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Xin Lü

Publications and source records attributed to Xin Lü.

11 recordsLinked to original sources
arXiv2025

Upper bounds on the genus of hyperelliptic Albanese fibrations

Let $S$ be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration $f:\,S \to B$ of genus $g$.We prove a quadratic upper bound on the genus $g$, i.e., $g\leq h\big(χ(\mathcal{O}_S)\big)$, where $h$ is a quadratic function. We also construct examples showing that the quadratic upper bounds can not be improved to the linear ones. In the special case when $p_g(S)=q(S)=1$, we show that $g\leq 14$.

math.AG↗
arXiv2024

Albanese fibrations of surfaces with low slope

Let $S$ be a minimal irregular surface of general type, whose Albanese map induces a fibration $f:\,S \to C$ of genus $g$.We prove a linear upper bound on the genus $g$ if $K_S^2\leq 4χ(\mathcal{O}_S)$. Examples are constructed showing that the above linear upper bound is sharp. We also give a characterization of the Albanese fibrations reaching the above upper bound when $χ(\mathcal{O}_S)\geq 5$.On the other hand, we will construct a sequence of surfaces $S_n$ of general type with $K_{S_n}^2/χ(\mathcal{O}_{S_n})>4$ and with an Albanese fibration $f_n$, such that the genus $g_n$ of a general fiber of $f_n$ increases quadratically with $χ(\mathcal{O}_{S_n})$,and that $K_{S_n}^2/χ(\mathcal{O}_{S_n})$ can be arbitrarily close to $4$.

math.AG↗
arXiv2024

Topological Hyperbolicity of Moduli spaces of Elliptic Surfaces

We introduce the notion of topological hyperbolicity to characterize the largeness of the topological fundamental group of a complex variety. Inspired by the Shafarevich conjecture, we propose to study the topological hyperbolicity of moduli spaces of polarized manifolds. We provide two pieces of supporting evidence: first, we show that moduli spaces where the infinitesimal Torelli theorem holds are very close to being topologically hyperbolic. Second, we establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one without multiple fibers, where the infinitesimal Torelli theorem generally does not hold.

math.AG↗
arXiv2024

The canonical map of a foliated surface of general type

Let $(S,\mathcal{F})$ be a foliated surface over the complex number of general type, i.e., the Kodaira dimension $\mathrm{Kod}(\mathcal{F})=2$. We study the geometry of the canonical map $φ$ of the foliated surface $(S,\mathcal{F})$, and prove several boundedness results on the canonical map $φ$, generalizing Beauville's beautiful work on the canonical maps of algebraic surfaces to foliated surfaces. As an application, we prove three Noether type inequalities for $(S,\mathcal{F})$ depending on the Kodaira dimension of the surface $S$.

math.AG↗
arXiv2024

The Poincaré Problem for a foliated surface

Let $\mathcal F$ be a foliation on a smooth projective surface $S$ over the complex number $\mathbb{C}$. We introduce three birational non-negative invariants $c_1^2(\mathcal F)$, $c_2(\mathcal F)$ and $χ(\mathcal F)$, called the Chern numbers. If the foliation $\mathcal F$ is not of general type, the first Chern number $c_1^2(\mathcal F)=0$, and $c_2(\mathcal F)=χ(\mathcal F)=0$ except when $\mathcal F$ is induced by a non-isotrivial fibration of genus $g=1$. If $\mathcal F$ is of general type, we obtain a slope inequality when $\mathcal F$ is algebraically integral. As a corollary, $\mathcal F$ is always transcendental if the slope is less than $2$. On the other hand, we also prove three sharp Noether type inequalities if $\mathcal F$ is of general type. As applications, we obtain a criterion for foliations to be transcendental using Noether type inequalities, and we also give a partial positive answer to the question on the lower bound on the volume of a foliation of general type.

math.AG↗
arXiv2020

Preliminary prediction of the basic reproduction number of the Wuhan novel coronavirus 2019-nCoV

Objectives.--To estimate the basic reproduction number of the Wuhan novel coronavirus (2019-nCoV). Methods.--Based on the susceptible-exposed-infected-removed (SEIR) compartment model and the assumption that the infectious cases with symptoms occurred before January 25, 2020 are resulted from free propagation without intervention, we estimate the basic reproduction number of 2019-nCoV according to the reported confirmed cases and suspected cases, as well as the theoretical estimated number of infected cases by other research teams, together with some epidemiological determinants learned from the severe acute respiratory syndrome. Results The basic reproduction number falls between 2.8 to 3.3 by using the real-time reports on the number of 2019-nCoV infected cases from People's Daily in China, and falls between 3.2 and 3.9 on the basis of the predicted number of infected cases from colleagues. Conclusions.--The early transmission ability of 2019-nCoV is closed to or slightly higher than SARS. It is a controllable disease with moderate-high transmissibility. Timely and effective control measures are needed to suppress the further transmissions. Notes Added.--Using a newly reported epidemiological determinants for early 2019-nCoV, the estimated basic reproduction number is in the range [2.2,3.0].

q-bio.PE↗
arXiv2012

Mutually unbiased bases for the rotor degree of freedom

We consider the existence of a continuous set of mutually unbiased bases for the continuous and periodic degree of freedom that describes motion on a circle (rotor degree of freedom). By a singular mapping of the circle to the line, we find a first, but somewhat unsatisfactory, continuous set which does not relate to an underlying Heisenberg pair of complementary observables. Then, by a nonsingular mapping of the discrete angular momentum basis of the rotor onto the Fock basis for linear motion, we construct such a Heisenberg pair for the rotor and use it to obtain a second, fully satisfactory, set of mutually unbiased bases.

quant-ph↗
arXiv2011

Mutually unbiased bases in dimension six: The four most distant bases

We consider the average distance between four bases in dimension six. The distance between two orthonormal bases vanishes when the bases are the same, and the distance reaches its maximal value of unity when the bases are unbiased. We perform a numerical search for the maximum average distance and find it to be strictly smaller than unity. This is strong evidence that no four mutually unbiased bases exist in dimension six. We also provide a two-parameter family of three bases which, together with the canonical basis, reach the numerically-found maximum of the average distance, and we conduct a detailed study of the structure of the extremal set of bases.

quant-ph↗
arXiv2008

Logic Functions and Quantum Error Correcting Codes

In this paper, based on the relationship between logic functions and quantum error correcting codes(QECCs), we unify the construction of QECCs via graphs, projectors and logic functions. A construction of QECCs over a prime field GF(p) is given, and one of the results given by Ref[8] can be viewed as a corollary of one theorem in this paper. With the help of Boolean functions, we give a clear proof of the existence of a graphical QECC in mathematical view, and find that the existence of an [[n,k,d]] QECC over GF(p) requires similar conditions with that depicted in Ref[9]. The result that under the correspondence defined in Ref[17], every [[n,0,d]] QECC over GF(2) corresponding to a simple undirected graph has a Boolean basis state, which is closely related to the adjacency matrix of the graph, is given. After a modification of the definition of operators, we find that some QECCs constructed via projectors depicted in Ref[11] can have Boolean basis states. A necessary condition for a Boolean function being used in the construction via projectors is given. We also give some examples to illustrate our results.

quant-ph↗
arXiv2004

Quantum Digital Signature based on Quantum One-way Functions

A quantum digital signature protocol based on quantum mechanics is proposed in this paper. The security of the protocol relies on the existence of quantum one-way functions by quantum information theorem. This protocol involves a so-called arbitrator who validates and authenticates the signed message. In this protocol, we use privacy key algorithm to ensure the security of quantum information on channel and use quantum public keys to sign message. To guarantee the authenticity of the message, a family of quantum stabilizer codes are employed. Our protocol presents a novel method to construct ultimately secure digital system in future secure communication.

quant-ph↗