The concepts of relative velocity and acceleration, deviation velocity and acceleration and relative momentum of point particles in spaces (manifolds), the tangent bundle of which is equipped with a transport along paths, are introduced. If the tangent bundle is endowed also with a metric, it gives rise also to the not…
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48 results for “Relative Quantities”
New definitions of conserved quantities at null infinity resolve ambiguities in general relativity.
problem Ambiguities in defining conserved quantities like angular momentum at null infinity.
method New definitions based on Chen-Wang-Yau quasilocal conserved quantities and optimal isometric embedding theory.
result These new definitions are free of supertranslation ambiguity and limit to classical Bondi mass.
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 -dimensional mass as an integral of geometric quantities.
Conserved quantities in General Relativity: the case of initial data sets with a noncompact boundarymath.DG
The study extends conserved quantities theory to non-compact boundary initial data sets.
problem Extending conserved quantities theory to initial data sets with non-compact boundaries.
method Analysis of scalar curvature and mean curvature in the interior and boundary.
result Rigidity/flexibility phenomena in positive mass theorems and Penrose inequalities.
Study on relative entropy for hypersurfaces in hyperbolic space.
problem Understanding relative entropy for hypersurfaces in hyperbolic space.
method Relate relative entropy to renormalized area and apply monotonicity formula to mean curvature flows.
result Obtained a monotonicity formula for relative entropy in hyperbolic space.
A new LSV model uses relative quantities for better trading and risk management.
problem Inability to use intuitive and stable parameters in LSV models.
method Develops a hybrid method using relative quantities for efficient derivative pricing and scenario generation.
result Shows improved stability and ease of use for model parameters.
Paper develops robust neural network sensors for fuel injection quantities.
problem Adversarial noise increases error in standard neural network models for fuel injection measurements.
method Apply provable robust network learning and verification methods to fuel injection measurements.
result Provable robust model reduces mean relative error to 16.5% under sensor noise.
The writers propose a mathematical Method for deriving risk weights which describe how a borrower's income, relative to their debt service obligations (serviceability) affects the probability of default of the loan. The Method considers the borrower's income not simply as a known quantity at the time the loan is made, …
In a coordinate free form are found the (deviation) equations satisfied by the (infinitesimal) deviation vector, relative velocity, relative momentum, relative acceleration and relative energy of two point particles in a differentiable manifold the tangent bundle of which is endowed with a linear transport along paths,…
For a spacelike 2-surface in spacetime, we propose a new definition of quasi-local angular momentum and quasi-local center of mass, as an element in the dual space of the Lie algebra of the Lorentz group. Together with previous defined quasi-local energy-momentum, this completes the definition of conserved quantities i…
We study a notion of relative entropy motivated by self-expanders of mean curvature flow. In particular, we obtain the existence of this quantity for arbitrary hypersurfaces trapped between two disjoint self-expanders asymptotic to the same cone. This allows us to begin to develop the variational theory for the relativ…
REMEDI improves neural entropy estimation across various tasks.
problem Challenges in estimating information theoretic quantities in high-dimensional data.
method Combines minimization of cross-entropy with estimation of deviation from data density.
result Improves accuracy in entropy estimation on synthetic and natural data.
From turbulence to financial time seriescond-mat.stat-mech
We develop a framework especially suited to the autocorrelation properties observed in financial times series, by borrowing from the physical picture of turbulence. The success of our approach as applied to high frequency foreign exchange data is demonstrated by the overlap of the curves in Figure (1), since we are abl…
These lecture notes provide some introduction to the 3+1 formalism of general relativity, which is the foundation of most modern numerical relativity. The text is rather self-contained, with detailed calculations and numerous examples. Contents: 1. Introduction, 2. Geometry of hypersurfaces, 3. Geometry of foliations, …
In Hamiltonian mechanics, a (continuous) symmetry leads to conserved quantity, which is a function on (extended) phase space. In Nambu mechanics, a straightforward consequence of symmetry is just a relative integral invariant, a differential form which only upon integration over a cycle provides a conserved real number…
A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem
problem Defining and analyzing a mass-type invariant for smooth metric measure spaces
method Defining a mass-type quantity and showing its geometric invariance properties
result The mass-type quantity has a close relation with the fractional Yamabe problem and the relevant Green's function
Many inference problems involving questions of optimality ask for the maximum or the minimum of a finite set of unknown quantities. This technical report derives the first two posterior moments of the maximum of two correlated Gaussian variables and the first two posterior moments of the two generating variables (corre…
Study on heat content for submanifolds in sub-Riemannian geometry.
problem Understanding heat content for submanifolds in sub-Riemannian geometry.
method Existence of smooth tubular neighborhood, definition of relative heat content, approximation via smooth neighborhoods, asymptotic expansion analysis.
result Approximation of relative heat content fails to recover the exact expansion.
Incorporating Expert Opinion on Observable Quantities into Statistical Models -- A General Frameworkstat.ML
The paper introduces a method to incorporate expert opinion on observable quantities into statistical models.
problem Tackling the challenge of integrating expert knowledge on observable quantities into statistical models.
method The approach involves updating a prior belief using a loss function that reflects expert opinion on observable quantities.
result The method allows for a flexible specification of expert opinion and is straightforward to implement.
The paper introduces a new method to characterize cosmological models using observer-based invariants.
problem Equivalence problem for cosmological models in four-dimensional gravity theories.
method Modified Cartan-Karlhede algorithm adapted to fundamental observers, including derivatives of the time-like vector field.
result A list of invariants that completely characterize cosmological models, independent of coordinates.
In two previous papers the author developed a second-order price adjustment (tâtonnement) process. This paper extends the approach to include both quantity and price adjustments. We demonstrate three results: a analogue to physical energy, called "activity" arises naturally in the model, and is not conserved in general…
Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvaturemath.DG
Study a relative aspherical conjecture and prove 3-manifold obstruction to positive scalar curvature.
problem Obstructing the existence of positive scalar curvature in higher dimensions.
method Introduced a relative aspherical condition and a new geometric quantity called spherical width.
result Proved results on how 3-manifolds obstruct the existence of positive scalar curvature.
We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in [26] to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group…
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing relative surface areas, proving invariance and inequalities, and using geometric interpretations.
result Established inequalities and a new notion of entropy for ball-convex bodies.
Extends specific relative entropy to multidimensional continuous martingales.
problem Mutual singularity of martingale laws in continuous time.
method Extension of specific relative entropy from one to multiple dimensions, including closed-form expressions for simple examples.
result Establishes that the lower bound on specific relative entropy from Gantert carries over to higher dimensions and is tight.
Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.
problem Understanding endperiodic maps on infinite graphs with finitely many ends.
method Adapting relative train track maps and combinatorial techniques to infinite type setting.
result Any generalized endperiodic map is homotopic to a relative train track map.
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 -ADM mass and using a monotonicity formula.
result Established a relative positive mass theorem for asymptotically flat 3-manifolds.
New method targets relative risk heterogeneity in clinical trials.
problem Identifying treatment effects across subgroups with absolute risk differences.
method Modified causal forests using a novel node-splitting procedure based on relative risk.
result Relative risk causal forests can capture heterogeneity not detected by absolute risk methods.
The center of mass in General Relativity is hard to define due to coordinate freedom.
problem Defining the center of mass in General Relativity rigorously and consistently.
method Analyzing the challenges in Newtonian Gravity and using Bartnik's asymptotic harmonic coordinates.
result Examples of initial data sets in General Relativity that do not satisfy center of mass definitions.
We present the general theory of curves in conformal geometry using tractor calculus. This primarily involves a tractorial determination of distinguished parametrizations and relative and absolute conformal invariants of generic curves. The absolute conformal invariants are defined via a tractor analogue of the classic…
In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previo…
Introduces relative information gain for improving Gaussian process regression rates.
problem Improving the sample complexity of estimating or maximizing unknown functions.
method Introduces relative information gain, interpolates between effective dimension and information gain, and proves PAC-Bayesian bounds.
result Obtains minimax-optimal rates of convergence through the relative information gain.
Study on Einstein deformations and desingularizations, finding some 4-orbifolds cannot be limits of smooth metrics.
problem Whether every Einstein 4-orbifold is a limit of smooth Einstein 4-manifolds.
method Analysis of integrability of deformations through variations of Schoen's Pohozaev identity and introduction of preserved integral quantities.
result Spherical and hyperbolic 4-orbifolds with the simplest singularities cannot be limits of smooth Einstein 4-manifolds.
The relative -entropy is the Rényi analog of relative entropy and arises prominently in information-theoretic problems. Recent information geometric investigations on this quantity have enabled the generalization of the Cramér-Rao inequality, which provides a lower bound for the variance of an estimator of an escort…
Meta-learning improves relative density-ratio estimation from limited data.
problem Estimating relative density-ratios from few instances.
method Meta-learning using neural networks to extract and embed dataset information for relative DRE.
result Meta-learning enables efficient and effective adaptation to few instances for relative DRE.
ROM-net framework applies to industrial design uncertainty quantification.
problem Uncertainty quantification in industrial design models.
method Dictionary-based ROM-net framework for reduced order modeling.
result ROM-net computes predictions in 2 hours with high accuracy.
The first passage event for sums of dependent Lévy processes with applications to insurance riskmath.PR
For the sum process of a bivariate Lévy process with possibly dependent components, we derive a quintuple law describing the first upwards passage event of over a fixed barrier, caused by a jump, by the joint distribution of five quantities: the time relative to the time of the previous maxi…
The paper analyzes the generalization performance of spectral clustering algorithms and proposes new methods to improve their effectiveness.
problem Theoretical analysis of spectral clustering's generalization performance.
method Theoretical analysis and development of new spectral clustering algorithms.
result The excess risk bounds of spectral clustering algorithms have a convergence rate.
We show how the Dixon's system of first order equations of motion for the particle with inner dipole structure together with the side Mathisson constraint follows from rather general construction of the 'Hamilton system' developed by Weyssenhoff, Rund and Grässer to describe the phase space counterpart of the evolution…
The paper is concerned with regularity properties of boundaries of causal pasts of points in a 3+1-dimensional Einstein-vacuum spacetime. In a Lorentzian manifold such boundaries play crucial role in propagation of linear and nonlinear waves. We prove a uniform lower bound on the radius of injectivity of these null bou…
Theory of Information Pricingq-fin.PR
In financial markets valuable information is rarely circulated homogeneously, because of time required for information to spread. However, advances in communication technology means that the 'lifetime' of important information is typically short. Hence, viewed as a tradable asset, information shares the characteristics…
Disease phenotyping algorithms process observational clinical data to identify patients with specific diseases. Supervised phenotyping methods require significant quantities of expert-labeled data, while unsupervised methods may learn non-disease phenotypes. To address these limitations, we propose the Semi-Supervised …
We introduce polynomial processes in the sense of [8] in the context of stochastic portfolio theory to model simultaneously companies' market capitalizations and the corresponding market weights. These models substantially extend volatility stabilized market models considered by Robert Fernholz and Ioannis Karatzas in …
New formulation tackles arbitrage in volatile markets using eigenvalue bounds.
problem Arbitrage opportunities in volatile markets beyond a certain time horizon.
method Formulated as a stochastic optimal control problem, solved via PDE.
result Characterized arbitrage time horizon through PDE solution.
Study BGG operators on homogeneous conformal geometries.
problem Finding solutions to BGG operators on homogeneous conformal geometries.
method Invariant calculus for algebraic computations.
result Explicit solutions found for conformal Killing tensors, forms, and spinors.
We consider an agent who needs to buy (or sell) a relatively small amount of asset over some fixed short time interval. We work at the highest frequency meaning that we wish to find the optimal tactic to execute our quantity using limit orders, market orders and cancellations. To solve the agent's control problem, we b…
Sharp estimates for p-capacity on manifolds with Ricci curvature bounds.
problem Estimating p-capacity on manifolds with Ricci curvature constraints.
method Sharp comparison inequalities, warped-product model ends, and scale-invariant quantities.
result Characterization of equality cases and optimal ranges for normalization parameters.
When eliciting judgements from humans for an unknown quantity, one often has the choice of making direct-scoring (cardinal) or comparative (ordinal) measurements. In this paper we study the relative merits of either choice, providing empirical and theoretical guidelines for the selection of a measurement scheme. We pro…