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

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

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

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

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

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 Gâteaux differential calculus on the space of functionals on \(\mathcal {S}^{\Phi ,g}_{\approx }\) in

We are concerned with the long-time asymptotic behavior of the solution for the focusing Hirota equation (also called third-order nonlinear Schrödinger 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

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

A Correction to this paper has been published: https://doi.org/10.1007/s11040-021-09380-8

Let m denote the number of quasielectrons (QEs) in a quantum Hall system containing N particles altogether. We show in several general cases that for systems containing m QEs in a single angular momentum shell above N − m Fermions in an incompressible quantum liquid (IQL) state having filling factor ν =1/3 that there always exists a configuration whose symmetric correlation function G is nonzero. This

An interesting feature of the finite-dimensional real spectral triple (A,H,D,J) of the Standard Model is that it satisfies a “second-order” condition: conjugation by J maps the Clifford algebra \(\mathcal {C}\ell _{D}(A)\) into its commutant, which in fact is isomorphic to the Clifford algebra itself (H is a self-Morita equivalence \(\mathcal {C}\ell _{D}(A)\)-bimodule). This resembles a property of

We consider the asymmetric simple exclusion process (ASEP) with open boundary condition at the left boundary, where particles exit at rate γ and enter at rate α = γτ2, and where τ is the asymmetry parameter in the bulk. At the right boundary, particles neither enter nor exit. By mapping the generator to the Hamiltonian of an XXZ quantum spin chain with reflection matrices, and using previously known

In this work, an efficient method for constructing a complete integral of the geodesic Hamilton–Jacobi equation on pseudo-Riemannian manifolds with simply transitive groups of motions is suggested. The method is based on using a special transition to canonical coordinates on coadjoint orbits of the group of motion. As a non-trivial example, we consider the problem of constructing a complete integral

In this paper, we study a two-point boundary value problem consisting of the heat equation on the open interval (0,1) with boundary conditions which relate first and second spatial derivatives at the boundary points. Moreover, the unique solution to this problem can be represented probabilistically in terms of a sticky Brownian motion. This probabilistic representation is attained from the stochastic

We discretise the Lax pair for the reduced Nahm systems and prove its equivalence with the Kahan–Hirota–Kimura discretisation procedure. We show that these Lax pairs guarantee the integrability of the discrete reduced Nahm systems providing an invariant. Also, we show that Nahm systems cannot solve the general problem of characterisation of the integrability for Kahan–Hirota–Kimura discretisations

We show the validity of the cluster expansion in the canonical ensemble for the Ising model. We compare the lower bound of its radius of convergence with the one computed by the virial expansion working in the grand-canonical ensemble. Using the cluster expansion we give direct proofs with quantification of the higher order error terms for the decay of correlations, central limit theorem and large

We find noncommutative analogs for well-known polynomial evolution systems with higher conservation laws and symmetries. The integrability of obtained non-Abelian systems is justified by explicit zero curvature representations with spectral parameter.

We contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin

We prove essential self-adjointness of the spatial part of the linear Klein-Gordon operator with external potential for a large class of globally hyperbolic manifolds. The proof is conducted by a fusion of new results concerning globally hyperbolic manifolds, the theory of weighted Hilbert spaces and related functional analytic advances.

We study discrete random Schrödinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green’s function and the smoothness of the local density of states either at weak disorder and at energies in proximity of the unperturbed spectrum or at strong disorder and at any energy

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in \(d \geqslant 2\) dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on \(\mathbb {Z}^{d}\) with parameter p ∈ (0,1]. For each occupied site v, and for each of the 2d possible coordinate directions, declare the entire

We give a new proof that the resolvent of the renormalised Nelson Hamiltonian at fixed total momentum P improves positivity in the (momentum) Fock-representation, for every P. The argument is based on an explicit representation of the renormalised operator and its domain using interior boundary conditions, which allows us to avoid the intermediate steps of regularisation and renormalisation used in

We characterize the Darboux integrability and the global dynamics of the 3-dimensional polynomial differential systems of degree 2 which are invariant under the D 2 symmetric group, which have been partially studied previously by several authors.

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in $L^{2}(\mathbb R)$ was considered by Gabardo and Nashed (J Funct. Anal. 158:209-241, 1998). In this setting, the associated translation set ${\Lambda } =\left \{ 0,r/N\right \}+2 \mathbb Z$ is no longer a discrete subgroup of $\mathbb R$ but a spectrum associated with a certain one-dimensional

As discussed in arXiv:1501.03298, Physics suggests that, close to cosmological singularities, the effective Planck length diverges, hence a "quantum point" becomes infinitely extended. We argue that, as a consequence, at the origin of times spacetime might reduce effectively to a single point and interactions disappear. This last point is supported by converging evidences in two different approaches

In 1983, Klaus studied a class of potentials with bumps and computed the essential spectrum of the associated Schr{o}dinger operator with the help of some localisations at infinity. A key hypothesis is that the distance between two consecutive bumps tends to infinity at infinity. In this article, we introduce a new class of graphs (with patterns) that mimics this situation, in the sense that the distance

In this paper, first, we investigate the commuting property between the normal Jacobi operator~${\bar R}_N$ and the structure Jacobi operator~$R_{\xi}$ for Hopf real hypersurfaces in the complex quadric~$Q^m = SO_{m+2}/SO_mSO_2$, $m \geq 3$, which is defined by ${\bar R}_N R_{\xi} = R_{\xi}{\bar R}_N$. Moreover, a new characterization of Hopf real hypersurfaces with $\mathfrak A$-principal singular

The two-time distribution gives the limiting joint distribution of the heights at two different times of a local 1D random growth model in the curved geometry. This distribution has been computed in a specific model but is expected to be universal in the KPZ universality class. Its marginals are the GUE Tracy-Widom distribution. In this paper we study two limits of the two-time distribution. The first

Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected

It is shown that when dependence on the second flow of the KP hierarchy is added, the resulting semi-stationary wave function of certain points in George Wilson's adelic Grassmannian are generating functions of the exceptional Hermite orthogonal polynomials. This surprising correspondence between different mathematical objects that were not previously known to be so closely related is interesting in

Let $X$ be a compact Riemann surface of genus $g \geq 2$, and let $D \subset X$ be a fixed finite subset. Let $\mathcal{M}(r,d,\alpha)$ denote the moduli space of stable parabolic $G$-bundles (where $G$ is an complex orthogonal or symplectic group) of rank $r$, degree $d$ and weight type $\alpha$ over $X$. Hitchin discovered that the cotangent bundle of the moduli space of stable bundles on an algebraic

In this paper, we prove the uniform-in-η estimates of the local strong solutions of the density-dependent incompressible MHD system in a bounded domain. Here η is the resistivity coefficient.

We propose a general procedure to construct noncommutative deformations of an algebraic submanifold $M$ of $\mathbb{R}^n$, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of $\mathbb{R}^n$, arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883-1911, 2006), whereby the commutative

Let $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ be self-adjoint and let $A:D(H)\to{\mathscr F}$ (playing the role of the annihilator operator) be $H$-bounded. Assuming some additional hypotheses on $A$ (so that the creation operator $A^{*}$ is a singular perturbation of $H$), by a twofold application of a resolvent Krein-type formula, we build self-adjoint realizations $\widehat H$ of the formal Hamiltonian

In this paper we consider Wronskian polynomials labeled by partitions that can be factorized via the combinatorial concepts of $p$-cores and $p$-quotients. We obtain the asymptotic behavior for these polynomials when the $p$-quotient is fixed while the size of the $p$-core grows to infinity. For this purpose, we associate the $p$-core with its characteristic vector and let all entries of this vector

We prove that Weyl quantization preserves constant of motion of the Harmonic Oscillator. We also prove that if $f$ is a classical constant of motion and $\mathfrak{Op}(f)$ is the corresponding operator, then $\mathfrak{Op}(f)$ maps the Schwartz class into itself and it defines an essentially selfadjoint operator on $L^2(\mathbb R^n)$. As a consequence, we provide detailed spectral information of $\mathfrak{Op}(f)$

The formation of singularity of strong solutions to the two-dimensional (2D) Cauchy problem of the compressible Navier-Stokes equations with variable viscosity is considered. It is shown that for the initial density allowing vacuum, the strong solution exists globally if the density is bounded from above. Some weighted estimates play a crucial role in the proof.

We consider a scalar quantum field $\phi$ with arbitrary polynomial self-interaction in perturbation theory. If the field variable $\phi$ is repaced by a local diffeomorphism $\phi(x) = \rho(x) + a_1 \rho^2(x) +\ldots$, this field $\rho$ obtains infinitely many additional interaction vertices. We show that the $S$-matrix of $\rho$ coincides with the one of $\phi$ without using path-integral arguments

We develop scattering theory for non-local Schrodinger operators defined by functions of the Laplacian that include its fractional power $(-\Delta)^\rho$ with $0<\rho\leqslant1$. In particular, our function belongs to a wider class than the set of Bernstein functions. By showing the existence and non-existence of the wave operators, we clarify the threshold between the short and long-range decay conditions

This paper provides an alternative, much simpler, definition for Li-Bland's LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs of 2-representations. We use the known geometric examples of LA-Courant algebroids

We construct a solution for the $1d$ integro-differential stationary equation derived from a finite-volume version of the mesoscopic model proposed in [G. B. Giacomin, J. L. Lebowitz, "Phase segregation dynamics in particle system with long range interactions", Journal of Statistical Physics 87(1) (1997)]. This is the continuous limit of an Ising spin chain interacting at long range through Kac potentials

We study spinful non-interacting electrons moving in two-dimensional materials which exhibit a spectral gap about the Fermi energy as well as time-reversal invariance. Using Fredholm theory we revisit the (known) bulk topological invariant, define a new one for the edge, and show their equivalence (the bulk-edge correspondence) via homotopy.

We propose an extended version of supersymmetric quantum mechanics which can be useful if the Hamiltonian of the physical system under investigation is not Hermitian. The method is based on the use of two, in general different, superpotentials. Bi-coherent states of the Gazeau-Klauder type are constructed and their properties are analyzed. Some examples are also discussed, including an application

We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to − ∞ $-\infty $ .

The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems.

In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperber's construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the

We consider the existence of global attractor for non-Newtonian fluids near the BCS-BEC crossover in this paper. By Gronwall’s inequality, interpolar inequality and the techniques for proving regularity theory in partial differential equations, etc., we establish suitable prior estimates and obtain the global attractor for non-Newtonian fluids.

We regard the Cauchy problem for a particular Whitham–Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The proof of well-posedness relies on energy estimates. However, due to the symmetry lack of the nonlinear part, in order to close the a priori estimates one has to modify

We study the motion of charged particle under a natural choice of electromagnetic field in a general class of compact homogeneous spaces. As a special case we describe the motion in homogeneous Riemannian spaces ( G / H , g ), where g is any deformation of a normal metric along the fibers of a homogeneous fibration K / H → G / H → G / K $K/H\rightarrow G/H\rightarrow G/K$ .

Given N absolutely continuous probabilities ρ 1 , … , ρ N $\rho _{1}, \dotsc , \rho _{N}$ over ℝ d ${\mathbb {R}}^{d}$ which have Sobolev regularity, and given a transport plan P with marginals ρ 1 , … , ρ N $\rho _{1}, \dotsc , \rho _{N}$ , we provide a universal technique to approximate P with Sobolev regular transport plans with the same marginals. Moreover, we prove a sharp control of the energy

We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type n − α for α > 0. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region α > 1 2 $\alpha >\frac 12$

For a system of N bosons in one space dimension with two-body δ -interactions the Hamiltonian can be defined in terms of the usual closed semi-bounded quadratic form. We approximate this Hamiltonian in norm resolvent sense by Schrödinger operators with rescaled two-body potentials, and we estimate the rate of this convergence.

We construct P(ϕ)1-processes indexed by the full time-line, separately derived from the functional integral representations of the relativistic and non-relativistic Nelson models in quantum field theory. These two cases differ essentially by sample path regularity. Associated with these processes we define a martingale which, under an appropriate scaling, allows to obtain a central limit theorem for

We analyse a Curie-Weiss model with two disjoint groups of spins with homogeneous coupling. We show that similarly to the single-group Curie-Weiss model a bivariate law of large numbers holds for the normed sums of both groups’ spin variables. We also show central limit theorem in the high temperature regime.

In this paper, by maximal regularity estimates for the Stokes equation, we establish the local existence of solutions to the three-dimensional magnetic Bénard system with Hall, ion-slip effects and zero thermal conductivity under the condition that the initial data *.

In this article, we study the mean curvature type flow of spacelike graphical hypersurfaces in Lorentzian warped product. This flow was introduced by Guan and Li in [ 6 ]. Under mild assumptions on the warping function and the Ricci curvature of the base manifold, we obtain the longtime existence and smooth convergence to an umbilic slice for this flow in Lorentzian setting. As an application of the

We study the structure of the Mather and Aubry sets for the family of Lagrangians given by the kinetic energy associated to a Riemannian metric g on a closed manifold M . In this case the Euler-Lagrange flow is the geodesic flow of ( M , g ). We prove that there exists a residual subset of the set of all conformal metrics to g, such that, if g ¯ ∈ G $ \overline g \in G$ then the corresponding geodesic

We give a direct proof for the positivity of Kirillov’s character on the convolution algebra of smooth, compactly supported functions on a connected, simply connected nilpotent Lie group G . Then we use this positivity result to construct a representation of G × G and establish a G × G -equivariant isometric isomorphism between our representation and the Hilbert–Schmidt operators on the underlying