Smooth even solutions found for a generalized convex geometry problem.
problem Dual Orlicz-Minkowski problem in convex geometry.
method Geometric flow involving Gauss curvature and normal vectors.
result Existence of smooth even solutions for smooth even measures.
Paper proves unique solutions for mean field equation on sphere.
problem Proving uniqueness of solutions for a specific equation on a sphere.
method Analyzing the mean field equation Δu=λ(1-e^u) on S^2, proving axially symmetry for even solutions.
result Zero is the only even solution for λ=6, implying rigidity of Hawking mass.
We show that a set of conformally invariant equations derived from the Fefferman-Graham tensor can be used to construct global solutions of the vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich's result on the future hyperboloidal stability of Minkowski space…
Study finds non-uniqueness for Minkowski problem solutions in higher dimensions, uniqueness in lower dimensions.
problem Minkowski problem of even dual curvature measures in higher dimensions.
method Logarithmic Brunn-Minkowski inequality for dual quermassintegrals.
result Non-uniqueness for q>n and uniqueness for 0<q<n in Rn. Proves existence of smooth convex solutions to capillary curvature equations.
problem Proving existence of smooth convex solutions to capillary curvature equations.
method Gradient estimate for capillary curvature equations in half-space.
result Existence of even, smooth, strictly convex solutions for all 1<p<k+1 and θ∈(0,π/2). Survey shows degenerate elliptic equations have analytic properties despite low regularity.
problem Understanding analytic properties of degenerate elliptic equations.
method Explains why solutions of a specific degenerate elliptic equation are analytic.
result Solutions of a specific degenerate elliptic equation are analytic despite low regularity.
Derives stability for curvature measure near constant density, proving dual Minkowski problem solutions.
problem Stability of curvature measure near constant density
method Derives stability result for curvature measure, proves existence and uniqueness of solutions to dual Minkowski problem.
result Existence and uniqueness of solutions to dual Minkowski problem for positive indices, stability result for curvature measure.
Unique solution found for even L^p Minkowski problem at p=p0, but fails for p<p0.
problem Uniqueness of solutions for even L^p Minkowski problem.
method Characterization of p0 through Hilbert operator eigenvalue, analysis of convex bodies.
result Existence and uniqueness of solutions for even L^p Minkowski problem at p=p0, failure for p<p0.
Establishes scattering theory for de Sitter vacuum solutions in even dimensions.
problem Quantitative nonlinear scattering theory for asymptotically de Sitter vacuum solutions in even spatial dimensions.
method Geometric Littlewood-Paley decomposition of the solution, constructing the scattering map.
result Existence and uniqueness of scattering states, asymptotic completeness, and invertible scattering map with quantitative control.
Study proves only origin-centered spheres solve certain curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Using the Heintze-Karcher inequality, the study proves the uniqueness of smooth, strictly convex solutions to a class of Minkowski type problems.
result Only origin-centered spheres solve isotropic and Lp-Gaussian-Minkowski problems. Paper finds smooth convex solutions to curvature problem.
problem Finding smooth, convex solutions to curvature problems.
method Established existence of solutions through mathematical analysis.
result Smooth, origin-symmetric, strictly convex solutions found.
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.
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.
Efficient algorithms find solutions in a rare well-connected cluster at low constraint densities.
problem Finding solutions in the symmetric binary perceptron at low density.
method Formal proof of existence of a subdominant connected cluster and application of an efficient multiscale majority algorithm.
result An efficient algorithm can find solutions in a subdominant connected cluster with high probability.
Solves Lp-Gaussian chord Minkowski problem using Gauss curvature flow.
problem Solving the Lp-Gaussian chord Minkowski problem. method Using Gauss curvature flow to obtain smooth even solutions.
result Obtains smooth even solutions to the Lp-Gaussian chord Minkowski problem. Paper shows non-existence of solutions for minimal surface problems.
problem Non-existence of solutions to Dirichlet problems for minimal surfaces.
method Analyzes minimal graphs of codimension ≥2 over domains with possibly non-C1 boundaries. result Proves non-existence of solutions in various scenarios.
We prove some related results concerning blow-up solutions for the Jang equation. First: it has been shown that, given an outermost marginally outer trapped surface (MOTS) Σ, there exists a solution to Jang's equation which blows up at Σ. Here we show that in addition, large classes of spherically symmetric initial dat…
Ancient solutions on bundles are found for non-abelian groups.
problem Finding ancient solutions on bundles with non-abelian structural groups.
method Generalized ancient solutions of Ricci flow on mSO(3) bundles to RP3 fibre bundles over quaternionic Kähler manifolds. result Ancient solutions of Type I, κ-noncollapsed, and positive Ricci curvature on bundles.
We prove that the signature of an even, symmetric form on a finite rank integral lattice, has signature divisible by 8, provided its associated linking form vanishes in the Witt group of linking forms. Our result generalizes the well know fact that an even, unimodular form has signature divisible by 8. We give applicat…
Researchers find explicit solutions to complex Monge-Ampère equation.
problem Solving complex Monge-Ampère equation with constant right-hand side.
method Explicit pluripotential and viscosity solutions.
result Presented solutions lie in Wloc1,2∩Wloc2,1 and are not Dini continuous. This paper presents a new method for solving systems with polynomial stiffness.
problem Finding analytical solutions to nonlinear differential equations with polynomial stiffness is challenging.
method The paper introduces a geometric/algebraic method using generating series and shuffle product.
result The method provides a recursive schematic that can be automated and applied to systems with polynomial stiffness.
Optimal transport is #P-hard when components are independent, even with approximate solutions.
problem Computational complexity of optimal transport with independent marginals.
method Proved #P-hardness and developed a pseudo-polynomial time approximation algorithm.
result Optimal transport is #P-hard even with independent components and approximate solutions.
Study shows that ridgeless Gaussian kernel regression overfits even with varying bandwidth or dimensionality.
problem Analyzing overfitting in Gaussian kernel ridgeless regression with varying bandwidth or dimensionality.
method Examined the behavior of minimum norm interpolating solutions for fixed and increasing dimensions under varying bandwidth and sample size.
result Ridgeless solutions are never consistent and can be worse than null predictor with large enough noise, even with varying bandwidth or dimensionality.
The paper finds manifolds with many Rarita-Schwinger fields.
problem Finding large spaces of Rarita-Schwinger fields.
method Construction of compact manifolds with specific properties.
result Existence of manifolds with dimension of Rarita-Schwinger field space tending to infinity.
Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with b_1 even. It is shown, however, that not all solutions of th…
Research shows no dual solutions for Lorentzian cost functions in general, but proves their existence under certain conditions.
problem Existence of dual solutions for Lorentzian cost functions in optimal transportation problems.
method Analyzes dual problem in Lorentz-Finsler geometry, proves existence under natural assumptions, and shows implications for optimal transport.
result Existence of dual solutions under specific conditions, implying timelike optimal transport on a set of full measure.
Motivated by optimal investment problems in mathematical finance, we consider a variational problem of Neyman-Pearson type for law-invariant robust utility functionals and convex risk measures. Explicit solutions are found for quantile-based coherent risk measures and related utility functionals. Typically, these solut…
Study on PDEs in Heston model with unique solution and convergence proof.
problem Analyzing PDEs in the Heston model for financial applications.
method Regularity results, verification theorem, unique viscosity solution, convergence proof.
result Unique viscosity solution for wide initial and source data.
Paper approximates solutions for complex decision processes with limited precision.
problem Approximating the set of all solutions for Multi-objective Markov Decision Processes.
method Limited precision approach based on White's multi-objective value-iteration dynamic programming algorithm.
result The number of calculated solutions is tractable and approximates the true Pareto front.
Study on ancient Ricci flows with positive curvature, proving noncollapsedness.
problem Characterizing ancient Ricci flows with positive sectional curvature.
method Analyzing complete and noncompact Type I ancient Ricci flows with positive sectional curvature.
result Ancient solutions are noncollapsed on all scales in complete and noncompact cases, and in even-dimensional closed cases.
Paper investigates curvature problems and existence of solutions.
problem Existence of admissible solutions to curvature problems.
method Investigates curvature problems with prescribed Lp quotient type, proving existence under specific conditions. result Proves existence of admissible solutions without additional conditions.
We study the problem of the existence and the holomorphicity of the Monge-Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a …
We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge-Ampere equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of s…
We revisit the task of learning a Euclidean metric from data. We approach this problem from first principles and formulate it as a surprisingly simple optimization problem. Indeed, our formulation even admits a closed form solution. This solution possesses several very attractive properties: (i) an innate geometric app…
New proof shows origin-centred balls are unique solutions to curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Local Brunn-Minkowski inequality and Alexandrov-Fenchel inequality.
result Origin-centred balls are the only solutions to curvature and related problems.
We simplify and extend a 6D conformal gravity theory to 8D, linking it to Q-curvature.
problem Constructing and understanding conformal gravity actions in different dimensions.
method Streamlined construction of 6D action, proving existence of 8D action, relating to Q-curvature.
result A unique 8D conformal gravity action exists with Einstein metrics as solutions.
Study of geodesics on SL(n) with Hilbert-Schmidt metric, revealing complex dynamics in higher dimensions.
problem Geodesics on SL(n) with Hilbert-Schmidt metric.
method Analysis of geodesics, use of Virial-identity-based criterion, study of explicit families of solutions, classification of geodesics.
result Complex dynamics in higher dimensions, existence of bounded geodesic motions in even dimensions, instability of swirling and shear flows in even dimensions.
New methods solve complex PDEs with mixed boundary conditions.
problem Solving inhomogeneous Robin type boundary value problems for linear PDEs.
method Odd and even Hilbert transforms.
result Non-standard solutions to various PDEs in finance, stochastic analysis, etc.
Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.
problem Solving special Lagrangian equations near infinity with specific conditions.
method Modified Kelvin transforms to characterize remainders in asymptotic expansions.
result Remainders in asymptotic expansions are characterized by a single smooth function in even dimensions and Cn−1,α in odd dimensions. In this paper we consider a variation of the Merton's problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumptions on the underlying process and the utility function. The resultin…
In this paper we study a general framework of American put option with stochastic volatility whose value function is associated with a 2-dimensional parabolic variational inequality with degenerate boundaries. We apply PDE methods to analyze the existences of the strong solution and the properties of the 2-dimensional …
Simplifies solving noisy SDPs for low rank matrix recovery problems.
problem Solving SDPs with noisy data for low rank matrix recovery problems.
method Identifies conditions called simplicity to limit error in noisy SDP solutions.
result Simple SDPs can be efficiently solved and their approximate solutions trusted.
This is primarily a survey of the developments in the theory of harmonic maps of finite uniton number (or unitons) which have taken place since the introduction of extended solutions by Uhlenbeck. Such maps include all harmonic maps from the two-sphere to a compact Lie group or symmetric space. Extended solutions are e…
New stable minimal surfaces generalize classical Henneberg surface.
problem Finding new stable minimal surfaces in 3D.
method Generalized Henneberg surface with infinite families of complete, non-orientable surfaces.
result Infinite families of complete, finitely branched, non-orientable, stable minimal surfaces.
Solves optimal control for trading multiple mean-reverting assets.
problem How to construct a portfolio from mean-reverting assets.
method Optimal control problem for power utility agent.
result Nearly explicit solution with properties of optimal solution.
Study of conformally compact metrics and Lovelock tensors in even dimensions.
problem Understanding conformally compact metrics satisfying Lovelock equations.
method Polyhomogeneous expansions and formal solutions to singular Yamabe-(2q) problem.
result Identification of a boundary obstruction in even dimensions that generalizes the ambient obstruction tensor.
Study well-posedness of fast diffusion equation on noncompact manifolds.
problem Investigate well-posedness of fast diffusion equation in noncompact Riemannian manifolds.
method Establish existence and uniqueness of solutions for globally integrable initial data.
result Global solutions exist for initial data in Lloc1 on general Riemannian manifolds. We show that asymptotically, completely asynchronous stochastic gradient procedures achieve optimal (even to constant factors) convergence rates for the solution of convex optimization problems under nearly the same conditions required for asymptotic optimality of standard stochastic gradient procedures. Roughly, the n…