175,228 research outputs found
Kochen-Specker theorem for von Neumann algebras
No full textThe Kochen-Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type I_n factor as algebra of observables, including I_infinity. Afterwards, we give a proof of the Kochen-Specker theorem for an arbitrary von Neumann algebra R without summands of types I_1 and I_2, using a known result on two-valued measures on the projection lattice P(R). Some connections with presheaf formulations as proposed by Isham and Butterfield are made
A comparison of deflation and the balancing Neumann-Neumann preconditioner
No full textIn this paper we compare various preconditioners for the numerical solution of partial differential equations. We compare the well-known balancing Neumann Neumann preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the deflated preconditioned system is always, i.e. for all deflation vectors and all restrictions and prolongations, below the condition number of the system preconditioned by the balancing Neumann-Neumann preconditioner. Even more, we establish that both preconditioners lead to almost the same spectra. The zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing Neumann-Neumann preconditioner is used. Moreover, we proved that the A-norm of the errors of the iterates build by the deflation preconditioner is always below the A-norm of the errors of the iterates build by the balancing Neumann-Neumann preconditioner. Additionally, the amount of work of one iteration of the de ation preconditioned system is less than the amount of work of one iteration of the balancing Neumann-Neumann preconditioned system. Finally, we establish that the deflation preconditioner and the balancing Neumann-Neumann preconditioner produces the same iterates if one uses certain starting vectors. Numerical results for porous media flows emphasize the theoretical results.Electrical Engineering, Mathematics and Computer Scienc
Three nontrivial solutions for the p-Laplacian Neumann problems with a concave nonlinearity near the origin
No full textWe consider a nonlinear Neumann problem driven by the p-
Laplacian, with a right-hand side nonlinearity which is concave near the
origin. Using variational techniques, combined with the method of upper-lower
solutions and with Morse theory, we show that the problem has at least three
nontrivial smooth solutions, two of which have a constant sign (one positive
and one negative).FCTPOCI/MAT/55524/200
The von Neumann Model and the Early Models of General Equilibrium
No full textThe paper reconstructs the von Neumann model, comments on its salient features and critically reviews some of its generalisations. The issues related to thetreatment of consumption, decomposability and uniqueness of the rate of growth and interest will be especially scrutinised. The most prominent models of general equilibrium that appeared before or roughly at the same time as von Neumann's model will be also reviewed in the paper and compared with it. It will be demonstrated that none of them had any noticeable influence on von Neumann's model, which is genuinely distinct, ideologically free and methodologically fresh and forward-looking. It will be argued that the model can be viewed as a brilliant mathematical metaphor of some deep-rooted old vision, pertaining to the core issues of commodity production
Introducing Formalism in Economics: The Growth Model of John von Neumann
No full textThe objective is to interpret John von Neumann's growth model as a decisive step of the forthcoming formalist revolution of the 1950s in economics. This model gave rise to an impressive variety of comments about its classical or neoclassical underpinnings. We go beyond this traditional criterion and interpret rather this model as the manifestation of von Neumann's involvement in the formalist programme of mathematician David Hilbert. We discuss the impact of Kurt Gödel’s discoveries on this programme. We show that the growth model reflects the pragmatic turn of the formalist programme after Gödel and proposes the extension of modern axiomatisation to economics..Von Neumann, Growth model, Formalist revolution, Mathematical formalism, Axiomatics
Some rigidity aspects in von Neumann algebras and C*-algebras arising from groups
No full textIn the first part of my talk I will discuss the problems of reconstructing a countable discrete group from its von Neumann algebra (W*-superrigidity) and its reduced C*-algebra (C*-superrigidity) and I will survey several recent results in this direction.
In the second part, using and interplay between von Neumann algebraic and C*-algebraic methods, I will introduce a new class of C*-superrigid groups which appear as wreath products with non-amenable core. As an application we obtain complete calculations of the symmetry groups of various group C*-algebras---a problem barely touched in the literature. This is based on a recent joint work with Alec Diaz-Arias.Non UBCUnreviewedAuthor affiliation: University of IowaResearche
Type II<sub>1</sub> factors satisfying the spatial isomorphism conjecture
No full textThis paper addresses a conjecture in the work by Kadison and Kastler [Kadison RV, Kastler D (1972) Am J Math 94:38–54] that a von Neumann algebra M on a Hilbert space H should be unitarily equivalent to each sufficiently close von Neumann algebra N, and, moreover, the implementing unitary can be chosen to be close to the identity operator. This conjecture is known to be true for amenable von Neumann algebras, and in this paper, we describe classes of nonamenable factors for which the conjecture is valid. These classes are based on tensor products of the hyperfinite II<sub>1</sub> factor with crossed products of abelian algebras by suitably chosen discrete groups
Mesh-based numerical implementation of the localized boundary-domain integral equation method to a variable-coefficient Neumann problem
No full textAn implementation of the localized boundary-domain integral-equation (LBDIE) method for the numerical solution of the Neumann boundary-value problem for a second-order linear elliptic PDE with variable coefficient is discussed. The LBDIE method uses a specially constructed localized parametrix (Levi function) to reduce the BVP to a LBDIE. After employing a mesh-based discretization, the integral equation is reduced to a sparse system of linear algebraic equations that is solved numerically. Since the Neumann BVP is not unconditionally and uniquely solvable, neither is the LBDIE. Numerical implementation of the finite-dimensional perturbation approach that reduces the integral equation to an unconditionally and uniquely solvable equation, is also discussed
Irrational behavior in the Brown-von Neumann-Nash dynamics
No full textWe present a class of games with a pure strategy being strictly dominated by another pure strategy such that the former survives along most solutions of the Brown-von Neumann-Nash dynamics.Nash map, BNN dynamics, Dominated strategies
A course space construction based on local Dirichlet-to-Neumann maps
No full textCoarse-grid correction is a key ingredient of scalable domain decomposition methods. In this work we construct coarse-grid space using the low-frequency modes of the subdomain Dirichlet-to-Neumann maps and apply the obtained two-level preconditioners to the extended or the original linear system arising from an overlapping domain decomposition. Our method is suitable for parallel implementation, and its efficiency is demonstrated by numerical examples on problems with large heterogeneities for both manual and automatic partitionings
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