A concise discussion of the axiomatic approach to the concept of parallel transport is presented. Attention is drawn to a bijective map between the sets of connections and (axiomatically defined) parallel transports. The transports along paths are pointed as a generalization of the (axiomatically defined) parallel tran…
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48 results for “axiomatization”
Paper axiomatizes interventional probability distributions.
problem Causal inference and intervention.
method Axiomatization of interventional families.
result Markovian property of intervened distributions.
Defines hierarchical clustering axioms for various densities.
problem Defining hierarchical clustering for different types of densities.
method An axiomatic approach to piecewise constant densities, then extending to general densities.
result Our axiomatic definition results in Hartigan's cluster tree under certain conditions.
We refurbish our axiomatics of differential geometry introduced in [Mathematics for Applications,, 1 (2012), 171-182]. Then the notion of Euclideaness can naturally be formulated. The principal objective in this paper is to present an adaptation of our theory of differential forms developed in [International Journal of…
In this paper we give an axiomatization of differential geometry comparable to model categories for homotopy theory. Weil functors play a predominant role.
The axiomatic approach to parallel transport theory is partially discussed. Bijective correspondences between the sets of connections, (axiomatically defined) parallel transports, and transports along paths satisfying some additional conditions, are constructed. In particular, the equivalence between the concepts "conn…
In our previous paper entitled "Axiomatic differential geometry -towards model categories of differential geometry-, we have given a category-theoretic framework of differential geometry. As the first part of our series of papers concerned with differential-geometric developments within the above axiomatic scheme, this…
An axiomatization of -quantilesq-fin.MF
Axiomatizes -quantiles, a generalization of quantiles.
problem Found an axiomatization for -quantiles.
method Characterized -quantiles using the locality property.
result Local changes in distribution do not affect -quantiles.
In this article we prove that any unitary, axiomatic topological quantum field theory in four-dimensions can not detect changes in the smooth structure of M, a simply connected, closed (compact without boundary), oriented smooth manifold. However, as Donaldson-Witten theory (a topological quantum field theory but not a…
We formulate and prove an axiomatic characterization of conditional information geometry, for both the normalized and the nonnormalized cases. This characterization extends the axiomatic derivation of the Fisher geometry by Cencov and Campbell to the cone of positive conditional models, and as a special case to the man…
Axiomatic Market Makingq-fin.MF
Axiomatizes the bid-ask market maker's quoting rule
problem Axiomatizing the quoting rule of a market maker
method Eight natural axioms and six environmental assumptions
result A unique three-parameter family emerges
In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of tangent-vector-valued forms from the general Jacobi identity.
Given a complete and (locally) cartesian closed category U, it is shown that the category of functors from the category of Weil algebras to the category U is (locally, resp.) cartesian closed. The corresponding axiomatization for differential geometry based upon Weil functors is then given.
As the fourth paper of our series of papers concerned with axiomatic differential geometry, this paper is devoted to the general Jacobi identity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as one of the few fundamental results belonging properly to smootheology.
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…
In this paper is proposed a kind of model theory for our axiomatic differential geometry. It is claimed that smooth manifolds, which have occupied the center stage in differential geometry, should be replaced by functors on the category of Weil algebras. Our model theory is geometrically natural and conceptually motiva…
Paper proves impossibility of three desirable properties in node embedding.
problem Understanding limitations of node embedding methods.
method Axiomatic approach to node embedding, proving impossibility of three properties.
result No node embedding method can satisfy all three desirable properties simultaneously.
New framework explains neural network behavior through geometric postulates.
problem Understanding neural network mechanisms and making them more transparent.
method Introducing the Pursuit of Subspaces (PoS) hypothesis as an axiomatic framework.
result Unified geometric perspective on neural network representation, computation, and generalization.
Paper introduces variance-based measures for second-order uncertainty quantification in classification problems.
problem Uncertainty in machine learning predictions and decision-making.
method Second-order uncertainty quantification using variance-based measures.
result Variance-based measures effectively quantify uncertainty on a class-based level and are competitive with entropy-based measures.
Diversification represents the idea of choosing variety over uniformity. Within the theory of choice, desirability of diversification is axiomatized as preference for a convex combination of choices that are equivalently ranked. This corresponds to the notion of risk aversion when one assumes the von-Neumann-Morgenster…
A framework for anonymized risk sharing without revealing identities or preferences.
problem Risk sharing without revealing individual identities or preferences.
method Axiomatic framework with four key axioms: actuarial fairness, risk fairness, risk anonymity, and operational anonymity.
result The conditional mean risk sharing rule is uniquely characterized by these axioms.
Defines globalness measure for explainers using optimal transport.
problem Challenges in evaluating and comparing explainability methods.
method Axiomatic definition and proof of Wasserstein Globalness measure.
result Wasserstein Globalness measure facilitates meaningful comparison and selection of explainers.
Geometric invariant theory introduces stability conditions mirroring abelian category theory.
problem Stability conditions in geometric invariant theory.
method Axiomatic notion of central charge and stability condition on schemes and stacks.
result Introduction of stability conditions for polarized schemes and smooth projective varieties.
New stability theorem for hyperbolic metrics without volume bounds.
problem Stability of finite volume hyperbolic metrics without upper volume bounds.
method Abstract axiomatic framework and bootstrap argument to extend stability result.
result Weaker exponential control of the metric allows for a broader application of the stability theorem.
We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the …
We provide an axiomatic approach to the theory of local tangent cones of regular sub-Riemannian manifolds and the differentiability of mappings between such spaces. This axiomatic approach relies on a notion of a dilation structure which is introduced in the general framework of quasimetric spaces. Considering quasimet…
Projection maps which appear in the theory of buildings and oriented matroids are closely related to the notion of shellability. This was first observed by Bj{ö}rner. In this paper, we give an axiomatic treatment of either concept and show their equivalence. We also axiomatize duality in this setting. As applications o…
New Gini indices capture more nuanced income inequality.
problem Measuring joint dispersion across multiple observations.
method Axiomatic approach to define and characterize n-th order Gini deviations.
result Higher-order Gini coefficients reveal more extreme income disparities.
The paper axiomatizes strong emergence in parameterized field theories and proves existence theorems.
problem Formalizing and proving existence of strong emergence in parameterized field theories.
method Axiomatization and proof of existence theorems for strong emergence between Lagrangian field theories.
result Existence of strong emergence phenomena between parameterized Lagrangian field theories.
The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a topological, not a metric concept. That leads quickly to the notion of connectedness with…
Milnor proved two uniqueness theorems for axiomatic (co)homology: one for pairs of compacta (1960) and another, in particular, for pairs of countable simplicial complexes (1961). We obtain their common generalization: the Eilenberg-Steenrod axioms along with Milnor's map excision axiom and a (non-obvious) common genera…
The paper develops axioms for uniquely decomposing functions with real arguments.
problem Decomposing functions with real arguments while preserving their overall structure.
method Developing axioms to uniquely decompose Borel measurable functions.
result Unique decompositions for all Borel measurable functions are achieved.
New axioms for singquandles simplify applications and reveal algebraic aspects.
problem Axiomatizing singular knots and links.
method Presented new axioms for singquandles, simplified existing ones, and reformulated for affine singquandles.
result Simplified applications and revealed new algebraic aspects of singquandles.
Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.
problem Rigidity and existence of discrete conformal structures on surfaces with boundary.
method Axiomatic framework and classification of discrete conformal structures.
result Extends results by Guo-Luo and Guo to a general context.
New method for interpreting financial model risks.
problem Fairly allocating risk in financial models.
method Extending Shapley value framework for axiomatic risk attribution.
result Risk can be well allocated in financial models.
This paper examines allocation mechanisms in markets with transfer costs, showing how these costs affect economic efficiency.
problem Transfer costs in decentralized exchange markets reduce economic efficiency.
method An axiomatic study of allocation mechanisms in the presence of transfer costs, providing robust and conditional mean allocation mechanisms.
result Robust and conditional mean allocation mechanisms are identified, relating to risk sharing in agent pools.
This note surveys axiomatic results for the Farrell-Jones Conjecture in terms of actions on Euclidean retracts and applications of these to GL_n(Z), relative hyperbolic groups and mapping class groups.
A new framework assesses financial and ESG risks for sustainable investing.
problem Measuring risk and reward in sustainable investing considering environmental, social, and governance factors.
method Proposes axiomatic definitions for ESG-coherent risk measures and reward-risk ratios based on bivariate random variables.
result Empirical analysis ranks stocks using the proposed measures.
Deep learning approximates geometric measures of planar curves.
problem Approximating differential invariants of planar curves.
method Utilizing deep neural networks to estimate geometric measures of planar curves.
result Deep neural networks can learn to overcome instabilities and sampling artifacts.
Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.
problem Proving the uniqueness of the mass center system in non-Euclidean geometries and deriving a generalized Pappus' theorem.
method Revisiting and simplifying G.A. Galperin's proof, extending the mass center system to manifolds, and deriving a generalized Pappus' theorem.
result Unified and simpler proofs for Pappus' theorem in Euclidean, spherical, and hyperbolic geometries.
A new method to break down insurance costs into risk and uncertainty.
problem Understanding and quantifying insurance costs in uncertain environments.
method An axiomatic approach to decompose premium principles into risk and deviation measures.
result Maximal risk and minimal deviation measures can be uniquely identified in decompositions.
Transports along path in fibre bundles are axiomatically introduced. Their general functional form and some their simple properties are investigated. The relationships of the transports along paths and lifting of paths are studied.
An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regressioncs.LG
This study introduces axioms to assess regression uncertainty measures.
problem Limited formal justification and evaluations of uncertainty measures in regression settings.
method Introduces axioms and analyzes entropy- and variance-based measures in a predictive exponential family context.
result Provides a principled foundation for reliable uncertainty assessment in regression.
The paper tackles fair sharing of exploration costs across groups in online learning.
problem Sharing the cost of exploration fairly across multiple groups in online learning.
method The paper introduces the 'grouped' bandit model and uses axiomatic bargaining theory, specifically the Nash bargaining solution, to formalize fairness.
result The paper derives policies that are optimally fair and regret-optimal, showing that regret-optimal policies can be unfair.
The parallel linear transports defined by flat linear connection are axiomatically described. On this basis a number of properties, some of which are new, of these transports and connections are derived.
On evaluation of joint risk for non-negative multivariate risks under dependence uncertaintyq-fin.RM
Paper proposes a new method to evaluate joint risk under uncertainty.
problem Evaluating joint risk of multiple insurance risks under dependence uncertainty.
method Axiomatic approach to scalar and vector-valued distortion joint risk measures.
result Established a new scalar distortion joint risk measure with positive homogeneity.
Defines Learning Analytics' foundational structure and scope.
problem Lack of theoretical foundation in Learning Analytics.
method Proposes an axiomatic theory based on psychological learning and LA methodology.
result Clarifies the epistemological stance of Learning Analytics and its limitations.
On the Correspondence and the Risk Contribution for Conditional Coherent and Deviation Risk Measuresq-fin.RM
The paper establishes a connection between different risk measures and their risk contributions.
problem Understanding the relationship between conditional coherent and deviation risk measures.
method Axiomatic framework and continuous-time risk contribution analysis.
result Risk contributions of time-consistent risk measures are also time-consistent.