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Multi-adjoint algebras versus non-commutative residuated structures

M. Eugenia Cornejo, Jesús Medina, Eloisa Ramírez-Poussa
2015 International Journal of Approximate Reasoning  
Extended-order 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.  ... 
doi:10.1016/j.ijar.2015.08.003 fatcat:q27ko7up75an7g3fjnxn4oogfi

Linear Logic Properly Displayed [article]

Giuseppe Greco, Alessandra Palmigiano
2016 arXiv   pre-print
Our proposal builds on an 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.  ... 
arXiv:1611.04181v1 fatcat:hdhlj5elvnfrzf6xuy7egi6yqy

Left and right adjoint operations on spaces and data types

F.William Lawvere
2004 Theoretical Computer Science  
Formal theories of higher types need to be augmented to permit treatment of some additional right 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  ... 
doi:10.1016/j.tcs.2004.01.026 fatcat:z4lzbv3wlfcxlez6a2yie3fmbe

On the bosonic spectrum of QCD with currents

Uwe Trittmann
2000 Nuclear Physics B  
In the case of 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  ... 
doi:10.1016/s0550-3213(00)00469-7 fatcat:uwkm24lofzflpkkibza6vjjcpa

Extending conceptualisation modes for generalised Formal Concept Analysis

Francisco J. Valverde-Albacete, Carmen Peláez-Moreno
2011 Information Sciences  
To provide a concrete example of the use of these techniques, we analyse the performance of 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.  ... 
doi:10.1016/j.ins.2010.04.014 fatcat:kdstbuyex5attiyvtq42bmkify

CCR- versus CAR-Quantization on Curved Spacetimes [chapter]

Christian Bär, Nicolas Ginoux
2012 Quantum Field Theory and Gravity  
This means that the odd parts of the 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.  ... 
doi:10.1007/978-3-0348-0043-3_10 fatcat:kojntwibsfaxrakrp7i4nf7b34

On the spectrum of(1+1)-dimensional QCD withSU(Nc)currents

Uwe Trittmann
2002 Physical Review D, Particles and fields  
We also present some new findings concerning the single-particle spectrum of the 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.  ... 
doi:10.1103/physrevd.66.025001 fatcat:jndbjjfsezdvxgmb4dey4fqg6q

DUALITY AND THE EQUIVALENCE PRINCIPLE OF QUANTUM MECHANICS

JOSÉ M. ISIDRO
2001 International Journal of Modern Physics A  
Our approach may be regarded as a manifestly 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.  ... 
doi:10.1142/s0217751x01005353 fatcat:txhhxjr6gzgu7hkbbiwwvz7tmm

Solving loop equations by Hitchin systems via holography in large-NQCD4

Marco Bochicchio
2003 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 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.  ... 
doi:10.1088/1126-6708/2003/06/026 fatcat:q7sgylwvznhlzmpwwcjfviowpy

Dynamical Systems On Weighted Lattices: General Theory [article]

Petros Maragos
2017 arXiv   pre-print
Their 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  ... 
arXiv:1606.07347v4 fatcat:fq7px7ex5vd3pddvg2zulldgxm

Positive polynomials in scalar and matrix variables, the spectral theorem and optimization [article]

John William Helton, Mihai Putinar
2006 arXiv   pre-print
These new applications have prompted a series of recent studies devoted to the 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.  ... 
arXiv:math/0612103v1 fatcat:4molh5aq2nfotbbpw7j3h3w6yq

Dynamical systems on weighted lattices: general theory

Petros Maragos
2017 MCSS. Mathematics of Control, Signals and Systems  
Their 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).  ... 
doi:10.1007/s00498-017-0207-8 fatcat:cff5jfdvpfbz7khca5lhhqumbi

Representations for Morphological Image Operators and Analogies with Linear Operators [chapter]

Petros Maragos
2013 Advances in Imaging and Electron Physics  
Then, it introduces a nonlinear signal space called complete weighted lattice, which generalizes both mathematical morphology and minimax 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 .  ... 
doi:10.1016/b978-0-12-407702-7.00002-4 fatcat:xeckgeughzemdpi632fqypmi7e

Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory

Vladimir S. Gerdjikov
2010 Symmetry, Integrability and Geometry: Methods and Applications  
Let us comment briefly on the 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 .  ... 
doi:10.3842/sigma.2010.044 fatcat:enob3fxsijhy5izh2qvd3vuise

Anomalous Bootstrap on the half line [article]

David Berenstein, George Hulsey
2022 arXiv   pre-print
CONCLUSION The power of the bootstrap approach to quantum mechanics is due to its reliance on the 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.  ... 
arXiv:2206.01765v3 fatcat:zthpbto2uvbcvldjzsfzv2euka
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