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Multi-adjoint algebras versus non-commutative residuated structures

2015
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International Journal of Approximate Reasoning
*

Extended-order

doi:10.1016/j.ijar.2015.08.003
fatcat:q27ko7up75an7g3fjnxn4oogfi
*algebras*are implicative*algebras*that generalize the integral*residuated**structures*. ...*Adjoint*triples are an interesting generalization of t-norms and their*residuated*implications, since they increase the flexibility in the framework where they are used. ... Similar*adjoint*triples can be obtained from the Gödel and Łukasiewicz t-norms. ⊓ ⊔ The*algebraic**structure*that considers these triples is the biresiduated*multi*-*adjoint**algebra*. ...##
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Linear Logic Properly Displayed
[article]

2016
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arXiv
*
pre-print

Our proposal builds on an

arXiv:1611.04181v1
fatcat:hdhlj5elvnfrzf6xuy7egi6yqy
*algebraic*and order-theoretic analysis of linear logic, and applies the guidelines of the*multi*-type methodology in the design of display calculi. ... the*non*-constructive behaviour of connectives, but also the linear*versus*the*non*-linear behaviour, using*structural*rules to capture each type of behaviour. ...*structures*, connected via suitable*adjoint*maps. ...##
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Left and right adjoint operations on spaces and data types

2004
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Theoretical Computer Science
*

Formal theories of higher types need to be augmented to permit treatment of some additional right

doi:10.1016/j.tcs.2004.01.026
fatcat:z4lzbv3wlfcxlez6a2yie3fmbe
*adjoints*which arise in mathematical practice. ... The concomitant mathematical scrutiny of supposedly well-established left*adjoints*resurrects a "critique of foundations" which may not be irrelevant for practice. ...*non*-*algebraically*closed ÿeld K of characteristic zero, i.e. presheaves on the opposite of the category of ÿnite-dimensional local K-*algebras*) the nature of the point inside an inÿnitesimal component varies ...##
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On the bosonic spectrum of QCD with currents

2000
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Nuclear Physics B
*

In the case of

doi:10.1016/s0550-3213(00)00469-7
fatcat:uwkm24lofzflpkkibza6vjjcpa
*adjoint*fermions we present some new results. ... The resulting numerical eigenvalue spectrum is free from trivial*multi*-particle states which obscured previous results. The well-known 't Hooft and large N_f spectra are reproduced. ... even a*multi*-Regge*structure*exists, as was suggested in Ref ...##
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Extending conceptualisation modes for generalised Formal Concept Analysis

2011
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Information Sciences
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To provide a concrete example of the use of these techniques, we analyse the performance of

doi:10.1016/j.ins.2010.04.014
fatcat:kdstbuyex5attiyvtq42bmkify
*multi*-class classifiers by conceptually analysing their confusion matrices. ... Second, we consider incidences with values in idempotent semirings-concretely the completed max-plus or schedule*algebra*R max,+ -and focus on generalising FCA to try and replicate the modes of conceptualisation ... multiplicative*structure*is*commutative*. ...##
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CCR- versus CAR-Quantization on Curved Spacetimes
[chapter]

2012
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Quantum Field Theory and Gravity
*

This means that the odd parts of the

doi:10.1007/978-3-0348-0043-3_10
fatcat:kojntwibsfaxrakrp7i4nf7b34
*algebras*anti-*commute*while the even parts*commute*with everything. instead of self-*adjoint*! ... They come in two flavors, bosonic*algebras*encoding the canonical*commutation*relations and fermionic*algebras*encoding the canonical anti-*commutation*relations. ...##
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On the spectrum of(1+1)-dimensional QCD withSU(Nc)currents

2002
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Physical Review D, Particles and fields
*

We also present some new findings concerning the single-particle spectrum of the

doi:10.1103/physrevd.66.025001
fatcat:jndbjjfsezdvxgmb4dey4fqg6q
*adjoint*theory. ... We construct the Hamiltonian matrix in DLCQ formulation as an*algebraic*function of the harmonic resolution K and the continuous parameter \lambda. ... Although the cyclic permutations are related*non*-trivially due to the Kac-Moody*structure*of the currents, all cyclic permutations of a given state have to be eliminated from the Fock basis. ...##
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DUALITY AND THE EQUIVALENCE PRINCIPLE OF QUANTUM MECHANICS

2001
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International Journal of Modern Physics A
*

Our approach may be regarded as a manifestly

doi:10.1142/s0217751x01005353
fatcat:txhhxjr6gzgu7hkbbiwwvz7tmm
*non*perturbative formulation of quantum mechanics, in that we take no classical phase space and no Poisson brackets as a starting point. ... One can nonetheless make sense out of a*non*self-*adjoint*operator Z. This is based on the fact that Z 2 admits a self-*adjoint*Friedrichs extension, whose square root is now self-*adjoint*. ... The point,*residual*and continuos spectra of P √ are σ p (P √ ) = φ, σ r (P √ ) = φ, σ c (P √ ) = [0, ∞). (4.16) We observe that taking the Friedrichs extension does not*commute*with the square root. ...##
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Solving loop equations by Hitchin systems via holography in large-NQCD4

2003
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Journal of High Energy Physics
*

The key point is that the contour integral that occurs in the loop equations for the de Rham local systems can be reduced to the computation of a

doi:10.1088/1126-6708/2003/06/026
fatcat:q7sgylwvznhlzmpwwcjfviowpy
*residue*in a certain regularization. ... In our case however the*algebra*of Euclidean matrix fields in the large-N limit is highly*non**commutative*. ... The theory of von Neumann*algebras*is a*non*-*commutative*far reaching generalisation of the integration theory and spectral theorem to which it reduces in the*commutative*case. ...##
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Dynamical Systems On Weighted Lattices: General Theory
[article]

2017
*
arXiv
*
pre-print

Their

arXiv:1606.07347v4
fatcat:fq7px7ex5vd3pddvg2zulldgxm
*algebraic**structure*has a polygonal geometry. Some of the special cases unified include max-plus, max-product, and probabilistic dynamical systems. ... The state vectors and input-output signals evolve on nonlinear spaces which we call complete weighted lattices and include as special cases the nonlinear vector spaces of minimax*algebra*. ... ' a*commutative*binary operation η(a, v) = a ⋆ v that is a dilation on K, its scalar*adjoint*erosion becomes ζ(a, w) = sup{v ∈ K : a ⋆ v ≤ w} (44) which is a (possibly*non*-*commutative*) binary operation ...##
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Positive polynomials in scalar and matrix variables, the spectral theorem and optimization
[article]

2006
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arXiv
*
pre-print

These new applications have prompted a series of recent studies devoted to the

arXiv:math/0612103v1
fatcat:4molh5aq2nfotbbpw7j3h3w6yq
*structure*of positivity and convexity in a free *-*algebra*, the appropriate setting for analyzing inequalities on polynomials ... We follow a stream of the history of positive matrices and positive functionals, as applied to*algebraic*sums of squares decompositions, with emphasis on the interaction between classical moment problems ...*Non*-*commutative**algebras*A direction in semi-*algebraic*geometry, recently blossoming still with many avenues to explore, concerns variables which do not*commute*. ...##
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Dynamical systems on weighted lattices: general theory

2017
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MCSS. Mathematics of Control, Signals and Systems
*

Their

doi:10.1007/s00498-017-0207-8
fatcat:cff5jfdvpfbz7khca5lhhqumbi
*algebraic**structure*has a polygonal geometry. Some of the special cases unified include max-plus, max-product, and probabilistic dynamical systems. ... The state vectors and input-output signals evolve on nonlinear spaces which we call complete weighted lattices and include as special cases the nonlinear vector spaces of minimax*algebra*. ... The last two results can be easily extended to*multi*-input*multi*-output (MIMO) systems by using an impulse response matrix as in (72). ...##
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Representations for Morphological Image Operators and Analogies with Linear Operators
[chapter]

2013
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Advances in Imaging and Electron Physics
*

Then, it introduces a nonlinear signal space called complete weighted lattice, which generalizes both mathematical morphology and minimax

doi:10.1016/b978-0-12-407702-7.00002-4
fatcat:xeckgeughzemdpi632fqypmi7e
*algebra*. ...*versus*discrete, local*versus*global, single-scale or*multi*-scale, functional*versus*graphical, the need also arises to study the representation of operators. ... it*commutes*with its*adjoint*: T T * = T * T . ...##
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Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory

2010
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Symmetry, Integrability and Geometry: Methods and Applications
*

Let us comment briefly on the

doi:10.3842/sigma.2010.044
fatcat:enob3fxsijhy5izh2qvd3vuise
*algebraic**structure*of the Lax pair, which is related to the symmetric space SO(n + 2)/(SO(n) × SO(2)). ... A The*adjoint*representations of so(5) and so (7) The*adjoint*representations of all orthogonal*algebras*so(2r + 1) and so(2r) are characterized by the fundamental weight ω 2 = e 1 + e 2 . ...##
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Anomalous Bootstrap on the half line
[article]

2022
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arXiv
*
pre-print

CONCLUSION The power of the bootstrap approach to quantum mechanics is due to its reliance on the

arXiv:2206.01765v3
fatcat:zthpbto2uvbcvldjzsfzv2euka
*algebraic**structure*of the problem. ... The benefit of this approach is that the*algebraic**structure*is usually known a priori, and no explicit knowledge of the dynamics is required. ... This means that such Hamiltonians are either essentially self-*adjoint*or admit self-*adjoint*extensions. ...
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