CyberSec Research

This paper considers a mean-field model of $n$ interacting particles whose state space is the unit circle, a generalization of the classical Kuramoto model. Global synchronization is said to occur if after starting from almost any initial state, all particles coalesce to a common point on the circle. We propose a general synchronization criterion in terms of $L_1$-norm of the third derivative of the particle interaction function. As an application we resolve a conjecture for the so-called self-attention dynamics (stylized model of transformers), by showing synchronization for all $\beta \ge -0.16$, which significantly extends the previous bound of $0\le \beta \le 1$ from Criscitiello, Rebjock, McRae, and Boumal (2024). We also show that global synchronization does not occur when $\beta < -2/3$.

This paper is concerned with solving the Helmholtz exterior Dirichlet and Neumann problems with large wavenumber $k$ and smooth obstacles using the standard second-kind boundary integral equations (BIEs) for these problems. We consider Galerkin and collocation methods - with subspaces consisting of $\textit{either}$ piecewise polynomials (in 2-d for collocation, in any dimension for Galerkin) $\textit{or}$ trigonometric polynomials (in 2-d) - as well as a fully discrete quadrature (a.k.a., Nystr\"om) method based on trigonometric polynomials (in 2-d). For each of these methods, we address the fundamental question: how quickly must $N$, the dimension of the approximation space, grow with $k$ to maintain accuracy as $k\to\infty$? For the methods involving piecewise-polynomials, we give sufficient conditions for $k$-uniform quasi-optimality. For the Galerkin method we prove that these are, in fact, necessary and sufficient. In particular, we prove that, when applied to the Neumann BIEs when the obstacle is a ball, the Galerkin method $\textit{suffers from the pollution effect}$; i.e., $N$ growing like $k^{d-1}$ is not sufficient for $k$-uniform quasi-optimality. For the Dirichlet BIEs, we prove that pollution occurs for the ball for certain choices of coupling parameter, and we give numerical experiments illustrating pollution for trapping domains with the standard coupling parameter. For all the methods involving trigonometric polynomials, we show that, up to potential factors of $k^\varepsilon$ for any $\varepsilon>0$, these methods do not suffer from the pollution effect (even for trapping obstacles). These are the first results about $k$-explicit convergence of collocation or Nystr\"om methods applied to the Dirichlet BIEs, and the first results about $k$-explicit convergence of any method used to solve the standard second-kind Neumann BIEs.

We develop a general distributional theory of fractional (an)isotropic Sobolev spaces associated with the non-degenerate symmetric $\alpha$-stable, $\alpha \in (1,2)$, probability measures on $\mathbb{R}^d$.

These notes address two problems. First, we investigate the question of ``how many'' are (in Baire sense) vector fields in $L^1_t L^q_x$, $q \in [1, \infty)$, for which existence and/or uniqueness of local, distributional solutions to the associated continuity equation holds. We show that, in certain regimes, existence of solutions (even locally in time, for at least one nonzero initial datum) is a meager property, whereas, on the contrary, uniqueness of solutions is a generic property. Secondly, despite the fact that non-uniqueness is a meager property, we prove that (Sobolev) counterexamples to uniqueness, both for the continuity equation and for the ODE, in the spirit of [Bru\`e, Colombo, Kumar 2024] and [Kumar 2024] respectively, form a dense subset of the natural ambient space they live in.

Motivated by inverse problems with a single passive measurement, we introduce and analyze a new class of inverse spectral problems on closed Riemannian manifolds. Specifically, we establish two general uniqueness results for the recovery of a potential in the stationary Schr\"odinger operator from partial spectral data, which consists of a possibly sparse subset of its eigenvalues and the restrictions of the corresponding eigenfunctions to a nonempty open subset of the manifold. Crucially, the eigenfunctions are not assumed to be orthogonal, and no information about global norming constants is required. The partial data formulation of our inverse spectral problems is naturally suited to the analysis of inverse problems with passive measurements, where only limited observational access to the solution is available. Leveraging this structure, we establish generic uniqueness results for a broad class of evolutionary PDEs, in which both the coefficients and the initial or source data are to be recovered from knowledge of the solution restricted to a subset of spacetime. These results introduce a spectral framework for passive imaging and extend inverse spectral theory into a regime characterized by highly incomplete, yet physically realistic, data.

In this paper we deal with the bounded critical points of a Riesz energy of attractive-repulsive type in dimension 1. Under suitable assumptions on the growth of the kernel in the origin, we are able to prove that they are continuous inside their support.

In this paper, we study solutions $u$ of parabolic systems in divergence form with zero Dirichlet boundary conditions in the upper-half cylinder $Q_1^+\subset \mathbb{R}^{n+1}$, where the coefficients are weighted by $x_n^\alpha$, $\alpha\in(-\infty,1)$. We establish higher-order boundary Schauder type estimates of $x_n^\alpha u$ under the assumption that the coefficients have partially Dini mean oscillation. As an application, we also achieve higher-order boundary Harnack principles for degenerate or singular equations with H\"older continuous coefficients.

Regularity properties of solutions for a class of quasi-stationary models in one spatial dimension for stress-modulated growth in the presence of a nutrient field are proven. At a given point in time the configuration of a body after pure growth is determined by means of a family of ordinary differential equations in every point in space. Subsequently, an elastic deformation, which is given by the minimizer of a hyperelastic variational integral, is applied in order to restore Dirichlet boundary conditions. While the ordinary differential equations governing the growth process depend on the elastic stress and the pullback of a nutrient concentration in the current configuration, the hyperelastic variational problem is solved on the intermediate configuration after pure growth. Additionally, the coefficients of the reaction-diffusion equation determining the nutrient concentration in the current configuration depend on the elastic deformation and the deformation due to pure growth.

We combine the construction of the canonical conservation law and the nonlocal cosymmetry to derive a collection of nonlocal conservation laws for the two-dimensional Euler equation in vorticity form. For computational convenience and simplicity of presentation of the results we perform a complex rotation of the independent variables.

In this paper, we study the fractional $p$-Laplacian Choquard equation $$ (-\Delta)_{p}^{s} u+h(x)|u|^{p-2} u=\left(R_{\alpha} *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^d$, where $s\in(0,1)$, $ p\geq 2$, $\alpha \in(0, d)$ and $R_\alpha$ represents the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under suitable assumptions on the potential function $h$, we first prove the existence of a strictly positive solution by the mountain-pass theorem for the nonlinearity $f$ satisfying some growth conditions. Moreover, if we add some monotonicity condition, we establish the existence of a positive ground state solution by the method of Nehari manifold.

We consider equations involving the truncated laplacians and having lower order terms with singular potentials posed in punctured balls. We study both the principal eigenvalue problem and the problem of classification of solutions, in dependence of their asymptotic behaviour near the origin, for equations having also superlinear absorbing lower order terms. In the case of the maximising truncated Laplacian "Pk+", owing to the mild degeneracy of the operator, we obtain results which are analogous to the results for the Laplacian in dimension k. On the other hand, for minimising operator "Pk-" we show that the strong degeneracy in ellipticity of the operator produces radically different results.

In this paper, we study the connectedness and compactness of the attainability set of weak solutions to the three-dimensional Navier--Stokes equations with damping. Depending on the value of the parameter \b{eta}, which controls the damping term, we establish these results with respect to either the weak or the strong topology of the phase space. In the latter case, we also prove that the global attractor is connected. Additionally, we establish results concerning the regularity of the global attractor and provide a new proof of its existence for strong solutions.

We study the fully-developed, time-periodic motion of a shear-dependent non-Newtonian fluid with variable exponent rheology through an infinite pipe $\Omega:= \mathbb{R}\times \Sigma\subseteq \mathbb{R}^d$, $d\in \{2,3\}$, of arbitrary cross-section $\Sigma\subseteq \mathbb{R}^{d-1}$. The focus is on a generalized $p(\cdot)$-fluid model, where the power-law index is position-dependent (with respect to $\Sigma$), $\textit{i.e.}$, a function $p\colon \Sigma\to (1,+\infty)$. We prove the existence of time-periodic solutions with either assigned time-periodic flow-rate or pressure-drop, generalizing known results for the Navier-Stokes and for $p$-fluid equations. In addition, we identify explicit solutions, relevant as benchmark cases, especially for electro-rheological fluids or, more generally, $\textit{`smart fluids'}$. To support practical applications, we present a fully-constructive existence proof for variational solutions by means of a fully-discrete finite-differences/-elements discretization, consistent with our numerical experiments. Our approach, which unifies the treatment of all values of $p(\overline{x})\in (1,+\infty)$, $\overline{x}\in \Sigma$, without requiring an auxiliary Newtonian term, provides new insights even in the constant exponent case. The theoretical findings are reviewed by means of numerical experiments.

We show the uniqueness of the cylindrical tangent cone $C(\mathbb{S}^2 \times \mathbb{S}^4) \times \mathbb{R}$ for area-minimizing hypersurfaces in $\mathbb{R}^9$, completing the uniqueness of all tangent cones of the form $C_{p,q} \times \mathbb{R}$ proved by Simon for dimensions at least 10 and Sz\'ekelyhidi for the Simons cone.

We study the rate of propagation of chaos for a McKea--Vlasov equation with conditional expectation terms in the drift. We use a (regularized) Nadaraya-Watson estimator at a particle level to approximate the conditional expectations; we then combine relative entropy methods in the spirit of Jabin and Wang (2018) with information theoretic inequalities to obtain the result. The nonparametric nature of the problem requires higher regularity for the density of the McKean-Vlasov limit, which we obtain with a bootstrap argument and energy estimates.

Since the seminal paper of Graham and Zworski (Invent. Math. 2003), conformal geometric problems are studied in the fractional setting. We consider the convergence of fractional Yamabe flow, which is previously known under small initial energy assumption. Inspired by the deep work of Brendle (J. Diff. Geom. 2005), we obtain the full convergence result for arbitrary initial energy, whenever the (fractional) positive mass conjecture is valid.

It is well known that derivatives of solutions to elliptic boundary value problems may become unbounded near the corner of a domain with a conical singularity, even if the data are smooth. When the corner domain is approximated by more regular domains, then higher order Sobolev norms of the solutions on these domains can blow up in the limit. We study this blow-up in the simple example of the Laplace operator with Dirichlet conditions in two situations: The rounding of a corner in any dimension, and the two-dimensional situation where a polygonal corner is replaced by two or more corners with smaller angles. We show how an inner expansion derived from a more general recent result on converging expansions into generalized power series can be employed to prove simple and explicit estimations for Sobolev norms and singularity coefficients by powers of the approximation scale.

We establish a new integration by parts formula for the regional fractional laplacian $(-\Delta)^s_\Omega$ in bounded open sets of class $C^2$. As a direct application, we prove that weak solutions to the corresponding Dirichlet problem satisfy a Pohozaev-like identity with an explicit remainder term. We apply the later to eigenvalue problems in the unit ball and discuss its potential use in establishing boundary-type unique continuation properties.