Study on periodic solutions for Keller-Segel system in various spaces.
problem Existence and uniqueness of periodic solutions for Keller-Segel system.
method Dispersion and smoothing estimates of heat semigroup, fixed point arguments.
result Existence and uniqueness of periodic solutions for Keller-Segel system on Rn and Hn. The study finds the minimum number of critical points for functionals on Frechet spaces and Finsler manifolds.
problem Finding the minimum number of critical points for functionals on Frechet spaces and Finsler manifolds.
method Applying the Lusternik-Schnirelmann category to evaluate the minimal number of critical points for Keller Cc1-functionals on Frechet spaces and Finsler manifolds. result The minimal number of critical points is determined by the Lusternik-Schnirelmann category.
In this paper we obtain generalized Keller-Osserman conditions for wide classes of differential inequalities on weighted Riemannian manifolds of the form Lu≥b(x)f(u)ℓ(∣∇u∣) and Lu≥b(x)f(u)ℓ(∣∇u∣)−g(u)h(∣∇u∣), where L is a non-linear diffusion-type operator. Prototypical ex…
This note is devoted to Keller-Lieb-Thirring spectral estimates for Schrödinger operators on infinite cylinders: the absolute value of the ground state level is bounded by a function of a norm of the potential. Optimal potentials with small norms are shown to depend on a single variable. The proof is a perturbation arg…
Abstract theory extends optimal transport to Banach lattices.
problem Generalize optimal transport theory to Banach lattices.
method Abstract framework, duality theory, Banach lattice, order unit.
result Characterization of dual elements for generalized optimal transport.
Global implicit function theorem for Fréchet spaces, solving derivative loss problems.
problem Solving initial value problems with derivative loss in Fréchet spaces.
method Global implicit function theorems for Keller's Cc1-mappings in Fréchet spaces, applied through submersions and transversality. result Global existence and uniqueness of solutions to initial value problems with derivative loss.
The paper introduces new metrics and stability criteria for complex manifolds.
problem Stability conditions for complex manifolds and metrics.
method Quantization of the J-flow, J-balanced metrics, Chow stability, uniform stability criteria.
result Existence of J-balanced metrics has a purely algebro-geometric characterization in terms of Chow stability.
New algebraic structures extend Courant algebroids to higher multi-Courant algebroids.
problem Extending Courant algebroid structures to higher multi-Courant algebroids.
method Constructing higher geometric versions of algebraic structures defined by Keller and Waldmann.
result Higher multi-Courant algebroids form a Poisson algebra.
The paper explores geometric influences on PDE solutions on manifolds.
problem Qualitative behavior of solutions to quasilinear PDEs on Riemannian manifolds.
method Investigates strong and weak maximum principles, compact support principles, and Liouville theorems.
result Identifies thresholds involving curvatures or volume growth to guarantee properties under Keller-Osserman conditions.
We extend the Palais-Smale condition to Keller's Cc1-functionals on Fréchet spaces. Using this condition together with Ekeland's variational principle, we obtain some results regarding the existence of minima. In this setting, we prove that the Palais-Smale condition for functionals bounded below implies the coerci…
The aim of this paper is to study the qualitative behaviour of non-negative entire solutions of certain differential inequalities involving gradient terms on the Heisenberg group. We focus our investigation on the two classes of inequalities of the form Δφu≥f(u)l(∣∇u∣) and $Δ^φu \ge f(u) - h(u) g(|\nabla u…
Estimates for Donaldson's Q-operator on symplectic manifolds.
problem Estimating the Q-operator on symplectic manifolds. method Proving an estimate for Donaldson's Q-operator on prequantized compact symplectic manifolds. result An estimate for the Q-operator is proven. In this work we define a deformation theory for the Coupled Kähler-Yang-Mills equations in arXiv:1102.0991, generalizing work of Székelyhidi on constant scalar curvature Kähler metrics. We use the theory to find new solutions of the equations via deformation of the complex structure of a polarised manifold endowed with…
Paper discusses solving generalized Hessian inequalities with various operators.
problem Finding global solutions to generalized Hessian inequalities.
method Analyzes various Hessian operators and provides conditions for global solvability.
result Provides necessary and sufficient conditions for global solvability of generalized Hessian inequalities.
Corrected an error in yield curve behavior models.
problem Error in the boundary expression for yield curve shapes.
method Revised the mathematical expression for yield curve behavior.
result Corrected the boundary expression for normal and humped yield curves.
We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of …
Study limits of quasi-local angular momentum at infinity of gravitating systems.
problem Understanding limits of quasi-local angular momentum at infinity of gravitating systems.
method Based on optimal isometric embedding and quasilocal mass theory, the study defines and analyzes the limits of quasi-local angular momentum at spatial and null infinity.
result Limits of quasi-local angular momentum are discussed at spatial and null infinity of an isolated gravitating system.
We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces ov…
Study finds weak solutions for complex map flows with optimal lifespan.
problem Existence of weak solutions for two-phase matrix-valued harmonic map flows.
method Modified minimizing movement scheme, discretizing time and interpolating solutions.
result Existence of weak solutions with optimal lifespan for the limiting system.
The paper proves critical point results for Frechet manifolds.
problem Finding critical points in the context of Frechet manifolds.
method Using a deformation result and sufficient conditions for the Palais-Smale condition.
result Proves a mountain pass theorem and three critical points theorem.
We present a flexible approach for the valuation of interest rate derivatives based on Affine Processes. We extend the methodology proposed in Keller-Ressel et al. (2009) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible…
We prove the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler-Ricci flow, and of Keller, Rubinstein on…
Survey revisits Bachelier and Dupire, highlighting optimal transport's role.
problem Finding arbitrage-free models calibrated to volatility surfaces.
method Revisits mathematical finance principles, uses optimal transport results.
result Optimal transport provides rigorous foundations for Dupire's model.
New Monte Carlo method for calculating sensitivities of barrier options.
problem Calculating sensitivities for discontinuous payoff functions in barrier options.
method Combining one-step survival idea with stable differentiation approach.
result Calculated sensitivities for different types of barrier options.
In stochastic volatility models based on time-homogeneous diffusions, we provide a simple necessary and sufficient condition for the discretely sampled fair strike of a variance swap to converge to the continuously sampled fair strike. It extends Theorem 3.8 of Jarrow, Kchia, Larsson and Protter (2013) and gives an aff…
Study minimizes Willmore energy with constraints on surface properties.
problem Minimizing Willmore energy under specific surface properties.
method Adapting Keller-Mondino-Rivière, Bauer-Kuwert, and Ndiaye-Schätzle methods.
result Existence of smooth minimizers for a broad range of constraints.
Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.
problem Minimizing the Willmore energy under isoperimetric constraints.
method Connected sum approach, building on previous work by Keller-Mondino-Rivière.
result Existence of minimizers for the isoperimetric constrained Willmore problem in every genus.
The Bergman metrics converge to symplectic forms at a rate of 1/p^2.
problem Understanding convergence rates of Bergman metrics on symplectic manifolds.
method Embedding symplectic manifolds in projective spaces and analyzing the convergence of induced Fubini-Study forms.
result The convergence rate of Bergman metrics to symplectic forms is 1/p^2.
We study the fair strike of a discrete variance swap for a general time-homogeneous stochastic volatility model. In the special cases of Heston, Hull-White and Schobel-Zhu stochastic volatility models we give simple explicit expressions (improving Broadie and Jain (2008a) in the case of the Heston model). We give condi…
New method resolves ambiguity in measuring black hole merger angular momentum.
problem Ambiguity in measuring angular momentum during black hole mergers.
method Quasilocal mass and optimal isometric embedding theory.
result New definition of angular momentum free of supertranslation ambiguity.
Study mass transport in low-diffusivity using Lagrangian coordinates.
problem Mass preserving transport of passive tracers in low-diffusivity limit.
method Lagrangian coordinates, time-averaged diffusion equation, weighted manifold structure.
result Leading order asymptotics extend to dominant nontrivial singular value in low-diffusivity limit.
Formula for α-Futaki character on toric manifolds.
problem Obstruction to Kähler-Yang-Mills equations existence.
method Provided a formula and computed values for α-Futaki character.
result No solutions with α>0 on certain ample line bundles over toric manifolds.
This work extends variance reduction for path-dependent derivatives to affine stochastic volatility models.
problem Pricing path-dependent derivatives in affine stochastic volatility models.
method Prove large deviations principle, apply Esscher transform, use Varadhan's lemma.
result Numerical efficiency demonstrated on Heston model with and without jumps.
Study of dHYM connections on ruled surfaces with variable background metrics.
problem Finding new dHYM connections on ruled surfaces with variable metrics.
method Using momentum construction and moment map partial differential equations, coupled to scalar curvature of the background.
result Provide many new examples of dHYM connections coupled to a variable background Kähler metric.
The paper examines mass aspects at future null infinity and limits of quasilocal mass.
problem Understanding mass aspects and limits of quasilocal mass at future null infinity.
method Review and extension of Bondi mass and mass loss formula in Bondi-Sachs coordinate system.
result New results about the limit of quasilocal mass of unit spheres at null infinity.
Study shows how heat leaks from material sets in low diffusivity scenarios.
problem Understanding heat leakage from material sets in low diffusivity limits.
method Generalized leading-order asymptotics for time-dependent diffusion processes.
result Diffusive transport out of a material set is proportional to the surface area of the set boundary.
The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space T of oriented affine lines in R3 with the tangent bundle of S2. Thus, the round metric on S2 induces a Kähler structure on T which turns out to h…
DeepMoD discovers equations from noisy data using sparse regression and neural networks.
problem Discovering equations from noisy spatio-temporal data.
method Sparse regression on neural network approximated function library.
result DeepMoD discovers equations with few samples and high noise levels.
Let D be an irreducible lattice in a connected, semisimple Lie group G with finite center. Assume that the real rank of G is at least two, that G/D is not compact, and that G has more than one noncompact simple factor. We show that D has no orientation-preserving actions on the real line. (In algebraic terms, this mean…
Delaunay tori minimize Willmore energy under isoperimetric constraints.
problem Finding minimizers of the Willmore energy under isoperimetric constraints.
method Constructing Delaunay tori using complete elliptic integrals and analyzing their Willmore energy.
result Existence of smoothly embedded tori minimizing the Willmore functional under isoperimetric constraints.
3D gravity shows phase transitions with scalar condensation.
problem Phase transitions in 3D gravity with higher genus boundaries.
method Analytical and numerical computations of Rényi entropies and critical dimensions.
result Rényi entropies of holographic CFTs undergo phase transitions.
New probabilistic approach to optimal transport using martingales.
problem Optimal transport between given distributions.
method Martingale formulation of the Benamou-Brenier problem.
result Unique solution mimics Brownian motion and provides time-consistent interpolations.
Motivated by a probabilistic approach to Kahler-Einstein metrics we consider a general non-equilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasi-convex N-particle interaction energy. We show that a deterministic "macroscopic" evolution…
Method identifies IPS governing equations from particle data efficiently.
problem Identify governing equations of interacting particle systems efficiently.
method Combines mean-field theory and WSINDy for large N and M. result Converges with rate O(N−1/2) for N≥100.