Combinatorial QFT and the Jacobian Conjecture

2021 ◽  
pp. 50-55
Author(s):  
Adrian Tanasa
Keyword(s):  
Linear Subspace ◽  
Polynomial System ◽  
Field Method ◽  
Jacobian Conjecture ◽  
Reduction Theorem ◽  
Degree Reduction ◽  
Intermediate Field ◽  
Partial Elimination ◽  
Elimination Of Variables

The Jacobian Conjecture states that any complex n-dimensional locally invertible polynomial system is globally invertible with polynomial inverse. In 1982, Bass et al. proved an important reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension n. We show in this chapter a result concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter 0≤n′≤n, parameter which represents the dimension of a linear subspace where some particular conditions on the system must hold. We exhibit a proof, in a QFT formulation, using the intermediate field method exposed in Chapter 3.

scholarly journals Colored Triangulations of Arbitrary Dimensions are Stuffed Walsh Maps

10.37236/5614 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Valentin Bonzom ◽  
Luca Lionni ◽  
Vincent Rivasseau
Keyword(s):  
Matrix Models ◽  
Building Blocks ◽  
Field Method ◽  
Fixed Number ◽  
Colored Graphs ◽  
Intermediate Field ◽  
Tensor Models ◽  
Combinatorial Maps ◽  
Arbitrary Dimensions ◽  
Number Of Faces

Regular edge-colored graphs encode colored triangulations of pseudo-manifolds. Here we study families of edge-colored graphs built from a finite but arbitrary set of building blocks, which extend the notion of $p$-angulations to arbitrary dimensions. We prove the existence of a bijection between any such family and some colored combinatorial maps which we call stuffed Walsh maps. Those maps generalize Walsh's representation of hypermaps as bipartite maps, by replacing the vertices which correspond to hyperedges with non-properly-edge-colored maps. This shows the equivalence of tensor models with multi-trace, multi-matrix models by extending the intermediate field method perturbatively to any model. We further use the bijection to study the graphs which maximize the number of faces at fixed number of vertices and provide examples where the corresponding stuffed Walsh maps can be completely characterized.

Evaluation of the dark field method for large association of spread DNA

1978 ◽  
Vol 36 (2) ◽  
pp. 190-191
Author(s):  
Douglas C. Barker
Keyword(s):  
Electron Microscopy ◽  
Molecular Weight ◽  
Mitochondrial Dna ◽  
Extraction Method ◽  
Field Method ◽  
Dark Field ◽  
Kinetoplast Dna ◽  
Field Electron ◽  
Field Electron Microscopy ◽  
Dark Field Electron

A number of satisfactory methods are available for the electron microscopy of nicleic acids. These methods concentrated on fragments of nuclear, viral and mitochondrial DNA less than 50 megadaltons, on denaturation and heteroduplex mapping (Davies et al 1971) or on the interaction between proteins and DNA (Brack and Delain 1975). Less attention has been paid to the experimental criteria necessary for spreading and visualisation by dark field electron microscopy of large intact issociations of DNA. This communication will report on those criteria in relation to the ultrastructure of the (approx. 1 x 10-14g) DNA component of the kinetoplast from Trypanosomes. An extraction method has been developed to eliminate native endonucleases and nuclear contamination and to isolate the kinetoplast DNA (KDNA) as a compact network of high molecular weight. In collaboration with Dr. Ch. Brack (Basel [nstitute of Immunology), we studied the conditions necessary to prepare this KDNA Tor dark field electron microscopy using the microdrop spreading technique.

A Many-Beam Technique for Enhanced Resolution of Partial Dislocations

1972 ◽  
Vol 30 ◽  
pp. 646-647
Author(s):  
J. M. Oblak ◽  
B. H. Kear
Keyword(s):  
Phase Shift ◽  
Field Method ◽  
Dark Field ◽  
Bright Field ◽  
Field Technique ◽  
Weak Beam ◽  
Partial Dislocations ◽  
Enhanced Resolution ◽  
Nickel Base ◽  
Beam Technique

The “weak-beam” and systematic many-beam techniques are the currently available methods for resolution of closely spaced dislocations or other inhomogeneities imaged through strain contrast. The former is a dark field technique and image intensities are usually very weak. The latter is a bright field technique, but generally use of a high voltage instrument is required. In what follows a bright field method for obtaining enhanced resolution of partial dislocations at 100 KV accelerating potential will be described.A brief discussion of an application will first be given. A study of intermediate temperature creep processes in commercial nickel-base alloys strengthened by the Ll2 Ni3 Al γ precipitate has suggested that partial dislocations such as those labelled 1 and 2 in Fig. 1(a) are in reality composed of two closely spaced a/6 <112> Shockley partials. Stacking fault contrast, when present, tends to obscure resolution of the partials; thus, conditions for resolution must be chosen such that the phase shift at the fault is 0 or a multiple of 2π.

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

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