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We study scattering and inverse scattering generated by sources and penetrable media with corner singularities. We introduce the notion of split corners, a local model that unifies geometric singularities and coefficient discontinuities arising in source and medium scattering. Within this framework, we establish scattering and inverse scattering results that extend the classical theory to split corners, allowing the source or medium contrast to approach different limiting values in distinct sectors meeting at the corner tip. For source scattering, we prove that every admissible two-split corner necessarily radiates in a general bounded inhomogeneous background, thereby generalizing the classical corner radiation principle to configurations involving both geometric corners and jump discontinuities. For penetrable media, we establish analogous results together with explicit compatibility conditions for incident waves of arbitrary vanishing order. As consequences, we obtain several uniqueness results in inverse source and inverse medium scattering, including recovery of polygonal convex hulls from a single far-field measurement.

In conventional Differential Geometry one studies manifolds, locally modelled on ${\mathbb R}^n$, manifolds with boundary, locally modelled on $[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. Manifolds with corners $X$ have boundaries $\partial X$, also manifolds with corners, with $\mathop{\rm dim}\partial X=\mathop{\rm dim} X-1$. We introduce a new notion of 'manifolds with generalized corners', or 'manifolds with g-corners', extending manifolds with corners, which form a category $\bf Man^{gc}$ with ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}\subset{\bf Man^{gc}}$. Manifolds with g-corners are locally modelled on $X_P=\mathop{\rm Hom}_{\bf Mon}(P,[0,\infty))$ for $P$ a weakly toric monoid, where $X_P\cong[0,\infty)^k\times{\mathbb R}^{n-k}$ for $P={\mathbb N}^k\times{\mathbb Z}^{n-k}$. Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries $\partial X$. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, tr

We investigate the localization and vanishing of $L^2$ interior transmission eigenfunctions at corners. Past numerical computations suggest that these eigenfunctions localize at non-convex corners. This phenomenon has, however, not been proven theoretically. We show that localization does indeed occur for some eigenfunctions at a non-convex corner. We also investigate the vanishing of interior transmission eigenfunctions at a convex corner. We prove that these eigenfunctions vanish at convex corners with reduced smoothness assumptions compared to earlier results.

The corner-based detection paradigm enjoys the potential to produce high-quality boxes. But the development is constrained by three factors: 1) Hard to match corners. Heuristic corner matching algorithms can lead to incorrect boxes, especially when similar-looking objects co-occur. 2) Poor instance context. Two separate corners preserve few instance semantics, so it is difficult to guarantee getting both two class-specific corners on the same heatmap channel. 3) Unfriendly backbone. The training cost of the hourglass network is high. Accordingly, we build a novel corner-based framework, named Corner2Net. To achieve the corner-matching-free manner, we devise the cascade corner pipeline which progressively predicts the associated corner pair in two steps instead of synchronously searching two independent corners via parallel heads. Corner2Net decouples corner localization and object classification. Both two corners are class-agnostic and the instance-specific bottom-right corner further simplifies its search space. Meanwhile, RoI features with rich semantics are extracted for classification. Popular backbones (e.g., ResNeXt) can be easily connected to Corner2Net. Experimental results

We define categories of stratified manifolds (s-manifolds) and stratified manifolds with corners (s-manifolds with corners). An s-manifold $\bf X$ of dimension $n$ is a Hausdorff, locally compact topological space $X$ with a stratification $X=\coprod_{i\in I}X^i$ into locally closed subsets $X^i$ which are smooth manifolds of dimension $\le n$, satisfying some conditions. S-manifolds can be very singular, but still share many good properties with ordinary manifolds, e.g. an oriented s-manifold $\bf X$ has a fundamental class $[\bf X]_{\rm fund}$ in Steenrod homology $H_n^{St}(X,\mathbb Z)$, and transverse fibre products exist in the category of s-manifolds. S-manifolds are designed for applications in Symplectic Geometry. In future work we hope to show that after suitable perturbations, the moduli spaces $\mathcal M$ of $J$-holomorphic curves used to define Gromov-Witten invariants, Lagrangian Floer cohomology, Fukaya categories, and so on, can be made into s-manifolds or s-manifolds with corners, and their fundamental classes used to define Gromov-Witten invariants, Lagrangian Floer cohomology, ....

In gravitational theories with boundaries, diffeomorphisms can become physical and acquire a non-vanishing Noether charge. Using the covariant phase space formalism, on shell of the gravitational constraints, the latter localizes on codimension-$2$ surfaces, the corners. The corner proposal asserts that these charges, and their algebras, must be important ingredients of any quantum gravity theory. In this manuscript, we continue the study of quantum corner symmetries and algebras by computing the entanglement entropy and quantum informational properties of quantum states abiding to the quantum representations of corners in the framework of $2$-dimensional gravity. We do so for two classes of states: the vacuum and coherent states, properly defined. We then apply our results to JT gravity, seen as the dimensional reduction of $4$d near extremal black holes. There, we demonstrate that the entanglement entropy of some coherent quantum gravity states -- states admitting a semiclassical description -- scales like the dilaton, reproducing the semiclassical area law behavior and further solidifying the quantum informational nature of entropy of quantum corners. We then study general state

Given a connected manifold with corners $X$ of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles, these conormal homology groups are denoted by $H^{cn}_*(X)$. Using our previous works we define an index morphism $$K^0(^bT^*X)\stackrel{Ind_{ev,cn}^X}{\longrightarrow}H_{ev}^{cn}(X)$$ for $X$ a manifold with corners of codimension less or equal to three and called here the even conormal index morphism. In the case that $X$ is compact and connected and $D$ is an elliptic $b-$pseudodifferential operator in the associated $b-$calculus of $X$ we know, by our previous works and other authors works, that, up to adding an identity operator, $D$ can be perturbed (with a regularizing operator in the calculus) to a Fredholm operator iff $Ind_{ev,cn}^X([σ_D])$ (where $[σ_D]\in K^0(^bT^*X)$ is the principal symbol class) vanishes in the even conormal homology group $H_{ev}^{cn}(X)$. The main result of this paper is the explicit computation of the even and odd conormal index morphisms $Ind_{ev/odd,cn}^X(σ)\in H_{ev/odd}^{cn}(

Let $Σ$ be a closed orientable hyperbolic surface. We introduce the notion of a \textit{geodesic current with corners} on $Σ$, which behaves like a geodesic current away from certain singularities (the "corners"). We topologize the space of all currents with corners and study its properties. We prove that the space of currents with corners shares many properties with the space of geodesic currents, although crucially, there is no canonical action of the mapping class group nor is there a continuous intersection form. To circumvent these difficulties, we focus on those currents with corners arising from harmonic maps of graphs into $Σ$. This leads to the space of \textit{marked harmonic currents with corners}, which admits a natural Borel action by the mapping class group, and an analog of Bonahon's\cite{Bonahon} compactness criterion for sub-level sets of the intersection form against a filling current. As an application, we consider an analog of a curve counting problem on $Σ$ for triangulations. Fixing an embedding $φ$ of a weighted graph $Γ$ into $Σ$ whose image $φ(Γ)$ is a triangulation of $Σ$, let $N_φ(L)$ denote the number of mapping classes $f$ so that a weighted-length mini

In this paper, we investigate the dynamics of solutions of the Muskat equation with initial interface consisting of multiple corners allowing for linear growth at infinity. Specifically, we prove that if the initial data contains a finite set of small corners then we can find a precise description of the solution showing how these corners desingularize and move at the same time. At the analytical level, we are solving a small data critical problem which requires renormalization. This is accomplished using a nonlinear change of variables which serves as a logarithmic correction and accurately describes the motion of the corners during the evolution.

Manifolds with boundary and with corners form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. A manifold with corners $X$ has two notions of tangent bundle: the tangent bundle $TX$, and the b-tangent bundle ${}^bTX$. The usual definition of smooth structure uses $TX$, as $f:X\to\mathbb{R}$ is defined to be smooth if $ abla^kf$ exists as a continuous section of $\bigotimes^kT^*X$ for all $k\ge 0$. We define 'manifolds with analytic corners', or 'manifolds with a-corners', with a different smooth structure, in which roughly $f:X\to\mathbb{R}$ is smooth if ${}^b abla^kf$ exists as a continuous section of $\bigotimes^k({}^bT^*X)$ for all $k\ge 0$. These are different from manifolds with corners even when $X=[0,\infty)$, for instance $x^α:[0,\infty)\to\mathbb{R}$ is smooth for all real $α\ge 0$ when $[0,\infty)$ has a-corners. Manifolds with a-boundary and with a-corners form categories ${\bf Man}\subset{\bf Man^{ab}}\subset{\bf Man^{ac}}$, with well behaved differential geometry. Partial differential equations on manifolds with boundary may have boundary conditions of two kinds: (i) 'at finite distance', e.g. Dirichlet or Neumann boundary conditions, or (ii) 'at infinity',

In this paper, we study tree--like tableaux and some of their probabilistic properties. Tree--like tableaux are in bijection with other combinatorial structures, including permutation tableaux, and have a connection to the partially asymmetric simple exclusion process (PASEP), an important model of an interacting particles system. In particular, in the context of tree-like tableaux, a corner corresponds to a node occupied by a particle that could jump to the right while inner corners indicate a particle with an empty node to its left. Thus, the total number of corners represents the number of nodes at which PASEP can move, i. e. the total current activity of the system. As the number of inner corners and regular corners is connected, we limit our discussion to just regular corners and show that asymptotically, the number of corners in a tableau of length $n$ is normally distributed.

A phase-field model for diffusion-limited crystal growth is formulated that is capable of handling highly anisotropic interfaces. It uses a Willmore regularization that yields corners of finite size. An asymptotic analysis reveals that Herring's law is recovered for the advancing surfaces. The model is validated by conducting simulations of dendritic growth for low anistorpies and comparing the results to the data from the literature. The model makes it possible to simulate high anisotropy dendrites for which the standard phase-field models are ill-posed. In this regime, the interplay between a Herring instability on the dendrite flanks and the corner regularization creates zig-zag shaped corrugations and leads to a non-monotonic trend of tip velocity as a function of anisotropy strength.

Manifolds with fibered corners arise as resolutions of stratified spaces, in many body compactifications of vector spaces, moduli spaces, and other settings. We define a category of fibered corners manifolds which has products and transverse fiber products, generalizing both the resolutions of products of stratified spaces and many body products, which are special cases. The product in the fibered corners category is a resolution of the cartesian product by blow-up which we call the 'ordered product'. This ordered product is a natural product for wedge (aka incomplete edge) metrics and quasi-fibered boundary metrics, a class which includes QAC and QALE metrics.

A piecewise smooth domain is said to have generic corners if the corners are generic CR manifolds. It is shown that a biholomorphic mapping from a piecewise smooth pseudoconvex domain with generic corners in complex Euclidean space that satisfies Condition R to another domain extends as a smooth diffeomorphism of the respective closures if and only if the target domain is also piecewise smooth with generic corners and satisfies Condition R. Further it is shown that a proper map from a domain with generic corners satisfying Condition R to a product domain of the same dimension extends continuously to the closure of the source domain in such a way that the extension is smooth on the smooth part of the boundary. In particular, the existence of such a proper mapping forces the smooth part of the boundary of the source to be Levi degenerate.

We study the equations in divergence form with piecewise $C^{\ga}$ coefficients. The domains contain corners and the discontinuity surfaces are attached to edges of the corners. We obtain piecewise $C^{1,\ga}$ estimates across the discontinuity surfaces and provide an example to illustrate the issue about the regularity at the corners.

In this paper, we confirm conjectures of Laborde-Zubieta on the enumeration of corners in tree-like tableaux and in symmetric tree-like tableaux. In the process, we also enumerate corners in (type $B$) permutation tableaux and (symmetric) alternative tableaux. The proof is based on Corteel and Nadeau's bijection between permutation tableaux and permutations. It allows us to interpret the number of corners as a statistic over permutations that is easier to count. The type $B$ case uses the bijection of Corteel and Kim between type $B$ permutation tableaux and signed permutations. Moreover, we give a bijection between corners and runs of size 1 in permutations, which gives an alternative proof of the enumeration of corners. Finally, we introduce conjectural polynomial analogues of these enumerations, and explain the implications on the PASEP.

Manifolds without boundary, and manifolds with boundary, are universally known in Differential Geometry, but manifolds with corners (locally modelled on [0,\infty)^k x R^{n-k}) have received comparatively little attention. The basic definitions in the subject are not agreed upon, there are several inequivalent definitions in use of manifolds with corners, of boundary, and of smooth map, depending on the applications in mind. We present a theory of manifolds with corners which includes a new notion of smooth map f : X --> Y. Compared to other definitions, our theory has the advantage of giving a category Man^c of manifolds with corners which is particularly well behaved as a category: it has products and direct products, boundaries behave in a functorial way, and there are simple conditions for the existence of fibre products X x_Z Y in Man^c. Our theory is tailored to future applications in Symplectic Geometry, and is part of a project to describe the geometric structure on moduli spaces of J-holomorphic curves in a new way. But we have written it as a separate paper as we believe it is of independent interest.

We prove that the presence or absence of corners is spectrally determined in the following sense: any simply connected domain with piecewise smooth Lipschitz boundary cannot be isospectral to any connected domain, of any genus, which has smooth boundary. Moreover, we prove that amongst all domains with Lipschitz, piecewise smooth boundary and fixed genus, the presence or absence of corners is uniquely determined by the spectrum. This means that corners are an elementary geometric spectral invariant; one can hear corners.