271 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

    A solution to the MV-spectrum Problem in size aleph one

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    Denote by IdcG_c G the lattice of all principal ℓ\ell-ideals of an Abelian ℓ\ell-group GG. Our main result is the following. Theorem. For every countable Abelian ℓ\ell-group GG, every countable completely normal distributive 0-lattice L,L, and every closed 0-lattice homomorphism φ:IdcG→L\varphi : {\rm Id}_c G \to L, there are a countable Abelian ℓ\ell-group HH, an ℓ\ell-homomorphism f:G→Hf: G \to H, and a lattice isomorphism ι:IdcH→L\iota: {\rm Id}_c H \to L such that φ=ι∘Idcf\varphi = \iota \circ {\rm Id}_c f. We record the following consequences of that result: (1) A 0-lattice homomorphism φ:K→L\varphi: K \to L, between countable completely normal distributive 0-lattices, can be represented, with respect to the functor Idc_c, by an ℓ\ell-homomorphism of Abelian ℓ\ell-groups iff it is closed. (2) A distributive 0-lattice DD of cardinality at most ℵ1\aleph_1 is isomorphic to some IdcG_c G iff DD is completely normal and for all a,b∈Da,b \in D the set {x∈D∣a≤b∨x\{x\in D | a \leq b \vee x has a countable coinitial subset. This solves Mundici's MV-spectrum Problem for cardinalities up to ℵ1\aleph_1. The bound ℵ1\aleph_1 is sharp. Item (1) is extended to commutative diagrams indexed by forests in which every node has countable height.All our results are stated in terms of vector lattices over any countable totally ordered division ring

    Selected Topics in Gravity, Field Theory and Quantum Mechanics

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    Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories

    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

    Acta Cybernetica : Volume 25. Number 3.

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    Loops, Knots, Gauge Theories

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    This volume provides a self-contained introduction to applications of loop representations, and the related topic of knot theory, in particle physics and quantum gravity. These topics are of considerable interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. The book begins with a detailed review of loop representation theory and then describes loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed, with the following chapters considering knot theories, braid theories and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research. First published in 1996, this title has been reissued as an Open Access publication

    Compatible topologies on mixed lattice vector spaces

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    A mixed lattice vector space is a partially ordered vector space with two partial orderings, generalizing the notion of a Riesz space. The purpose of this paper is to develop the basic topological theory of mixed lattice spaces. A vector topology is said to be compatible with the mixed lattice structure if the mixed lattice operations are continuous. We give a characterization of compatible mixed lattice topologies, similar to the well known Roberts-Namioka characterization of locally solid Riesz spaces. We then study locally convex topologies and the associated seminorms, as well as connections between mixed lattice topologies and locally solid topologies on Riesz spaces. In the locally convex case, we obtain a more complete characterization of compatible mixed lattice topologies. We also briefly discuss asymmetric norms and cone norms on mixed lattice spaces with a particular application to finite dimensional spaces.publishedVersionPeer reviewe

    Three Risky Decades: A Time for Econophysics?

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    Our Special Issue we publish at a turning point, which we have not dealt with since World War II. The interconnected long-term global shocks such as the coronavirus pandemic, the war in Ukraine, and catastrophic climate change have imposed significant humanitary, socio-economic, political, and environmental restrictions on the globalization process and all aspects of economic and social life including the existence of individual people. The planet is trapped—the current situation seems to be the prelude to an apocalypse whose long-term effects we will have for decades. Therefore, it urgently requires a concept of the planet's survival to be built—only on this basis can the conditions for its development be created. The Special Issue gives evidence of the state of econophysics before the current situation. Therefore, it can provide excellent econophysics or an inter-and cross-disciplinary starting point of a rational approach to a new era
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