New framework analyzes regret in guided diffusion for optimizing structured inputs.
problem Understanding regret behavior in guided-diffusion black-box optimization for structured design problems.
method Developed a certificate-based expected simple-regret framework that avoids assumptions breaking down in modern diffusion BO pipelines.
result Explains how exponential and polynomial convergence can arise from mass lift in near-optimal designs.
Study connects symmetries in dynamical systems to phase plane representations.
problem Understanding symmetries in dynamical systems and their phase plane realizations.
method Analysis of symmetries in differential equations and phase plane representations, establishing correspondence and lifting conditions.
result Every symmetry generator in one formulation corresponds uniquely to a generator in the other, with a lifting condition to solve.
Existence and rigidity results for lifts in Carnot groups.
problem Existence and properties of lifts for maps between Carnot groups.
method Use central extensions to define lifts and prove existence and rigidity results for Lipschitz, Sobolev, and quasiconformal maps.
result Quasiconformal maps admit contact lifts that are bi-Lipschitz.
Study lift metrics and connections on tangent bundles of Riemannian manifolds.
problem Investigate geometric properties of tangent bundles and their lifts.
method Analyze lift metrics and connections on TM of (M,g), and study statistical and Codazzi couples. result Prove a result on 1-Stein and Osserman structures on TM. New formulations for comparing metric measure spaces with arbitrary positive measures.
problem Comparing metric measure spaces with arbitrary positive measures.
method Two novel formulations: a divergence and a conic lifting approach.
result Efficiently solvable formulations for comparing metric spaces with arbitrary positive measures.
In this paper, we define a complete lift for semisprays. If S is a semispray on a manifold M, its complete lift is a new semispray Sc on TM. The motivation for this lift is two-fold: First, geodesics for Sc correspond to the Jacobi fields for S, and second, this complete lift generalizes and unifies previ…
Unified approach to constructing integrable systems using Stäckel lifts.
problem Constructing new integrable Hamiltonian systems.
method Generalized Stäckel geometry and Haantjes structure.
result Hamiltonian systems with momentum-dependent Stäckel matrices exhibit symplectic-Haantjes structures.
Geometric structures are lifted to higher tangent bundles preserving statistical properties.
problem Lifting statistical structures to higher tangent bundles while maintaining their properties.
method Natural lifts of geometric objects and potentials to higher tangent bundles, preserving statistical manifold structures.
result Lifted statistical structures on higher tangent bundles maintain pseudo-Riemannian metrics and are again statistical manifolds.
The X-ADM mass is shown to be equivalent to the ADM mass, proving the X-positive mass theorem in all dimensions.
problem Proving the X-positive mass theorem for all dimensions.
method Conformal reduction argument.
result The X-ADM mass is equivalent to the ADM mass, proving the X-positive mass theorem in all dimensions.
The article classifies liftings of connections on differential manifolds for geodesic modeling.
problem Classifying liftings of connections on differential manifolds.
method Liftings of connections on frame bundles, induced and adjust liftings.
result Developed a method for geodesic modeling of differential equations.
Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
problem Embedding maps in higher dimensions without self-intersections.
method Lifting maps to embeddings in product spaces.
result Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold M to its Weil bundle TAM defined by a Frobenius Weil algebra A. For a Poisson manifold (M,w), we show that the complete lift wC and the vertical lift wV of the Poisson tensor w are Poisson tensors on $T^…
The paper studies nonlinear mass concepts in 3-manifolds with nonnegative scalar curvature.
problem Nonlinear isocapacitary mass in 3-manifolds with nonnegative scalar curvature.
method Derives positive mass theorems and shows mass coincides with ADM mass under mild conditions.
result Nonlinear masses coincide with ADM mass and prove the Penrose inequality.
In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays. Firstly we consider basic examples. We give a projective line of diff(M)-equivariant pencil liftings for first order operators,…
Study covers of sphere with homeomorphisms lifting property.
problem Finite abelian covers of sphere with lifting homeomorphisms.
method Completely determined covers with specific lifting property.
result Properties of finite abelian covers with lifting homeomorphisms.
The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral ζ-invariants using lifted …
The paper constructs Sasakian lifts from Kähler manifolds and studies their properties.
problem Constructing Sasakian structures from Kähler manifolds and analyzing their geometric properties.
method Local construction of Sasakian manifolds from Kähler base using vector field operations.
result Existence and properties of α-Sasakian Ricci solitons in Sasakian lifts. New quantum code lacks sparse lift.
problem Existence of sparse lifts for quantum codes.
method Constructed a sparse Z2 chain complex without a sparse lift. result Found a quantum code without a sparse lift.
Equivalence proven for isocapacitary mass notions.
problem Proving equivalence of isocapacitary mass notions.
method Proof of equivalence for G. Huisken's and J. L. Jauregui's isocapacitary mass.
result Equivalence of isocapacitary mass notions proven.
Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.
problem Lifting diffeomorphisms to vector bundles.
method Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
result Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
Simple lifts of non-simple curves on surfaces.
problem Finding simple lifts of non-simple closed curves on surfaces.
method Constructing finite covers and curves on surfaces.
result Explicit non-simple closed curves with simple lifts on degree n covers.
This paper extends braid lifting to coloured braid groupoids for all simple disc covers.
problem Lifting braids to homeomorphisms on branched covers of the disc.
method Defines a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc.
result Characterizes the lift of every coloured braid, recovering classical lifting on liftable braids.
We construct some lift of an almost complex structure to the cotangent bundle, using a connection on the base manifold. This generalizes the complete lift defined by I.Sato and the horizontal lift introduced by K.Yano and S.Ishihara. We study some geometric properties of this lift and its compatibility with symplectic …
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.
Proves harmonicity equivalence on manifold metrics.
problem Harmonicity of metrics on pseudo-Riemannian manifolds.
method Equivalence of harmonicity on manifold metrics and their lifts.
result Harmonicity of a metric is equivalent to its lifts' harmonicity.
In Finsler geometry the complete lift vector fields have distinguished geometric significance. For example a vector field on a Finsler manifold is said to be conformal if its complete lift is conformal in usual sense. In this work we define a new Riemannian or Pseudo-Riemannian metric on TM derived from a Finsler metri…
Introduce new boundary mass for asymptotically flat half-manifolds
problem Define boundary mass for asymptotically flat half-manifolds
method Introduce new boundary mass
result Define boundary mass for asymptotically flat half-manifolds
The Eisenhart lift connects Hamiltonian systems to geodesics in pp-wave spacetimes.
problem Studying the stability and dynamics of Hamiltonian systems.
method Eisenhart lift to pp-wave spacetimes and conformal classes of ODEs.
result Existence of a constant of the motion generalizing conservation of energy.
Survey of twistor lifts of surfaces in 4-dimensional spaces.
problem Understanding the properties of surfaces in 4-dimensional Euclidean space.
method Definitions of Gauss maps and twistor lifts using orthogonal complex structures.
result Holomorphicity and isotropicity of minimal surfaces in E4. We prove directly without using a density theorem that (i) the ADM mass defined in the usual way on an asymptotically flat manifold is equal to the mass defined intrinsically using Ricci tensor; (ii) the Hamiltonian formulation of center of mass and the center of mass defined intrinsically using Ricci tensor are the sa…
Study of lifting maps in branched covers of 3-manifolds, showing non-injectivity.
problem Injectivity of lifting maps in branched covers of 3-manifolds.
method Analysis of branched covers and mapping class groups.
result Lifting map is generally not injective for most regular branched covers of 3-manifolds.
In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate …
Develops a new bidding system to maximize advertiser profit.
problem Inaccurate prediction of ad lift-effect due to biased log data.
method Unbiased Lift-based Bidding System that predicts lift-effect from biased log data.
result Demonstrates superior and practical high-performing lift-based bidding strategy.
Continuous metrics on R^3 with specific properties have non-negative harmonic mass.
problem Proving non-negativity of mass for continuous metrics.
method Defining harmonic mass and using properties of approximating smooth metrics.
result The harmonic mass of continuous metrics is non-negative.
The paper examines mass aspects at future null infinity and limits of quasilocal mass.
problem Understanding mass aspects and limits of quasilocal mass at future null infinity.
method Review and extension of Bondi mass and mass loss formula in Bondi-Sachs coordinate system.
result New results about the limit of quasilocal mass of unit spheres at null infinity.
We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection a…
Study the mass of flat 3-manifolds with boundary using specific methods.
problem Calculate the mass of asymptotically flat 3-manifolds with boundary.
method Use the method of Bray-Kazaras-Khuri-Stern to derive a mass formula.
result Derive sufficient conditions for the positivity of the mass.
Unified definition of mass aspect function for weakly regular hyperbolic manifolds.
problem Ambiguity in mass definition for asymptotically hyperbolic manifolds.
method Introduced an ADM-style mass aspect function for broad asymptotics and low regularity.
result Unified mass aspect function exhibits favorable covariance properties.
Local mass perspective on Bayesian inference
problem Measuring distributional discrepancy in Bayesian inference
method Introducing Mass Index and Regularised Extended KL
result Proving inequalities for comparing local small-ball masses
On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.
Simple proof for sphere mass calculation.
problem Computing the ADM mass of static sphere extensions.
method Uses mass formula for static asymptotically flat manifolds.
result Validated mass formula for small spheres.
We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.
Develops a new algebraic framework for differential geometry of infinite dimensional spaces.
problem Creating a differential geometry for infinite dimensional spaces without topology or local coordinates.
method Introduces a general algebraic framework for lifted geometry applicable to various infinite dimensional spaces.
result Stokes' Theorem appears as a form of differentiability in the lifted geometry of spaces of submanifolds.
New ADM mass definition for weakly regular manifolds.
problem Defining ADM mass for non-smooth manifolds.
method Proposed a new definition for metrics with local Sobolev regularity.
result Finite mass, invariance under coordinate changes, and agreement with smooth case.
Researchers lift knot coloring polynomial to Habiro ring.
problem Lifting colored Jones polynomial of knots to Habiro ring.
method Introduced new Habiro ring and map, used Alexander polynomial.
result Existence of loop expansion at roots of unity confirmed.
New theorem for spacetime mass in noncompact regions.
problem Mass in noncompact spacetime regions.
method Developed mass type invariant and boundary conditions; proof based on spinors.
result Proved positive mass theorem for noncompact boundaries.
We show the total space of the canonical line bundle L of a Kahler-Einstein manifold Xn supports integrable SU(n+1) structures, or Calabi-Yau structures. The canonical real line bundle L⊂L over a minimal Lagrangian submanifold M⊂X is calibrated in this setting and hence can …
Huisken's isoperimetric mass is always nonnegative.
problem Understanding the nonnegativity of Huisken's isoperimetric mass.
method Simple reasoning based on basic properties.
result Huisken's isoperimetric mass is nonnegative.