Despite the fact that the Euler allocation principle has been adopted by many financial institutions for their internal capital allocation process, a comprehensive description of Euler allocation seems still to be missing. We try to fill this gap by presenting the theoretical background as well as practical aspects. In…
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48 results for “Euler's Principle”
Geometric framework for dissipative systems on Lie algebroids.
problem Formulating geometric framework for dissipative systems.
method Herglotz-type variational principle on Lie algebroids.
result Recover classical equations as special cases.
New variational principle found for PDEs with symmetries and conservation laws.
problem Finding variational principles for PDEs with symmetries and conservation laws.
method Proving existence of a variational principle for PDEs with symmetries and conservation laws.
result A differential equation with sufficient symmetries and conservation laws leads to a variational functional.
New principle for optimal control with higher order differential constraints.
problem Optimal control problems with higher order differential constraints.
method Derivation of the Principle of Minimal Labour and generalization of Pontryagin Maximum Principle.
result Generalized Pontryagin Maximum Principle for higher order constraints.
New variational and multisymplectic formulations for soliton equations derived using the inverse map.
problem Formulating variational principles and multisymplectic formulations for Euler-Poincaré equations on the Virasoro-Bott group.
method Deriving new momentum map and multisymplectic formulation using the inverse map.
result New Clebsch momentum map with 2-cocycles for investigating soliton equations.
In this note we survey some recent results for the Euler equations in compressible and incompressible fluid dynamics. The main point of all these theorems is the surprising fact that a suitable variant of Gromov's -principle holds in several cases.
Study Loday algebroids, prove splitting theorem, and linearize problems.
problem Splitting and linearization of Loday algebroids.
method Local splitting-type results, Euler-like derivations.
result Established a general linearization principle.
Normal forms and isotropic embeddings via Euler-like vector fields.
problem Proving normal forms results for geometric structures.
method Construction of Euler-like vector fields compatible with geometric structures.
result Illustrated in various examples, including Morse-Bott, Weinstein, and Zung's theorems.
Extends Newton's minimal resistance problem to Lorentz-Minkowski space.
problem Minimal resistance in Lorentz-Minkowski space.
method Derived functional energy, determined Euler-Lagrange equation, analyzed maximum principle, found separable and radial solutions.
result Obtained solutions with conical singularities at the origin and analyzed the Single Shock Condition.
Mirror Principle Imath.AG
We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced b…
A variational principle connects fields to principal bundles and Einstein-Yang-Mills systems.
problem Constructing variational problems on fields related to principal bundles and Einstein-Yang-Mills systems.
method Constructing a variational problem on fields defined on a manifold, leading to spontaneous symmetry breaking and identification with principal bundles.
result Global solutions of Euler-Lagrange equations lead to principal bundles and Einstein-Yang-Mills systems with a cosmological constant.
Dirac structures on tangent bundles provide a unified framework for Lagrange--Dirac dynamical systems.
problem Unified geometric framework for Lagrange--Dirac dynamical systems
method Introducing a Lagrange--Dirac structure on the tangent bundle
result Unified framework for nonholonomic, degenerate Lagrangian, and symmetric systems
Develops higher-order Euler-Poincaré field equations for principal G-bundles.
problem Formulating field equations for higher-order jet bundles of principal G-bundles.
method Reduction theory applied to -invariant Lagrangian field theories on jet bundles, transferring Hamilton's principle to reduced configuration bundles.
result Higher-order Euler-Poincaré field equations are equivalent to conservation of Noether current.
The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Leg…
In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of U, find the shapes of the curves corresponding to these critical points and sho…
In recent years, total variation (TV) and Euler's elastica (EE) have been successfully applied to image processing tasks such as denoising and inpainting. This paper investigates how to extend TV and EE to the supervised learning settings on high dimensional data. The supervised learning problem can be formulated as an…
Variationality of conformal geodesics fails in higher dimensions.
problem The variationality of conformal geodesics in higher dimensions.
method Analysis of conformal geodesics in three and higher dimensions.
result Variationality fails in both parametrized and un-parametrized conformal geodesics in higher dimensions.
Paper proposes a capital allocation formula for insurance companies compliant with Solvency II.
problem No specific capital allocation formula is provided for insurance companies using the Solvency II Standard Formula.
method Develops a closed formula for capital allocation that is coherent with Solvency II requirements.
result Demonstrates that the proposed allocation formula is consistent with the Euler's allocation principle.
Study extremals on Lie groups with asymmetric polyhedral Finsler structures using Pontryagin's Maximal Principle.
problem Finding extremals on Lie groups with asymmetric polyhedral Finsler structures.
method Using Pontryagin's Maximal Principle and control systems of Euler-Arnold type to find extremals on the cotangent bundle of the group.
result Uniqueness of the control can be studied through the asymptotic curvature of the vertical part of the Pontryagin extremal.
Immersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic 3-manifoldsmath.GT
In this paper, it is shown that every closed hyperbolic 3-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two fram…
The paper integrates dissipative and curl forces using geometric methods.
problem Incorporating dissipative forces into curl forces for non-conservative systems.
method Geometric metriplectic approach, Herglotz principle, generalized Euler-Lagrange equation, Galley's method.
result Natural formulations for Lagrangian and Hamiltonian dynamics of non-conservative systems.
Derives stochastic and dissipative dynamics preserving Gibbs measure.
problem Understanding and deriving structure-preserving stochastic systems.
method Extension of Hamilton-Pontryagin principle, symmetry reduction, and inclusion of dissipation.
result New derivation of double-bracket dissipation.
Analyzes a finite set of metrics and functions to determine manifold torsion.
problem Determining the torsion of a manifold from a finite set of metrics and functions.
method Introduces a finite set of analytic quantities derived from a Riemannian metric and Morse function, which determine the torsion of the manifold.
result The virtually small spectral package determines the torsion of the manifold, analogous to calculating the Euler-Poincaré characteristic.
Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.
problem Deriving critical metrics for the Einstein-Hilbert functional on compact Riemannian manifolds.
method Restricts the variational problem to an infinite-dimensional subspace.
result Derives a novel structural characterization of critical metrics.
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
Geometric derivation of Einstein equations from causal fermion systems.
problem Deriving Einstein's equations from a new theoretical framework.
method Analysis of causal fermion systems and causal action principle.
result Einstein equations derived from causal action principle.
The aim of this paper is to write explicit expression in terms of a given principal connection of the Lagrange-d'Alembert-Poincarè equations in several stages. This is obtained by using a reduced Lagrange-d'Alembert's Principle in several stages, extending methods introduced for the case of two stages by one of the aut…
Energy and momentum conservation in the Euler-Poincaré formulation of local Vlasov-Maxwell-type systemsphysics.plasm-ph
Derives energy and momentum conservation laws for Vlasov-Maxwell systems using Euler-Poincaré formulation.
problem Challenges in deriving energy and momentum conservation laws for Vlasov-Maxwell systems due to mixed Eulerian and Lagrangian variables.
method Uses Euler-Poincaré formulation to derive conservation laws for Vlasov-Maxwell-type systems, focusing on symmetries generated by isometries and time translation.
result Derives energy and momentum conservation laws for a generic class of Vlasov-Maxwell-type systems, providing a new derivation in the spirit of the Euler-Poincaré machinery.
Study on risk contributions of portfolios using lambda quantile risk measures.
problem No known allocation rule for non-positively homogeneous risk measures.
method Defined lambda quantiles on portfolio compositions, derived derivatives, and introduced generalized Euler contributions.
result Explicit formulae for the derivatives of lambda quantiles, showing their homogeneity properties.
Surface tension of multi-phase flow with multiple junctions governed by the variational principlemath.NA
We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie, Nardone, Scardovelli, …
Complex Structures on Jet Spaces and Bosonic Fock Space Dynamics for Causal Variational Principlesmath-ph
New complex structures on jet spaces help explain Fock space dynamics.
problem Understanding dynamics of Fock spaces from variational principles.
method Endowing jet spaces with almost-complex structures, integrating to canonical complex structures, and analyzing Fock spaces.
result Holomorphic approximation explains dynamics of Fock spaces from variational principles.
This research improves forecasting and testing of risk contributions using Expected Shortfall.
problem Improving risk allocation and testing methods for regulatory standards.
method Developed a comprehensive framework for backtesting and forecasting Expected Shortfall contributions.
result Proposed a novel semiparametric model for forecasting dynamic Expected Shortfall contributions.
The paper explores the geometry and topology of DNN decision boundaries.
problem Understanding the geometric and topological properties of DNN decision boundaries.
method Differential geometry and the Gauss-Bonnet-Chern theorem.
result Computed the Euler characteristics of compact decision boundaries.
In 3D, conformal geodesics are variational.
problem Whether conformal geodesics are variational in 3D.
method Demonstrated that the equation for unparametrized conformal geodesics is variational.
result Variationality of conformal geodesics in 3D.
The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are t…
Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principlesnlin.SI
The paper introduces log-aesthetic curves and their integrable discretization.
problem Characterizing and discretizing log-aesthetic curves.
method Similarity geometry, integrable Burgers equation, variational principles.
result Proposed variational principle and discretization preserving integrable structure.
We identify the Variational Principle governing inifinity-Harmonic maps, that is solutions to the Infinity-Laplacian. The system was first derived in the limit of the p-Laplacian as p->inifinity in [K2] and is recently studied in [K3]. Here we show that it is the "Euler-Lagrange PDE" of vector-valued Calculus of Variat…
Unified approach classifies stable and minimal elastic curves.
problem Classifying stable and minimal elastic curves under various conditions.
method Unified geometric approach using a `cut-and-paste` trick.
result Complete classification of stable closed -elasticae and stable pinned -elasticae.
In this paper we study quasi-linear system of partial differential equations which describes the existence of the polynomial in momenta first integral of the integrable geodesic flow on 2-torus. We proved in [3] that this is a semi-Hamiltonian system and we show here that the metric associated with the system is a metr…
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with nonpositive Euler number.
method Discrete conformal theory and variational principles with constraints.
result Each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a metric with a specific discrete curvature constant.
Study of curves in Lie sphere geometry using moving frames and variational principles.
problem Characterize curves in Lie sphere geometry using Lie curvatures.
method Moving frames, exterior differential systems, and calculus of variations.
result Critical curves are uniquely determined by Lie curvatures.
We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature , therefore, the causal structure appears as in the general relativity theory. By means of the Pontryagin's m…
New frame method simplifies solving variational problems with Euclidean symmetry.
problem Solving variational problems with Euclidean symmetry.
method Rotation Minimising frame and symbolic invariant calculus.
result Noether's conservation laws and Euler-Lagrange equations derived directly.
Study manifolds with positive intermediate Ricci curvature and large symmetry rank.
problem Closed, simply connected manifolds with positive 2nd-intermediate Ricci curvature and large symmetry rank.
method New tools for studying isometric actions on closed manifolds with positive kth-intermediate Ricci curvature, including isotropy rank lemma, symmetry rank bound, and connectedness principle.
result Even dimensional manifolds with large symmetry rank must have trivial odd degree integral cohomology and are either spheres or complex projective spaces.
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 paper we assume a multivariate risk model has been developed for a portfolio and its capital derived as a homogeneous risk measure. The Euler (or gradient) principle, then, states that the capital to be allocated to each component of the portfolio has to be calculated as an expectation conditional to a rare eve…
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.
It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian dynamical system in a manifold, one can consider it as geometric equipment of this manif…