We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.
We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.
We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.
We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere (hat {mathbb {C}}). Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.
In this note, a Wegner estimate for random divergence-type operators that are monotone in the randomness is proven. The proof is based on a recently shown unique continuation estimate for the gradient and the ensuing eigenvalue liftings. The random model which is studied here contains quite general random perturbations, among others, some that have a non-linear dependence on the random parameters.
We establish a general method for obtaining set-theoretical solutions to the 3D reflection equation by using known ones to the Zamolodchikov tetrahedron equation, where the former equation was proposed by Isaev and Kulish as a boundary analog of the latter. By applying our method to Sergeev’s electrical solution and a two-component solution associated with the discrete modified KP equation, we obtain new solutions to the 3D reflection equation. Our approach is closely related to a relation between the transition maps of Lusztig’s parametrizations of the totally positive part of SL3 and SO5, which is obtained via folding the Dynkin diagram of A3 into one of B2.
For a manifold M with an integral closed 3-form ω, we construct a PU(H)-bundle and a Lie groupoid over its total space, together with a curving in the sense of gerbes. If the form is non-degenerate, we furthermore give a natural Lie 2-algebra quasi-isomorphism from the observables of (M, ω) to the weak symmetries of the above geometric structure, generalising the prequantisation map of Kostant and Souriau.
Renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity.
Feynman graphon representations of Feynman diagrams lead us to build a new separable Banach space (mathcal {S}^{Phi ,g}_{approx }) originated from the collection of all Dyson–Schwinger equations in a given (strongly coupled) gauge field theory Φ with the bare coupling constant g. We study the Gateaux differential calculus on the space of functionals on (mathcal {S}^{Phi ,g}_{approx }) in terms of a new class of homomorphism densities. We then show that Taylor series representations of smooth functionals on (mathcal {S}^{Phi ,g}_{approx }) provide a new analytic description for solutions of combinatorial Dyson–Schwinger equations.
We are concerned with the long-time asymptotic behavior of the solution for the focusing Hirota equation (also called third-order nonlinear Schr?dinger equation) with symmetric, non-zero boundary conditions (NZBCs) at infinity. Firstly, based on the Lax pair with NZBCs, the direct and inverse scattering problems are used to establish the oscillatory Riemann-Hilbert (RH) problem with distinct jump curves. Secondly, the Deift-Zhou nonlinear steepest-descent method is employed to analyze the oscillatory RH problem such that the long-time asymptotic solutions are proposed in two distinct domains of space-time plane (i.e., the plane-wave and modulated elliptic-wave domains), respectively. Finally, the modulation instability of the considered Hirota equation is also investigated.
In the present paper we show that it is possible to obtain the well known Pauli group P = 〈X,Y,Z | X2 = Y2 = Z2 =?1,(Y Z)4 = (ZX)4 = (XY )4 =?1〉 of order 16 as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere S3. The first of these spaces of orbits is realized via an action of the quaternion group Q8 on S3; the second one via an action of the cyclic group of order four (mathbb {Z}(4)) on S3. We deduce a result of decomposition of P of topological nature and then we find, in connection with the theory of pseudo-fermions, a possible physical interpretation of this decomposition.
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