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The notion of independence in categories of algebraic structures, part I: Basic properties

Gabriel Srour
1988 Annals of Pure and Applied Logic  
And also that we can de&~ (say, in a given class of structures) an independence relation with respect to a set of 'equations' in very much the same manner algebraic independence is in the class of gelds  ...  The following are some examples. (a) In the category of fields F with field homomorphisms, we have the notion of independence.  ...  PO analyze the independence relatioti in these ex s, we resort to the notion of type: Let L be a tit-order language and K a class of L-structures where a certain set A of formulas in L with .A = cl(A),  ... 
doi:10.1016/0168-0072(88)90053-x fatcat:6iubugynsvhj5mpyatlwkt2mby

On C-class equations [article]

Andreas Cap, Boris Doubrov, Dennis The
2020 arXiv   pre-print
to the equations in the class descends to the space of solutions.  ...  The concept of a C-class of differential equations goes back to E. Cartan with the upshot that generic equations in a C-class can be solved without integration.  ...  Acknowledgements The first and third authors were respectively supported by projects P27072-N25 and M1884-N35 of the Austrian Science Fund (FWF).  ... 
arXiv:1709.01130v3 fatcat:m5367ttxnfeorfhmvyt4jl6x4y

A class of quadratic difference equations on a finite graph [article]

Paul Baird
2011 arXiv   pre-print
We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure.  ...  A set of discrete parameters for which there exist non-trivial solutions leads to the construction of a polynomial invariant and the notion of a geometric spectrum.  ...  thanks Mike Eastwood for many helpful conversations in relation to this work.  ... 
arXiv:1109.3286v1 fatcat:2u276xxp4vekxbkqkuhd7zptny

The Hyperbolic Yang–Mills Equation for Connections in an Arbitrary Topological Class

Sung-Jin Oh, Daniel Tataru
2018 Communications in Mathematical Physics  
Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest.  ...  We generalize the standard notion of a topological class of connections on ℝ^d, defined via a pullback to the one-point compactification 𝕊^d = ℝ^d∪{∞}, to rough connections with curvature in the critical  ...  Moreover, in whole space case, we use the asymptotics of the good global gauge potential to extend the notion of topological classes of connections to the rough setting.  ... 
doi:10.1007/s00220-018-3205-x fatcat:sxlik3zgbjahjbnm77j562nzv4

On a class of numbers generated by differencial equations related with algebraic groups

Hiroshi Umemura
1994 Nagoya mathematical journal  
In this paper we propose a new category Qclof complex numbers which containsπ, eand the set of algebraic numbers. In fact this category contains most of the numbers studied so far in number theory.  ...  An element of the category is here called a classical number. The category of the classical numbers forms an algebraically closed field and consists of countably many numbers.  ...  In fact this category contains most of the numbers studied so far in number theory. An element of the category is here called a classical number.  ... 
doi:10.1017/s0027763000004736 fatcat:lxfvgbp55vbt5lsgt35445oos4

Characteristic cohomology of differential systems II: Conservation laws for a class of parabolic equations

Robert L. Bryant, Phillip A. Griffiths
1995 Duke mathematical journal  
We consider the evolution equation PROPOSITION. This equation is locally contact equivalent to an equation (ii) v, F(y, v, vy)(vy r + A(y, v, vr)), F > 0. Proof.  ...  We retain the notations 2 =dt, o du p dx, 1 c(x, u, p) dx + e(x, u, p) dp and 01 a(x, u, p) dx + b(x, u, p) dp with do -1^1 mod o  ...  To illustrate this, suppose that our parabolic system is given in the category of real algebraic varieties.  ... 
doi:10.1215/s0012-7094-95-07824-7 fatcat:wgwf7ix4tvat5bctg7cwyh5w6q

A Local Resolution of the Problem of Time. III. The other classical facets piecemeal [article]

Edward Anderson
2019 arXiv   pre-print
, 2') Assignment of Spacetime Observables, 3') Spacetime Construction from Less Structured Spacetime, and B) Foliation Independence.  ...  Secondly, on a coordinate-free Multi-tensor Calculus suitable for joint treatment of Background Independence aspects, presenting in particular the Dirac Algorithm, and ensuing theory of observables and  ...  Structure 2 Spanning, independence and bases for observables algebras (explained to be well-defined in Article XIV) is also a significant part of the theory of observables.  ... 
arXiv:1905.06212v4 fatcat:3ok6wrmzfvbnfdvacjz65fgufq

A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges

Ellery Ames, Florian Beyer, James Isenberg, Philippe G. LeFloch
2017 Journal of Geometry and Physics  
Our proof is based on an analysis of the singular initial value problem for the Einstein vacuum equations in the generalized wave gauge formalism, and provides a framework which we anticipate to be useful  ...  We establish the existence of smooth vacuum Gowdy solutions, which are asymptotically velocity term dominated (AVTD) and have T3-spatial topology, in an infinite dimensional family of generalized wave  ...  of our introducing the convenient notion of function operators (see Section 2.1 below) and partly because we focus here on the smooth category rather than working with less regular solutions.  ... 
doi:10.1016/j.geomphys.2017.06.005 fatcat:ilwsprpcffhovh33lgnknksz5e

Classification of Lie algebras of differential operators [article]

Helge Øystein Maakestad
2020 arXiv   pre-print
In a previous paper we introduced the notion of a D-Lie algebra L̃.  ...  We used this notion to define the universal enveloping algebra of the category of L̃-connections and to define the cohomology and homology of an arbitrary connection.  ...  In the paper we introduce the category of D-Lie algebras, connections on D-Lie algebras using the module of principal parts.  ... 
arXiv:1905.09630v12 fatcat:ttsk7yqeybhf3kkrgz5ahd7flu

Quantum stochastic convolution cocycles III [article]

J. Martin Lindsay, Adam G. Skalski
2011 arXiv   pre-print
Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic  ...  Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous  ...  , in particular, extended it to other quantum notions of independence (free, boolean and monotone), still in the algebraic context.  ... 
arXiv:0905.2410v2 fatcat:5gnmnpgxz5bhjprtynkp2sesye

Quantum stochastic convolution cocycles III

J. Martin Lindsay, Adam G. Skalski
2011 Mathematische Annalen  
Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive) quantum stochastic  ...  Every Markov-regular quantum Lévy process on a multiplier C *bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous  ...  The equivalence of (ii) and (iii) is the general form of an ǫ-structure map (see (1.12)). In view of (a), the last part is easily seen from the representation (iii). This completes the proof. Remark.  ... 
doi:10.1007/s00208-011-0656-1 fatcat:254irz6ksvdw5laobiyfeffkca

Stability of a class of nonlinear reaction–diffusion equations and stochastic homogenization

Omar Anza Hafsa, Jean Philippe Mandallena, Gérard Michaille
2019 Asymptotic Analysis  
We establish a convergence theorem for a class of nonlinear reaction-diffusion equations when the diffusion term is the subdifferential of a convex functional in a class of functionals of the calculus  ...  of variations equipped with the Mosco-convergence.  ...  The class of CP-structured reaction functionals.  ... 
doi:10.3233/asy-191531 fatcat:kabo3xb5pvcufi5oflfsur72xu

Division Algebras and Supersymmetry III [article]

John Huerta
2011 arXiv   pre-print
In the previous paper in this series, we constructed this Lie 2-superalgebra using the normed division algebras.  ...  Simple examples of Lie 2-superalgebras arise from 3-cocycles on Lie superalgebras, and it is in this way that we constructed the Lie 2-superalgebra above.  ...  Finally, we thank John Baez for his untiring help as he supervised the thesis on which this paper is based. This work was partially supported by FQXi grant RFP2-08-04.  ... 
arXiv:1109.3574v3 fatcat:75wabt5acbg37fy7smyfyqnoji

Shape Theory. III. Comparative Theory of Backgound Independence [article]

Edward Anderson
2019 arXiv   pre-print
Independence is the modern form of the relational side of the Absolute versus Relational Debate. Difficulties with its implementation form the Problem of Time.  ...  Closure of 1-2)'s constraints. 4) Expression in terms of observables: commutants with constraints. 5) Reconstructing spacetime from constraint algebra rigidity. 6) is the spacetime counterpart of 2-4),  ...  Acknowledgments I thank Chris Isham and Don Page for past discussions about configuration space topology, geometry and quantization, and they and Julian Barbour for past discussions of Background Independence  ... 
arXiv:1812.08771v3 fatcat:hffcviojybemjk2d6qbkmf5d5y

Division algebras and supersymmetry III

John Huerta
2012 Advances in Theoretical and Mathematical Physics  
In the first paper [5], we reviewed the known story of how the division algebras give rise to the supersymmetry of super-Yang-Mills theory.  ...  This is the third in a series of papers exploring the relationship between supersymmetry and division algebras [5, 6] , the first two of which were coauthored with John Baez.  ...  Finally, we thank John Baez for his untiring help as he supervised the thesis on which this paper is based.  ... 
doi:10.4310/atmp.2012.v16.n5.a4 fatcat:5qdzxwouo5cttni5zy5yxems2y
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