arXiv research

Corners can be identified by a drum's sound spectrum.

New method reveals corners of drum shapes.

Inverse Drum Machine separates drum mixes using transcription and synthesis.

New formula calculates volumes of ideal hyperbolic drums.

DRUM discovers interpretable rules from knowledge graphs for unseen entities.

DeepDrummer generates drum loops with human preferences via active learning.

DRUM transfers cardiac arrest models across registries with missing covariates.

We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, ``can one hear the shape of a drum?'' In order to construct simply connected examples, we exploit the observation that an orbifold whose unde…

Considering music as a sequence of events with multiple complex dependencies, the Long Short-Term Memory (LSTM) architecture has proven very efficient in learning and reproducing musical styles. However, the generation of rhythms requires additional information regarding musical structure and accompanying instruments. …

Generalizes complex manifolds to manifolds with corners and generalized corners.

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}\sub…

Proves spacetime positive mass theorem with corners.

Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 c…

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 …

New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.

Defines products for fibered corners manifolds, generalizing resolutions.

The study defines differential forms and currents on orbifolds with corners.

Extends 4D cornered skein theory to surfaces, proving gluing formulas.

Solves relative isoperimetric problem on polygonal domains, focusing on corners.

Paper proves corner connection tiles can represent knots with fewer tiles.

Inverse spectral theory reveals shapes from sound.

Fourth-order problem on half-ball with corner behavior.

Extends algebraic geometry to include spaces with corners.

Extends corner structure study to general case, constructs normal Trans-Sasakian structures.

We introduce the notions of the caustic-equivalence and the weak caustic-equivalence relations of reticular Lagrangian maps in order to give a generic classification of caustics on a corner. We give the figures of all generic caustics on a corner in a smooth manifold of dimension 2 and 3.

We construct a smooth Lie group structure on the group of real analytic diffeomorphisms of a compact analytic manifold with corners. This generalises the known analogous results in the situation where the real analytic manifold has no corners. Additionally our approach uses a different construction.

Currents with corners help count triangulations on surfaces.

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary. As a main application, a normal fo…

New tile types for knots and links reduce complexity.

New heat trace coefficients reveal curvature effects in polygonal domains.

We explore models for translating abstract musical ideas (scores, rhythms) into expressive performances using Seq2Seq and recurrent Variational Information Bottleneck (VIB) models. Though Seq2Seq models usually require painstakingly aligned corpora, we show that it is possible to adapt an approach from the Generative A…

We study metrics with positive scalar curvatures in domains with corners and suggest possible extensions of the concept of positive scalar curvature to singular spaces.

In this paper we present another notion of a smooth manifold with corners and relate it to the commonly used concept in the literature. Afterwards we introduce complex manifolds with corners and show that if $M$ is a compact (respectively complex) manifold with corners and $K$ is a smooth (respectively complex) Lie gro…

Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.

Study uses deep reinforcement learning for real-time control of nuclear microreactors, achieving similar or superior performance to traditional PID controllers.

John Conway created pairs of domains that sound the same for a special kind of music.

M-theory can be defined on closed manifolds as well as on manifolds with boundary. As an extension, we show that manifolds with corners appear naturally in M-theory. We illustrate this with four situations: The lift to bounding twelve dimensions of M-theory on Anti de Sitter spaces, ten-dimensional heterotic string the…

This paper makes a formal study of asymptotically hyperbolic Einstein metrics given, as conformal infinity, a conformal manifold with boundary. The space on which such an Einstein metric exists thus has a finite boundary in addition to the usual infinite boundary and a corner where the two meet. On the finite boundary …

Positive mass theorem for non-smooth metrics on flat manifolds with corners.

We give a number of examples of isospectral pairs of plane domains, and a particularly simple method of proving isospectrality. One of our examples is a pair of domains that are not only isospectral but homophonic: Each domain has a distinguished point such that corresponding normalized Dirichlet eigenfunctions take eq…

For every connected manifold with corners we use a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space),…

We introduce the notion of reticular Legendrian unfoldings in order to investigate stabilities of bifurcations of wavefronts generated by a hypersurface germ with a boundary, a corner, or an r-corner in a smooth n dimensional manifold. We define several stabilities of reticular Legendrian unfoldings and prove that they…

The study classifies graphs on surfaces with positive curvature properties.

The paper constructs submanifolds with corners in Delzant polytopes from affine subspaces.

We answer Mark Kac's famous question, "can one hear the shape of a drum?" in the positive for orbifolds that are 3-dimensional and 4-dimensional lens spaces; we thus complete the answer to this question for orbifold lens spaces in all dimensions. We also show that the coefficients of the asymptotic expansion of the tra…

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 definiti…

Given a connected manifold with corners 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. Our main theorem is that, for any manifold with cor…

New framework for logarithmically divergent integrals on manifolds with corners.