arXiv research

Inequality found for a specific equation on 5D manifolds.

We prove an a priori estimate of type sup*inf on Riemannian manifold of dimension 3 (not necessarily compact).

We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf fo…

The paper explores numerical characteristics of compact Riemannian manifolds and proves inequalities.

Algorithmic solutions to the conjugacy problem in the braid groups B_n were given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions yield two conjugacy class invariants which are known as `inf' and `sup'. A problem which was left unsolved in both papers was the number m of times one must `cycle' (res…

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe…

On a filtered probability space $(Ω,\mathcal{F},P,\mathbb{F}=(\mathcal{F}_t)_{t=0,\dotso,T})$, we consider stopper-stopper games $\overline V:=\inf_{\Rho\in\bT^{ii}}\sup_{τ\in\T}\E[U(\Rho(τ),τ)]$ and $\underline V:=\sup_{\Tau\in\bT^i}\inf_{ρ\in\T}\E[U(\Rho(τ),τ)]$ in discrete time, where $U(s,t)$ is $\mathcal{F}_{s\vee…

We study the existence of optimal actions in a zero-sum game $\inf_τ\sup_PE^P[X_τ]$ between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem $\inf_τ\mathcal{E}(X_τ)$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ such as the $G$-expectation. We show that …

We define a new differential invariant a compact manifold by $V_{\mathcal M}(M)=\inf_g V_c(M,[g])$, where $V_c(M,[g])$ is the conformal volume of $M$ for the conformal class $[g]$, and prove that it is uniformly bounded above. The main motivation is that this bound provides a upper bound of the Friedlander-Nadirashvili…

We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of bounded sectional curvature. More specifically, among other things we show that…

Lewis and Mordecki have computed the Wiener-Hopf factorization of a Lévy process whose restriction on $]0,+\infty[$ of their Lévy measure has a rational Laplace transform. That allows to compute the distribution of $(X_t,\inf_{0\leq s\leq t}X_s)$. For the same class of Lévy processes, we compute the distribution of $ (…

Harmonic maps between specific metric spaces are studied with Lipschitz estimates and variational principles.

Let $M,N$ be two smooth compact hypersurfaces of $\mathbb{R}^n$ which bound strictly convex domains equipped with two absolutely continuous measures $μ$ and $ν$ (with respect to the volume measures of $M$ and $N$). We consider the optimal transportation from $μ$ to $ν$ for the quadratic cost. Let $(φ:m \to \mathbb{R},ψ…

Let $(Σ,g)$ be a closed Riemannian surface, $W^{1,2}(Σ,g)$ be the usual Sobolev space, $\textbf{G}$ be a finite isometric group acting on $(Σ,g)$, and $\mathscr{H}_\textbf{G}$ be a function space including all functions $u\in W^{1,2}(Σ,g)$ with $\int_Σudv_g=0$ and $u(σ(x))=u(x)$ for all $σ\in \textbf{G}$ and all $x\inΣ…

Given a complete $n$-dimensional Riemannian manifold $M$, we study the existence of vertical graphs in $M\times\mathbb{R}$ with prescribed mean curvature $H=H(x,z)$. Precisely, we prove that the Dirichlet problem for the vertical mean curvature equation in a smooth bounded domain $Ω\subset M$ has solution for arbitrary…

This paper concerns the recursive utility maximization problem. We assume that the coefficients of the wealth equation and the recursive utility are concave. Then some interesting and important cases with nonlinear and nonsmooth coefficients satisfy our assumption. After given an equivalent backward formulation of our …

Let $M$ be a connected, noncompact, complete Riemannian manifold, consider the operator $L=\DD +\nn V$ for some $V\in C^2(M)$ with $\exp[V]$ integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of $L$. As a consequence of the main result, let $\rr$ be the distance functi…

Let $M$ be a compact manifold with a metric $g$ and with a fixed spin structure $χ$. Let $λ\_1^+(g)$ be the first non-negative eigenvalue of the Dirac operator on $(M,g,χ)$. We set $τ(M,χ):= \sup \inf λ\_1^+(g)$ where the infimum runs over all metrics $g$ of volume 1 in a conformal class $[g\_0]$ on $M$ and where the…

We characterize the communication complexity of the following distributed estimation problem. Alice and Bob observe infinitely many iid copies of $ρ$-correlated unit-variance (Gaussian or $\pm1$ binary) random variables, with unknown $ρ\in[-1,1]$. By interactively exchanging $k$ bits, Bob wants to produce an estimate $…

Researchers develop methods to recover agent behavior from sparse data using Gaussian processes.

The paper tackles robust control with uncertain dependence using data-driven methods.

In high dimensions, find paths connecting points with intermediate steps in a dense set.

Let M be a compact manifold with a spin structure χand a Riemannian metric g. Let λ_g^2 be the smallest eigenvalue of the square of the Dirac operator with respect to g and χ. The τ-invariant is defined as τ(M,χ):= sup inf \sqrt{λ_g^2} Vol(M,g)^{1/n} where the supremum runs over the set of all conformal classes on M, a…

Assuming that the stock price $Z=(Z_t)_{0\leq t\leq T}$ follows a geometric Brownian motion with drift $μ\in\mathbb{R}$ and volatility $σ>0$, and letting $M_t=\max_{0\leq s\leq t}Z_s$ for $t\in[0,T]$, we consider the optimal prediction problems \[V_1=\inf_{0\leqτ\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_τ}\biggr)\quadand\qu…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. For a metric $g$ on $M$, we let $\la_2(g)$ be the second eigenvalue of the Yamabe operator $L_g:= \frac{4(n-1)}{n-2} Δ_g + \scal_g$. Then, the second Yamabe invariant is defined as $$ \si_2(M) \definedas \sup \inf_{h \in [g]} \la_2(h) \Vol(M,h)^{2/n}.…

The paper proves a Moser-Trudinger inequality for zero-mean functions in 2D.

Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\geq 2$. For any metric $\tilde g$ conformal to $g$, we denote by $\tildeλ$ the first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. We show that $$\inf_{\tilde{g} \in [g]} \tildeλ\Vol(M,\tilde g)^{1/n} \leq (n/2) \Vol(S^n)^{1/n}.$$ T…

We consider as given a discrete time financial market with a risky asset and options written on that asset and determine both the sub- and super-hedging prices of an American option in the model independent framework of ArXiv:1305.6008. We obtain the duality of results for the sub- and super-hedging prices. For the sub…

We prove that on a compact $n$-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue $λ$ of the Dirac operator satisfies the inequality $λ^2 \geq \frac{n-1}{4(n-2)}\inf_M Scal$. In the limiting case the universal cover of the manifold is isometric to $R\times N$ where $N…

New comparison theorem for submanifolds with geometric inequalities.

Let G be a two generator subgroup of PSL(2,C). The Jorgensen number J(G) of G is defined by J(G)=inf{ |tr^2 A-4|+|tr[A,B]-2| ; G=<A,B>}. If G is a non-elementary Kleinian group, then J(G) >= 1. This inequality is called Jorgensen's inequality. In this paper, we show that, for any r >= 1, there exists a non-elementary K…

INF-clip optimizes heavy-tailed MAB problems with improved performance.

In this paper we will discuss the optimal risk transfer problems when risk measures are generated by G-expectations, and we present the relationship between inf-convolution of G-expectations and the inf-convolution of drivers G.

This paper tightens the law of the iterated logarithm for empirical KL_inf, applicable to unbounded data.

The paper estimates solutions to a heat inequality on Riemannian manifolds with specific initial data.

Sharp inequalities for weighted log canonical thresholds derived.

We investigate the adaptive robust control framework for portfolio optimization and loss-based hedging under drift and volatility uncertainty. Adaptive robust problems offer many advantages but require handling a double optimization problem (infimum over market measures, supremum over the control) at each instance. Mor…

We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+F(L_t)$, where $B_t$ is Brownian motion, $L_t$ its local time at zero and $M_t$ a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive …

Study risk sharing among agents with varying risk preferences.

SurvLIME-Inf simplifies explanation of survival models using a linear programming approach.

Simplified proof for Tsallis-INF algorithm without conjugate functions.

New spectral Dehn function characterizes word-hyperbolic groups.

Let $(M,g,\si)$ be a compact spin manifold of dimension $n \geq 2$. Let $λ_1^+(\tilde{g})$ be the smallest positive eigenvalue of the Dirac operator in the metric $\tilde{g} \in [g]$ conformal to $g$. We then define $\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) \Vol(M,\tilde{g})^{1/n} $. We show that $…

Extends inf-convolution to countable risk measures for risk sharing.

Deep neural networks with adversarial training achieve sup-norm convergence for nonparametric regression.

The paper studies properties of optimal metrics associated to curves on surfaces.

It is known that the hard Lefschetz action, together with Kähler identities for Kähler (resp. hyperkähler) manifolds, determines a $\mathfrak{su}(1,1)_{sup}$ (resp. $\mathfrak{sp}(1,1)_{sup}$) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also gener…

The paper examines eigenvalues and inequalities on Riemannian manifolds.