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Tao Zheng

Publications and source records attributed to Tao Zheng.

47 records · Page 3Linked to original sources
arXiv2019

An Effective Framework for Constructing Exponent Lattice Basis of Nonzero Algebraic Numbers

Computing a basis for the exponent lattice of algebraic numbers is a basic problem in the field of computational number theory with applications to many other areas. The main cost of a well-known algorithm \cite{ge1993algorithms,kauers2005algorithms} solving the problem is on computing the primitive element of the extended field generated by the given algebraic numbers. When the extended field is of large degree, the problem seems intractable by the tool implementing the algorithm. In this paper, a special kind of exponent lattice basis is introduced. An important feature of the basis is that it can be inductively constructed, which allows us to deal with the given algebraic numbers one by one when computing the basis. Based on this, an effective framework for constructing exponent lattice basis is proposed. Through computing a so-called pre-basis first and then solving some linear Diophantine equations, the basis can be efficiently constructed. A new certificate for multiplicative independence and some techniques for decreasing degrees of algebraic numbers are provided to speed up the computation. The new algorithm has been implemented with Mathematica and its effectiveness is verified by testing various examples. Moreover, the algorithm is applied to program verification for finding invariants of linear loops.

cs.SC↗
arXiv2018

A Performance Analysis Model of TCP over Multiple Heterogeneous Paths for 5G Mobile Services

Driven by the primary requirement of emerging 5G mobile services, the demand for concurrent multipath transfer (CMT) is still prominent. Yet, multipath transport protocols are not widely adopted and TCP-based CMT schemes will still be in dominant position in 5G. However, the performance of TCP flow transferred over multiple heterogeneous paths is prone to the link quality asymmetry, the extent of which was revealed to be significant by our field investigation. In this paper, we present a performance analysis model for TCP over multiple heterogeneous paths in 5G scenarios, where both bandwidth and delay asymmetry are taken into consideration. The evaluation adopting parameters from field investigation shows that the proposed model can achieve high accuracy in practical environments. Some interesting inferences can be drawn from the proposed model, such as the dominant factor that affect the performance of TCP over heterogeneous networks, and the criteria of determining the appropriate number of links to be used under different circumstances of path heterogeneity. Thus, the proposed model can provide a guidance to the design of TCP-based CMT solutions for 5G mobile services.

cs.NI↗
arXiv2017

A parabolic Monge-Ampère type equation of Gauduchon metrics

We prove the long time existence and uniqueness of solution to a parabolic Monge-Ampère type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth topology as $t$ approaches infinity which, up to scaling, is the solution to a Monge-Ampère type equation. This gives a parabolic proof of the Gauduchon conjecture based on the solution of Székelyhidi, Tosatti and Weinkove to this conjecture.

math.DG↗
arXiv2017

An almost complex Chern-Ricci flow

We consider the evolution of an almost Hermitian metric by the $(1,1)$ part of its Chern-Ricci form on almost complex manifolds. This is an evolution equation first studied by Chu and coincides with the Chern-Ricci flow if the complex structure is integrable and with the Kähler-Ricci flow if moreover the initial metric is Kähler. We find the maximal existence time for the flow in term of the initial data and also give some convergence results. As an example, we study this flow on the (locally) homogeneous manifolds in more detail.

math.DG↗
arXiv2017

The (logarithmic) Sobolev inequalities along geometric flow and applications

For some class of geometric flows, we obtain the (logarithmic) Sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non-collapsing and non-inflated properties, which generalize the results in the case of Ricci flow or List-Ricci flow or harmonic-Ricci flow. As applications, for mean curvature flow in Lorentzian space with nonnegative sectional curvature and twisted Kähler-Ricci flow on Fano manifolds, we get the results above.

math.DG↗
arXiv2017

The Chern-Ricci flow on Oeljeklaus-Toma manifolds

We study the Chern-Ricci flow, an evolution equation of Hermitian metrics, on a family of Oeljeklaus-Toma (OT-) manifolds which are non-Kähler compact complex manifolds with negative Kodaira dimension. We prove that, after an initial conformal change, the flow converges, in the Gromov-Hausdorff sense, to a torus with a flat Riemannian metric determined by the OT-manifolds themselves.

math.DG↗
arXiv2016

Higher dimensional generalizations of twistor spaces

We construct a generalization of twistor spaces of hypercomplex manifolds and hyper-Kahler manifolds $M$, by generalizing the twistor $\mathbb{P}^{1}$ to a more general complex manifold $Q$. The resulting manifold $X$ is complex if and only if $Q$ admits a holomorphic map to $\mathbb{P}^1$. We make branched double covers of these manifolds. Some class of these branched double covers can give rise to non-Kahler Calabi-Yau manifolds. We show that these manifolds $X$ and their branched double covers are non-Kahler. In the cases that $Q$ is a balanced manifold, the resulting manifold $X$ and its special branched double cover have balanced Hermitian metrics.

math.DG↗
arXiv2015

An upper bound of the heat kernel along the harmonic-Ricci flow

In this paper, we first derive a Sobolev inequality along the harmonic-Ricci flow. We then prove a linear parabolic estimate based on the Sobolev inequality and Moser's iteration. As an application, we will obtain an upper bound estimate for the heat kernel under the flow.

math.DG↗
arXiv2015

Inoue surfaces and the Chern-Ricci flow

We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics, on Inoue surfaces. These are non-Kahler compact complex surfaces of type Class VII. We show that, after an initial conformal change, the flow always collapses the Inoue surface to a circle at infinite time, in the sense of Gromov-Hausdorff.

math.DG↗
arXiv2013

Estimates of the gaps between consecutive eigenvalues of Laplacian

By the calculation of the gap of the consecutive eigenvalues of $\Bbb S^n$ with standard metric, using the Weyl's asymptotic formula, we know the order of the upper bound of this gap is $k^{\frac{1}{n}}.$ We conjecture that this order is also right for general Dirichlet problem of the Laplace operator, which is optimal if this conjecture holds, obviously. In this paper, using new method, we solve this conjecture in the Euclidean space case intrinsically. We think our method is valid for the case of general Riemannian manifolds and give some examples directly.

math.DG↗