Geometry and representation learning in deep generative models by Mathieu, E

“…As hyperbolic spaces are perfectly suited to embed tree-like data, in contrast to Euclidean geometry, we endow the latent space with hyperbolic geometry. We do so by deriving the necessary methods to work with two main Gaussian generalisations and geometry-aware architectures for the encoder and decoder networks. …”

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics by Torsten Asselmeyer-Maluga

“…Here, we will use Thurston&#8217;s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mrow> <mo>(</mo> <mi>K</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>S</mi> <mn>3</mn> </msup> <mo>\</mo> <mrow> <mo>(</mo> <mi>K</mi> <mo>&#215;</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> of a knot <i>K</i> carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. …”
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Representation learning for recommender systems by Lucas, Vinh Tran

“…Overall, this work aims to bridge the gap between Euclidean and hyperbolic geometry in recommender systems through metric learning approach. …”
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In-Vitro Investigation of Fatigue and Fracture Behavior of Transmucosal versus Submerged Bone Level Implants Used in Fixed Prosthesis by Saverio Cosola, Paolo Toti, Enrico Babetto, Ugo Covani, Miguel Peñarrocha-Diago, David Peñarrocha-Oltra

“…Two types of emergence profiles (Premium sub-crestal straight implant with a cylindrical-shaped coronal emergence or Prama one-piece cylindrical-shape implant with transmucosal convergent neck and hyperbolic geometry) were tested, and dynamic fatigue were run to failure. …”
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On the Maldacena Type Conjecture in Relation with Scale Relativity Theory by Alexandra Saviuc, Emma Birleanu, Mihail Frasila

“…In this way, it is shown that any dynamics which reflects SL (2R) “processes” has intrinsic hyperbolic geometries, harmonic mappings, from usual and scales spaces to the hyperbolic one, implying correlations between the General Relativity, in Ernst sense, and Skyrme Theory.…”
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Harmonic parameterization of geodesic quardangles on surfaces of constant curvaturess and 2-D quasi-isometric grids by Gennadii A. Chumakov, Sergei G. Chumakov

“…A method for the generation of quasi-isometric boundary-fitted curvilinear coordinates for arbitrary domains is developed on the basis of the quasi-isometric mappings theory and conformal representation of spherical and hyperbolic geometries. A one-parameter family of Riemannian metrics with some attractive invariant properties is analytically described. …”
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Hyperbolically symmetric sources in Palatini f(R) gravity by M. Z. Bhatti, Z. Yousaf, Z. Tariq

“…Furthermore, we evaluated the algebraic expressions for the mass of interior hyperbolical geometry and total energy budget, i.e., the Tolman mass of the considered source. …”
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A Century of Turbulent Cascades and the Emergence of Multifractal Operators by Daniel Schertzer, Ioulia Tchiguirinskaia

“…At the same time, we obtain new results on the entanglement of spherical and hyperbolic geometries, as well as on the existence of finite statistics of these cascades.…”
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The multi-field, rapid-turn inflationary solution by Vikas Aragam, Sonia Paban, Robert Rosati

“…In particular, simple results arise when the covariant Hessian of the potential has an eigenvector in close alignment with the gradient — a situation we find to be common and we prove generic in two-field hyperbolic geometries. We verify our methods on several known rapid-turn models and search two type-IIA constructions for rapid-turn trajectories. …”
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Boundary theories of critical matchgate tensor networks by A. Jahn, M. Gluza, C. Verhoeven, S. Singh, J. Eisert

“…Furthermore, our numerical probes of the bulk parameters corresponding to boundary excited states constitute a first step towards a tensor network bulk-boundary dictionary between regular hyperbolic geometries and critical boundary states.…”
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Label-free multi-step microfluidic device for mechanical characterization of blood cells: Diabetes type II by Diana Pinho, Vera Faustino, Susana O. Catarino, Ana I. Pereira, Graça Minas, Fernando T. Pinho, Rui Lima

“…At the outlets, hyperbolic geometries allow single-cell deformability analysis. …”
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