Brakke flow definitions proven equal.
problem Proving equality of various Brakke flow definitions.
method Corrected proof of Brakke's §3.5 estimate.
result Most Brakke flow definitions are equivalent.
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48 results for “equal definitions”
The paper compares different fairness definitions under various worldviews.
problem Avoiding disparity amplification under different worldviews.
method Mathematical comparison of four fairness definitions using a theoretical framework.
result Different worldviews require different fairness definitions to avoid disparity amplification.
In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygo…
We modify previous quasi-local mass definition. The new definition provides expressions of the quasi-local energy, the quasi-local linear momentum and the quasi-local mass. And they are equal to the ADM expressions at spatial infinity. Moreover, the new quasi-local energy has the positivity property.
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.
We classify semi-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to three and the metrics induced on fibres are negative definite. Also, we obtain the classificat…
A new geometric definition of integration for differential forms.
problem Standard integration definitions are coordinate-dependent and not suitable for certain contexts.
method Uses triangulations and cochains on the pair groupoid to define integration.
result Natural definition in Lie algebroids, stochastic integration, and quantum field theory.
The paper proposes a method to measure fairness through equality of effort using algorithmic recourse.
problem Measuring fairness through equality of effort in automated systems.
method Applying algorithmic recourse to quantify equality of effort, overcoming previous limitations.
result An algorithm for assessing equality of effort has been developed and validated.
An empirical method to determine kernel definiteness.
problem Determining definiteness of kernels is time-consuming and requires effort.
method Empirical approach using sampling and optimization with evolutionary algorithms.
result Empirical method can disprove definiteness and estimate likelihood of indefinite matrices.
New methods ensure fair rankings in web-scale recommender systems.
problem Ensuring fairness in recommender systems, especially in web-scale applications.
method Scalable methods for achieving fairness in rankings, addressing position bias.
result Our methods effectively achieve fairness in various recommender systems.
For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-Müller the…
Smooth maps show Gromoll filtration for spheres.
problem Understanding Gromoll filtration for special generic maps.
method Analyzing smooth maps with definite folds and Gromoll filtration.
result Gromoll filtration equals p for special generic maps.
In this paper, we define the rectangle condition on the bridge sphere for a -bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of -manifolds. We show that the satisfaction of the rectangle condition for a -bridge decomposition can …
New definition of twisted 1-loop invariant using Ptolemy coordinates.
problem Defining and proving properties of twisted 1-loop invariants.
method Alternative definition via Jacobian of Ptolemy coordinates.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial for hyperbolic once-punctured torus bundles.
Abstract reviews mathematical fairness in machine learning.
problem Ensuring fairness in machine learning models.
method Independence-based approach to fairness definitions and methodologies.
result Optimal fair classifiers and predictors under equality of odds are derived.
We define and calculate the HOMFLY polynomial for a specific type of quiver.
problem Calculating the HOMFLY polynomial for forest quivers.
method Recursive definition and closed-form expression for forest quivers.
result Closed-form expression for the HOMFLY polynomial of a forest quiver.
We prove in this paper that the weighted volume of the set of integral transportation matrices between two integral histograms r and c of equal sum is a positive definite kernel of r and c when the set of considered weights forms a positive definite matrix. The computation of this quantity, despite being the subject of…
Paper introduces threshold invariant fairness to ensure equitable predictions across different groups.
problem Machine learning models can be unfair to certain groups based on sensitive attributes.
method Proposes threshold invariant fairness and uses two approximation methods to equalize risk distributions.
result Demonstrates effectiveness in alleviating threshold sensitivity in fairness models.
Study shows pooling scores for conformal prediction distorts group coverage.
problem Pooling scores for conformal prediction distorts group coverage.
method Derived conservation law and lower bound, demonstrated tension between fairness definitions, quantified trade-off between policies.
result Pooling scores for conformal prediction distorts group coverage.
New distances defined between space-times, proving some definite.
problem Defining distances between space-times.
method Introducing causal-null-compactifiable space-times and using cosmological time and null distance.
result Various definite distances defined, proving convergence of space-times.
A Moral Framework for Understanding of Fair ML through Economic Models of Equality of Opportunitycs.LG
Abstract: Maps fairness concepts to EOP, unifies them, and proposes new measures.
problem Understanding and interpreting fairness in machine learning.
method Mapping fairness concepts to EOP, formalizing existing fairness definitions, proposing new measures.
result Existing fairness definitions can be seen as special cases of EOP, and new measures proposed.
New method uses causal thinking to make AI fairer decisions.
problem Designing fair machine learning models that treat equal individuals equally and unequals unequally.
method Rank-preserving interventional distributions and warping method.
result Warping method effectively identifies discriminated individuals and mitigates unfairness.
Paper proves edge-connectivity equals minimum degree for graphs with non-negative curvature.
problem Edge-connectivity vs. minimum degree in graphs with non-negative curvature.
method Analyzes finite connected graphs with non-negative Lin-Lu-Yau curvature.
result Edge-connectivity equals minimum degree for graphs with non-negative curvature.
Researchers define and analyze the renormalized volume of 4D Ricci-flat ALE spaces.
problem Understanding the properties of 4D Ricci-flat ALE spaces.
method Introducing a new definition of renormalized volume and proving its properties.
result The renormalized volume is always less than or equal to zero, with equality for specific spaces.
For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic…
An algorithm calculates Gabai width for thousands of knots.
problem Calculating Gabai width for many knots.
method Algorithmic definition of Wirtinger width leading to efficient Gabai width bounds.
result Proved Wirtinger width equals Gabai width for knots.
We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov …
The main goal of this paper is presentation a modern axiomatic approach to financial arithmetic. At the first, the axiomatic financial arithmetic theory was proposed by Peccati who has introduced the axiomatic definition of the future value. This theory has been extensively developed in past years. Proposed approach to…
We study the problem posed by F. Burstall of developing a theory of isothermic Euclidean submanifolds of dimension greater than or equal to three. As a natural extension of the definition in the surface case, we call a Euclidean submanifold {\it isothermic} if it is locally the image of a conformal immersion of a Riema…
In this paper, we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure; we prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kahler differentials; therefore, it induces a differential complex from which we derive the notio…
Study projective symmetries in Finsler spaces, showing reductions and constant flag curvature.
problem Exploring projective symmetries in Finsler spaces and their properties.
method Analyzing algebraic sub-algebras and curvature invariants of projective vector fields.
result Closed Finsler spaces with negative Ricci curvature reduce to Killing vector fields.
We combine and extend the work of Alexander & Antman \cite{alexander.82} and Fuller \cite{fuller.71,fuller.78} to give a framework within which precise definitions can be given of topological and geometrical quantities characterising the contortion of open rods undergoing large deformations under end loading. We use th…
We show how the cost of funding the collateral in a particular set up can be equal to the Bilateral Valuation Adjustment with the "funded" probability of default, leading to the definition of a Funded Bilateral Valuation Adjustment (FBVA). That set up can also be viewed by an investor as an effective way to restructure…
Considering a non-constant smooth solution of the Tanno equation on a closed, connected Kähler manifold with positively definite metric , Tanno showed that the manifold can be finitely covered by $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS})$, where denotes the Fubini-Study metric of constant hol…
Agnostic Risk Parity improves diversification by minimizing unknown-unknown risks.
problem Markowitz' optimal portfolio theory fails to deliver diversification out-of-sample.
method Symmetry arguments lead to Eigenrisk Parity portfolios, then specialize to AGP minimizing unknown-unknown risks.
result AGP portfolios minimize unknown-unknown risks and perform well with technical strategies.
Using the Fourier expansion of Markov traces for Ariki-Koike algebras over , we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite simple …
In the previous paper we constructed the local system of Khovanov complexes on the Vassiliev space of knots and extended it to the singular locus. In this paper we introduce the definition of the homology theory (local system) of finite type and prove the first finiteness result: the Khovanov local system restricted to…
Fair Pipelinescs.CY
This research tackles fairness in multi-stage decision-making processes.
problem Fairness in multi-stage decision-making processes like hiring.
method Analyzes fairness in compound decisions and identifies conditions for fairness.
result Fairness in multi-stage decisions may not be guaranteed by fair components alone.
Warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifoldmath.DG
We introduce warped product skew semi-invariant submanifolds of order of a locally product Riemannian manifold. We give a necessary and sufficient condition for skew semi-invariant submanifold of order 1 to be a locally warped product. We also prove that the invariant distribution which is involved in the definitio…
Suppose M is a compact manifold with boundary. Let N be a normal covering of M. Suppose (A,T) is an elliptic differential boundary value problem on M with lift (\tilde A,\tilde T) to N. Then the von Neumann dimension of kernel and cokernel of this lift are defined. The main result of this paper is: these numbers are fi…
In a recent work [2] with Datta, we introduced the mu vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu vectors. With the new definition, most results of [2] become correct without the hypothesis of 2-neighbourli…
In Thurston's notes, he gives two different definitions of the Gromov norm (also called simplicial volume) of a manifold and states that they are equal but does not prove it. Gromov proves it in the special case of hyperbolic manifolds as a consequence of his proof that simplicial volume is proportional to volume. We g…
Optimization algorithms affect fairness in deep learning models, especially with adaptive methods like RMSProp.
problem The impact of optimization algorithms on fairness in deep learning models, particularly under imbalance.
method Stochastic differential equation analysis of optimization dynamics in an analytically tractable setup.
result RMSProp, an adaptive optimizer, converges to fairer minima than SGD under severe imbalance.
The Prescribed Metric on Convex Subsets of Anti-de Sitter Space with Quasi-Circle Ideal Boundariesmath.DG
The paper constructs convex subsets in anti-de Sitter space with specific metrics on boundaries.
problem Creating convex subsets with prescribed metrics on boundaries in anti-de Sitter space.
method Using quasi-symmetric maps and properties of hyperbolic metrics, the paper constructs convex subsets with specific metrics on boundaries.
result Existence of globally hyperbolic convex subsets with prescribed metrics on boundaries.
Study CR-statistical submanifolds in holomorphic statistical spaces.
problem Characterize CR-statistical submanifolds and their properties.
method Optimization technique to relate Ricci curvature and mean curvature.
result Established relationship between Ricci curvature and mean curvature.
Defines Alexandrov spaces via axioms, focusing on curvature bounds.
problem Understanding Alexandrov spaces with curvature constraints.
method Formalizes spaces via axioms, distinguishing between curvature above and below.
result Comprehensive structure theory of Alexandrov spaces with curvature bounds.
Study classifies special metrics for which the isometry group of 3D Lie groups is larger than expected.
problem Classifying metrics for which the isometry group of 3D Lie groups is larger than expected.
method Lie-theoretical methods to classify pairs (G, g)
result Determines the dimension of the isometry group for every pair (G, g)
In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition, see section 2) generalizing the conformal Laplacian, and their associated conforma…