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.
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.
Paper extends positive energy theorem to anti-de Sitter spacetimes.
problem Proving positive energy theorem for weighted anti-de Sitter spacetimes.
method Generalized positive energy theorem for 3D anti-de Sitter initial data sets.
result Positive energy theorem proved for weighted anti-de Sitter spacetimes.
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. Generalizes Simon's theorem to spacetime and hyperbolic manifolds.
problem Proving uniqueness of black holes in various spacetime settings.
method Extending Simon's conformal positive mass theorem to spacetime and hyperbolic manifolds.
result Proves a conformal positive mass theorem on asymptotically hyperbolic manifolds.
A method to generate multi-label data from single positive annotations.
problem Generating multi-label datasets is costly and impractical.
method Single-to-multi-label (S2M) sampling using Markov chain Monte Carlo.
result S2M sampling enables high-quality multi-label data with minimal annotation cost.
Curved metrics on Wallach spaces bounded by curves under flow.
problem Properties of positively curved Riemannian metrics on Wallach spaces.
method Normalized Ricci flow analysis on specific Wallach spaces.
result Set of metrics forms bounded curves asymptotically.
The paper extends the spacetime positive mass theorem to multiple time dimensions.
problem Proving the nonnegativity of mass in spacetimes with multiple time dimensions.
method Generalizing the spacetime positive mass theorem to include multiple time dimensions and showing mass nonnegativity through energy inequalities.
result Equality in the energy inequality implies a foliation by flat submanifolds.
The paper proves a spacetime positive mass theorem for singular initial data sets.
problem Proving the positive mass theorem for initial data sets with corners.
method Extending Hirsch-Kazaras-Khuri's method to singular cases using Hirsch-Miao-Tsang ideas.
result Integral lower bound on spacetime mass and characterisation of zero mass.
Continuous selections for optimal portfolios under convex risk measures fail in finite-dimensional settings.
problem Finding optimal financial positions with continuous selections under convex risk measures.
method Analyzing set-valued maps in finite-dimensional settings with convex risk measures.
result Continuous selections do not always exist for optimal portfolios under convex risk measures.
Let K be a polygonal knot in general position with vertex set V. A \emph{generic quadrisecant} of K is a line that is disjoint from the set V and intersects K in exactly four distinct points. We give an upper bound for the number of generic quadrisecants of a polygonal knot K in general position. This upper…
The study sets limits on scalar curvature in positive curvature manifolds.
problem Bounding scalar curvature in positive curvature manifolds.
method Establishing inequalities for manifolds with positive scalar curvature and scalar curvature bounded from below.
result Established inequalities for manifolds with positive scalar curvature.
The study proves umbilical points have positive measure in 3D space forms.
problem Characterizing umbilical points in 3D space forms.
method Analyzing mean curvature and proving measure properties.
result Umbilical points have positive measure in 3D space forms.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
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.
Study on sets with positive reach in Euclidean and Riemannian spaces.
problem Understanding sets with positive reach in various spaces.
method Structural results on subsets of positive reach.
result New insights into sets with positive reach in Euclidean and Riemannian spaces.
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.
A positive space is a space with a positive atlas, i.e. a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical compactification of the latter. We show that it generalizes the Thurston compactificat…
The paper shows dense and residual sets of continuous maps with positive metric mean dimension.
problem Understanding the genericity of continuous maps with positive metric mean dimension.
method Analyzing continuous maps on compact Riemannian manifolds and Cantor sets.
result The set of continuous maps with metric mean dimension equal to a given value is dense and, for the dimension, residual in the space of continuous maps.
The Positive Mass Theorem for special singular initial data.
problem Proving the positive mass theorem for data with a codimension one singularity.
method Using asymptotically flat spin initial data sets with matching Bartnik data condition involving spacetime rotations.
result Established a spacetime positive mass theorem and rigidity statement.
Extends conformal prediction to contrastive learning for better coverage of positive samples.
problem Lack of principled guarantees on coverage in contrastive learning.
method Introduces minimum-volume covering sets with learnable constraints.
result Improves inclusion-exclusion trade-offs in positive and negative samples.
The paper proves a theorem for generalized p-Kähler manifolds.
problem Characterization of compact generalized p-Kähler manifolds.
method Proof based on duality between closed and exact positive forms and currents.
result Complete unified proof of Characterization Theorem for compact generalized p-Kähler manifolds.
This work generalizes Log-Determinant divergences to infinite-dimensional settings.
problem Generalizing Log-Determinant divergences to infinite-dimensional spaces.
method Introducing a parametrized family of divergences, Alpha-Beta Log-Determinant divergences, for positive definite unitized trace class operators.
result The Alpha-Beta Log-Det divergences encompass various divergences and metrics, including the affine-invariant Riemannian distance and symmetric Stein divergence.
Proves positive mass theorem for 3-manifolds with a boundary.
problem Proving the positive mass theorem for specific 3-manifolds.
method Uses harmonic level set approach.
result Validates the positive mass theorem for new class of manifolds.
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. Study proves a conjecture for certain spin manifolds.
problem Proving a conjecture about positive scalar curvature metrics.
method Connected sum construction and Gromov-Lawson area enlargement.
result Connected sum of spin manifolds admits no complete metric of positive scalar curvature.
The visibility transformation embeds data position into signature features for efficient pattern recognition.
problem Embedding absolute position into signature features for efficient pattern recognition.
method The visibility transformation is put on a theoretical footing and used to embed absolute position into signature features efficiently.
result The generated feature set simplifies pattern recognition by accommodating nonlinear functions of absolute and relative values.
Proves positive mass theorem for spin initial data sets with arbitrary ends and dominant energy shields.
problem Proving the positive mass theorem for spin initial data sets with various ends and energy shields.
method Modification of Witten's approach involving an additional independent timelike direction in the spinor bundle.
result Positive mass theorem for spin initial data sets with arbitrary ends and dominant energy shields.
CNNs improve positioning accuracy in massive MIMO systems.
problem Improving positioning accuracy in massive MIMO systems.
method Applying CNNs to learn sparse massive MIMO channel fingerprints.
result Moderately deep CNNs achieve fractional-wavelength positioning accuracy.
New metric found on a complex space with positive curvature properties.
problem Finding metrics with positive curvature on complex spaces.
method Used Lie group G2 and octonions to construct a metric on the Grassmannian of oriented 2-planes in R7. result First example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1.
The study shows that ergodic measures are not generic on non-positively curved manifolds.
problem Determining the genericity of ergodic measures on non-positively curved Riemannian manifolds.
method Investigates the existence of an open isometric embedding of a product manifold with a factor isometric to S1. result The closure of the set of ergodic measures does not encompass all invariant measures, indicating the failure of genericity.
In this paper, we develop a general study of contributions at infinity of Bochner-Weitzenböck-type formulas on asymptotically flat manifolds, inspired by Witten's proof of the positive mass theorem. As an application, we show that similar proofs can be obtained in a much more general setting as any choice of an irreduc…
Improved bounds on Z2-torus actions on positively curved manifolds.
problem Bounding the rank of Z2-tori for fixed point set components. method Lowered the rank bound and classified cohomology rings.
result Fixed point set components are classified by integral or Z2-cohomology rings. Extends classical model of transaction costs to convex costs and multivariate positions.
problem Risk arbitrage and hedging under transaction costs with convex costs and multivariate positions.
method Extends classical model to convex transaction costs and multivariate acceptable positions, using results for unbounded and non-closed random sets.
result Formulates no arbitrage conditions and explores their connections, leading to a decrease in superhedging prices.
We construct large families of initial data sets for the vacuum Einstein equations with positive cosmological constant which contain exactly Delaunay ends; these are non-trivial initial data sets which coincide with those for the Kottler-Schwarzschild-de Sitter metrics in regions of infinite extent. From the purely Rie…
Extends results on marginally outer trapped surfaces to general null expansion.
problem Analyzing geometry and topology of expanding horizons.
method Introduces g-stability and proves conditions for positive Yamabe type and scalar curvature. result Initial data sets with compact boundary of positive null expansion have positive mass.
The study characterizes sets for which one-layer neural networks are positive.
problem Characterizing sets of points for which one-layer neural networks are positive.
method Investigation of subsets of the real plane for one-layer ReLU neural networks.
result Full characterization of such sets for cones, and necessary condition for any subset of \(\mathbb{R}^d\).
Proposes new deviation measures using Minkowski gauges.
problem Lack of suitable acceptance sets for deviation measures.
method Derives deviation measures through Minkowski gauges of acceptable sets.
result Any positive homogeneous deviation measure can be accommodated in the framework.
Proves existence of solutions to Poisson equation on manifolds with positive spectrum.
problem Existence of solutions to Poisson equation on manifolds with positive essential spectrum.
method Sharp pointwise decay on source function, unbounded Ricci curvature, general spectrum and curvature bounds.
result Existence of solutions on manifolds with positive essential spectrum and unbounded Ricci curvature.
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.
Positive definite kernels are an important tool in machine learning that enable efficient solutions to otherwise difficult or intractable problems by implicitly linearizing the problem geometry. In this paper we develop a set-theoretic interpretation of the Earth Mover's Distance (EMD) and propose Earth Mover's Interse…
The study generalizes Santalo's formula and shows stability of trapping sets in Riemannian manifolds.
problem Stability of trapping sets in Riemannian manifolds under smooth perturbations of obstacles.
method Generalization of Santalo's formula applied to billiard trajectories in the exterior of obstacles.
result The measure of the set of trapped points depends continuously on perturbations of the obstacle.
Study examines solutions to Jang equation on anti-de Sitter spacetimes.
problem Existence and properties of solutions to the generalized Jang equation.
method Rigorous analysis in asymptotically anti-de Sitter setting.
result Provides solutions for a broad class of asymptotic conditions.
Paper extends methods to prove positive scalar curvature in all dimensions.
problem Proving positive scalar curvature in all dimensions without spin assumption.
method Develops minimal hypersurface approach to extend positive mass theorem and structure of manifolds.
result Shows singular sets in slices are closed with Hausdorff codimension at least three.
We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow. When the basic first Chern class is positive, we prove a uniform bound on the scalar curvature, the diameter and a uniform C1 bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler…
Study functional confounders in causal inference, enabling estimable effects.
problem Causal inference challenges with functional confounders violating positivity.
method Functional interventions, functional positivity, gradient fields, Level-set Orthogonal Descent Estimation (LODE).
result Valid causal effect estimation under certain conditions.
The paper defines a new mass quantity for 3-manifolds and proves a positive mass theorem.
problem Proving the positive mass theorem for a new geometric quantity.
method Defining X-ADM mass and using a monotonicity formula. result Established a relative positive mass theorem for asymptotically flat 3-manifolds.
Stability of positive mass theorem proven under Ricci curvature bounds.
problem Stability of positive mass theorem under Ricci curvature lower bounds.
method Harmonic level set approach combined with techniques from almost splitting theorem.
result Proves Gromov-Hausdorff stability of positive mass theorem.