New pushforward operation on vector pseudo-bundles creates new examples.
problem Creating new objects from vector bundle theory in diffeology.
method Introducing pushforward operation on diffeological vector pseudo-bundles.
result Pushforward operation produces new projective diffeological vector spaces.
Study projective symmetries in Finsler spaces, showing reductions and constant flag curvature.
problem Exploring projective symmetries in Finsler spaces and their properties.
method Analyzing algebraic sub-algebras and curvature invariants of projective vector fields.
result Closed Finsler spaces with negative Ricci curvature reduce to Killing vector fields.
3D manifolds with certain projective fields are projectively flat.
problem Geodesic rigidity of Levi-Civita connections with essential projective vector fields.
method Proved projective flatness for specific manifolds with projective vector fields.
result Connected 3D Riemannian and closed semi-Riemannian manifolds with non-linearizable projective singularities are projectively flat.
In 2D, Finsler metrics with 3+ projective fields are projectively equivalent to Randers.
problem Characterizing Finsler metrics with multiple projective vector fields.
method Analyzing the Lie algebra of projective vector fields and showing equivalence to Randers metrics.
result A complete list of 2D Finsler metrics with at least 3 projective fields up to equivalence.
Characterizes C-projective vector fields on Randers spaces.
problem Characterizing C-projective vector fields on Randers spaces.
method Using a non-Riemannian quantity ${fΞ}$, it is shown that ${fΞ}$ is invariant for C-projective vector fields and the dimension of the algebra of C-projective vector fields is at most n(n+2). result An n-dimensional Randers space has a C-projective algebra of maximum dimension n(n+2) if and only if it is locally Minkowskian or (up to re-scaling) locally isometric to the generalized Funk metric. Solves Lie's 3D metric problem for projective vector fields.
problem Describing 3D Levi-Civita metrics with non-trivial projective vector fields.
method Analyzes Riemannian and Levi-Civita metrics of arbitrary signature.
result Solves the analog of Lie's problem in 3D.
Researchers compute c-projective symmetry algebras for Kähler surfaces.
problem Understanding symmetries in Kähler surfaces.
method Defined and analyzed c-projective vector fields and computed their symmetries.
result Computed c-projective symmetry algebras for Kähler surfaces with essential c-projective vector fields.
Classifies flat projective structures with specific symmetries.
problem Classifying local projective structures with non-trivial Lie symmetries.
method Analyzes flat projective structures with positive-dimensional Lie algebra of projective vector fields.
result Obtained a classification of flat projective structures.
The paper studies how norms of random vectors are preserved by random projections.
problem Understanding how random matrix affects norms of random vectors.
method Proved the distribution of the norm of random vector is preserved by random projection.
result Random matrix preserves the distribution of the norm of random vectors with i.i.d. entries.
Computes the decomposition of rank-three bundles over the projective line with three marked points.
problem Decomposing rank-three bundles over the projective line with three marked points.
method Using the monodromy derivative to compute the roots of the bundles.
result Computes the exact decomposition of rank-three bundles for m=3. In this paper, it is proved that any conformal vector field is homothetic on a locally projectively flat (α,β)-space of non-Randers type in dimension n≥3, and the local solutions of such a vector field are determined. While on a locally projectively flat Randers space, examples showthat the conformal vector fiel…
Listed normal forms for metrics with one essential projective vector field.
problem Classifying metrics with a single essential projective vector field.
method Provided a complete list of normal forms for metrics with one essential projective vector field.
result Extended and revised previous classifications by Lie (1882), Bryant & Manno & Matveev (2008), and Matveev (2012).
The paper classifies vector fields on 5D nilpotent Lie groups.
problem Classifying left-invariant affine and projective vector fields on 5D nilpotent Lie groups.
method Algebraic characterization and case-by-case analysis of vector fields.
result All projective vector fields are affine, extending classical results.
The study proves rationality of complex projective varieties with holomorphic vector fields.
problem Rationality of complex projective varieties with holomorphic vector fields.
method Key technique by Harvey-Lawson on finite volume flows.
result Uniform upper bound on Betti numbers for varieties with holomorphic vector fields.
Two Kähler metrics on a complex manifold are called c-projectively equivalent if their J-planar curves coincide. These curves are defined by the property that the acceleration is complex proportional to the velocity. We give an explicit local description of all pairs of c-projectively equivalent Kähler metrics of arb…
A vector field on a Kähler manifold is called c-projective if its flow preserves the J-planar curves. We give a complete local classification of Kähler real 4-dimensional manifolds that admit an essential c-projective vector field. An important technical step is a local description of 4-dimensional c-projectively equiv…
Paper shows biharmonic Hermitian vector bundles are harmonic on Kaehler Einstein manifolds.
problem Characterizing biharmonic Hermitian vector bundles.
method Analyzes properties of biharmonic projections over Kaehler Einstein manifolds.
result Biharmonic projections are harmonic.
Three new efficient algorithms project vectors onto weighted l1 ball.
problem Sparse system identification and feature selection.
method Projected gradient descent algorithms with linear or highly competitive quadratic worst case complexities.
result Efficient tools for machine learning methods like compress sensing and feature selection.
Explains differences and similarities of strictly nef and ample vector bundles.
problem Characterizing geometry of projective manifolds with strictly nef bundles.
method Brief exposition on strictly nef and ample vector bundles.
result Differences and similarities between strictly nef and ample vector bundles.
We give a complete list of two-dimensional metrics that admit an essential projective vector field. This solves a problem explicitly posed by Sophus Lie in 1882.
Proves complex Finsler bundles with positive curvature are ample and biholomorphic to projective space.
problem Solving a 1975 problem posed by S. Kobayashi about complex Finsler vector bundles with positive Kobayashi curvature.
method Analyzes properties of complex Finsler vector bundles with positive Kobayashi curvature.
result Complex Finsler vector bundles with positive Kobayashi curvature are ample and biholomorphic to projective space.
Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.
problem Understanding projective flat holomorphic vector bundles over Riemann surfaces.
method Assigning Wronskian line bundles to vector bundles and interpreting Abel's identity.
result Abel's identity is the first Chern class of the Wronskian line bundle.
Classifies and constructs intertwining differential operators between line and vector bundles over real projective space.
problem Classifying and constructing intertwining differential operators between line and vector bundles over real projective space.
method F-method for classification and construction of intertwining differential operators.
result Generalizes a classical result of Bol for SL(2,R) and classifies intertwining operators for SL(n,R). Researchers found indecomposable Killing tensor fields on quaternionic projective spaces.
problem Characterizing Killing tensor fields on projective spaces.
method Analyzing Killing tensor fields on quaternionic and complex projective spaces, proving algebraic properties.
result Generated algebras of Killing tensor fields on quaternionic and complex projective spaces.
Defines super projective modules and explores their properties.
problem Exploring the geometric-algebraic link in super geometry.
method Defined and explored super projective modules over supersmooth functions.
result Module of vector fields over a supersphere is a super projective module.
New method sets explicit sparsity for groups of vectors in deep learning and NMF.
problem Tackles the challenge of achieving a desired average sparsity level in vector groups.
method Designs a new sparse projection method that sets the sparsity level for the whole set explicitly and automatically tunes the sparsity of each vector.
result Shows significant improvements in accuracy and reconstruction errors compared to existing methods in deep learning and NMF.
Random projections enhance neural networks by reducing dimensions and speeding up training.
problem Training and expressive power of neural networks with high-dimensional inputs.
method Random projections to embed sparse vectors or low-dimensional manifolds into a smaller space, reducing the number of parameters and speeding up training.
result The number of neurons required for approximating a function depends on sparsity or manifold dimension, not the input vector dimension.
The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.
problem Classifying and constructing differential symmetry breaking operators.
method Utilizing factorization identities and branching laws of generalized Verma modules.
result Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces are classified and constructed.
Geometric structures on surfaces relate to 2-plane distributions in 5D.
problem Understanding geometric properties of vector bundles and distributions.
method Study of horizontal 2-plane distributions on 5-manifolds.
result Established a connection between surface projective differential geometry and 2-plane distribution growth.
We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this …
Paper studies binary random projections with controllable sparsity patterns for computational and accuracy advantages.
problem Improving computational efficiency and accuracy in random projections.
method Proposes two sparse binary projection models with controllable sparsity patterns.
result Significant computational advantages and improved accuracies in empirical evaluations.
We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a r…
This paper classifies superintegrable systems on 2D geometries with projective symmetries.
problem Classifying superintegrable systems on 2D geometries with projective symmetries.
method Combining metric projective differential geometry and superintegrability, defining projective equivalence, and applying transformation rules.
result Potentials of projectively equivalent Hamiltonians follow a linear superimposition rule.
Conditions for forming nontrivial knots from vector collections.
problem Conditions for forming nontrivial knots from vector collections.
method Analyzing vector collections and their reordering to form knots.
result For n≥7, it's always possible to reorder vectors to form a nontrivial knot. Researchers found non-Killing tensor fields on certain symmetric spaces.
problem Understanding Killing tensors on all Riemannian symmetric spaces.
method Constructed explicit examples of quadratic Killing tensors on quaternionic and Cayley projective spaces.
result Quadratic Killing tensors can be non-Killing on some symmetric spaces.
In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety Xn (n>4) has strictly nef Λ2TX, then it is isomorphic to Pn or quadric Qn. We also prove that on elliptic curves, strict…
New framework generalizes complex projective structures, proving non-existence of certain structures on Calabi-Yau manifolds.
problem Proving non-existence of certain holomorphic structures on Calabi-Yau manifolds.
method Introducing branched holomorphic Cartan geometries, proving independence of results, and using vector bundle properties.
result Non-projective compact simply connected Kähler Calabi-Yau manifolds do not admit branched holomorphic projective structures.
New discriminant analysis using GDS projection improves face recognition.
problem Improving face recognition accuracy with limited data.
method GDS projection onto generalized difference subspace, simplified Fisher criterion, normalization.
result GDS projection and gFDA are equivalent, inheriting FDA's discriminant ability.
Analyzes projections of test configurations to vector fields, proving moment convergence.
problem Analyzing moment convergence in projections of test configurations.
method Analytic approach involving moment convergence and weak geodesic ray.
result Proves moment convergence of weight distributions in projections.
Classifies and constructs intertwining differential operators between vector bundles over real projective space.
problem Classifying and constructing intertwining differential operators between vector bundles over RP2. method Utilizes SL(3,R)-intertwining differential operators, BGG resolution, and representation theory. result Irreducible unitary highest weight modules of SU(1,2) at reduction points classified by Cartan and PRV operators. Characterizes K-polystability for projective bundles over curves.
problem Existence of extremal Kähler metrics on projective bundles.
method Relative K-polystability and decomposition of vector bundles.
result Existence of extremal Kähler metrics linked to K-polystability.
We extend the mixtures of Gaussians (MOG) model to the projected mixture of Gaussians (PMOG) model. In the PMOG model, we assume that q dimensional input data points z_i are projected by a q dimensional vector w into 1-D variables u_i. The projected variables u_i are assumed to follow a 1-D MOG model. In the PMOG model…
Develops optimal approximations for SDEs on submanifolds.
problem Approximating high-dimensional SDEs on lower-dimensional submanifolds.
method Ito-vector and Ito-jet projections based on differential geometry.
result Optimal approximations in mean-square sense for SDEs.
Using twistor methods, we explicitly construct all local forms of four--dimensional real analytic neutral signature anti--self--dual conformal structures (M,[g]) with a null conformal Killing vector. We show that M is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural pr…
We construct an infinite sequence of projectively flat manifolds by using castling transformations of prehomogeneous vector spaces. We also give a classification of manifolds equipped with a flat projective structure obtained by a finite number of castling transformations, and describe these flat projective structures …
Study compact Kähler manifolds with pseudo-effective tangent bundles.
problem Characterize compact Kähler manifolds with strongly pseudo-effective tangent bundles.
method New proofs and characterizations of properties of vector bundles.
result Only projective spaces have big tangent bundles.
Computes L∞-algebroid for linear foliations on vector spaces.
problem Invariants of singular foliations on vector spaces induced by Lie subalgebras.
method Explicitly constructs projective resolutions and computes L∞-algebroid structure. result Provides invariants and constant-rank replacements of singular foliations.
The study characterizes finite vector bundles on specific complex manifolds.
problem Characterizing finite vector bundles on complex manifolds.
method Proves a theorem for compact complex manifolds with Gauduchon astheno-Kahler metrics.
result Establishes a condition for finite vector bundles similar to Nori's theorem.