A method is provided to resolve Lie algebroids with singularities.
problem Resolving Lie algebroids with singular points.
method Nash-type blow-up construction for Lie algebroids.
result A short exact sequence is established linking the blow-up to Lie algebroids.
A generic compact real codimension two submanifold X of C^(n+2) will have a CR structure at all but a finite number of points (failing at the complex jump points J). The main theorem of this paper gives a method of extending the CR structure on the non-jump points X-J to the jump points. We examine a Gauss map from X-J…
We prove that for every compact, connected, differentiable 3--manifold M there is a compact complex manifold X which can be obtained from projective 3--space by a sequence of smooth, real blow ups and downs such that M is diffeomorphic to the set of real points of X. By earlier results, such an X can almost n…
Study network equilibria in saturated systems, revealing how small shocks can trigger major losses.
problem Understanding how small shocks can lead to major losses in financial networks and games.
method Derived explicit expressions for network equilibria, proved conditions for their uniqueness, and analyzed discontinuities.
result Bifurcation phenomenon in network equilibria, showing sensitivity to small shocks.
Let M and N be Nash manifolds, and f and g Nash maps from M to N. If M and N are compact and if f and g are analytically R-L equivalent, then they are Nash R-L equivalent. In the local case, Cinfty R-L equivalence of two Nash map germs implies Nash R-L equivalence. This shows a difference of Nash…
Nash's theorem proved with Günther's trick
problem Proving Nash's smooth embedding theorem
method Using Günther's trick
result Nash's theorem proved
We propose local symplectic surgery, a two-timescale procedure for finding local Nash equilibria in two-player zero-sum games. We first show that previous gradient-based algorithms cannot guarantee convergence to local Nash equilibria due to the existence of non-Nash stationary points. By taking advantage of the differ…
Machine learning detects NASH patients from medical claims data.
problem Detecting undiagnosed NASH patients for screening and management.
method Gradient-boosted decision trees trained on administrative medical claims data.
result Model precision for NASH detection is significantly higher than NASH incidence.
Optimal geometric estimates for Kähler manifolds with bounded Nash entropy
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded q-Nash entropy The paper examines Nash equilibrium in GANs for stationary Gaussian processes.
problem Existence and uniqueness of Nash equilibrium in GANs for stationary Gaussian processes.
method Analyzes the existence of Nash equilibrium in GANs for stationary Gaussian processes, considering different discriminator families.
result The existence of Nash equilibrium depends on the discriminator family and symmetry properties of the generator family.
New method finds all Nash equilibria via vector optimization.
problem Finding all Nash equilibria in games.
method Formulate vector optimization problem to find Pareto optimal solutions.
result Characterize set of all Nash equilibria as Pareto optimal solutions.
The h-cobordism theorem is a noted theorem in differential and PL topology. A generalization of the h-cobordism theorem for possibly non simply connected manifolds is the so called s-cobordism theorem. In this paper, we prove semialgebraic and Nash versions of these theorems. That is, starting with semialgebraic or Nas…
Algorithm learns Nash equilibria in stochastic games using entropy-regularized policies.
problem Learning Nash equilibria in zero-sum stochastic games is computationally expensive.
method Entropy-regularized soft policies for Q-function updates.
result Algorithm converges to Nash equilibrium under certain conditions.
The study characterizes Nash maps between semialgebraic sets and their properties.
problem Existence of surjective Nash maps between semialgebraic sets.
method Characterization of semialgebraic subsets and their images under Nash maps.
result Characterization of semialgebraic sets that are Nash images of the unit ball.
A Nash game theory approach allocates capital requirements among financial institutions.
problem Allocating systemic risk measures among financial institutions.
method Proposes a Nash allocation rule inspired by game theory.
result Provides sufficient conditions for the existence and uniqueness of Nash allocation rules.
In this paper we review our earlier work on quantum computing and the Nash Equilibrium, in particular, tracing the history of the discovery of new Nash Equilibria and then reviewing the ways in which quantum computing may be expected to generate new classes of Nash equilibria. We then extend this work through a substan…
This paper simplifies the Nash Bargaining Solution for use in intellectual property cases.
problem Limited application of Nash Bargaining Solution in assigning intellectual property damages.
method Normalizes the Nash Bargaining Solution and provides a methodology for determining bargaining weight.
result Clarifies the application of Nash Bargaining Solution to specific case facts.
Study on half spheres solves Nirenberg problem with complex blow-up analysis.
problem Prescribing scalar curvature on half spheres.
method Refined blow-up analysis of finite energy approximated solutions.
result Complex blow-up points and vortex problems reveal new connections.
The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.
problem Existence and uniqueness of almost complex blow-ups on almost complex manifolds.
method Definition and construction of almost complex blow-ups, proving their existence and uniqueness.
result Existence and uniqueness of almost complex blow-ups on 4D almost complex manifolds.
Proposes a new criterion for selecting Nash equilibria considering both utility and inequality.
problem Finding a fair Nash equilibrium in group decision-making.
method Introduces entropy-norm space for geometric selection of strict Nash equilibria.
result The closest entropy-norm pair to the largest entropy-norm pair in rescaled space is the most suitable equilibrium.
A new method for RLHF using proximal point Nash learning.
problem Capturing real human preferences in RLHF.
method Proximal point Nash learning, embedding self-play updates into a proximal point framework.
result High-probability last-iterate convergence for the combined method.
Paper refines royalty determination using Bayesian methods.
problem Determining a reasonable royalty with risk and uncertainty.
method Bayesian Cost approach to refine Nash Bargaining Solution.
result Nash Bargaining Solution emerges as more reliable.
Formula derived for holomorphic Poisson blow-ups.
problem Invariance of Koszul-Brylinski homology under Poisson blow-ups.
method Blow-up formula derivation for holomorphic Koszul-Brylinski homologies.
result Invariance of E1-degeneracy of Dolbeault-Koszul-Brylinski spectral sequence. Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.
problem Bounding Nash entropy in ancient Ricci flows.
method Uniformly bounded Nash entropy implies uniform bounds on the ν-functional, leading to uniform logarithmic and Sobolev inequalities.
result Uniform logarithmic and Sobolev inequalities on ancient Ricci flows with bounded Nash entropy.
An complete exposition of Matthias Gunther's elementary proof of Nash's isometric embedding theorem.
Study identifies numerical signs of blow-up in hydrodynamic equations.
problem Determining if numerical results of blow-up are genuine or artifacts.
method Geometrically consistent spatiotemporal discretization of complexified Euler equations.
result Identification of a signature based on supremum norm growth rates of vorticity.
Formula derived for Bott-Chern classes in complex blow-ups.
problem Calculating Bott-Chern classes in blow-ups of complex manifolds.
method Proved blow-up formula for Bott-Chern classes, established Riemann-Roch without denominators.
result Formula for Bott-Chern classes in blow-ups.
We prove that the infinite family of homotopy 4-spheres constructed by Daniel Nash are all diffeomorphic to 4-sphere.
Explains blow-ups for Lie groupoids and algebroids, comparing different methods.
problem Blow-ups of Lie groupoids and algebroids.
method Detailed explanation of various blow-up constructions.
result Different blow-up constructions for Lie groupoids and algebroids are shown to fit into a general geometric framework.
We give geometrical conditions under which there exist extremal functions for the sharp L2-Nash inequality.
The paper derives a formula for Chow weights of toric blow-ups.
problem Chow weights of toric blow-ups.
method Combinatorial formula derived from toric manifold and Delzant polytope.
result Explicit formula for Chow weights of blow-ups.
Study on curvature blow-up rates in black hole interiors from gravitational collapse.
problem Understanding curvature blow-up rates in black hole interiors during gravitational collapse.
method Investigation of spherically symmetric Einstein-scalar field spacetimes, focusing on blow-up rates of curvature and mass.
result Kretschmann scalar blows up faster than in Schwarzschild setting, indicating a new blow-up phenomenon.
The study examines Nash equilibria in utility maximization games with multiplicative performance criteria.
problem Existence and uniqueness of Nash equilibria in multiplicative performance criteria games.
method General characterization of Nash equilibria for a large class of utility functions.
result Existence and uniqueness of Nash equilibria for arbitrary initial wealth vectors.
The paper examines the blow-up of Ricci curvatures in conformal metrics.
problem Characterizing the blow-up set of Ricci curvatures in conformal metrics.
method Analyzing the blow-up phenomena of Ricci curvatures on domains close to a limit set of lower dimension.
result Characterization of the blow-up set according to the Yamabe invariant of the manifold.
We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a b…
This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.
problem Determining the RLCT of sum-of-products polynomials through blow-up.
method Investigates a specific blow-up algorithm for sop polynomials to resolve their singularities.
result It is possible to resolve the singularities of sop polynomials using a specific blow-up algorithm.
Study Nash equilibrium between broker and trader in a lit exchange with price impact.
problem Optimizing trading strategies between informed and uninformed traders with broker's inventory penalties.
method Characterized Nash equilibrium through FBSDEs, solved explicitly.
result Explicit solution to trading strategies of broker and informed trader.
Kuranishi's proof of complex deformation theory revisited
problem Existence of complex deformations on compact complex manifolds
method Hamilton-Nash-Moser implicit function theorem
result Revisits classical proof with modern tools
We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisso…
We obtain global extensions of the celebrated Nash-Kuiper theorem for C1,θ isometric immersions of compact manifolds with optimal Hölder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent $θ<1…
PAPAL algorithm finds mixed Nash equilibria in continuous games.
problem Finding mixed Nash equilibria in non-convex, non-concave games.
method Particle-based Primal-Dual Algorithm (PAPAL) for weakly entropy-regularized min-max optimization.
result PAPAL offers non-asymptotic convergence guarantees for ε-mixed Nash equilibrium. Improved SEG method converges to Nash equilibrium in bilinear games.
problem Stochastic bilinear minimax optimization problem
method Stochastic ExtraGradient (SEG) method with constant step size, iteration averaging, and scheduled restarting.
result Provable convergence to Nash equilibrium under standard settings, optimal convergence rate in interpolation setting.
This research uses reinforcement learning to find optimal emission offsets in greenhouse gas markets.
problem Finding optimal emission offsets in greenhouse gas markets to control excess emissions.
method Utilized reinforcement learning, specifically Nash-DQN, to estimate market Nash equilibria.
result Emitting firms can achieve significant financial savings by abiding by the Nash equilibria found in the market.
Study shows how multiple traders can trade together without excessive price impact.
problem Coordination issues in trading to exploit a common signal.
method Closed-loop Nash competition model for stochastic differential games.
result Excessive trading reduced but not significantly for practical parameters.
The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.
problem The Nirenberg problem on high-dimensional spheres with residual mass.
method Analysis of subcritical approximations and blowing up solutions.
result Comprehensive description of blowing up solutions, including blow-up points and rates.
The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up …
The Yamabe flow can blow up in infinite time with small perturbations.
problem Understanding the behavior of the Yamabe flow under small perturbations.
method Constructive proof using solutions of the Yamabe problem on the unit sphere as blow-up profiles.
result The Yamabe flow can blow up at multiple points on a Riemannian manifold in infinite time with small perturbations.
In this paper, we study the blow-up of a locally conformal symplectic manifold.We show that there exists a locally conformal symplectic structure on the blow-up of a locally conformal symplectic manifold along a compact induced symplectic submanifold.