This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz -radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz -radial addition of two star bodies, we derive a f…
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Dual spherical conchoidal motion has been defined by Yapar. In this work, we define this motion on a dual hyperbolic unit sphere in the dual Lorentzian space with dual signature, and the results carried to the Lorentzian lines space by means of the Study s mapping. We also obtain the study maps of the orbits drawn on t…
Study of curves in dual space with constant curvature and torsion.
problem Classifying curves in dual space with specific geometric properties.
method Defined curvature and torsion for curves in dual space, classified curves with constant properties, and proved existence theorems.
result Established fundamental theorem of existence for dual curves with prescribed curvature and torsion.
The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.
problem Classifying curves with constant curvature in dual affine and Lorentz-Minkowski planes.
method Investigation of invariants under equiaffine transformations and explicit equations for curves with constant curvature.
result Curves with constant curvature in dual affine and Lorentz-Minkowski planes are classified.
In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the …
Geodesic currents on hyperbolic surfaces have dual spaces that are metric trees.
problem Understanding the dual spaces of geodesic currents on hyperbolic surfaces.
method Analyzing the geometric properties of dual spaces, including their hyperbolicity and completeness.
result The dual spaces of geodesic currents are Gromov hyperbolic metric tree-graded spaces.
The paper explores dual learning, a technique that improves machine translation and image transformation.
problem Understanding and improving dual learning's effectiveness and conditions.
method Theoretical analysis and algorithmic extension of dual learning.
result Multi-step dual learning boosts performance under mild conditions.
New construction of self-dual black holes using quadrics.
problem Hidden features of self-dual black holes obscured by twistor theory.
method Holomorphic quadrics in dual twistor space.
result Directly encoded geometry of self-dual black holes in quadrics.
Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology cla…
New formula for dual knots using involutions.
problem Understanding dual knots and their transformations.
method Involutive analog of knot surgery formula.
result Computed local equivalence class for involutive dual knots.
Notions of self-dual and anti self-dual almost quaternionic structures are introduced. The complete classification of self-dual and anti self-dual generalized Kaehler manifolds is obtained.
Construct dual F-manifolds for regular F-manifolds.
problem Constructing dual F-manifolds for non-semi-simple F-manifolds.
method Define eventual identity to ensure dual F-manifold, construct dual coordinate system.
result Construct families of Nijenhuis operators as an application.
Dual martingales improve primal optimal stopping problem efficiency.
problem Optimal stopping problem in the primal formulation.
method Investigation of dual martingales to improve primal methods.
result Accurate dual martingale approximations reduce primal problem variance.
We consider empirical risk minimization of linear predictors with convex loss functions. Such problems can be reformulated as convex-concave saddle point problems, and thus are well suitable for primal-dual first-order algorithms. However, primal-dual algorithms often require explicit strongly convex regularization in …
We introduce the dual isoperimetrix which solves the isoperimetric problem in the dual Brunn-Minkowski theory. We then show how the dual isoperimetrix is related to the isoperimetrix from the Brunn-Minkowski theory.
This paper solves the dual Minkowski problem for q-torsional rigidity.
problem The dual Minkowski problem for q-torsional rigidity.
method Introduced the p-th dual q-torsional measure and solved the p-th dual Minkowski problem for q-torsional rigidity using a Gauss curvature flow.
result Existence of smooth even and non-even solutions to the p-th dual Minkowski problem for q-torsional rigidity.
Weak dual pairs defined in Dirac-Jacobi geometry, proving equivalence and leaf correspondence theorems.
problem Defining and studying weak dual pairs in Dirac-Jacobi structures.
method Adopting omni-Lie algebroid approach, proving equivalence and leaf correspondence theorems.
result Existence of self-dual pairs and alternative proof of normal form theorem.
We provide a dual representation of quasiconvex maps between two lattices of random variables in terms of conditional expectations. This generalizes the dual representation of quasiconvex real valued functions and the dual representation of conditional convex maps.
Develops torsion dual connections for statistical manifolds.
problem Defining statistical manifolds using dual connections.
method Introduces torsion dual connections and proves their properties.
result Curvature tensor of torsion dual connections has specific divergence.
In this paper, we study Mannheim surface offsets in dual space. By the aid of the E. Study Mapping, we consider ruled surfaces as dual unit spherical curves and define the Mannheim offsets of the ruled surfaces by means of dual geodesic trihedron (dual Darboux frame). We obtain the relationships between the invariants …
Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.
problem Proving uniqueness and continuity of solution to L_p dual Minkowski problem.
method Established new Minkowski-type inequalities related to optimization problem.
result Uniqueness and continuity of solution for general convex bodies when .
Dual representations for robust risk measures and uncertainty sets.
problem Characterizing continuity of robust risk measures and their uncertainty sets.
method Develop dual representations for robust risk measures and uncertainty sets based on distinct geometric assumptions.
result Two dual frameworks for consolidated uncertainty sets are complementary, not interchangeable.
Solves a generalized dual Minkowski problem for specific values of q.
problem Finding solutions to the generalized dual Minkowski problem for given q and star bodies.
method Variational methods
result Existence of solutions for q<0 and 0≤q≤1, sufficient condition for q>1.
Let be a quadruplet of points in . We define a ``dual'' quadruplet of it in a conformal geometric way. We show that the dual of a dual quadruplet coincides with the original one. We also show that the cross ratio of the dual quadruplet is equal to the complex conjugate of that of the orig…
Classifies self-dual almost-Kähler 4-manifolds, proving uniqueness up to rescaling.
problem Classifying self-dual almost-Kähler four-manifolds.
method Using LeBrun's result and properties of Ricci tensor, the authors classify manifolds of different types.
result Any self-dual almost-Kähler metric on is the Fubini-Study metric up to rescaling.
In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim …
Dual averaging-type methods are widely used in industrial machine learning applications due to their ability to promoting solution structure (e.g., sparsity) efficiently. In this paper, we propose a novel accelerated dual-averaging primal-dual algorithm for minimizing a composite convex function. We also derive a stoch…
Contact Dual Pairsmath.DG
We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. Contact groupoids and contact reduction are the main sources of …
Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.
problem Solving dual Orlicz Minkowski problems for anisotropic flows.
method Anisotropic inverse Gauss curvature flows and stationary solutions.
result New existence results for dual Orlicz Minkowski problems for smooth measures.
Many supervised learning tasks are emerged in dual forms, e.g., English-to-French translation vs. French-to-English translation, speech recognition vs. text to speech, and image classification vs. image generation. Two dual tasks have intrinsic connections with each other due to the probabilistic correlation between th…
Totally geodesic dual leaves on curved manifolds are also curved.
problem Characterizing dual leaves of nonnegatively curved polar manifolds.
method Proving dual leaves are totally geodesic and closed, and inducing a Riemannian submersion.
result Dual leaves of nonnegatively curved polar manifolds are themselves nonnegatively curved and totally geodesic.
The first aim of this paper is to define the dual timelike Mannheim partner curves in Dual Lorentzian Space D3 1, the second aim of this paper is to obtain the relationships between the curvatures and the torsions of the dual timelike Mannheim partner curves with respect to each other and the final aim of this paper is…
Classification of toric self-dual Einstein gravitational instantons with negative cosmological constant
problem Classification of toric self-dual Einstein gravitational instantons
method Proving that if the conformal Kähler structure associated to one of the torus Killing fields is global and extends to an ALE manifold with no additional fixed points, then the corresponding self-dual Einstein instanton is precisely given by the infinite class of multipole solutions
result The classification of toric self-dual Einstein gravitational instantons is precisely given by the infinite class of multipole solutions
Exact relationships found between ATM slope, volatility swap, and zero vanna.
problem Understanding relationships between implied volatilities and swaps.
method Analyzes exact relationships between ATM slope, volatility swap, and zero vanna.
result Exact relationships between ATM slope, volatility swap, and zero vanna.
The first aim of this paper is to define the dual timelike - spacelike Mannheim partner curves in Dual Lorentzian Space ID3 1, the second aim of this paper is to obtain the relationships between the curvatures and the torsions of the dual timelike - spacelike Mannheim partner curves with respect to each other and the f…
This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is…
Study curvature and torsion in Gaussian distribution's dual coordinate system.
problem Characterize geometric invariants of Gaussian distribution.
method Investigate Riemannian curvature and torsion in a dual coordinate system of Gaussian distribution.
result Explicitly give Amari formulas in the new coordinate system.
The paper studies dual catenaries in the dual plane, deriving equations and characterizations.
problem Understanding catenaries in the dual plane.
method Introduced -catenaries as stationary points of a potential energy functional, derived Euler-Lagrange equations, and geometrically characterized them.
result Explicit equations and geometric characterization of -catenaries.
In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the r…
Generalizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with eq…
Dual optimization connects ERM-fDR to normalization function.
problem Empirical risk minimization with f-divergence regularization.
method Dual formulation, Legendre-Fenchel transform, implicit function theorem, nonlinear ODE.
result Computational method to calculate normalization function efficiently.
Constrained Markov Decision Process (CMDP) is a natural framework for reinforcement learning tasks with safety constraints, where agents learn a policy that maximizes the long-term reward while satisfying the constraints on the long-term cost. A canonical approach for solving CMDPs is the primal-dual method which updat…
We study projectively self-dual polygons and curves in the projective plane. Our results provide a partial answer to problem No 1994-17 in the book of Arnold's problems.
Study of self-dual polygons in higher dimensions, including explicit constructions and dimension calculations.
problem Understanding self-dual polygons in projective spaces of higher dimensions.
method Explicit construction and dimension calculation of moduli spaces of self-dual polygons.
result Provides the dimension of the moduli space for specific cases of n and m.
New dual approach for hedging Bermudan options efficiently.
problem Computing efficient hedging portfolios for Bermudan options.
method Pure dual approach, rewriting dual pricing formula as excess reward representation, strict convexification, Monte Carlo method.
result Convergence and effectiveness of the new algorithm tested on various Bermudan options.
In this paper, we investigate some characterizations of involute -- evolute curves in dual space. Then the relationships between dual frenet frame and darboux vectors of these curves are found.
Proves conjecture about foliations on curved spaces.
problem Completeness of dual foliations on curved spaces.
method Analyzes Riemannian foliations on nonnegatively curved symmetric spaces.
result Foliations split into trivial and single dual leaf foliations.
In this paper, we propose a dual memory structure for reinforcement learning algorithms with replay memory. The dual memory consists of a main memory that stores various data and a cache memory that manages the data and trains the reinforcement learning agent efficiently. Experimental results show that the dual memory …