Sharp bounds found for energy in projective space mappings.
problem Finding bounds for energy in mappings of real projective spaces.
method Sharp lower and upper bounds for energy in homotopy classes of mappings from real projective space to Riemannian manifolds.
result Characterization of maps that achieve the lower bound for energy and determination of the infimum of energy in a homotopy class.
The study classifies projectively flat metrics in a specific Finsler class.
problem Classifying projectively flat metrics in a particular Finsler class.
method Solving nonlinear PDEs to classify projectively flat metrics.
result New projectively flat Finsler metrics are discovered.
Hong Kong uses reference class forecasting to improve roadwork project cost and duration estimates.
problem Optimism bias and strategic misrepresentation in infrastructure project forecasts.
method Reference class forecasting applied to 25 roadwork projects in Hong Kong.
result Forecast accuracy distribution and project benchmarking established.
In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…
Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.
problem Holomorphic projective structures and bundles on surfaces.
method Generalization of principal bundle of projective 2-frames to branched projective structures.
result Affine spaces of branched projective structures with given branching classes.
Paper finds a divisibility property of first Pontrjagin classes for even-dimensional homotopy complex projective spaces.
problem Understanding the first Pontrjagin classes of homotopy complex projective spaces.
method Analyzing the difference of first Pontrjagin classes for even-dimensional manifolds homotopy equivalent to CP(n). result The difference of the first Pontrjagin classes is divisible by 16 for even n. Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.
problem Characterizing projectively flat metrics on Hopf manifolds.
method Partitioning projectively flat metrics into classes based on Chern scalar curvature sign and proving properties of each class.
result Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.
Optimizes energy of mappings from complex projective spaces.
problem Finding energy-minimizing mappings between complex projective spaces and Riemannian manifolds.
method Establishes optimal lower bounds for energy functionals and characterizes optimal mappings.
result Optimal lower bounds for energy functionals are characterized for mappings from real and complex projective spaces.
Stability results for complex Monge-Ampère equations in various classes.
problem Stability of solutions to complex Monge-Ampère equations.
method Weak stability results followed by Ck,α stability proofs. result Proves stability of solutions in relative full mass classes and on quasi-projective varieties.
The paper defines a new equivalence relation for knot projections and finds an infinite number of distinct classes.
problem Classifying knot projections based on weak homotopy equivalence.
method Defining weak (1, 2, 3) homotopy and using it to find an invariant.
result There are an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections.
Study shows zero Pontrjagin classes for sprays.
problem Characterization of locally projectively flat metrics.
method Analysis of Pontrjagin classes for sprays.
result Manifolds with locally projectively flat metrics have zero Pontrjagin classes.
The equation determining whether a projective structure admits a connection in its given projective class that has skew-symmetric Ricci tensor is an overdetermined system of semi-linear partial differential equations which we call the projective Einstein-Weyl (pEW) equation. In 2-dimensions, we give local obstructions …
The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions f…
A major source of risk in project management is inaccurate forecasts of project costs, demand, and other impacts. The paper presents a promising new approach to mitigating such risk, based on theories of decision making under uncertainty which won the 2002 Nobel prize in economics. First, the paper documents inaccuracy…
Paper introduces new Finsler metrics preserved under projective transformations.
problem Developing new projective invariant in Finsler geometry.
method Formulated weakly generalized Douglas-Weyl (W−GDW) equation to generalize Finsler metrics. result Introduces new subclasses of Finsler metrics: generalized weakly-Weyl and generalized ildeD-metrics. PPF uses projections to improve classification accuracy.
problem Improving classification accuracy in multi-class problems.
method PPF constructs trees using projections of variables, enhancing traditional random forest.
result PPF outperforms traditional random forest in multi-class problems.
FROCC uses random projections for fast one-class classification.
problem Efficient one-class classification for large datasets.
method Random projection of data onto random unit vectors, with region bounding.
result Significant improvement in ROC performance with substantial speedup.
Study non-vanishing ℓ2-Betti numbers for specific groups.
problem Calculating non-vanishing ℓ2-Betti numbers for certain groups. method Using Euler characteristics, higher Kazhdan projections, and Baum-Connes assembly map.
result Non-vanishing calculations for delocalised ℓ2-Betti numbers. The paper stratifies projective measured laminations and identifies a group of transformations.
problem Stratifying the space of projective measured laminations.
method Introducing a natural stratification and proving rigidity results.
result The group of self-homeomorphisms preserving the stratification is identified with the extended mapping class group.
A result about projections of Gibbs measures from a particular class arising in economic modeling is proved.
Extends K-stability theory to projective klt pairs with a big anticanonical class.
problem Behavioral pathologies in K-stability for projective klt pairs with a big anticanonical class.
method Extends K-stability theory to projective klt pairs with a big anticanonical class, observing that K-semistability forces a klt anticanonical model with the same stability property.
result K-semistability forces projective klt pairs with a big anticanonical class to have a klt anticanonical model with the same stability property.
Study projective and almost conformally symplectic structures on manifolds.
problem Relations between projective and almost conformally symplectic structures.
method Single almost conformally symplectic connection with totally trace-free torsion.
result Generalizes Fedosov structures and encodes variability of connections in projective class.
New classes generalize Chern classes in supergeometry.
problem Generalizing Chern classes to supergeometry.
method Introduced analytic representatives for cohomology elements.
result Cohomology elements called ν classes describe supergeometric projective spaces.
Paper tackles multi-label zero-shot learning, improving label embedding projection for unseen classes.
problem Challenges in transferring knowledge from seen to unseen classes in multi-label zero-shot learning.
method Proposes a transfer-aware embedding projection approach to project label embeddings into a low-dimensional space for better inter-label relationships and explicit information transfer.
result Demonstrates the efficacy of the proposed approach through experiments on zero-shot multi-label image classification.
PCA outperforms random projections in retaining second order signals from latent groups.
problem Preserving second order structure in latent groups under unsupervised linear projections.
method Theoretical framework and quasi-exhaustive enumeration of projections.
result PCA outperforms random projections in retaining second order signals across a broad range of data-generating parameters.
Only products of projective lines have vanishing Futaki invariants for all Kähler classes.
problem Finding Bott manifolds with vanishing Futaki invariants for all Kähler classes.
method Proving isomorphism to products of projective lines.
result Only products of projective lines satisfy the condition.
Characterizes projective submanifolds in high dimensions.
problem Characterizing projective submanifolds of specific codimensions.
method Uses Chern classes and ample line bundles.
result Generalizes earlier findings for hypersurfaces.
Projective varieties remain stable under close polarizations, extending to Kähler cones.
problem Maintaining stability of projective varieties under close polarizations.
method Uniformly valuative stability definition and extension to Kähler cones.
result Openness of uniformly valuative stability on the Kähler cone of projective manifolds.
Holonomy map is a local biholomorphism for parabolic projective structures on non-compact surfaces.
problem Holonomy map properties for parabolic projective structures on non-compact surfaces.
method Proving the holonomy map is a local biholomorphism.
result Holonomy map is a local biholomorphism for parabolic projective structures on non-compact surfaces.
It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential equation. This equation is a special case of a so-called first BGG equation. The ge…
A projection maps geodesic currents to Teichmüller space.
problem Mapping geodesic currents to Teichmüller space.
method Equivariant, length-minimizing projection from filling currents to Teichmüller space.
result The projection is well-behaved and maps geodesic currents to Teichmüller space.
Unique maximal curve systems found for up to 5 punctures.
problem Finding unique maximal curve systems in punctured projective planes.
method Analyzing mapping class group action on maximal 1-systems of loops. result Maximal 1-system is unique for up to 5 punctures. Enhances projection pursuit tree classifier with visual diagnostics for better multi-class classification.
problem Rigidity of original algorithm limits performance in complex high-dimensional classification problems.
method Allowing more splits and flexible class groupings in projection pursuit computation, and developing visual diagnostics.
result Demonstrates enhanced classifier performs as intended through interactive visual diagnostics.
Study on moduli spaces of branched projective structures on surfaces.
problem Characterizing and understanding moduli spaces of branched projective structures.
method Analytic and geometric methods to study the moduli spaces of branched projective structures.
result The moduli space of marked branched projective structures is a complex analytic space with specific dimensions and singular points.
Sharp-SSL uses random projections to identify important variables for semi-supervised learning.
problem High-dimensional semi-supervised learning problems.
method Careful aggregation of low-dimensional results from many axis-aligned random projections.
result Sharp-SSL algorithm can recover signal coordinates with high probability.
Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality.
Study estimates risks of nuclear waste storage projects.
problem Cost and schedule risks in nuclear waste storage projects.
method Reference class of 216 past projects for cost risk, 200 for schedule risk.
result Cost and schedule risks are substantial for nuclear waste storage projects.
The curve complex of a 3-holed projective plane is shown to be quasi-isometric to a tree and its automorphisms are isomorphic to the mapping class group.
problem Characterizing the automorphisms of the curve complex of a 3-holed projective plane.
method Exhaustion of the curve complex by finite rigid sets, quasi-isometry to a tree, and isomorphism to the mapping class group.
result The group of simplicial automorphisms of the curve complex is isomorphic to the mapping class group.
Researchers calculate alpha invariant for certain complex projective spaces.
problem Computing alpha invariant for specific types of complex projective spaces.
method Computed alpha invariant using total degree and Pontryagin classes.
result Alpha invariant depends only on total degree and Pontryagin classes.
Singular Finsler metrics, such as Kropina metrics and m-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of singular Finsler metrics defined by a Riemann metric α and 1-form β and characterize those which are respectively Douglasian and locally projectively flat in di…
Exploits class similarity for better machine learning models with confidence labels and projective loss functions.
problem Poor model performance due to confusing similar classes.
method Exploits class similarity with confidence labels and projective loss functions.
result Improved model performance on noisy labels.
We prove Chern class equalities for abelian families with a holomorphic normal projective connection.
Researchers confirm a 15-vertex triangulation of the quaternionic projective plane is minimal.
problem Prove the quaternionic projective plane has a minimal triangulation with 15 vertices.
method Implemented Gaifullin's algorithm to compute the first rational Pontryagin class of the combinatorial manifold.
result The 15-vertex triangulation is indeed a minimal triangulation of the quaternionic projective plane.
Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
Study on existence of weighted extremal Kähler metrics on projective bundles.
problem Existence of weighted extremal Kähler metrics on projective bundles.
method General existence result for weighted extremal metrics on admissible projective bundles.
result Existence of conformally Kähler, Einstein-Maxwell metrics of complex dimension m>2. We extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for arbitrary projective mor…
Exam project proof of Markov's Theorem.
problem Markov's Theorem in knot theory.
method Based on Birman's book and Petronio's classes.
result Detailed proof of Markov's Theorem.
The paper calculates the mapping class group of specific complex projective plane bundles.
problem Computing the mapping class group of complex projective plane bundles.
method Analyzes sphere bundles of real vector bundles over P2. result Calculates the mapping class group for specific examples like Milnor hypersurfaces.