Introduces Θ-positivity in Lie groups, generalizing Lusztig's positivity.
problem Generalizing Lusztig's total positivity to a broader class of Lie groups.
method Introduces and studies Θ-positivity in real simple Lie groups. result Four families of Lie groups admit Θ-positive structures. Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.
problem Generalizing Lusztig's total positivity to the setting of general real semisimple Lie groups.
method Classifying Lie groups admitting a positive structure and establishing key properties of unipotent positive semigroups.
result Establishes key geometric properties of elements in the positive semigroup.
In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…
Generalized Nakano positivity for certain singular cases.
problem Nakano positivity of direct image sheaves for singular cases.
method Generalization of Berndtsson's result to singular cases.
result Nakano positivity for direct image sheaves in special singular cases.
Theory of Θ-positive representations for real closed fields.
problem Generalizing positive representations to real closed fields.
method Developing theory for Fuchsian groups to linear groups over real closed fields.
result Theory encompasses many generalizations of positive or Anosov representations.
We prove new results about unknotting fibered positive knots and braids.
problem Proving the unknotting number equals genus for fibered positive knots and braids.
method Analyzing positive braid diagrams and fibered positive knots, proving new constraints and conjectures.
result We found fibered positive knots that cannot be unknotted optimally, contradicting Stoimenow's conjecture.
Generalizes Nakano-positivity to Hilbert space fields.
problem Extending Nakano-positivity to Hilbert space fields.
method Exhaustion arguments to generalize the theorem.
result Log-plurisubharmonic variation results for Stein manifolds.
In this short note, we prove positivity of Brown-York mass under quasi-positive boundary data which generalize some previous results by the authors. The corresponding rigidity result is obtained.
The paper defines new types of positivity and proves properties of Schur forms for vector bundles.
problem Defining and characterizing new types of positivity for vector bundles.
method Introducing and characterizing two types of strongly decomposable positivity, proving properties of Schur forms.
result Schur forms of strongly decomposable positive vector bundles are positive or weakly positive, answering a question of Griffiths.
Study ranking in generalized linear bandits with position and item dependencies.
problem Complex reward function due to position and item dependencies in recommendation systems.
method Model position and item dependencies, design UCB and Thompson Sampling algorithms.
result Generalizes existing studies in position dependencies and graph theory.
Paper proposes a new approach to stabilize GAN training by treating generated data as unlabeled.
problem Traditional GAN training treats generated data as negative, ignoring their potential quality.
method Defines positive and unlabeled classification for GANs, treating generated data as unlabeled.
result PUGAN achieves comparable or better performance than sophisticated discriminator stabilization methods.
Study proves positivity of quasi-local masses in general relativity using spinors.
problem Proving the positivity of quasi-local masses in general relativity.
method Using spinors and solving Dirac equation on compact Riemannian manifolds with boundary conditions.
result Gravitational mass bounded by a spacelike topological 2-sphere is non-negative, vanishing only in Minkowski space.
Randomized positional encodings boost transformer performance on longer sequences.
problem Transformers struggle with generalizing to sequences of arbitrary length.
method Introduced randomized positional encodings that simulate longer sequences and randomly select positions.
result Randomized positional encodings increase test accuracy by 12.0% on average for sequences of unseen length.
Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.
problem Positivity conjecture for Roger--Yang skein algebras.
method Used explicit polynomials like Chebyshev polynomials of the first kind to give candidates of positive bases.
result Polynomials form a lower bound in the sense of [Lê18] and [LTY21].
Extends Perelman's theorem to positive intermediate curvature conditions.
problem Positive intermediate curvature conditions and their implications.
method Generalization of Perelman's gluing theorem to positive intermediate curvature conditions.
result Observer moduli space can have non-trivial higher homotopy groups.
Proves a higher-order positive energy theorem for stationary solutions in fourth-order gravity.
problem Proving a positive energy theorem for fourth-order gravitational theories.
method Analyzes geometric analysis intersections and links to Q-curvature. result Establishes a positive energy theorem for stationary solutions in fourth-order gravity, similar to the classical ADM theorem.
Generalizes surgery theorem for positive Ricci curvature metrics.
problem Preserving positive Ricci curvature under surgery.
method Gluing a sphere bundle with a core metric.
result Constructs new metrics of positive Ricci curvature.
Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.
problem Non-semi-positivity of line bundles
method Introduce generalized Ueda obstruction classes and use Dolbeault resolution
result Recover classical examples and provide new examples of nef but not semi-positive line bundles
Researchers parametrize spaces of positive representations for Lie groups.
problem Tackling spaces of positive representations for Lie groups.
method Generalizing Lusztig's total positivity, they introduce spaces of positive framed representations and parametrize them.
result The number of connected components of the space of framed positive representations agrees with the number of positive representations.
It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative links). It was shown by L.Rudolph that positive braids have positive signature (if the…
Workshop notes on positivity in Lie groups and its applications.
problem Understanding total and Θ-positivity in semisimple Lie groups. method Discussion and analysis of existing theories and recent developments.
result Progress in classifying higher Teichmüller spaces through Θ-positivity. New positivity condition for Hermitian manifold curvature.
problem Generalizing curvature positivity to non-Kähler manifolds.
method Introducing a new positivity condition and deriving a Bochner formula.
result Positivity condition on certain generalized Hopf and Vaisman manifolds.
We introduce Θ-positivity, a new notion of positivity in real semisimple Lie groups. The notion of Θ-positivity generalizes at the same time Lusztig's total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of …
Study k-positive surface group representations and their degenerations.
problem Understanding the behavior of surface group representations under degenerations.
method Introduced k-positive representations and studied their degenerations using a limit theorem for positively ratioed representations.
result Degenerations of k-positive representations can lead to limits that are at least (k-3)-positive and irreducible limits are (k-1)-positive.
We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizat…
Study dynamic assortment and positioning of products with varying display effects.
problem Dynamic assortment and positioning of products with varying display effects.
method Design round-based learning algorithms for both multiplicative and general position effects models, and develop efficient subroutines for optimization.
result First regret-optimal characterization for both models, with matching upper and lower bounds.
In this paper, position vector of a spacelike general helix with respect to standard frame in Minkowski space E13 are studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike general helix. In terms of solution, …
W. Simon proved a conformal positive mass theorem, which was used to prove uniqueness of black holes later. In this note, we will generalize Simon's conformal positive mass theorem in two directions. First we will consider spacetime version of conformal positive mass theorems on asymptotically flat initial data set. Ne…
The study confirms most Cantor sets are in general position for all projections.
problem Understanding the general position of Cantor sets under various projections.
method Proof of the theorem stated in the title.
result Most Cantor sets are in general position with respect to all projections.
Generalizes symmetries of curved manifolds.
problem Maximizing symmetries in curved manifolds.
method Replaces torus with abelian group, generalizes results.
result Generalizes symmetry results for positively curved manifolds.
Establishes a concavity property for positive Hessian quotient operators.
problem Analyzing positive Hessian quotient operators on Riemannian manifolds.
method Proves a special concavity property and a Jacobi inequality.
result Proves a Jacobi inequality for symmetric tensors.
The book is devoted to constructing embedding finite-dimensional maps into trivial bundles and investigating the corresponding general position properties.
New Θ-positive representations of surface groups discovered.
problem Generalizing Lusztig's total positivity to surface groups.
method Introducing Θ-positivity and proving properties of Θ-positive representations. result Discrete and faithful Θ-positive representations exist and form open sets in representation varieties. Study of new link types and their invariants, extending previous results.
problem Properties of polynomial invariants and signatures of weakly successively almost positive links.
method Analysis of minimal genus and fibering properties, extending known theorems.
result Extension of Scharlemann-Thompson's theorem to weakly successively almost positive links.
It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend t…
The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.
problem Conditions for positive scalar curvature on manifolds with boundaries and their doubles.
method Analyzes the relationship between boundary conditions and positive scalar curvature metrics on manifolds and their doubles.
result Provides conditions for positive scalar curvature metrics on manifolds with boundaries and their doubles.
Proves positive mass theorem for non-spin weighted manifolds.
problem Proving the positive mass theorem for non-spin weighted manifolds.
method Establishing density theorem and generalizing Geroch conjecture.
result Proves positive weighted mass theorem for non-spin weighted manifolds.
The study classifies spinc manifolds with positive generalized scalar curvature.
problem Classifying spinc manifolds with positive generalized scalar curvature. method Using generalized scalar curvature, index of spinc Dirac operator, and surgery techniques. result Positive generalized scalar curvature metrics exist if and only if a specific invariant vanishes.
We generalize the definition of thin position of Scharlemann and Thompson for compact orientable 3-manifolds with torus boundary components and introduce α-sloped generalized Heegaard splittings. We examine its relationship to generalized Heegaard splittings of manifolds resulting from Dehn filling. We compare alpha-…
A positive path in the linear symplectic group $\Sp(2n)$ is a smooth path which is everywhere tangent to the positive cone. These paths are generated by negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean space. A special case are autonomous positive paths, which are generated by time-indepe…
Generates positive examples from noisy data streams.
problem Learning from noisy example streams in hypothesis classes.
method Extending results from previous studies to account for noise.
result Conditions for noisily generatable binary hypothesis classes.
In this paper, we prove that the position vector of every space curve satisfies a vector differential equation of fourth order. Also, we determine the parametric representation of the position vector ψ=(ψ1,ψ2,ψ3) of general helices from the intrinsic equations κ=κ(s) and τ=τ(s) where κ and τ are th…
Positive factorization for pseudoperiodic homeomorphisms on surfaces.
problem Factorization of pseudoperiodic homeomorphisms on surfaces.
method Generalization of classical results on smooth germs of surfaces, topological characterization of monodromies, and use of positive factorization criteria.
result Pseudoperiodic homeomorphisms on surfaces with positive fractional Dehn twist coefficients and screw numbers admit a positive factorization.
New methods learn from PU data with non-representative positives.
problem Learning from PU data with non-representative positive classes.
method Integrates negative-unlabeled and unlabeled-unlabeled learning, or uses a recursive risk estimator.
result Effective across various real-world datasets and forms of positive bias.
The paper studies invariant metrics with positive scalar curvature on 3-manifolds.
problem Classifying G-invariant 3-manifolds with positive scalar curvature. method Analyzes the space of G-invariant Riemannian metrics with positive scalar curvature on closed 3-manifolds. result The space of G-invariant PSC metrics is either empty or contractible. The paper proves rational connectedness for certain Kähler manifolds.
problem Rational connectedness of compact Kähler manifolds.
method Uniform weak RC-positivity of the tangent bundle.
result Compact Kähler manifolds with uniformly weakly RC-positive tangent bundles are projective and rationally connected.
Study the intersection of positive closed currents using tangent currents and King's residue formula.
problem Investigate the intersection of positive closed currents in complex manifolds.
method Employ tangent currents and King's residue formula to establish a natural condition for intersection.
result Derive an integral representation of the intersection of positive closed currents.
Study classifies Halin graphs with positive curvature.
problem Classifying Halin graphs with specific curvature.
method Analyzing generalized Halin graphs formed by connecting tree leaves.
result Identified all generalized Halin graphs with positive Lin-Lu-Yau curvature.