arXiv research

We use contact handle decompositions and a stabilization process to compute the cylindrical contact homology of a subcritical Stein-fillable contact manifold with vanishing first Chern class, and show that it is completely determined by the homology of a subcritical Stein-filling of the contact manifold.

The study examines Hawkes processes and their long-term behavior.

We study asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, whi…

We consider a stable Cox--Ingersoll--Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, c…

The paper establishes a central limit theorem for estimating the influence parameter in a partially observed Hawkes process system.

We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case the fundamental group is finite cyclic, and we show that on the 5-sphere the standard contact structure is the unique s…

It is shown that every subcritical Stein manifold is deformation equivalent to the product of a Stein manifold with $\C$.

We consider a jump-type Cox--Ingersoll--Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak …

Study on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.

We prove strong consistency and asymptotic normality of least squares estimators for the subcritical Heston model based on continuous time observations. We also present some numerical illustrations of our results.

We study asymptotic properties of some (essentially conditional least squares) parameter estimators for the subcritical Heston model based on discrete time observations derived from conditional least squares estimators of some modified parameters.

Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.

Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.

For an affine two factor model, we study the asymptotic properties of the maximum likelihood and least squares estimators of some appearing parameters in the so-called subcritical (ergodic) case based on continuous time observations. We prove strong consistency and asymptotic normality of the estimators in question.

We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.

We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetriza…

Simplified proof for Frank and Lieb's inequality on Heisenberg group.

Adapting \cite{strz3}, we define generalized $p$-harmonic maps into Riemannian homogeneous targets, a notion of solutions not belonging to the energy space. Restricting our attention to the subcritical range $p$ greater than the domain dimension $n$, we show a uniform $C^{1,α}$-regularity result for a sequence of such …

The paper classifies solutions to semilinear equations on curved spaces.

Framework simulates market microstructure with stable Hawkes processes.

The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudo gradient flows. W…

Let $L_g$ be the subcritical GJMS operator on an even-dimensional compact manifold $(X, g)$ and consider the zeta-regularized trace $\mathrm{Tr}_ζ(L_g^{-1})$ of its inverse. We show that if $\ker L_g = 0$, then the supremum of this quantity, taken over all metrics $g$ of fixed volume in the conformal class, is always g…

We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a …

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot ν(p,t)$, for $t\in [0,T)$. We show that at the first singular time of the mean curvature flow, certain subcritic…

New Brownian motion defined in Minkowski normed spaces.

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-3$. This result complements existing classifications for automorphism group dimension $n^2-2$ (which is in some sense critical) and greater.

We study the set of trapped photons of a subcritical (a<M) Kerr spacetime as a subset of the phase space. First, we present an explicit proof that the photons of constant Boyer--Lindquist coordinate radius are the only photons in the Kerr exterior region that are trapped in the sense that they stay away both from the h…

New theory explains signal propagation in normalization-free transformers.

Researchers prove zero solutions for certain p-Laplacian equations in convex cones.

We define an activity dependent branching ratio that allows comparison of different time series $X_{t}$. The branching ratio $b_x$ is defined as $b_x= E[ξ_x/x]$. The random variable $ξ_x$ is the value of the next signal given that the previous one is equal to $x$, so $ξ_x=\{X_{t+1}|X_t=x\}$. If $b_x>1$, the process is …

We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform $ε-$regularity estimates whic…

The paper proves constant rank theorems for special Lagrangian equations.

Develops methods to solve complex and real Hessian equations.

Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.

We define a family of functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for the critical dimensions. Consequently, we have an alternate proof of the convergence of…

We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected a…

Given a family of smooth immersions $F_t: M^n\to N^{n+1}$ of closed hypersurfaces in a locally symmetric Riemannian manifold $N^{n+1}$ with bounded geometry, moving by the mean curvature flow, we show that at the first finite singular time of the mean curvature flow, certain subcritical quantities concerning the second…

The study finds multiple solutions for constant Q-curvature metrics.

Proves smoothness and estimates for special Lagrangian solutions with semi-convexity.

We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at infinity. We connect the behavior of non-negative solutions to the interplay betwe…

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplec…

We prove that, for every closed (not necessarily convex) hypersurface $Σ$ in $\mathbb{R}^{n+1}$ and every $p>n$, the $L^p$-norm of the trace-free part of the anisotropic second fundamental form controls from above the $W^{2,p}$-closeness of $Σ$ to the Wulff shape. In the isotropic setting, we provide a simpler proof. T…

Study free energy in spherical spin glasses, proving universality dichotomy.

According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an …

Contact surgeries yield algebraically overtwisted manifolds.

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, …

Study of integral flows on Riemannian manifolds with focus on blow-up profiles and concentration-compactness.

Study on hemisphere threshold for Escobar functional on Riemannian manifolds, revealing mass and boundary invariant behaviors.