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Supernova remnants (SNRs) remain prime candidates for hadronic particle acceleration within our galaxy, accounting for much of the Cosmic Ray flux. Next-generation instruments such as the Southern Wide-field Gamma-ray Observatory (SWGO) will be of crucial importance in identifying new candidate SNRs. SWGO will observe two-thirds of the gamma-ray sky, covering the energy range between a few hundreds GeV and a PeV. In this work, we apply a model of SNR evolution to predict their gamma-ray spectra. Furthermore, we use our model in combination with the target SWGO sensitivity range to explore the SNR emission phase space and quantify detection prospects for SWGO. Finally, we validate our model for sources observed with current-generation instruments, fitting it using a Monte-Carlo Markov Chain technique to the observed gamma-ray emission from four SNRs. We anticipate that at least 8 SNRs will be detected by SWGO within 1 year., Comment: 24 pages, 16 figures. Submitted to JCAP

The energetically most efficient way how a deformed red blood cell regains equilibrium is mathematically described by the gradient flow of the Canham-Helfrich functional, including a spontaneous curvature and the conservation of surface area and enclosed volume. Using a recently discovered multiplicity inequality, we prove global existence and convergence of smooth solutions for spheres and axisymmetric tori, provided the initial energy lies below explicit thresholds., Comment: 33 pages. Comments welcome!

Accurate parameter dependent electro-chemical numerical models for lithium-ion batteries are essential in industrial application. The exact parameters of each battery cell are unknown and a process of estimation is necessary to infer them. The parameter estimation generates an accurate model able to reproduce real cell data. The field of optimal input/experimental design deals with creating the experimental settings facilitating the estimation problem. Here we apply two different input design algorithms that aim at maximizing the observability of the true, unknown parameters: in the first algorithm, we design the applied current and the starting voltage. This lets the algorithm collect information on different states of charge, but requires long experimental times (60 000 s). In the second algorithm, we generate a continuous current, composed of concatenated optimal intervals. In this case, the experimental time is shorter (7000 s) and numerical experiments with virtual data give an even better accuracy results, but experiments with real battery data reveal that the accuracy could decrease hundredfold. As the design algorithms are built independent of the model, the same results and motivation are applicable to more complex battery cell models and, moreover, to other applications., Comment: 29 pages, 8 figures

We provide sharp sufficient criteria for an integral $2$-varifold to be induced by a $W^{2,2}$-conformal immersion of a smooth surface. Our approach is based on a fine analysis of the Hausdorff density for $2$-varifolds with critical integrability of the mean curvature and a recent local regularity result by Bi-Zhou. In codimension one, there are only three possible density values below $2$, each of which can be attained with equality in the Li--Yau inequality for the Willmore functional by the unit sphere, the double bubble, and the triple bubble. We show that below an optimal threshold for the Willmore energy, a varifold induced by a current without boundary is in fact a curvature varifold with a uniform bound on its second fundamental form. Consequently, the minimization of the Willmore functional in the class of curvature varifolds with prescribed even Euler characteristic provides smooth solutions for the Willmore problem. In particular, the "ambient" varifold approach and the "parametric" approach are equivalent for minimizing the Willmore energy., Comment: 28 pages, 3 figures. Comments welcome!

Motivated by a model for lipid bilayer cell membranes, we study the minimization of the Willmore functional in the class of oriented closed surfaces with prescribed total mean curvature, prescribed area, and prescribed genus. Adapting methods previously developed by Keller-Mondino-Rivi\`ere, Bauer-Kuwert, and Ndiaye-Sch\"atzle, we prove existence of smooth minimizers for a large class of constraints. Moreover, we analyze the asymptotic behaviour of the energy profile close to the unit sphere and consider the total mean curvature of axisymmetric surfaces., Comment: 63 pages, 4 figures. Comments are welcome!

A model describing cell membranes as optimal shapes with regard to the $L^2$-deficit of their mean curvature to a given constant called spontaneous curvature is considered. It is shown that the corresponding energy functional is lower semi-continuous with respect to oriented varifold convergence on a space of surfaces whose second fundamental form is uniformly bounded in $L^2$. Elementary examples are presented showing that the latter condition is necessary. As a consequence, smoothly embedded minimisers among surfaces of higher genus are obtained. Moreover, it is shown that the diameter of a connected surface is controlled by the $L^1$-deficit of its mean curvature to the spontaneous curvature leading to an improved condition for the existence of minimisers. Finally, the diameter bound can be applied to obtain an isoperimetric inequality., Comment: 37 pages, no figures

We prove a lower bound on the length of closed geodesics for spherical surfaces with Willmore energy below $6\pi$. The energy threshold is optimal and there is no comparable result for surfaces of higher genus. We also discuss consequences for the injectivity radius., Comment: 23 pages, 4 figures. Comments are welcome!

The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of its boundary. Of particular interest is the application of the inequality to minimal surfaces in the three-sphere and in the hyperbolic space. Here the result implies an a priori estimate for connected solutions of Plateau's problem, as well as a necessary condition on the boundary data for the existence of such solutions. The proof follows a construction of Miura and uses a diameter bound for closed surfaces obtained by Topping and Wu-Zheng., Comment: 8 pages, no figures

Topological defects play a central role in the physics of many materials, including magnets, superconductors and liquid crystals. In active fluids, defects become autonomous particles that spontaneously propel from internal active stresses and drive chaotic flows stirring the fluid. The intimate connection between defect textures and active flow suggests that properties of active materials can be engineered by controlling defects, but design principles for their spatiotemporal control remain elusive. Here we propose a symmetry-based additive strategy for using elementary activity patterns, as active topological tweezers, to create, move and braid such defects. By combining theory and simulations, we demonstrate how, at the collective level, spatial activity gradients act like electric fields which, when strong enough, induce an inverted topological polarization of defects, akin to a negative susceptibility dielectric. We harness this feature in a dynamic setting to collectively pattern and transport interacting active defects. Our work establishes an additive framework to sculpt flows and manipulate active defects in both space and time, paving the way to design programmable active and living materials for transport, memory and logic., Comment: 11 pages (including Methods), 5 figures. Changed title and format, final version

For varifolds whose first variation is representable by integration, we introduce the notion of indecomposability with respect to locally Lipschitzian real valued functions. Unlike indecomposability, this weaker connectedness property is inherited by varifolds associated with solutions to geometric variational problems phrased in terms of sets, $G$ chains, and immersions; yet it is strong enough for the subsequent deduction of substantial geometric consequences therefrom. Our present study is based on several further concepts for varifolds put forward in this paper: real valued functions of generalised bounded variation thereon, partitions thereof in general, partition thereof along a real valued generalised weakly differentiable function in particular, and local finiteness of decompositions., Comment: This is the second paper of a series of three papers which shall supersede arXiv:1709.05504 by the same authors. 58 pages (v1: 43 pages), no figures, added treatment of varifolds with free boundary which includes an isoperimetric lower density ratio bound in 9.22 and results in Theorem E and Corollary 3

As service to the community, we provide - for Euclidean space - a basic treatment of locally rectifiable chains and of the complex of locally integral chains. In this setting, we may beneficially develop the idea of a complete normed commutative group bundle over the Grassmann manifold whose fibre is the coefficient group of the chains. Our exposition also sheds new light on some algebraic aspects of the theory. Finally, we indicate an extension to a geometric approach to locally flat chains centring on locally rectifiable chains rather than completion procedures., Comment: 41 pages, five diagrams. This is the first paper of a series of three papers which shall supersede arXiv:1709.05504 by the same authors. v2: Amended acknowledgement with Max Planck Institute for Mathematics in the Sciences, updated list of references

We prove a general Li-Yau inequality for the Helfrich functional where the spontaneous curvature enters with a singular volume type integral. In the physically relevant cases, this term can be converted into an explicit energy threshold that guarantees embeddedness. We then apply our result to the spherical case of the variational Canham-Helfrich model. If the infimum energy is not too large, we show existence of smoothly embedded minimizers. Previously, existence of minimizers was only known in the classes of immersed bubble trees or curvature varifolds., Comment: 46 Pages, 10 Figures. Fixed some typos

Using Rauch's comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li-Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch's comparison theorem, the proofs mainly rely on the first variation formula, thus are valid for general varifolds., Comment: 33 pages. Final version to appear in Annals of Global Analysis and Geometry

A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below $8\pi$. Combining such a strict inequality with previous works by Keller-Mondino-Rivi\`ere and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Similarly, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Moreover, it is shown that the tori degenerate in the moduli space which gives an application also to the conformally-constrained Willmore problem. Finally, because of their symmetry, the Delaunay tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case., Comment: 28 pages. Final version to appear in Nonlinear Analysis (TMA)

This White Paper summarizes the authors' discussion regarding objectionable content for the University of Houston (UH) Research Team to outline a strategy for building an extensive repository of online videos to support research into automated multimodal approaches to detect objectionable content. The workshop focused on defining what harmful content is, to whom it is harmful, and why it is harmful.

Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by Keller-Mondino-Rivi\`ere, our strict inequality leads to existence of minimisers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below $8\pi$. Besides the geometric interest, such a minimisation problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes., Comment: 16 pages. Final version to appear in Advances in Calculus of Variations

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main results we establish the lower semicontinuity of the Canham-Helfrich energy under weak convergence of (possibly branched and bubbled) weak immersions., Comment: 29 pages. Final version published in Archive for Rational Mechanics and Analysis