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The aim of the article is to give a conceptual understanding of Kontsevich’s construction of the universal element of the cohomology of the coarse moduli space of smooth algebraic curves with given genus and punctures. In a first step the author presents a toy model of tree graphs coloured by an operad $𝒫$ for which the graph complex and the universal cycle will be constructed. The universal cycle has coefficients in the operad for $Ω (𝒫^{*})$ -algebras with trivial differential over the (dual) cobar construction...

Remarks on effect-tribes

Sylvia Pulmannová, Elena Vinceková (2015)

Kybernetika

We show that an effect tribe of fuzzy sets $𝒯 \subseteq {[0, 1]}^{X}$ with the property that every $f \in 𝒯$ is $ℬ_{0} (𝒯)$ -measurable, where $ℬ_{0} (𝒯)$ is the family of subsets of $X$ whose characteristic functions are central elements in $𝒯$ , is a tribe. Moreover, a monotone $σ$ -complete effect algebra with RDP with a Loomis-Sikorski representation $(X, 𝒯, h)$ , where the tribe $𝒯$ has the property that every $f \in 𝒯$ is $ℬ_{0} (𝒯)$ -measurable, is a $σ$ -MV-algebra.

Remarks on integration over Lie algebras

Daniel Altschuler, Claude Itzykson (1991)

Annales de l'I.H.P. Physique théorique

Remarks on joint distributions of observables

Kazimierz Urbanik (1987)

Colloquium Mathematicum

Remarks on q-CCR relations for |q| > 1

Marek Bożejko (2007)

Banach Center Publications

In this paper we give a construction of operators satisfying q-CCR relations for q > 1: $A (f) A * (g) - A * (g) A (f) = q^{N} ⟨ f, g ⟩ I$ and also q-CAR relations for q < -1: $B (f) B * (g) + B * (g) B (f) = {| q |}^{N} ⟨ f, g ⟩ I$ , where N is the number operator on a suitable Fock space $ℱ_{q} ()$ acting as Nx₁ ⊗ ⋯ ⊗ xₙ = nx₁ ⊗ ⋯ ⊗xₙ. Some applications to combinatorial problems are also given.

Remarks on relativistic Schrödinger operators and their extensions

Matania Ben-Artzi, Jonathan Nemirovsky (1997)

Annales de l'I.H.P. Physique théorique

Remarks on representation of fuzzy quantum posets

Anatolij Dvurečenskij (1994)

Mathematica Slovaca

Remarks on tensor products and their applications in quantum theory I. General considerations

J. Blank, P. Exner (1976)

Acta Universitatis Carolinae. Mathematica et Physica

Remarks on tensor proucts and their applications in quantum theory - II. Spectral properties

J. Blank, P. Exner (1977)

Acta Universitatis Carolinae. Mathematica et Physica

Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

Kenichi Ito, Shu Nakamura (2012)

Annales de l’institut Fourier

We consider Schrödinger operators $H$ on $ℝ^{n}$ with variable coefficients. Let $H_{0} = - \frac{1}{2} ▵$ be the free Schrödinger operator and we suppose $H$ is a “short-range” perturbation of $H_{0}$ . Then, under the nontrapping condition, we show that the time evolution operator: $e^{- i t H}$ can be written as a product of the free evolution operator $e^{- i t H_{0}}$ and a Fourier integral operator $W (t)$ which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results...

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Relativistic Toda chain with boundary interaction at root of unity.

Relativistic wave equations with fractional derivatives and pseudodifferential operators.

Relativity and irreversibility.

Remark on Essential Selfadjointness of Dirac Operators with Coulomb Potentials.

Remark on two papers concerning axiomatics of quantum mechanics

Remark to the problem of eigenvalues of Schrödinger equation

Remarks on CR-manifolds of codimension $2$ in $C^{4}$