arXiv subjects

Ke Zhu

Publications and source records attributed to Ke Zhu.

82 records · Page 5Linked to original sources

arXiv2016

Solvability of Dirac type equations

This paper develops a weighted $L^2$-method for the (half) Dirac equation. For Dirac bundles over closed Riemann surfaces, we give a sufficient condition for the solvability of the (half) Dirac equation in terms of a curvature integral. Applying this to the Dolbeault-Dirac operator, we establish an automatic transversality criteria for holomorphic curves in Kähler manifolds. On compact Riemannian manifolds, we give a new perspective on some well-known results about the first eigenvalue of the Dirac operator, and improve the estimates when the Dirac bundle has a $Z_2$-grading. On Riemannian manifolds with cylindrical ends, we obtain solvability in the $L^2$-space with suitable exponential weights while allowing mild negativity of the curvature.

math.DG↗

arXiv2015

A Double AR Model Without Intercept: an Alternative to Modeling Nonstationarity and Heteroscedasticity

This paper presents a double AR model without intercept (DARWIN model) and provides us a new way to study the non-stationary heteroskedastic time series. It is shown that the DARWIN model is always non-stationary and heteroskedastic, and its sample properties depends on the Lyapunov exponent. An easy-to-implement estimator is proposed for the Lyapunov exponent, and it is unbiased, strongly consistent and asymptotically normal. Based on this estimator, a powerful test is constructed for testing the stability of the model. Moreover, this paper proposes the quasi-maximum likelihood estimator (QMLE) for the DARWIN model, which has an explicit form. The strong consistency and asymptotical normality of the QMLE are established regardless of the sign of the Lyapunov exponent. Simulation studies are conducted to assess the performance of the estimation and testing and an empirical example is given for illustrating the usefulness of the DARWIN model.

math.ST↗

arXiv2013

Isometric embeddings via heat kernel

For any n-dimensional compact Riemannian manifold (M,g), we construct a canonical t-family of isometric embeddings I_{t}: M->R^{q(t)}, with t>0 sufficiently small and q(t)>>t^{-n/2}. This is done by intrinsically perturbing the heat kernel embedding introduced in [BBG]. As t->0, asymptotic geometry of the embedded images is discussed.

math.DG↗

arXiv2013

High-jet relations of the heat kernel embedding map and applications

For any compact Riemannian manifold $(M,g)$ and its heat kernel embedding map $psi_t$ from M into $l^2$ constructed in [BBG], we study the higher derivatives of $psi_t$ with respect to an orthonormal basis at $x$ on $M$. As the heat flow time $t$ goes to 0, it turns out the limiting angles between these derivative vectors are universal constants independent on $g$, $x$ and the choice of orthonormal basis. Geometric applications to the mean curvature and the Riemannian curvature are given. Some algebraic structures of the infinite jet space of $psi_t$ are explored.

math.DG↗

arXiv2013

Thin instantons in G_2-manifolds and Seiberg-Witten invariants

For two nearby disjoint coassociative submanifolds C and C' in a G_2-manifold, we construct thin instantons with boundaries lying on C and C' from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.

math.DG↗

arXiv2013

Instantons in G2 manifolds from J-holomorphic curves in coassociative submanifolds

In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 geometry, we explain the main ingredients in the proofs in [21]. We also construct new examples of instantons.

math.DG↗

arXiv2012

Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA--GARCH/IGARCH models

This paper investigates the asymptotic theory of the quasi-maximum exponential likelihood estimators (QMELE) for ARMA--GARCH models. Under only a fractional moment condition, the strong consistency and the asymptotic normality of the global self-weighted QMELE are obtained. Based on this self-weighted QMELE, the local QMELE is showed to be asymptotically normal for the ARMA model with GARCH (finite variance) and IGARCH errors. A formal comparison of two estimators is given for some cases. A simulation study is carried out to assess the performance of these estimators, and a real example on the world crude oil price is given.

math.ST↗

arXiv2010

Floer trajectories with immersed nodes and scale-dependent gluing

We define an enhanced compactification of Floer trajectories under Morse background using the adiabatic degeneration and the scale-dependent gluing techniques. The compactification reflects the 1-jet datum of the smooth Floer trajectories nearby the limiting nodal Floer trajectories arising from adiabatic degeneration of the background Morse function. This paper studies the gluing problem when the limiting gradient trajectories has length zero through a renomalization process. The case with limiting gradient trajectories of non-zero length will be treated elsewhere. An immediate application of our result is a proof of the isomorphism property of the PSS map : A proof of this isomorphism property was first outlined by Pünihikin-Salamon-Schwarz \cite{PSS} in a way somewhat different from the current proof in its details. This kind of scale-dependent gluing techniques was initiated in [FOOO07] in relation to the metamorphosis of holomorphic polygons under Lagrangian surgery and is expected to appear in other gluing and compactification problem of pseudo-holomorphic curves that involves `adiabatic' parameters or rescales the targets.

math.SG↗

arXiv2009

Moduli Spaces of $J$-holomorphic Curves with General Jet Constraints

In this paper, we prove that the tagent map of the holomorphic $k$- jet evaluation $j^k_{hol}$ from the mapping space to holomorphic $k$-jet bundle, when restricted on the universal moduli space of simple J-holomorphic curves with one marked point, is surjective. From this we derive that for generic $J$, the moduli space of simple $J$-holomorphic curves in class $β\in H_2(M)$ with general jet constraints at marked points is a smooth manifold of expected dimension.

math.SG↗

arXiv2009

Embedding property of $J$-holomorphic curves in Calabi-Yau manifolds for generic $J$

In this paper, we prove that for a generic choice of tame (or compatible) almost complex structures $J$ on a symplectic manifold $(M^{2n},ω)$ with $n \geq 3$ and with its first Chern class $c_1(M,ω) = 0$, all somewhere injective $J$-holomorphic maps from any closed smooth Riemann surface into $M$ are \emph{embedded}. We derive this result as a consequence of the general optimal 1-jet evaluation transversality result of $J$-holomorphic maps in general symplectic manifolds that we also prove in this paper.

math.SG↗