4,491 research outputs found
Arithmetic properties of eigenvalues of generalized Harper operators on graphs
Let \Qbar denote the field of complex algebraic numbers. A discrete group
is said to have the -multiplier algebraic eigenvalue property, if
for every matrix with entries in the twisted group ring over the complex
algebraic numbers M_d(\Qbar(G,\sigma)), regarded as an operator on
, the eigenvalues of are algebraic numbers, where is an
algebraic multiplier. Such operators include the Harper operator and the
discrete magnetic Laplacian that occur in solid state physics. We prove that
any finitely generated amenable, free or surface group has this property for
any algebraic multiplier . In the special case when is
rational (=1 for some positive integer ) this property holds for a
larger class of groups, containing free groups and amenable groups, and closed
under taking directed unions and extensions with amenable quotients. Included
in the paper are proofs of other spectral properties of such operators.Comment: 28 pages, latex2e, paper revise
Two-dimensional models as testing ground for principles and concepts of local quantum physics
In the past two-dimensional models of QFT have served as theoretical
laboratories for testing new concepts under mathematically controllable
condition. In more recent times low-dimensional models (e.g. chiral models,
factorizing models) often have been treated by special recipes in a way which
sometimes led to a loss of unity of QFT. In the present work I try to
counteract this apartheid tendency by reviewing past results within the setting
of the general principles of QFT. To this I add two new ideas: (1) a modular
interpretation of the chiral model Diff(S)-covariance with a close connection
to the recently formulated local covariance principle for QFT in curved
spacetime and (2) a derivation of the chiral model temperature duality from a
suitable operator formulation of the angular Wick rotation (in analogy to the
Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational
chiral theories. The SL(2,Z) modular Verlinde relation is a special case of
this thermal duality and (within the family of rational models) the matrix S
appearing in the thermal duality relation becomes identified with the
statistics character matrix S. The relevant angular Euclideanization'' is done
in the setting of the Tomita-Takesaki modular formalism of operator algebras.
I find it appropriate to dedicate this work to the memory of J. A. Swieca
with whom I shared the interest in two-dimensional models as a testing ground
for QFT for more than one decade.
This is a significantly extended version of an ``Encyclopedia of Mathematical
Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
Nonlinear Models of Neural and Genetic Network Dynamics:\ud \ud Natural Transformations of Łukasiewicz Logic LM-Algebras in a Łukasiewicz-Topos as Representations of Neural Network Development and Neoplastic Transformations \ud
A categorical and Łukasiewicz-Topos framework for Algebraic Logic models of nonlinear dynamics in complex functional systems such as Neural Networks, Cell Genome and Interactome Networks is introduced. Łukasiewicz Algebraic Logic models of both neural and genetic networks and signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable next-state/transfer functions is extended to a Łukasiewicz Topos with an N-valued Łukasiewicz Algebraic Logic subobject classifier description that represents non-random and nonlinear network activities as well as their transformations in developmental processes and carcinogenesis.\u
Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism
This essay examines the philosophical significance of -logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of -logical validity can then be countenanced within a coalgebraic logic, and -logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of -logical validity correspond to those of second-order logical consequence, -logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets
M-theory FDA, Twisted Tori and Chevalley Cohomology
The FDA algebras emerging from twisted tori compactifications of M-theory
with fluxes are discussed within the general classification scheme provided by
Sullivan's theorems and by Chevalley cohomology. It is shown that the
generalized Maurer Cartan equations which have appeared in the literature, in
spite of their complicated appearance, once suitably decoded within cohomology,
lead to trivial FDA.s, all new p--form generators being contractible when the
4--form flux is cohomologically trivial. Non trivial D=4 FDA.s can emerge from
non trivial fluxes but only if the cohomology class of the flux satisfies an
additional algebraic condition which appears not to be satisfied in general and
has to be studied for each algebra separately. As an illustration an exhaustive
study of Chevalley cohomology for the simplest class of SS algebras is
presented but a general formalism is developed, based on the structure of a
double elliptic complex, which, besides providing the presented results, makes
possible the quick analysis of compactification on any other twisted torus.Comment: LaTeX, 40 pages, article. Few typos corrected, one formula improve
Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models
A categorical and Łukasiewicz-Topos framework for Algebraic Logic models of nonlinear dynamics in complex functional systems such as Neural Networks, Cell Genome and Interactome Networks is introduced. Łukasiewicz Algebraic Logic models of both neural and genetic networks and signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable 'next-state functions' is extended to a Łukasiewicz Topos with an n-valued Łukasiewicz Algebraic Logic subobject classifier description that represents non-random and nonlinear network activities as well as their transformations in developmental processes and carcinogenesis
Twenty-five years of two-dimensional rational conformal field theory
In this article we try to give a condensed panoramic view of the development
of two-dimensional rational conformal field theory in the last twenty-five
years.Comment: A review for the 50th anniversary of the Journal of Mathematical
Physics. Some references added, typos correcte
Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models
Operational logic and bioinformatics models of nonlinear dynamics in complex functional systems such as neural networks, genomes and cell interactomes are proposed. Łukasiewicz Algebraic Logic models of genetic networks and signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable 'next-state functions' is extended to a Łukasiewicz Topos with an n-valued Łukasiewicz Algebraic Logic subobject classifier description that represents non-random and nonlinear network activities as well as their transformations in developmental processes and carcinogenesis
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