In this paper, we study Basmajian-type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence…
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48 results for “Basmajian-type identities”
The paper counts primes in complex orthospectra and polynomial orbits.
problem Counting primes in complex orthospectra and polynomial orbits.
method Analytic number theory and Thermodynamic Formalism.
result Equidistribution of holonomy in dynamical systems.
For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian's identity from Teichmueller theory. Any equality characterizes diagonal embeddings.
The paper proves a Basmajian identity for non-Archimedean local fields.
problem Proving Basmajian's identity over non-Archimedean local fields.
method Projective Anosov representations and Berkovich hyperbolic geometry.
result A signed finite sum series identity for Basmajian's identity.
New inequality links surface orthospectrum to boundary length.
problem Establishing a relationship between orthospectrum and boundary length for compact Riemannian surfaces.
method Analyzing compact Riemannian surfaces with a single closed geodesic, establishing a uniform lower bound on boundary length in terms of orthospectrum.
result A uniform lower bound on boundary length in terms of orthospectrum, akin to Basmajian's identity.
This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.
problem Understanding relationships between geodesic and orthogeodesic lengths on hyperbolic surfaces.
method Investigates a broad family of identities involving lengths of all closed geodesics and orthogeodesics.
result Introduces new identities that include lengths of all closed geodesics, contrasting with previous identities.
Establishes a correspondence between two mathematical identities.
problem None explicitly stated; focuses on identity correspondence.
method Establishes correspondence between Pestov and Weitzenböck identities.
result Established correspondence between Pestov and Weitzenböck identities.
Quandle homology was defined from rack homology as the quotient by a subcomplex corresponding to the idempotency, for invariance under the type I Reidemeister move. Similar subcomplexes have been considered for various identities of racks and moves on diagrams. We observe common aspects of these identities and subcompl…
The paper derives curvature identities for 5D and 6D Einstein manifolds.
problem Deriving curvature identities for specific dimensions of Einstein manifolds.
method Using Patterson's curvature identities and the Chern-Gauss-Bonnet Theorem, the paper provides explicit formulae for 5D and 6D Einstein manifolds.
result The curvature identities for 5D and 6D Einstein manifolds are confirmed to be consistent with previous work.
Global Pestov identity proved on frame bundle and related fibrations.
problem Global Pestov identity on frame bundles and fibrations.
method Global Pestov identity on frame bundles and fibrations.
result Global Pestov identity on frame bundles and fibrations.
RLINK uses deep reinforcement learning to improve user identity linkage across social networks.
problem Recognizing the same user across different social networks.
method Converts user identity linkage into a sequence decision problem and uses deep reinforcement learning to optimize the linkage strategy.
result Achieves better performance than state-of-the-art methods in experiments on various datasets.
Bochner's Identity on Graphsmath.DG
Proves Bochner's identity on graphs using a new auxiliary graph.
problem Extending Bochner's identity to graph theory.
method Introduces a complete tangent graph to prove the identity.
result Validates Bochner's identity on graphs.
Doodles link to commutator identities in a 2-sphere.
problem Understanding commutator identities in free groups via doodles.
method Analyzing doodles with proper noose systems and establishing bijections.
result A bijection between doodles and commutator identities.
The paper studies harmonic identity maps on Riemannian manifolds.
problem Understanding harmonicity of identity maps on Riemannian manifolds.
method Constructing new examples and defining a symmetric tensor field.
result New examples of identity harmonic maps are constructed.
The importance of Einstein's geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar …
We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence of the defining identities for Lie-Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit…
Discover new identities linking hypersurface mean curvatures.
problem Understanding mean curvatures of hypersurfaces in Riemannian manifolds.
method Developed three most general Minkowski or Hsiung-Minkowski identities.
result Classical Minkowski identity is natural to all Riemannian manifolds.
In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of tangent-vector-valued forms from the general Jacobi identity.
Graded identities for hyperbolic surfaces with cusps and cone points.
problem Understanding dilogarithm identities on hyperbolic surfaces.
method Establishing graded versions of Bridgeman's dilogarithm identity.
result Applications to the study of orthogeodesics.
We give a curvature identity derived from the generalized Gauss-Bonnet formula for 4-dimensional compact oriented Riemannian manifolds. We prove that the curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact. We also provide some applications of the identity.
New energy identity found for biharmonic maps into spheres.
problem Establishing energy identity for biharmonic maps in supercritical dimensions.
method Adapting Lin-Rivière's strategy for sphere-valued maps.
result Energy identity for stationary biharmonic maps into spheres in supercritical dimensions .
New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.
problem Understanding relationships between Frobenius elements and Jones-Wenzl projectors at roots of unity.
method Obtained skein identities relating Frobenius elements to Jones-Wenzl projectors in the Kauffman bracket skein module.
result Skein identities provide new proofs of the existence of the Chebyshev-Frobenius homomorphism.
New proof of Minkowski identities for hypersurfaces in curved spaces.
problem Proving Minkowski identities for hypersurfaces in constant curvature manifolds.
method Using a differential system and position vector field.
result New proof of Minkowski identities for hypersurfaces.
The paper proves curvature identities for symplectic connections.
problem Curvature tensor identities on symplectic connections.
method Invariant theory of the symplectic group, analogous to Riemannian or Kahler geometry.
result Describes the first space of p-covariant curvature identities.
In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…
The study finds new infinite dilogarithm identities related to number sequences and continued fractions.
problem Finding new infinite dilogarithm identities.
method Demonstrating families of identities associated with specific number sequences and continued fractions.
result New infinite dilogarithm identities related to Fibonacci, Lucas numbers, convergents of even period continued fractions, and recurrence relations.
Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.
Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergentsmath.GT
The paper connects dilogarithm identities to Pell's equation solutions via continued fractions.
problem Connecting dilogarithm identities to solutions of Pell's equation.
method Using continued fraction expansions of units in the ring of integers of quadratic fields.
result Ramanujan's dilogarithm identities correspond to a hexagonal identity.
Most existing word embedding approaches do not distinguish the same words in different contexts, therefore ignoring their contextual meanings. As a result, the learned embeddings of these words are usually a mixture of multiple meanings. In this paper, we acknowledge multiple identities of the same word in different co…
Study on functional ellipsoids to decompose the identity.
problem Decompose the identity for functional ellipsoids.
method Construct a decomposition similar to Fritz John's theorem.
result Developed a new approach to functional ellipsoids.
The paper proves identities for hyperconvex Anosov representations and their applications to Cantor sets.
problem Establishing identities for hyperconvex Anosov representations.
method Analyzing holomorphic families of Cantor non-conformal repellers and studying series identities.
result The series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is less than 1.
Elastic-InfoGAN learns object identity in class-imbalanced data.
problem Learning disentangled representations in class-imbalanced data.
method Invariance to identity-preserving transformations to learn object identity.
result Effectiveness in disentangling object identity in imbalanced datasets.
Super McShane identitymath.GT
Authors derive McShane identity for super tori.
problem Deriving McShane identity for super tori.
method Super Teichmüller theory and supergeometry.
result Asymptotic growth rate of length spectra established.
The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions both identities are captur…
In this paper we study McShane's identity in real and complex hyperbolic spaces and obtain various generalizations of the identity for representations of surface groups into the isometry groups of rank one symmetric spaces. Our methods unify most of the existing methods used in the existing literature for proving this …
Greg McShane introduced a remarkable identity for the lengths of simple closed geodesics on cusped hyperbolic surfaces. This was subsequently generalized by the authors to hyperbolic cone-surfaces, possibly with cusps and/or geodesic boundary. In this paper, we generalize the identity further to the case of classical S…
The paper extends McShane identities to higher Teichmüller theory and convex real projective surfaces.
problem Deriving McShane identities for higher Teichmüller theory.
method By studying mapping class group invariant functions and generalizing horocycle lengths.
result Established McShane-type identities for various surface group representations.
We study curvature identities on contact metric manifolds on the geometry of the corresponding almost Käehler cones, and we provide applications of the derived curvature identities.
On the basis of the generalizations of the Jacobi identity found by the author some identities satisfied by the curvature and torsion of a covariant differentiation are derived. A kind of the generalized covariant differentiation is proposed and a method of finding some of satisfied by them identities is given in the c…
Weak Bianchi identity found for spacetimes with timelike singularities.
problem Weak Bianchi identity for spacetimes with curvature singularities.
method Found sufficient conditions for a weak version of the second Bianchi identity.
result Identified a class of spherically symmetric static spacetimes with timelike singularities for which the identity holds.
We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to hyperbolic cone-surfaces (with all cone angles ), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotien…
A vector field E on an F-manifold (M, o, e) is an eventual identity if it is invertible and the multiplication X*Y := X o Y o E^{-1} defines a new F-manifold structure on M. We give a characterization of such eventual identities, this being a problem raised by Manin. We develop a duality between F-manifolds with eventu…
The identity map of certain Einstein manifolds is stable in both energy and bienergy.
problem Stability of the identity map in Einstein manifolds.
method Investigation of conformal-biharmonic stability compared to harmonic stability.
result The conformal-biharmonic index coincides with the harmonic index, except for the 4D Euclidean sphere.
Almost Hermitian Identitiesmath.DG
Study local commutation relation on almost complex manifolds.
problem Local commutation relation between Lefschetz operator and exterior differential.
method Analyzing almost complex manifolds with compatible metrics.
result Generalizes local Kähler identities to almost Hermitian manifolds.
A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-rec…
Study evaluates impact of probabilistic identity data in lookalike targeting campaigns.
problem Evaluate the impact of probabilistic identity data in lookalike targeting campaigns.
method Employ off-policy techniques to evaluate without risking large ad spend or A/B tests.
result Significant lift in conversion rate with identity-powered lookalikes.
Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two…
We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results …