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This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity $\omega_{\varepsilon}(x,t)$ concentrated around points $\xi_{j}(t)$ that converge to a sum of Dirac delta masses as $\varepsilon\to0$. These solutions are associated with the Kirchhoff-Routh point-vortex system, and the points $\xi_{j}(t)$ follow an expanding self similar trajectory of spirals, with the support of the vorticities contained in balls of radius $3\varepsilon$ around each $\xi_{j}$.

We consider the Ginzburg-Landau equation in the plane linearized around the standard degree-one vortex solution $W(x)=w(r)e^{i\theta}$. Using explicit representation formulae for the Fourier modes in $\theta$, we obtain sharp estimates for the inverse of the linearized operator which hold for a large class of right-hand sides. This theory can be applied, for example, to estimate the inverse after dropping the usual orthogonality conditions., Comment: 21 pages

The paper focuses on the conformal Lorentz geometry of quasi-umbilical timelike surfaces in the $(1+2)$-Einstein universe, the conformal compactification of Minkowski 3-space realized as the space of oriented null lines through the origin of $\mathbb{R}^{2,3}$. A timelike immersion of a surface $X$ in the Einstein universe is quasi-umbilical if its shape operator at any point of $X$ is non-diagonalizable over the complex numbers. We prove that quasi-umbilical surfaces are isothermic, that their conformal deformations depend on one arbitrary function in one variable, and show that their conformal Gauss map is harmonic. We then investigate their geometric structure and show how to construct all quasi-umbilical surfaces from null curves in the 4-dimensional neutral space form $S^{2,2}$., Comment: 37 pages, 11 figures

Dark matter haloes form from the collapse of matter around special positions in the initial field, those where the local matter flows converge to a point. For such a triaxial collapse to take place, the energy shear tensor -- the source of the evolution of the inertia tensor -- must be positive definite. It has been shown that this is indeed the case for the energy shear tensor of the vast majority of protohaloes. At generic positions in a Gaussian random field, the trace and traceless parts of the tensor are independent of one another. Here we show that, on the contrary, in positive definite matrices they correlate strongly, and these correlations are very similar to those exhibited by protohaloes. Moreover, while positive-definiteness ensures that an object will collapse, it does not specify when. Previous work has shown that the trace of the energy tensor -- the energy overdensity -- exhibits significant scatter in its values, but must lie above a critical `threshold' value for the halo to collapse by today. We show that suitable combinations of the eigenvalues of the traceless part are able to explain a substantial part of the scatter of the trace. These variables provide an efficient way to parameterise the initial value of the energy overdensity, allowing us to formulate an educated guess for the threshold of collapse. We validate our ansatz by measuring the distribution of several secondary properties of protohaloes, finding good agreement with our analytical predictions., Comment: 12 pages, 11 figures

The liquid drop model was introduced by Gamow in 1928 and Bohr-Wheeler in 1938 to model atomic nuclei. The model describes the competition between the surface tension, which keeps the nuclei together, and the Coulomb force, corresponding to repulsion among protons. More precisely, the problem consists of finding a surface $\Sigma =\partial \Omega$ in $\mathbb{R}^3$ that is critical for the energy $$ E(\Omega) = {\rm Per\,} (\Omega ) + \frac 12 \int_\Omega\int_\Omega \frac {dxdy}{|x-y|} $$ under the volume constraint $|\Omega| = m$. The term ${\rm Per\,} (\Omega ) $ corresponds to the surface area of $\Sigma$. The associated Euler-Lagrange equation is $$ H_\Sigma (x) + \int_{\Omega } \frac {dy}{|x-y|} = \lambda \quad \hbox{ for all } x\in \Sigma, \quad $$ where $H_\Sigma$ stands for the mean curvature of the surface, and where $\lambda\in\mathbb{R}$ is the Lagrange multiplier associated to the constraint $|\Omega|=m$. Round spheres enclosing balls of volume $m$ are always solutions. They are minimizers for sufficiently small $m$. Since the two terms in the energy compete, finding non-minimizing solutions can be challenging. We find a new class of compact, embedded solutions with large volumes, whose geometry resembles a "pearl necklace" with an axis located on a large circle, with a shape close to a Delaunay's unduloid surface of constant mean curvature. The existence of such equilibria is not at all obvious, since for the closely related constant mean curvature problem $H_\Sigma = \lambda$, the only compact embedded solutions are spheres, as stated by the classical Alexandrov result.

We provide the first construction of overhanging gravity water waves having the approximate form of a disk joined to a strip by a thin neck. The waves are solitary with constant vorticity, and exist when an appropriate dimensionless gravitational constant $g>0$ is sufficiently small. Our construction involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an exceptional domain discovered by Hauswirth, H\'elein, and Pacard \cite{hauswirth-helein-pacard}. The method developed here is related to the construction of constant mean curvature surfaces through gluing., Comment: 93 pages

Instabilities in relativistic magnetized jets are thought to be deeply connected to their energy dissipation properties and to the consequent acceleration of the non-thermal emitting relativistic particles. Instabilities lead to the development of small scale dissipative structures, in which magnetic energy is converted in other forms. In this paper we present three-dimensional numerical simulations of the instability evolution in highly magnetized plasma columns, considering different kinds of equilibria. In fact, the hoop stresses related to the azimuthal component of magnetic field can be balanced either by the magnetic pressure gradient (force-free equilibria, FF) or by the thermal pressure gradient (pressure-balanced equilibria, PB) or by a combination of the two. FF equilibria are prone to current-driven instabilities (CDI), while PB equilibria are prone to pressure-driven instabilities (PDI). We perform a global linear stability analysis, from which we derive the different instability properties in the two regimes, showing that PDI have larger growth rates and are also unstable for high wavenumbers. The numerical simulations of the non-linear instability evolution show similar phases of evolution in which the formation of strong current sheets is followed by a turbulent quasi steady-state. PDI are however characterized by a faster evolution, by the formation of smaller scale dissipative structures and larger magnetic energy dissipation., Comment: 16 pages, 11 figures, acceped for pubblication in MNRAS

The large volume and variety of imaging data from high-throughput screening (HTS) in the pharmaceutical industry present an excellent resource for training virtual staining models. However, the potential of models trained under one set of experimental conditions to generalize to other conditions remains underexplored. This study systematically investigates whether data from three cell types (lung, ovarian, and breast) and two phenotypes (toxic and non-toxic conditions) commonly found in HTS can effectively train virtual staining models to generalize across three typical HTS distribution shifts: unseen phenotypes, unseen cell types, and the combination of both. Utilizing a dataset of 772,416 paired bright-field, cytoplasm, nuclei, and DNA-damage stain images, we evaluate the generalization capabilities of models across pixel-based, instance-wise, and biological-feature-based levels. Our findings indicate that training virtual nuclei and cytoplasm models on non-toxic condition samples not only generalizes to toxic condition samples but leads to improved performance across all evaluation levels compared to training on toxic condition samples. Generalization to unseen cell types shows variability depending on the cell type; models trained on ovarian or lung cell samples often perform well under other conditions, while those trained on breast cell samples consistently show poor generalization. Generalization to unseen cell types and phenotypes shows good generalization across all levels of evaluation compared to addressing unseen cell types alone. This study represents the first large-scale, data-centric analysis of the generalization capability of virtual staining models trained on diverse HTS datasets, providing valuable strategies for experimental training data generation.

We study the stability of fracton gravity, a variant of linearized gravity where the gauge symmetry is restricted to longitudinal diffeomorphisms. These transformations can be connected to a spacetime generalization of dipole symmetry, hence the tag fracton. We find that fracton gravity features an instability in the spin-1 sector corresponding to solutions with growing amplitude. This dynamical instability can be removed by tuning the couplings of the theory. Nonetheless, the Hamiltonian for the spin-1 modes remains always unbounded from below when evaluated on the classical solutions. We find no other sources of instability in the spin-2 or spin-0 sectors. We analyze in detail the canonical formulation and the constraints arguing that neither auxiliary fields nor gauge-fixing conditions can be employed to remove the problematic vector modes or stabilize them., Comment: 21 pages

As it collapses to form a halo, the shape of a protohalo patch is deformed by the initial shear field. This deformation is often modeled using the "deformation" tensor, constructed from second derivatives of the gravitational potential, whose trace gives the initial overdensity. However, especially for lower mass protohalos, this matrix is not always positive definite: one of its eigenvalues has a different sign from the others. We show that the evolution of a patch is better described by the "energy shear" tensor, which is positive definite and plays a direct role in the evolution. This positive-definiteness simplifies models of halo abundances, assembly and of the cosmic web., Comment: 9 pages, 10 figures

In this paper, we present a novel formulation to model the effects of a locked differential on the lateral dynamics of an autonomous open-wheel racecar. The model is used in a Model Predictive Controller in which we included a micro-steps discretization approach to accurately linearize the dynamics and produce a prediction suitable for real-time implementation. The stability analysis of the model is presented, as well as a brief description of the overall planning and control scheme which includes an offline trajectory generation pipeline, an online local speed profile planner, and a low-level longitudinal controller. An improvement of the lateral path tracking is demonstrated in preliminary experimental results that have been produced on a Dallara AV-21 during the first Indy Autonomous Challenge event on the Monza F1 racetrack. Final adjustments and tuning have been performed in a high-fidelity simulator demonstrating the effectiveness of the solution when performing close to the tire limits., Comment: Presented at the 2023 IEEE 11th International Conference on Systems and Control, Sousse, Tunisia, December 18-20, 2023

We provide a geometric transformation on null curves in the anti-de Sitter $3$-space (AdS) which induces the B\"{a}cklund transformation for the KdV equation. In addition, we show that this geometric transformation satisfies a suitable permutability theorem. We also illustrate how to implement it when the original null curve has constant bending., Comment: The present paper is a continuation of the work [8], which is submitted for a possible publication but it still in the review process. As a consequence, this work will not be submitted to a journal until [8] is already accepted/published. arXiv admin note: text overlap with arXiv:2311.11137

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =\Delta u - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v = (-\Delta_{\mathbb{R}^2})^{-1} u := \displaystyle\frac {1}{2\pi} \displaystyle\int_{\mathbb{R}^2} \log \frac {1}{|x-z|}u(z,t) dz, \ \ \ \ \ \ \ \ \ (\star)\\[5pt] u(\cdot ,0) = u_{0}^{\star} \ge 0 \ \ \ \text{in } \mathbb{R}^2. \end{cases} \end{equation}$ We show that there exists $\varepsilon>0$ such that for any $m$ satisfying $8\pi

We show that the classical Brezis-Nirenberg problem $$\Delta u + |u|^{4 \over N-2} u + \varepsilon u = 0 ,\quad {\mbox {in}} \quad \Omega, \quad u= 0 , \quad {\mbox {on}} \quad \partial \Omega$$ admits nodal solutions clustering around a point on the boundary of $\Omega$ as $\varepsilon \to 0$, for smooth bounded domains $\Omega \subset \mathbb{R}^N $ in dimensions $N\geq 7$.

We formulate integrable flows related to the KdV hierarchy on null curves in the anti-de Sitter 3-space (${\rm AdS}$). Exploiting the specific properties of the geometry of ${\rm AdS}$, we analyze their interrelationships with Pinkall flows in centro-affine geometry. We show that closed stationary solutions of the lower order flow can be explicitly found in terms of periodic solutions of a Lam\'e equation. In addition, we study the evolution of non-stationary curves arising from a 3-parameter family of periodic solutions of the KdV equation.

A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ \omega_t + \nabla^\perp \Psi\cdot \nabla \omega = 0 , \quad \Psi = (-\Delta)^{-1} \omega, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a travelling wave solution of the form $\omega(x,t) = W(x_1-ct,x_2 )$ where $W(x)$ is compactly supported and odd in $x_2$. We revisit the problem of constructing solutions which are highly $\varepsilon$-concentrated around points $ (0, \pm q)$, more precisely with approximately radially symmetric, compactly supported bumps with radius $\varepsilon$ and masses $\pm m$. Fine asymptotic expressions are obtained, and the smooth dependence on the parameters $q$ and $\varepsilon$ for the solution and its propagation speed $c$ are established. These results improve constructions through variational methods in [14] and in [5] for the case of a bounded domain., Comment: To appear in Journal of Differential Equations. arXiv admin note: substantial text overlap with arXiv:2310.07238

Dipole charge conservation forces isolated charges to be immobile fractons. These couple naturally to spatial two-index symmetric tensor gauge fields that resemble a spatial metric. We propose a spacetime Lorentz covariant version of dipole symmetry and study the theory of the associated gauge fields. In the presence of a suitable background field, these contain a massive anti-symmetric and a massless symmetric two-index tensors. The latter transforms only under longitudinal diffeomorphisms, making the massless sector similar to linearized gravity, but with additional modes of lower spin. We show that the theory can be consistently coupled to a curved background metric and study its possible interaction terms with itself and with matter. In addition, we construct a map between solutions of linearized gravity in Kerr-Schild form and solutions of fracton gravity coupled to matter., Comment: 30 pages, extended appendix, published version

We consider the problem of finding a solution to the incompressible Euler equations $$ \omega_t + v\cdot \nabla \omega = 0 \quad \hbox{ in } \mathbb{R}^2 \times (0,\infty), \quad v(x,t) = \frac 1{2\pi} \int_{{\mathbb R}^2} \frac {(y-x)^\perp}{|y-x|^2} \omega (y,t)\, dy $$ that is close to a superposition of traveling vortices as $t\to \infty$. We employ a constructive approach by gluing classical traveling waves: two vortex-antivortex pairs traveling at main order with constant speed in opposite directions. More precisely, we find an initial condition that leads to a 4-vortex solution of the form $$ \omega (x,t) = \omega_0(x-ct\, e ) - \omega_0 ( x+ ct \, e) + o(1) \ \hbox{ as } t\to\infty $$ where $$ \omega_0( x ) = \frac 1{\varepsilon^{2}} W \left ( \frac {x-q} \varepsilon \right ) - \frac 1{\varepsilon^{2}}W \left ( \frac {x+q} \varepsilon \right ) + o(1) \ \hbox{ as } \varepsilon \to 0 $$ and $W(y)$ is a certain fixed smooth profile, radially symmetric, positive in the unit disc zero outside.

We investigate geometric evolution equations for Legendrian curves in the 3-sphere which are invariant under the action of the unitary group ${\rm U}(2)$. We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associated hierarchy, are realized as curvature evolutions induced by a sequence of Hamiltonian flows. For the flow among these that induces the mKdV equation, we investigate the geometry of solutions which evolve by rigid motions in ${\rm U}(2)$. Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.

We investigate the total CR twist functional on transversal curves in the standard CR 3-sphere $\mathrm S^3 \subset \mathbb C^2$. The question of the integration by quadratures of the critical curves and the problem of existence and properties of closed critical curves are addressed. A procedure for the explicit integration of general critical curves is provided and a characterization of closed curves within a specific class of general critical curves is given. Experimental evidence of the existence of infinite countably many closed critical curves is provided.

In an inviscid and incompressible fluid in dimension 3, we prove the existence of several helical filaments, or vortex helices, collapsing into each others.

We introduce a family of quantum field theories for fields carrying monopole and dipole charges. In contrast to previous realizations, fields have quadratic two-derivative kinetic terms. The dipole symmetry algebra is realized in a discretized internal space and connected to the physical space through a background gauge field. We study spontaneous symmetry breaking of dipole symmetry in 1+1 dimensions in a large-$N$ limit. The trivial classical vacuum is lifted by quantum corrections into a vacuum which breaks dipole symmetry while preserving monopole charge. By means of a Hubbard-Stratonovich transformation, heat-kernel and large-$N$ techniques, we compute the effective action for the low-energy modes. We encounter a fractonic immobile Nambu-Goldstone mode whose dispersion characteristics avoid Coleman-Hohenberg-Mermin-Wagner theorem independently of the large-$N$ limit., Comment: 33 pages, 2 figures. V2: minor editings in response to Sci-Post Core peer-review process