Expected centre of mass for random embeddings is constant.
problem Understanding the expected centre of mass for random embeddings.
method Analyzing the Haar measure and Gaussian unitary ensemble on SL(N, C).
result The expectation of the centre of mass is a constant multiple of the identity matrix.
Adaptive HMC improves sampling efficiency by optimizing mass matrix.
problem Inefficient HMC performance due to mass matrix choice.
method Gradient-based adaptation of mass matrix to maximize proposal entropy.
result Adaptation method outperforms HMC variants by optimizing mass matrix.
Gradient flow solves optimal mass transport for covariance matrices.
problem Optimal mass transport for covariance matrices.
method Gradient flow on fiber bundle structure.
result Global convergence to polar decomposition.
QHMC improves HMC for sampling from complex distributions.
problem Inefficiency of HMC in sampling from spiky and multimodal distributions.
method Proposes QHMC, a quantum-inspired version of HMC with a random mass matrix.
result QHMC and QSGNHT achieve more stable and accurate sampling results.
Using the notion of vacuum pairs we show how the (square of the) mass matrix of the fermions can be considered geometrically as curvature. This curvature together with the curvature of space-time, defines the total curvature of the Clifford module bundle representing a ``free'' fermion within the geometrical setup of s…
Paper proposes a new approach to optimal transport for vector and matrix densities.
problem Optimal transport for vector and matrix densities with positivity and action transitivity constraints.
method Gauge-theoretic approach using semi-direct product groups of diffeomorphisms and gauge transformations.
result Bures-type metrics on semi-direct product groups relate to Wasserstein-type metrics on vector and matrix densities via Riemannian submersions.
SML improves pancreatic mass diagnosis accuracy using CT images.
problem Improving accuracy in pancreatic mass screening using CT imaging.
method Spectral machine learning method trained on 30,000 images, choosing fundamental images based on eigenvectors and removing irrelevant pixels.
result Achieved 94.6% test accuracy in diagnosing 113 patients.
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 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.
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.
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.
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
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…
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.
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.
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.
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.
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.
GT estimator shows convergence for Markov samples, improving i.i.d. results.
problem Estimating missing mass in Markov samples.
method Analyzed convergence of Good-Turing estimator for Markov samples, considering spectral properties of transition matrices.
result The convergence of the GT estimator for Markov samples depends on the spectral properties of the transition matrices, leading to a new minimax rate of 1/(nβ5) for rank-2 Markov chains. 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.
New optimal transport method handles mass creation and destruction.
problem Optimal re-balancing of portfolios with mass creation or destruction.
method Formalizes an optimal transport problem with mass-change factor.
result Existence of optimal transport plans and maps established.
Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.
problem Estimating the residual Monge-Ampère mass of symmetric plurisubharmonic functions with isolated singularities.
method Utilized Sasakian geometry to derive estimates on the residual mass in relation to Lelong numbers.
result Partially resolved the zero mass conjecture by Guedj and Rashkovskii.
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.
Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.
problem Analyzing the residual Monge-Ampère mass of symmetric plurisubharmonic functions.
method Proved zero mass for functions with zero Lelong number at origin and S1-invariance. result Zero mass conjecture answered for symmetric functions.
Surveying mass in 2D hyperbolic geometry, overcoming challenges via minimisation.
problem Defining mass in 2D hyperbolic geometry.
method Minimisation using positive energy theorem and gluing theorems.
result Construction of novel initial data sets with controlled mass.
The paper connects mass, harmonic functions, and capacity in asymptotically flat 3-manifolds.
problem Connections among ADM mass, harmonic functions, and capacity in asymptotically flat 3-manifolds.
method New formulae for ADM mass via harmonic functions, monotone quantities, and geometric inequalities.
result The mass-to-capacity ratio is bounded below by 1 - sqrt(normalized Willmore functional of the boundary).
New mass definition linked to ADM mass for general metrics.
problem Defining mass for metrics with low regularity.
method Using isocapacitary inequality to define total mass.
result Inequality between new mass and ADM mass proved.
Refines geometric center of mass analysis for Einstein field equations.
problem Analyzing the geometric center of mass of Willmore surfaces in initial data for Einstein field equations.
method Refined Lyapunov-Schmidt analysis to study geometric center of mass of area-constrained Willmore surfaces.
result The geometric center of mass agrees with the Hamiltonian center of mass under specific conditions.
Mass in relativity linked to polyhedra geometry.
problem Understanding ADM mass in general relativity.
method Relating ADM mass to the total mean curvature and defect of dihedral angles of Riemannian polyhedra.
result Expressed n-dimensional mass as an integral of geometric quantities. Proves spacetime positive mass theorem in all dimensions.
problem Proving the spacetime positive mass theorem in arbitrary dimensions.
method Using Brendle--Wang's Riemannian positive mass theorem approach.
result Proves the spacetime positive mass theorem for all dimensions.
New mass inequalities and proofs for causal variational principles.
problem Proving new mass inequalities for causal variational principles.
method Proved a new inequality for minimizers of causal variational principles and applied it to prove the positive mass theorem.
result Introduced a positive quasilocal mass and proved new mass inequalities.
Defines mass for non-smooth hyperbolic spaces using a modified flow.
problem Defining mass for non-smooth, asymptotically hyperbolic spaces.
method Normalized Ricci-DeTurck flow with scalar curvature lower bound.
result Mass function well-defined for continuous metrics.
New solutions found with negative mass in general relativity.
problem Finding metrics with negative mass in general relativity.
method Constructing families of metrics with specific properties.
result Obtained new classes of solutions with negative mass.
Defines a new geometric quantity for hyperbolic manifolds, showing it's well-defined and invariant.
problem Defining a mass for asymptotically hyperbolic manifolds under weaker conditions.
method Volume-renormalized mass defined as a linear combination of ADM mass and renormalized volume.
result Volume-renormalized mass is well-defined and diffeomorphism invariant under weaker conditions.
Bartnik mass is positive and non-decreasing for black holes
problem Quasilocal mass for black holes
method Defining a Bartnik mass and proving positivity and monotonicity
result Positive and non-decreasing Bartnik mass for black holes
New formula shows how causal vectors relate to mass-minimizing data.
problem Understanding mass-minimizing initial data sets and their geometry.
method Developed a new monotonicity formula for causal Killing vectors.
result Established strong maximum principles for the Lorentzian length.
New mass definition for negative cosmological constant spacetimes.
problem Defining quasilocal mass for spacetimes with negative cosmological constant.
method Spinorial approach based on previous work for vanishing cosmological constant.
result Non-negative mass, equal to Misner-Sharp mass in spherical symmetry, zero for AdS.
Schoen-Yau's zero mass theorem stability remains an open question.
problem Geometric stability of Schoen-Yau's zero mass theorem.
method Review of geometric stability, examples, and convergence notions.
result Open question on geometric stability of Schoen-Yau's zero mass theorem.
For a closed surface M with metric g, the Robin mass m(p) at the point p is the value of the Green function G(p,q) at p=q after the logarithmic singularity has been removed. The Laplacian-mass is the average value of the Robin mass, minus the value of the Robin mass for the round sphere of the same area. The Laplacian-…
Paper defines Bartnik mass for hyperbolic extensions and proves staticity.
problem Defining and proving staticity of asymptotically hyperbolic minimal mass extensions.
method Definition of Bartnik mass, construction of metrics, one-parameter family analysis.
result Static potential for asymptotically hyperbolic admissible extensions achieving Bartnik mass.
We study Hawking mass and the Huisken's isoperimetric mass evaluated on surfaces with boundary. The convergence to an ADM mass defined on asymptotically flat manifold with a non-compact boundary are proved.
Paper introduces OMD for ordered state transitions in SSMs.
problem Modeling ordered latent states in dynamic systems.
method Ordered Matrix Dirichlet (OMD) prior over ordered stochastic matrices.
result OMD models recover interpretable ordered latent structure without sacrificing predictive performance.
Study shows a mass quantity for C0 metrics that agrees with ADM mass.
problem Understanding ADM mass for C0 metrics and its behavior under Ricci-DeTurck flow. method Developed a C0 mass quantity and analyzed its behavior under Ricci-DeTurck flow. result The C0 mass at infinity is independent of coordinate charts and has controlled distortion under Ricci-DeTurck flow. Study proves inequalities for mass-capacity on curved spaces.
problem Proving nonnegativity and positive lower bounds of mass on curved spaces.
method Applying mass-capacity inequalities from \cite{M22} to manifolds with nonnegative scalar curvature.
result Sufficient conditions for nonnegativity and positive lower bounds of mass.
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…