We extend the approach from [arXiv:2110.15301] to prove windowed spectral projection estimates and a generalized Weyl law for the (Weyl) quantized baker's map on the torus. The spectral window is allowed to shrink. As a consequence, we obtain a strengthening of the quantum ergodic theorem from [arXiv:math-ph/0412058] to hold in shrinking spectral windows, a Weyl law on uniform spreading of eigenvalues, and statistics of random band-limited waves. Using similar techniques, we also investigate generic random eigenbases of a different (non-Weyl) quantization, the Walsh-quantized $D$-baker map, which has high degeneracies in its spectrum. For generic random eigenbases, we prove gaussian eigenstate statistics and QUE with high probability in the semiclassical limit.

The present notes provide an extended version of a small lecture course given at the Humboldt Universit\"at zu Berlin in the Winter Term 2022/23 (of 36 hours). The material starting in Section 5.4 was added afterwards. The aim of these notes to give an introductory overview on the analytical approaches for gradient-flow equations in Hilbert spaces, Banach spaces, and metric spaces and to show that on the first entry level these theories have a lot in common. The theories and their specific setups are illustrated by suitable examples and counterexamples.

In this paper, the propagation of the blast (shock) waves in non-ideal gas atmosphere in rotational medium is studied using a power series method in cylindrical geometry. The flow variables are assumed to be varying according to the power law in the undisturbed medium with distance from the symmetry axis. To obtain the similarity solution, the initial density is considered as constant in the undisturbed medium. Approximate analytical solutions are obtained using Sakurai's method by extending the power series of the flow variables in power of ${\left( {\frac{{{a_0}}}{U}} \right)^2}$, where $U$ and $a_0$ are the speeds of the shock and sound, respectively, in undisturbed fluid. The strong shock wave is considered for the ratio ${\left( {\frac{{{a_0}}}{U}} \right)^2}$ which is considered to be a small quantity. With the aid of that method, the closed-form solutions for the zeroth-order approximation is given as well as first-order approximate solutions are discussed. Also, with the help of graphs behind the blast wave for the zeroth-order approximation, the distributions of variables such as density, radial velocity, pressure and azimuthal fluid velocity are analyzed. The results for the rotationally axisymmetric non-ideal gas environment are compared to those for the ideal gas atmosphere.

We derive the action and symmetries of the bosonic sector of non-Lorentzian IIB supergravity by taking the non-relativistic string limit. We find that the bosonic field content is extended by a Lagrange multiplier that implements a restriction on the Ramond-Ramond fluxes. We show that the SL(2,R) transformation rules of non-Lorentzian IIB supergravity form a novel, nonlinear polynomial realization. Using classical invariant theory of polynomial equations and binary forms, we will develop a general formalism describing the polynomial realization of SL(2,R) and apply it to the special case of non-Lorentzian IIB supergravity. Using the same formalism, we classify all the relevant SL(2,R) invariants. Invoking other bosonic symmetries, such as the local boost and dilatation symmetry, we show how the bosonic part of the non-Lorentzian IIB supergravity action is formed uniquely from these SL(2,R) invariants. This work also points towards the concept of a non-Lorentzian bootstrap, where bosonic symmetries in non-Lorentzian supergravity are used to bootstrap the bosonic dynamics in Lorentzian supergravity, without considering the fermions.

The R\"ossler System is one of the best known chaotic dynamical systems, exhibiting a plethora of complex phenomena - and yet, only a few studies tackled its complexity analytically. In this paper we find sufficient conditions for the existence of chaotic dynamics for the R\"ossler System. This will allow us to prove the existence of infinitely many periodic trajectories for the flow, and determine their knot types.

The forced Korteweg-de Vries (fKdV) equation describes incompressible inviscid free surface flows over some arbitrary topography. We investigate solitary and hydraulic fall solutions to the fKdV equation. Numerical results show that the calculation of exponentially decaying solitary waves at the critical Froude number is a nonlinear eigenvalue problem. Furthermore we show how exponential decaying solitary waves evolve into the continuous spectrum of algebraic decaying solitary waves. A novel and stable numerical approach using the wave-resistance coefficient and tabletop solutions is used to generate the hydraulic fall parametric space. We show how hydraulic fall solutions periodically evolve into exponential decaying solitary waves.

We consider the one-parametric family of self-adjoint realizations of the two-dimensional massive Dirac operator with a Lorentz scalar $\delta$-shell interaction of strength $\tau\in\mathbb{R}\setminus\{-2,0,2\}$ supported on a broken line of opening angle $2\omega$ with $\omega\in(0,\frac{\pi}{2})$. The essential spectrum of any such self-adjoint realization is symmetric with respect to the origin with a gap around zero whose size depends on the mass and, for $\tau < 0$, also on the strength of the interaction, but does not depend on $\omega$. As the main result, we prove that for any $N\in\mathbb{N}$ and strength $\tau\in(-\infty,0)\setminus\{-2\}$ the discrete spectrum of any such self-adjoint realization has at least $N$ discrete eigenvalues, with multiplicities taken into account, in the gap of the essential spectrum provided that $\omega$ is sufficiently small. Moreover, we obtain an explicit estimate on $\omega$ sufficient for this property to hold. For $\tau\in(0,\infty)\setminus\{2\}$, the discrete spectrum consists of at most one simple eigenvalue.

The behavior of simple eigenvalues of Aharonov-Bohm operators with many coalescing poles is discussed. In the case of half-integer circulation, a gauge transformation makes the problem equivalent to an eigenvalue problem for the Laplacian in a domain with straight cracks, laying along the moving directions of poles. For this problem, we obtain an asymptotic expansion for eigenvalues, in which the dominant term is related to the minimum of an energy functional associated with the configuration of poles and defined on a space of functions suitably jumping through the cracks. Concerning configurations with an odd number of poles, an accurate blow-up analysis identifies the exact asymptotic behaviour of eigenvalues and the sign of the variation in some cases. An application to the special case of two poles is also discussed.

The uniform even subgraph is intimately related to the Ising model, the random-cluster model, the random current model and the loop $\mathrm{O}$(1) model. In this paper, we first prove that the uniform even subgraph of $\mathbb{Z}^d$ percolates for $d \geq 2$ using its characterisation as the Haar measure on the group of even graphs. We then tighten the result by showing that the loop $\mathrm{O}$(1) model on $\mathbb{Z}^d$ percolates for $d \geq 2$ on some interval $(1-\varepsilon,1]$. Finally, our main theorem is that the loop $\mathrm{O}$(1) model and random current models corresponding to a supercritical Ising model are always at least critical, in the sense that their two-point correlation functions decay at most polynomially and the expected cluster sizes are infinite.

We propose a probabilistic derivation of the so-called chemical diffusion master equation (CDME) and describe an infinite dimensional moment generating function method for finding its analytical solution. CDMEs model by means of an infinite system of coupled Fokker-Planck equations the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles; here, we focus an creation and mutual annihilation chemical reactions combined with Brownian diffusion of the single particles. Our probabilistic approach mimics the derivation of backward Kolmogorov equations for birth-death continuous time Markov chains. Moreover, the proposed infinite dimensional moment generating function method links certain finite dimensional projections of the solution of the CDME to the solution of a single linear fourth order partial differential equation containing as many variables as the dimension of the aforementioned projection space.

In this paper, we continue the study of large $N$ problems for the Wick renormalized linear sigma model, i.e. $N$-component $\Phi^4$ model, in two spatial dimensions, using stochastic quantization methods and Dyson--Schwinger equations. We identify the large $N$ limiting law of a collection of Wick renormalized $O(N)$ invariant observables. In particular, under a suitable scaling, the quadratic observables converge in the large $N$ limit to a mean-zero (singular) Gaussian field denoted by $\mathcal{Q}$ with an explicit covariance; and the observables which are renormalized powers of order $2n$ converge in the large $N$ limit to suitably renormalized $n$-th powers of $\mathcal{Q}$. The quartic interaction term of the model has no effect on the large $N$ limit of the field, but has nontrivial contributions to the limiting law of the observables, and the renormalization of the $n$-th powers of $\mathcal{Q}$ in the limit has an interesting finite shift from the standard one. Furthermore, we derive the $1/N$ asymtotic expansion for the $k$-point functions of the quadratic observables by employing graph representations and analyzing the order of each graph from Dyson--Schwinger equations. Finally, turning to the stationary solutions to the stochastic quantization equations, with the Ornstein--Uhlenbeck process being the large $N$ limit, we derive here its next order correction in stationarity, as described by an SPDE with the right-hand side having explicit marginal law which involves the above field $\mathcal{Q}$.

This article presents a novel application of the Hirota bilinear formalism to the $N=2$ supersymmetric KdV and Burgers equations. This new approach avoids splitting N=2 equations into two $N=1$ equations. We use the super Bell polynomials to obtain bilinear representations and present multi-soliton solutions.

We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidal algebra $\mathcal E_{m|n}$ associated to Lie superalgebra $\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutative subalgebras $\mathcal B_{m|n}(s)$ depending on parameters $s=(s_1,\dots,s_{m+n})$, $\prod_i s_i=1$, given by appropriate regularity conditions. We show that $\mathcal B_{m|n}(s)$ is a free polynomial algebra and give explicit generators which conjecturally correspond to the traces of the $s$-weighted $R$-matrix computed on the degree zero part of $\mathcal E_{m|n}$ modules of levels $\pm 1$.

We investigate different problems regarding wave turbulence for the Benjamin-Bona-Mahony (BBM) equation in the context of discrete turbulence regime. In the part I, we investigate the behaviour of the correlations between the solution to the BBM equation at latter times with the initial datum.

We prove the large deviation principle for several entropy and cross entropy estimators based on return times and waiting times on shift spaces over finite alphabets. In the case of standard return times, we obtain a nonconvex large-deviation rate function. We consider shift-invariant probability measures satisfying some decoupling conditions which imply no form of mixing nor ergodicity. We establish precise relations between the rate functions of the different estimators, and between these rate functions and the corresponding pressures, one of which is the R\'enyi entropy function. The results apply in particular to irreducible Markov chains, equilibrium measures for Bowen-regular potentials, g-measures, invariant Gibbs states for summable interactions in statistical mechanics, and also to probability measures that may be far from Gibbsian, including some hidden Markov models and repeated quantum measurement processes.

We present two different arguments using stochastic analysis to construct super-renormalizable tensor field theories, namely the $\mathrm{T}^4_3$ and $\mathrm{T}^4_4$ models. The first approach is the construction of a Langevin dynamic combined with a PDE energy estimate while the second is an application of the variational approach of Barashkov and Gubinelli. By leveraging the melonic structure of divergences, regularising properties of non-local products, and controlling certain random operators, we demonstrate that for tensor field theories these arguments can be significantly simplified in comparison to what is required for $\Phi^4_d$ models.

We study the anisotropic Forchheimer-typed flows for compressible fluids in porous media. The first half of the paper is devoted to understanding the nonlinear structure of the anisotropic momentum equations. Unlike the isotropic flows, the important monotonicity properties are not automatically satisfied in this case. Therefore, various sufficient conditions for them are derived and applied to the experimental data. In the second half of the paper, we prove the existence and uniqueness of the steady state flows subject to a nonhomogeneous Dirichlet boundary condition. It is also established that these steady states, in appropriate functional spaces, have local H\"older continuous dependence on the forcing function and the boundary data.

Starting from a torus knot $\mathcal{K}$ in the lens space $L(p,-1)$, we construct a Lagrangian sub-manifold $L_{\mathcal{K}}$ in $\mathcal{X}=\big(\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)\big)/\mathbb{Z}_p$ under the conifold transition. We prove a mirror theorem which relates the all genus open-closed Gromov-Witten invariants of $(\mathcal{X},L_{\mathcal{K}})$ to the topological recursion on the B-model spectral curve. This verifies a conjecture in \cite{Bor-Bri} in the case of lens space.