so-called theta bodies. For real multipartite tensor products we obtain theta bodies which are close to the projective unit ball, leading to a generalisation of the Schmidt decomposition. In a second step the method is applied to complex tensor products, in a third step to separable states. In a more general context, the projective norm can be related to binomial ideals, especially to so-called Hibi relations. In this respect, we also focus on a generalisation of the projective unit ball, here called Hibi body, and its theta bodies. It turns out that many statements also hold in this general context

    Polyhedra, lattice structures, and extensions of semigroups

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    For an arbitrary rational polyhedron, we consider its decompositions into Minkowski summands and, dual to this, the so-called free extensions of the associated pair of semigroups. Being free for a pair of semigroups is equivalent to flatness for the corresponding algebras. The main result is phrased in this dual setup: the category of free extensions always contains an initial object, which we describe explicitly. This provides a canonical free extension of the original pair of semigroups provided by the given polyhedron. Our motivation comes from the deformation theory of the associated toric singularity
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