The paper defines general-affine invariants for plane and space curves.
problem Determining curves up to general-affine motions.
method Defining general-affine length parameter and curvatures, studying extremal problems.
result General-affine invariants uniquely determine curves up to motions.
In this paper we study the general affine differential geometry of surfaces in affine space A3. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to aff…
In this paper we study the general affine geometry of curves in affine space A2. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…
Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for Lφ affine surface areas are established.
The paper explores the connection between 3d gravity and Chern-Simons theory using affine group connections.
problem Exploring the relationship between 3d gravity and Chern-Simons theory.
method A variational problem of Chern-Simons type on a principal fiber bundle with general affine group structure is studied. The connection is established through a generalized notion of extension and reduction of connections.
result Established a correspondence between 3d gravity and Chern-Simons theory using affine group connections.
Estimates box dimension of fractal interpolation surfaces using oscillation vectors.
problem Estimating the complexity of fractal interpolation surfaces.
method Defined vertical scaling matrices and used them to relate oscillation vectors of different levels.
result Obtained the box dimension of generalized affine fractal interpolation surfaces.
Classifies connections on Galilei manifolds, generalizing known results.
problem Classifying general affine connections on Galilei manifolds.
method Classification through tensor fields, extending known Galilei connections.
result Additional freedom in connections not metric-compatible, linked to clock form and space metric.
Proposes a deep hedging method for robust pricing and hedging under parameter uncertainty.
problem Pricing and hedging under parameter uncertainty for generalized affine processes.
method Deep learning approach linked to variational form of Kolmogorov equation.
result Robust deep hedging outperforms existing methods in volatile periods.
Solitons are special polygon midpoints under affine transformations.
problem Characterizing polygons whose midpoints under affine transformations form a new polygon.
method Analyzing midpoints polygons and their relationship to affine transformations and differential equations.
result A large class of polygons are on an orbit of a one-parameter subgroup of the affine group, and these curves are solutions to a specific differential equation.
In this paper we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain close-form solutions for some relevant affine …
Gromov showed that for fixed, arbitrarily large C, any uniformly C-Lipschitz affine action of a random group in his graph model on a Hilbert space has a fixed point. We announce a theorem stating that more general affine actions of the same random group on a Hilbert space have a fixed point. We discuss some aspects of …
The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the Lp-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz Lφ affine and geominimal surface areas for single convex body as well as for multiple convex bod…
Parallel transport is an important step in many discrete algorithms for statistical computing on manifolds. Numerical methods based on Jacobi fields or geodesics parallelograms are currently used in geometric data processing. In this last class, pole ladder is a simplification of Schild's ladder for the parallel transp…
Study affine models for alternative risk-free rates and derive caplet pricing formulas.
problem Valuation of caplets/floorlets in models for alternative risk-free rates.
method Affine process for RFRs, explicit valuation formulas for various derivatives.
result Explicit formulas for caplet/floorlet pricing in affine models for RFRs.
Builds geometric structures for algebraic groups over real closed fields.
problem Characterizing and decomposing algebraic groups over specific valued fields.
method Real algebraic geometry to construct and analyze affine buildings.
result Computed stabilizers and obtained group decompositions.
Study path-dependent affine models under uncertain parameters for financial applications.
problem Valuation of path-dependent financial derivatives under parameter uncertainty.
method Developed path-dependent setting for value function, established dynamic programming principle, approximated functional derivatives with neural networks.
result Efficient numerical methods for valuation of complex financial derivatives under parameter uncertainty.
The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations…
This is a review with examples concerning the concepts of affine (in particular, constant and linear) vector fields and fundamental vector fields on a manifold. The affine, linear and constant vector fields on a manifold are shown to be in a bijective correspondence with the fundamental vector fields on it of respectiv…
New connections found for quaternionic and para-quaternionic structures.
problem Characterizing integrability of generalized quaternionic and para-quaternionic structures.
method Defined and characterized integrability with respect to a abla-bracket on the generalized tangent bundle. result Existence of a canonical connection for these structures.
The Funk metric connects billiards, projective geometry, and convex geometry.
problem Exploring the Funk metric's invariants and inequalities.
method Using the Funk metric, extending results from projective geometry and convex geometry.
result General affine inequalities and volume maximizers in Funk geometry.
We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ⊳. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry. Assume that Q is normal as an algebraic variety and that the a…
New measures quantify how data augmentation improves model performance.
problem Understanding the effectiveness of data augmentation in deep learning.
method Introduced Affinity and Diversity measures to quantify augmentation performance.
result Augmentation performance is best achieved by optimizing both Affinity and Diversity.
Post-Lie algebra structure found on non-flat manifolds with curvature and torsion.
problem Understanding vector fields and endomorphisms on manifolds with curvature and torsion.
method Analyzing the post-Lie algebra structure of vector fields and endomorphisms for non-flat connections.
result A universal Lie algebra is constructed for the post-Lie algebra of vector fields and endomorphisms.
Develops method to compute Chern-Simons potentials from higher-dimensional Pontryagin densities.
problem Computing Chern-Simons potentials from higher-dimensional Pontryagin densities.
method Systematic approach using a generic affine connection with non-vanishing torsion and non-metricity.
result Algorithm and code for determining Chern-Simons potential from Pontryagin density in arbitrary even dimensions.
This work deals with the simulation of Wishart processes and affine diffusions on positive semidefinite matrices. To do so, we focus on the splitting of the infinitesimal generator, in order to use composition techniques as Ninomiya and Victoir or Alfonsi. Doing so, we have found a remarkable splitting for Wishart proc…
Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.
problem Understanding the Lipschitz geometry of conic singular sub-manifolds.
method Analyzing the metric properties of conic singular sub-manifolds in compact non-Euclidean manifolds.
result Connected conic singular sub-manifolds are Lipschitz Normally Embedded.
Improved SGD with AdaGrad stepsizes adapts to unknown parameters and unbounded gradients.
problem Adaptive optimization with unknown parameters and unbounded gradients.
method Stochastic Gradient Descent with AdaGrad stepsizes, without assuming problem parameters or strong global Lipschitz conditions.
result Sharp rates of convergence in both low-noise and high-noise regimes, supporting an affine variance noise model.
SL(3,Z) contains subgroups whose intersection is not finitely generated.
problem Identifying subgroups of SL(3,Z) whose intersection is not finitely generated.
method Explicit construction of subgroups H and K, using Schreier graph of an affine action of a free group on Z^2.
result Intersection of two 2-generated subgroups H and K in SL(3,Z) is not finitely generated.
New framework discovers non-affine continuous symmetries in neural networks.
problem Lack of efficient methods for detecting non-affine continuous symmetries in neural networks.
method Computational framework for discovering infinitesimal generators of multi-parameter group actions.
result Framework can discover non-affine continuous symmetries in neural networks.
In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D_4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingo…
Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite dimensional generalization: Affine Kac-Moody symmetric spaces. Affine Kac-Moody …
The paper extends ternary algebra concepts using cube roots of unity.
problem Extending algebraic structures from binary to ternary multiplication.
method Introducing ternary associator, commutator, and Lie algebra at cube roots of unity.
result Derived an identity for ternary commutator based on GA(1,5). Geodesic connectedness proved for statistical manifolds with divisible cubic forms.
problem Geodesic connectedness of affine connections on statistical manifolds with divisible cubic forms.
method Analogy with Hopf-Rinow theorem in Riemannian geometry, establishing geodesic completeness.
result Geodesic connectedness established for statistical manifolds with divisible cubic forms.
Develops a unified framework for valuing insurance products with guarantees.
problem Valuing insurance products with guarantees in an affine setting.
method General affine approach to model financial markets, mortality, and policyholder behavior.
result Explicit valuation formulas for variable annuities and related contracts derived.
Unified physics-informed learning method improves generalization performance.
problem Lack of theoretical analysis for hybrid settings with incomplete physical constraints.
method Unified residual form unifying collocation and variational methods, establishing generalization performance governed by affine variety dimension.
result Generalization performance is determined by affine variety dimension, not just the number of parameters.
We propose a general framework for the simultaneous modeling of equity, government bonds, corporate bonds and derivatives. Uncertainty is generated by a general affine Markov process. The setting allows for stochastic volatility, jumps, the possibility of default and correlation between different assets. We show how to…
Top-down information plays a central role in human perception, but plays relatively little role in many current state-of-the-art deep networks, such as Convolutional Neural Networks (CNNs). This work seeks to explore a path by which top-down information can have a direct impact within current deep networks. We explore …
This work explores algebraic structures from curvature and torsion in affine connections.
problem Understanding algebraic structures from curvature and torsion in affine connections.
method Post-Lie algebra, D-algebra, and special polynomials.
result A particular class of geometrically special polynomials is generated by torsion and curvature.
The paper studies affine models driven by independent Lévy processes and their calibration.
problem Characterizing and classifying affine models driven by Lévy processes.
method Analyzing the short rate equation with independent Lévy processes and characterizing the generator.
result A precise form of the generator and classification of affine models with canonical representations.
Abstract invariant cannot be expressed using various slice-torus invariants.
problem Cannot express Iida-Taniguchi's slice-torus invariant using other known invariants.
method Analysis of various known invariants and their properties.
result Iida-Taniguchi's slice-torus invariant cannot be realized as a linear combination of other invariants.
The θ invariant encompasses the Rozansky-Overbay invariant.
problem None explicitly stated in the abstract.
method Generalization of the Rozansky-Overbay invariant using the θ invariant. result The θ invariant recovers the Rozansky-Overbay invariant. Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
Study of Bauer-Furuta invariants under Lie group actions and Galois coverings.
problem Investigating invariants of 4-manifolds under group actions and Galois coverings.
method Functorial approach to equivariant invariants and study in Galois covering situations.
result Ordinary invariants of quotients are determined by equivariant invariants of the covering manifold.
The Kuperberg invariant is shown to be gauge invariant for certain framed 3-manifolds.
problem Exploring gauge invariance of the Kuperberg invariant for specific 3-manifolds.
method Using hyperbolic 3-manifolds and finite-dimensional Hopf algebras.
result First examples of gauge invariants of general finite-dimensional Hopf algebras via topological methods.
Paper introduces new invariant for pairs of immersions.
problem Understanding behavior of immersions through tangencies and triple points.
method Introduces J2+-invariant for oriented pairs of immersions, invariant under inverse tangencies and triple points. result Invariant changes under direct tangencies but remains invariant under orientation change and inverse tangencies.
New polynomial invariant distinguishes singular links.
problem Distinguishing singular links using existing invariants.
method Generalized quandle polynomial to singquandles and constructed a singular link invariant.
result New polynomial invariant distinguishes singular links with same counting invariant.
We show that the perturbative g invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra g, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …
Constructs universal link invariants from intersections in configuration spaces.
problem Globalise topologically all coloured Jones polynomials and ADO polynomials.
method Defines new link invariants from graded intersections in configuration spaces.
result Recover all coloured Jones polynomials and ADO polynomials for links.