National Institute for Pure and Applied Mathematics

We establish the existence of one-parameter families of helicoidal surfaces of $\mathbb H^2\times \mathbb R$ which, under mean curvature flow, simultaneously rotate about a vertical axis and translate vertically.

The method of sound propagation modeling based on the mode parabolic equations (MPEs) theory is applied to the verification scenarios for environmental impact assessment. The results for selected scenarios from the 2022 Cambridge Joint Industry Programme Acoustic Modelling Workshop and the configuration of the computational programs AMPLE and MPE for these scenarios is discussed. Furthermore, it is revealed how the results for these scenarios change in the case of the bottom slope across and along the propagation path. It is observed that for the cross-slope propagation scenario, the distribution of acoustic energy over decidecade frequency bands does not depend on the slope angle and is practically the same as that for range-independent environment. At the same time, the dependence of energy distribution is noticeable for up- and downslope propagation scenarios, where greater slope angles result in higher propagation loss. It is also shown that MPEs are capable of adequately handling typical sound propagation problems related to the environmental impact assessment for frequencies up to 1000 Hz. A possibility of using frequency-dependent mesh size and number of modes must be implemented in codes based on this approach.

We develop techniques to construct isomorphisms between simple affine W -algebras and affine vertex algebras at admissible levels. We then apply these techniques to obtain many new, and conjecturally all, admissible collapsing levels for affine W -algebras. In short, if a simple affine W -algebra at a given level is equal to its affine vertex subalgebra generated by the centraliser of an ${\mathfrak {sl}}_2$ -triple associated with the underlying nilpotent orbit, then that level is said to be collapsing . Collapsing levels are important both in representation theory and in theoretical physics. Our approach relies on two fundamental invariants of vertex algebras. The first one is the associated variety , which, in the context of admissible level simple affine W -algebras, leads to the Poisson varieties known as nilpotent Slodowy slices. We exploit the singularities of these varieties to detect possible collapsing levels. The second invariant is the asymptotic datum . We prove a general result asserting that, under appropriate hypotheses, equality of asymptotic data implies isomorphism at the level of vertex algebras. Then we use this to give a sufficient criterion, of combinatorial nature, for an admissible level to be collapsing. Our methods also allow us to study isomorphisms between quotients of W -algebras and extensions of simple affine vertex algebras at admissible levels. Based on such examples, we are led to formulate a general conjecture: for any finite extension of vertex algebras, the induced morphism between associated Poisson varieties is dominant.

Matrix-valued (multivariate) correlation functions are increasingly used within both the statistics and machine learning communities, but their properties have been studied to a limited extent. The motivation of this paper comes from the fact that the celebrated local stationarity construction for scalar-valued correlations has not been considered for the matrix-valued case. The main reason is a lack of theoretical support for such a construction. We explore the problem of extending a matrix-valued correlation from a d-dimensional ball with arbitrary radius into the d-dimensional Euclidean space. We also consider such a problem over product spaces involving the d-dimensional ball with arbitrary radius. We then provide a useful architecture to matrix-valued local stationarity by defining the class of p-exponentially convex matrix-valued functions, and characterize such a class as scale mixtures of the d-Schoenberg kernels against certain families of measures. We exhibit bijections from such a class into the class of positive semidefinite matrix-valued functions and we extend exponentially convex matrix-valued functions from d-dimensional balls into the d-dimensional Euclidean space. We finally provide similar results for the case of function-valued correlations defined over certain Hilbert spaces.

The statistical behavior of scalars passively advected by random flows exhibits intermittency in the form of anomalous multiscaling, in many ways similar to the patterns commonly observed in incompressible high-Reynolds fluids. This similarity suggests a generic dynamical mechanism underlying intermittency, though its specific nature remains unclear. Scalar turbulence is framed in a linear setting that points towards a zero-mode scenario connecting anomalous scaling to the presence of statistical conservation laws; the duality is fully substantiated within Kraichnan theory of random flows. However, extending the zero-mode scenario to nonlinear settings faces formidable technical challenges. Here, we revisit the scalar problem in the light of a hidden symmetry scenario introduced in recent deterministic turbulence studies addressing the Sabra shell model and the Navier–Stokes equations. Hidden symmetry uses a rescaling strategy based entirely on symmetry considerations, transforming the original dynamics into a rescaled (hidden) system; It yields the universality of Kolmogorov multipliers and ultimately identifies the scaling exponents as the eigenvalues of Perron-Frobenius operators. Considering a minimal shell model of scalar advection of the Kraichnan type that was previously studied by Biferale & Wirth, the present work extends the hidden symmetry approach to a stochastic setting, in order to explicitly contrast it with the zero-mode scenario. Our study indicates that the zero-mode and the multiplicative scenarios are intrinsically related. While the zero-mode approach solves the eigenvalue problem for pth$p {{\text {th}}}$ order correlation functions, Perron-Frobenius (multiplicative) scenario defines a similar eigenvalue problem in terms of pth$p{\text {th}}$ order measures. For systems of the Kraichnan type, the first approach provides a quantitative chararacterization of intermittency, while the second approach highlights the universal connection between the scalar case and a larger class of hydrodynamic models.

In 1981, Frisch and Morf (1981 Phys. Rev. A 23 2673–705) postulated the existence of complex singularities in solutions of Navier–Stokes equations. Present progress on this conjecture is hindered by the computational burden involved in simulations of the Euler equations or the Navier–Stokes equations at high Reynolds numbers. We investigate this conjecture in the case of fluid dynamics on log-lattices, where the computational burden is logarithmic concerning ordinary fluid simulations. We analyze properties of potential complex singularities in both 1D and 3D models for lattices of different spacings. Dominant complex singularities are tracked using the singularity strip method to obtain new scalings regarding the approach to the real axis and the influence of normal, hypo and hyper dissipation.

We examine the vanishing adsorption limit of solutions of Riemann problems for the Glimm–Isaacson model of chemical flooding of a petroleum reservoir. A contact discontinuity is deemed admissible if it is the limit of traveling waves or rarefaction waves for an augmented system that accounts for weak chemical adsorption onto the rock. We prove that this criterion justifies the admissibility criteria adopted previously by Keyfitz–Kranzer, Isaacson–Temple and de Souza–Marchesin, provided that the fractional flow function depends monotonically on chemical concentration. We also demonstrate that the adsorption criterion selects the undercompressive contact discontinuities required to solve the general Riemann problem in an example model with non-monotone dependence.

In this paper, we extend the well‐known concentration–compactness principle of P.L. Lions to Orlicz spaces. As an application, we show an existence result to some critical elliptic problem with nonstandard growth.

We investigate the Ising and Heisenberg models using the block renormalization group method (BRGM), focusing on its behavior across different system sizes. The BRGM reduces the number of spins by a factor of 1/2 (1/3) for the Ising (Heisenberg) model, effectively preserving essential physical features of the model while using only a fraction of the spins. Through a comparative analysis, we demonstrate that as the system size increases, there is an exponential convergence between results obtained from the original and renormalized Ising Hamiltonians, provided the coupling constants are redefined accordingly. Remarkably, for a spin chain with 24 spins, all physical features, including magnetization, correlation function, and entanglement entropy, exhibit an exact correspondence with the results from the original Hamiltonian. The study of the Heisenberg model also shows this tendency, although complete convergence may appear for a size much larger than 24 spins, and is therefore beyond our computational capabilities. The success of BRGM in accurately characterizing the Ising model, even with a relatively small number of spins, underscores its robustness and utility in studying complex physical systems, and facilitates its simulation on current NISQ computers, where the available number of qubits is largely constrained.

We consider a percolation model, the vacant set of random interlacements on , , in the regime of parameters in which it is strongly percolative. By definition, such values of pinpoint a robust subset of the supercritical phase, with strong quantitative controls on large local clusters. In the present work, we give a new characterization of this regime in terms of a single property, monotone in , involving a disconnection estimate for . A key aspect is to exhibit a gluing property for large local clusters from this information alone, and a major challenge in this undertaking is the fact that the conditional law of exhibits degeneracies. As one of the main novelties of this work, the gluing technique we develop to merge large clusters accounts for such effects. In particular, our methods do not rely on the widely assumed finite‐energy property, which the set does not possess. The characterization we derive plays a decisive role in the proof of a lasting conjecture regarding the coincidence of various critical parameters naturally associated to in the companion article [Duminil‐Copin, Goswami, Rodriguez, Severo, and Teixeira, Phase transition for the vacant set of random walk and random interlacements , arXiv:2308.07919, 2023].

We simulated a turbulent pipe flow within the lattice Boltzmann method using a multiple-relaxation-time collision operator with Maxwell–Boltzmann equilibrium distribution expanded, for the sake of a more accurate description, up to the sixth order in Hermite polynomials. The moderately turbulent flow ( Reτ≈181.3) is able to reproduce up to the fourth statistical moment with great accuracy compared with other numerical schemes and with experimental data. A coherent structure identification was performed based on the most energetic streamwise turbulent mode, which revealed a surprising memory effect related to the large-scale forcing scheme that triggered the pipe's turbulent state. We observe that the existence of large-scale motions that are out of the pipe's stationary regime does not affect the flow's detailed single-point statistical features. Furthermore, the transitions between the coherent structures of different topological modes were analyzed as a stochastic process. We find that for finely resolved data, the transitions are effectively Markovian, but for larger decimation time lags, due to topological mode degeneracy, non-Markovian behavior emerges, in agreement with previous experimental studies.

Пусть $\mathbb H$ - алгебра кватернионов, порожденная I,J и K. Будем говорить, что гиперкомплексная нильпотентная алгебра Ли $\mathfrak g$ является $\mathbb H$-разрешимой, если существует последовательность $\mathbb H$-инвариантных подалгебр, содержащих $\mathfrak g_{i+1}=[\mathfrak g_i,\mathfrak g_i]$, $\mathfrak g=\mathfrak g_0\supset\mathfrak g_1^{\mathbb H}\supset\mathfrak g_2^{\mathbb H}\supset\cdots\supset\mathfrak g_{k-1}^{\mathbb H}\supset\mathfrak g_k^{\mathbb H}=0$ такая, что $[\mathfrak g_i^{\mathbb H},\mathfrak g_i^{\mathbb H}]\subset\mathfrak g^{\mathbb H}_{i+1}$ и $\mathfrak g_{i+1}^{\mathbb H}=\mathbb H[\mathfrak g_i^{\mathbb H},\mathfrak g_i^{\mathbb H}]$. Пусть $N=\Gamma\setminus G$ - гиперкомплексное нильмногообразие с плоской связностью Обаты и $\mathfrak g=\operatorname{Lie}(G)$. Тогда алгебра Ли $\mathfrak g=\operatorname{Lie}(G)$ является $\mathbb H$-разрешимой.

We study the first chiral homology group of elliptic curves with coefficients in vacuum insertions of a conformal vertex algebra V. We find finiteness conditions on V guaranteeing that these homologies are finite dimensional, generalizing the C2$C_2$-cofinite, or quasi-lisse condition in the degree 0 case. We determine explicitly the flat connections that these homologies acquire under smooth variation of the elliptic curve, as insertions of the conformal vector and the Weierstrass ζ$\zeta$ function. We construct linear functionals associated to self-extensions of V-modules and prove their convergence under said finiteness conditions. These linear functionals turn out to be degree 1 analogs of the n-point functions in the degree 0 case. As a corollary we prove the vanishing of the first chiral homology group of an elliptic curve with values in several rational vertex algebras, including affine sl2$\mathfrak {sl}_2$ at non-negative integral level, the (2,2k+1)(2,2k+1)-minimal models and arbitrary simple affine vertex algebras at level 1. Of independent interest, we prove a Fourier space version of the Borcherds formula.

We study the scaling properties of the non-equilibrium stationary states (NESS) of a reaction-diffusion model. Under a suitable smallness condition, we show that the density of particles satisfies a law of large numbers with respect to the NESS, with an explicit rate of convergence, and we also show that at mesoscopic scales the NESS is well approximated by a local equilibrium (product) measure, in the total variation distance. In addition, in dimensions d≤3$d \le 3$ we show a central limit theorem for the density of particles under the NESS. The corresponding Gaussian limit can be represented as an independent sum of a white noise and a massive Gaussian free field, and in particular it presents macroscopic correlations.

For an elliptic surface π:X→P1$\pi :X\rightarrow \mathbb {P}^1$ defined over a number field K, a theorem of Silverman shows that for all but finitely many fibres above K-rational points, the resulting elliptic curve over K has Mordell-Weil rank at least as large as the rank of the group of sections of π$\pi$. When X is a K3 surface with two distinct elliptic fibrations, we show that the set of K-rational points of P1$\mathbb {P}^1$ for which this rank inequality is strict, is not a thin set, under certain hypothesis on the fibrations. Our results provide one of the first cases of this phenomenon beyond that of rational elliptic surfaces.

It is known that the space of holomorphic foliations on some compact holomorphic manifold is a union of algebraic spaces. Here we intend to explain some results concerning the irreducible components of the space of holomorphic foliations of codimension one on projective spaces of dimension $n\ge 3$.