Ref HAL: hal-04173618_v1
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In the following pages we take a fresh look at the ancient Indian Chakravāla or Cyclic algorithm for solving the Brahmagupta-Fermat-Pell quadratic Diophantine equation in integers taking account of recent developments. This is the oldest general algorithm (1150 CE) for solving this equation. The algorithm can be proved directly in its own terms, following a recent work by A. Bauval, to always lead to a periodic solution in a finite number of steps. We review a slightly modified version of this work. It forms the basis of a reinterpretation of this algorithm in the framework of a reduction theory of binary indefinite integer valued quadratic forms on which the modular group SL(2, Z) acts. The reduction condition of this algorithm are restated in terms of the roots of the quadratic form. The SL(2, Z) action on (reduced) roots of this form furnish semi-regular continued fractions which are periodic and which furnish SL(2, Z) automorphisms of the quadratic form. Very much as in the classical theory of Gauss, this gives solutions of the Brahmagupta-Fermat-Pell equation. We give a proof that the solution at the end of the first cycle is fundamental (positive and least). We also give the conversion to regular continued fractions which involve larger periods. A number of worked out examples are given to illustrate the main points and this for the benefit of the uninitiated reader.

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In these lectures I will introduce and explore properties of the Renormalisation Group (RG) both in its discrete and continuous forms. Topics to be discussed include multi scale expansions, the RG, Gaussian fixed points, Wick products from the RG point of view, ultraviolet cutoff removal, flow equations and perturbative flows, trivial and non-trivial fixed points, and critical field theories. Rigorous Discrete RG analysis. I will discuss a rigorous application for a non-trivial fixed point.

Ref HAL: hal-01492858_v1
Ref Arxiv: 1701.04111
DOI: 10.1007/s10955-017-1828-5
WoS: WOS:000407382100003
Ref. & Cit.: NASA ADS
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3 Citations
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In previous papers, [M1, M2], [M3], we proved the existence as well as regularity of a finite range decomposition for the resolvent $G_{\alpha} (x-y,m^2) = ((-\Delta)^{\alpha\over 2} + m^{2})^{-1} (x-y) $, for $0<\alpha <2$ and all real $m$, in the lattice ${\bf Z}^{d}$ for dimension $d\ge 2$. In this paper, which is a continuation of the previous one, we extend those results by proving the existence as well as regularity of a finite range decomposition for the same resolvent but now on the lattice torus ${\bf Z}^{d}/L^{N+1}{\bf Z}^{d} $ for $d\ge 2$ provided $m\neq 0$ and $0<\alpha <2$. We also prove differentiability and uniform continuity properties with respect to the resolvent parameter $m^{2}$. Here $L$ is any odd positive integer and $N\ge 2$ is any positive integer.

Commentaires: 17 pages

Ref HAL: hal-01492689_v1
DOI: 10.1007/s10955-016-1687-5
WoS: WOS:000392332700011
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Erratum to: On a Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian

Ref HAL: hal-01338274_v1
Ref Arxiv: 1512.02877
DOI: 10.1007/s10955-016-1507-y
WoS: WOS:000375579300009
Ref. & Cit.: NASA ADS
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3 Citations
Résumé:

We prove the existence as well as regularity of a finite range decomposition for the resolvent $G_{\alpha} (x-y,m^2) = ((-\Delta)^{\alpha\over 2} + m^{2})^{-1} (x-y) $, for $0<\alpha<2$ and all real $m$, in the lattice ${\mathbf Z}^{d}$ as well as in the continuum ${\mathbf R}^{d}$ for dimension $d\ge 2$. This resolvent occurs as the covariance of the Gaussian measure underlying weakly self- avoiding walks with long range jumps (stable L\'evy walks) as well as continuous spin ferromagnets with long range interactions in the long wavelength or field theoretic approximation. The finite range decomposition should be useful for the rigorous analysis of both critical and off-critical renormalisation group trajectories. The decomposition for the special case $m=0$ was known and used earlier in the renormalisation group analysis of critical trajectories for the above models below the critical dimension $d_c =2\alpha$.

Commentaires: 16 pages, 1 figure, typos corrected, corrected, reference added, one sentence added

Ref HAL: hal-01203777_v1
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There is considerable interest at present in Singular Stochastic PDEs especially in connection with rough path theory in different guises. The mathematical work started with results by Giovanni Jona-Lasinio and myself a long time ago. I will review what was accomplished at the time and the progress that has been made since then. One reason for presenting this to this audience is that there is, in my opinion, considerable scope for rigorous renormalsation group work in this subject as I will explain. I will also attempt to prepare the ground for Jona-Lasinio's talk on our common work on large deviations for singular SPDEs.

Ref HAL: cel-01239463_v1
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In this lecture I will first review rigorous results in the theory of ferromagnets with continuous spins and long range interactions on a lattice. In the long wave length approximation we get a lattice field theory of continuous spins with a non-linear perturbation.I will then give a rigorous version of Wilson's renormalisation group to take the continuum limit as a scaling limit in a finite volume. We construct a critical theory which has a non trivial fixed point. Moreover we prove the existence of correlation functions and obtain the true scaling dimension of fields.