Solves characteristic problem in general relativity for null data.
problem Characteristic Cauchy problem in Einstein vacuum field equations.
method Abstract data formalism and tangential components of the ambient Ricci tensor.
result Formulates and solves the characteristic problem completely abstractly.
Solves Einstein vacuum equations gluing problem for close Minkowski data.
problem Solving the characteristic gluing problem for Einstein vacuum equations.
method Derived infinite-dimensional and 10-dimensional gauge-dependent and gauge-invariant charges; constructed null lapse function and conformal geometry.
result Obstructions to gluing problem are gauge-dependent charges, modulo gauge-invariant charges.
Proves a generalized table theorem for odd Euler characteristic surfaces.
problem Proving a generalized table theorem for surfaces with odd Euler characteristic.
method Using the square peg problem for smooth curves, the result is generalized to real valued functions on Riemannian surfaces with odd Euler characteristic.
result Proves the table conjecture for even functions on the two sphere.
Maximally hyperbolic solutions contain future neighborhoods of intersecting hypersurfaces.
problem Maximally globally hyperbolic solutions of higher-dimensional vacuum Einstein equations.
method Analyzing intersections of characteristic hypersurfaces.
result Contains a future neighborhood of intersecting hypersurfaces.
What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and …
Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics …
Study robust utility maximization with uncertain continuous semimartingales.
problem Maximizing utility in continuous time under model uncertainty.
method Duality and conjugate problems for logarithmic, exponential, and power utilities.
result Existence of optimal portfolios for various utilities.
We give a local integral formula, valid on general curved space-times, for the characteristic Cauchy problem for the Dirac equation with arbitrary spin using the method developed by Friedlander in his book "the wave equation on a curved spacetime" (1975). The results obtained by Penrose in the flat case in "Null hypers…
Paper solves Einstein vacuum equations gluing problem with applications.
problem Solving characteristic gluing problem for Einstein vacuum equations.
method Codimension-10 gluing construction, relating charges to ADM energy, and localized version.
result Asymptotically flat data can be glued to Kerr spacetime with 10 charges.
Paper defines and proves geometric uniqueness of Einstein field equations.
problem Einstein field equations characteristic Cauchy problem
method Covariant definition of double null data, proving geometric uniqueness
result Double null data fully covariant and geometrically unique
A new method learns noise characteristics for better state estimation in real-time systems.
problem Challenges in accurately estimating states due to uncertainty in process and measurement models.
method Proposes a learning-based approach with different loss functions to identify noise characteristics.
result Demonstrates improved performance in real-time vehicle state estimation.
Paper glues characteristic data to Kerr spacetime, proving spacelike gluing.
problem Solving characteristic gluing problem for Einstein vacuum equations.
method Detailed characteristic gluing of strongly asymptotically flat data to Kerr spacetime.
result Alternative proof of spacelike gluing construction for strongly asymptotically flat spacelike initial data.
Decides undecidability of equations and first-order theory for Seifert 3-manifold groups.
problem Decidability of equations and first-order theory in Seifert 3-manifold groups.
method Encoding Hilbert's tenth problem and using it to show undecidability.
result Undecidability of equations and first-order theory in Seifert 3-manifold groups with non-negative Euler characteristic.
We study the interplay among Wall's D(2) problem, normal generation conjecture (the Wiegold Conjecture) of perfect groups and Swan's problem on partial Euler characteristic and deficiency of groups. In particular, for a 3-dimensional complex X of cohomological dimension 2 with a finite fundamental group, assuming t…
The paper explores conditions for homothetic Killing vectors on spacetime hypersurfaces.
problem Conditions for the existence of homothetic Killing vectors on spacetime hypersurfaces.
method General identities relating deformation tensor and tensor on hypersurfaces, applied to specific settings.
result Necessary and sufficient conditions for homothetic Killing vectors on spacetime hypersurfaces.
Study on simplicial volume and Euler characteristic of aspherical manifolds.
problem Whether vanishing simplicial volume implies vanishing Euler characteristic.
method Various strategies for both affirmative and negative answers, context with other problems, and comparative analysis of additivity properties.
result Found counterexamples among aspherical spaces that are homology equivalent to manifolds but not manifolds themselves.
Integrability of mean curvature near degenerate points in Heisenberg group.
problem Integrability of sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group.
method Introduction of mildly degenerate characteristic points and use of perimeter measure.
result The sub-Riemannian mean curvature is integrable in a neighborhood of these points.
We assign to a finite CW-complex and an element in its first cohomology group a twisted version of the L2-Euler characteristic and study its main properties. In the case of an irreducible orientable 3-manifold with empty or toroidal boundary and infinite fundamental group we identify it with the Thurston norm. W…
The abstract discusses braided surfaces and their characteristic maps, linking them to algebraic and geometric properties.
problem Characterizing braided surfaces and their characteristic maps.
method Using branched coverings and factorization through free groups, the abstract explores the algebraic and geometric properties of braided surfaces.
result The abstract establishes a connection between braided surfaces and characteristic maps, providing new insights into their structure.
Proves stability of Minkowski space for specific initial data.
problem Stability of Minkowski space under spacelike-characteristic initial data.
method Vectorfield method and bootstrapping argument, with new geometric constructions.
result Global nonlinear stability of Minkowski space proved for the spacelike-characteristic Cauchy problem.
We use some basic properties of binomial and Stirling numbers to prove that the Euler characteristic is, essentially, the unique numerical topological invariant for compact polyhedra which can be expressed as a linear combination of the numbers of faces of triangulations. We obtain this result converting it into an eig…
This paper classifies regular maps with Euler characteristic -p^4 for a prime p≥5.
problem Classify regular maps on surfaces with Euler characteristic -p^4.
method Use inductive method and properties of Sylow p-subgroups to classify.
result Closed surfaces with Euler characteristic -p^4 support no regular maps if p∉{2,3,5,7,13}.
The paper introduces branched α-flows on surfaces with negative Euler characteristic and proves their long-term existence and convergence.
problem Long-term behavior and convergence of branched α-flows on surfaces with negative Euler characteristic.
method Introducing branched α-flows and proving their long-term existence and convergence based on the strict convexity of branched α-potentials.
result Established the long time existence and convergence of branched α-flows on closed surfaces with \( \chi \leq 0 \).
A multi-task network avoids indirect discrimination in insurance pricing.
problem Indirect discrimination in insurance pricing models based on protected characteristics.
method Multi-task neural network architecture trained with partial protected characteristic information.
result Multi-task network produces discrimination-free insurance prices with comparable accuracy to conventional models.
A new method assigns anomaly scores to features for better interpretation.
problem Interpreting anomaly scores from feature attributions.
method Proposes a characteristic function to attribute anomaly scores using Shapley value.
result Demonstrates the potential utility of the proposed attribution methods.
Introduces a new characteristic class for vector bundles with a connection.
problem Tackles the classification of vector bundles with algebraic connections.
method Defines a new characteristic class using a connection and proves its independence of the choice of connection.
result The class c(E) is an invariant of the vector bundle E and is stronger than the Chern and Euler classes. We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler cha…
Study characteristic classes for manifold bundles, focusing on fiber families.
problem Understanding characteristic classes for families of bundles.
method Construction of relative Sullivan models and explicit cocycle representatives for characteristic classes.
result Tools for computing the rational cohomology ring of Baut(ξ) and subring of generalized Miller-Morita-Mumford classes. New Euler characteristics for groupoids generalize orbifold Euler characteristics.
problem Generalizing orbifold Euler characteristics to non-orbifold groupoids.
method Introducing two Euler characteristics for groupoids, using o-minimal structures, and relating them to orbifold Euler characteristics.
result The two new Euler characteristics coincide and generalize orbifold Euler characteristics.
In this work we introduce a mixture of GPs to address the data association problem, i.e. to label a group of observations according to the sources that generated them. Unlike several previously proposed GP mixtures, the novel mixture has the distinct characteristic of using no gating function to determine the associati…
The paper defines and calculates Euler characteristics for quandles.
problem Defining and calculating Euler characteristics for quandles.
method Definition and calculation of Euler characteristics for quandles.
result The quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic.
The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-spac…
Given an S1-manifold with isolated fixed points, some recent papers are concerned with the relationship between the least number of fixed points and the characteristic numbers of this manifold, and their proofs have some similar features. The main purpose of this short survey article is, by using the language of equ…
Study of Markov-modulated affine processes for richer models in finance.
problem Richer models in various applications.
method Martingale problem approach, characteristic function derivation, mathematical properties study.
result Existence and characteristic function of Markov-modulated affine processes.
Proposes a probabilistic model to improve hydrology predictions and trust.
problem Noisy or missing basin characteristics impact streamflow prediction.
method Probabilistic inverse modeling framework to reconstruct basin characteristics.
result 6% improvement in R2 for streamflow prediction, 17% reduction in uncertainty. The paper develops a theory of Ehresmann structures in positive characteristic.
problem Developing a theory for Ehresmann structures in positive characteristic.
method Comparing Frobenius-Ehresmann structures with Cartan geometries and studying their equivalence.
result Formulating and proving the Ehresmann-Weil-Thurston principle for Frobenius-Ehresmann structures.
Study characteristic classes of a specific type of determinantal varieties.
problem Understanding the geometric properties of a special class of determinantal varieties.
method Used Schubert calculus to derive explicit formulas for Chern-Schwartz-MacPherson and Chern-Mather classes.
result Explicit formulas for sectional Euler characteristics, characteristic cycles, and polar classes were obtained.
Proves Euler characteristic of collapsing Alexandrov spaces.
problem Euler characteristic of collapsing Alexandrov spaces.
method Analyzes strata and fibers of the limit space.
result Euler characteristic equals sum of products of strata and fiber Euler characteristics.
Odd-dimensional orbifolds' Euler characteristic equals half of their boundary's.
problem Calculating the Euler characteristic of odd-dimensional orbifolds.
method Proved through mathematical analysis of orbifolds and their boundaries.
result The Euler characteristic of an odd-dimensional orbifold is half of its boundary's.
Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and demonstrate that…
CCVAE captures label characteristics in VAEs for better representation learning.
problem Capturing rich label characteristics in VAEs without conflating them with label values.
method Developed CCVAE, a novel VAE model that explicitly captures label characteristics in latent space.
result CCVAE allows for effective and general interventions like smooth traversals and diverse conditional generation.
Calibrating a Lévy process usually requires characterizing its jump distribution. Traditionally this problem can be solved with nonparametric estimation using the empirical characteristic functions (ECF), assuming certain regularity, and results to date are mostly in 1D. For multivariate Lévy processes and less smooth …
Paper resolves decades-old problem about L-spectra.
problem Identifying L-spectra local information with geometric data. method Proved equivalence of L-orientations and characteristic classes. result Levitt-Ranicki's theory equivalent to Brumfiel-Morgan's classes.
The paper tackles drift identification in Lévy α-stable stochastic systems, proposing a Fourier space approach.
problem Estimating the drift field of a stochastic differential equation driven by Lévy α-stable noise.
method Fourier space approach, parameterizing the drift field using Fourier series, minimizing a loss function with gradients computed via the adjoint method.
result The method is capable of learning drift fields in qualitative and/or quantitative agreement with ground truth fields.
Optimal portfolios are formed by combining momentum, size, and volatility characteristics, enhancing utility for all investors.
problem Estimation error in forming optimal portfolios from characteristics.
method Maximizing an in-sample loss function that is more concave than the utility function, linking weights to characteristics.
result Optimal portfolios with significantly higher certainty equivalents than benchmarks for all investors.
Summarizes connections between Euler characteristic theorems and conjectures.
problem Vanishing of the Euler characteristic
method Diagrammatic summary of connections
result Connections between various theorems and conjectures
Paper optimizes industrial refrigeration using adaptive exploration.
problem Challenges in optimizing real-time industrial processes with unknown characteristics and safety constraints.
method Adaptive and explorative real-time optimization framework with Gaussian process uncertainty quantification.
result Approach increases energy efficiency of refrigeration process, approximating complete information solutions.
Expands Euler-Poincare characteristic to supergeometry.
problem No new problem introduced.
method Introduced Euler-Poincare characteristic pair in supergeometry.
result Examined transversality in Π-symmetric supermanifolds.