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

1988
*
Annals of Pure and Applied Logic
*

And also that we can de&~ (say,

doi:10.1016/0168-0072(88)90053-x
fatcat:6iubugynsvhj5mpyatlwkt2mby
*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), ...##
###
On C-class equations
[article]

2020
*
arXiv
*
pre-print

to

arXiv:1709.01130v3
fatcat:m5367ttxnfeorfhmvyt4jl6x4y
*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). ...##
###
A class of quadratic difference equations on a finite graph
[article]

2011
*
arXiv
*
pre-print

We study a

arXiv:1109.3286v1
fatcat:2u276xxp4vekxbkqkuhd7zptny
*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. ...##
###
The Hyperbolic Yang–Mills Equation for Connections in an Arbitrary Topological Class

2018
*
Communications in Mathematical Physics
*

Although this paper is a

doi:10.1007/s00220-018-3205-x
fatcat:sxlik3zgbjahjbnm77j562nzv4
*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. ...##
###
On a class of numbers generated by differencial equations related with algebraic groups

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. ...

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

1995
*
Duke mathematical journal
*

We consider

doi:10.1215/s0012-7094-95-07824-7
fatcat:wgwf7ix4tvat5bctg7cwyh5w6q
*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. ...##
###
A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
[article]

2019
*
arXiv
*
pre-print

, 2') Assignment

arXiv:1905.06212v4
fatcat:3ok6wrmzfvbnfdvacjz65fgufq
*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. ...##
###
A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges

2017
*
Journal of Geometry and Physics
*

Our proof is based on an analysis

doi:10.1016/j.geomphys.2017.06.005
fatcat:ilwsprpcffhovh33lgnknksz5e
*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. ...##
###
Classification of Lie algebras of differential operators
[article]

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*. ...

##
###
Quantum stochastic convolution cocycles III
[article]

2011
*
arXiv
*
pre-print

Then, working

arXiv:0905.2410v2
fatcat:5gnmnpgxz5bhjprtynkp2sesye
*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. ...##
###
Quantum stochastic convolution cocycles III

2011
*
Mathematische Annalen
*

Then, working

doi:10.1007/s00208-011-0656-1
fatcat:254irz6ksvdw5laobiyfeffkca
*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. ...##
###
Stability of a class of nonlinear reaction–diffusion equations and stochastic homogenization

2019
*
Asymptotic Analysis
*

We establish a convergence theorem for a

doi:10.3233/asy-191531
fatcat:kabo3xb5pvcufi5oflfsur72xu
*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. ...##
###
Division Algebras and Supersymmetry III
[article]

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. ...

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

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*...

##
###
Division algebras and supersymmetry III

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. ...

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