arXiv subjects
Publications and source records attributed to Long Li.
In 1987 Greene introduced the notion of the finite field analogue of hypergeometric series. In this paper we give a finite field analogue of Appell series and obtain some transformation and reduction formulas. We also establish the generating functions for Appell series over finite fields.
The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kaehler (cscK) metrics, when the cone angle is less than $π$. We introduce a new Hölder space called $\cC^{4,\a,\b}$ to study the regularities of this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$ conic cscK metric is indeed of class $\cC^{4,\a,\b}$. Finally, the reductivity is established by a careful study of the conic Lichnerowicz operator.
Analogous to time signals that can be composed of multiple frequency functions, we use uniquely structured orthogonal spatial modes to create different beam shapes. We tailor the spatial structure by judiciously choosing a weighted combination of multiple modal states within an orthogonal basis set, and we can tunably create beam phase and intensity "shapes" that are not otherwise readily achievable. As an example shape, we use a series of orbital-angular-momentum (OAM) functions with adjustable complex weights to create a reconfigurable spatial region of higher localized power as compared to traditional beam combining. We simulate a structured beam created by coherently combining several orthogonal OAM beams with different complex weights, and we achieve a >10X localized power density enhancement with 19 beams. Additionally, we can create unique shapes by passing a single beam through a specially designed phase and intensity mask that contains the combination of multiple OAM functions each with complex weights. Using this approach, we experimentally demonstrate a ~2.5X localized power density increase when utilizing 9 functions.
To increase system capacity of underwater optical communications, we employ the spatial domain to simultaneously transmit multiple orthogonal spatial beams, each carrying an independent data channel. In this paper, we multiplex and transmit four green orbital angular momentum (OAM) beams through a single aperture. Moreover, we investigate the degrading effects of scattering/turbidity, water current, and thermal gradient-induced turbulence, and we find that thermal gradients cause the most distortions and turbidity causes the most loss. We show systems results using two different data generation techniques, one at 1064 nm for 10-Gbit/s/beam and one at 520 nm for 1-Gbit/s/beam, we use both techniques since present data-modulation technologies are faster for infrared (IR) than for green. For the higher-rate link, data is modulated in the IR, and OAM imprinting is performed in the green using a specially-designed metasurface phase mask. For the lower rates, a green laser diode is directly modulated. Finally, we show that inter-channel crosstalk induced by thermal gradients can be mitigated using multi-channel equalisation processing.
The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic cscK metrics are its critical points. Finally we proved the convexity of the conic Mabuchi functional along any conic geodesics.
We are analysing the convexity and continuity properties of the Mabuchi functional along weak geodesics. The key technical point in our paper is the global approximation of weak geodesics obtained via a well-chosen family of Monge-Ampère equations.
We study the design parameters for an orbital angular momentum (OAM) multiplexed free-space data link. Power loss, channel crosstalk and power penalty of the link are analyzed in the case of misalignment between the transmitter and receiver (lateral displacement, receiver angular error, or transmitter pointing error). The relationship among the system power loss and link distance, transmitted beam size and receiver aperture size are discussed based on the beam divergence due to free space propagation. We also describe the trade-offs for different receiver aperture sizes and mode spacing of the transmitted OAM beams under given lateral displacements or receiver angular errors. Through simulations and some experiments, we show that (1) a system with a larger transmitted beam size and a larger receiver aperture is more tolerant to the lateral displacement but less tolerant to the receiver angular error; (2) a system with a larger mode spacing, which uses larger OAM charges, suffers more system power loss but less channel crosstalk; thus, a system with a small mode spacing shows lower system power penalty when system power loss dominates (e.g., small lateral displacement or receiver angular error) while that with a larger mode spacing shows lower power penalty when channel crosstalk dominates (e.g., larger lateral displacement or receiver angular error); (3) the effects of lateral displacement and receiver angular error are not necessarily independent; as an example of them combined, the effects of the transmitter pointing error on the system are also investigated.
The purpose of this paper is to prove the uniqueness of conical Kähler-Einstein metrics, under the condition that the twisted $Ding$-functional is proper. This is a generalization of the author's previous work, and we shall first investigate the uniqueness of twisted Kähler-Einstein metrics, and then use these smooth perturbed solutions to approximate the actual conical one. Finally, it will bring the uniqueness of the limiting singular metric.
The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of Kähler Einstein metrics on Fano manifolds, based on Chen's weak C^{1,1} geodesic without using any further regularities. Unlike the smooth case, the lack of regularities on the geodesic forbids us to use spectral formula of the weighed Laplacian operator directly. However, we can use smooth geodesics to approximate the weak one, then prove that a sequence of eigenfunctions will converge into the first eigenspace of the weighted Laplacian operator.
For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\sum_{k=0}^{[n/2]}\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$ can be viewed as a generalization of Euler numbers. Let $k$ and $m$ be positive integers, and let $b$ be a nonnegative integer. In this paper, we determine $E_{2^mk+b}^{(a)}$ modulo $ 2^{m+10}$ for $m\ge 5$. For $m\ge 5$ we also establish congruences for $U_{kφ{(5^m)}+b},\; E_{kφ{(5^m)}+b},\; S_{kφ{(5^m)}+b}\pmod{5^{m+5}}$ and $S_{kφ{(3^m)}+b}\pmod{3^{m+5}},$ where $U_{2n}=E_{2n}^{(3/2)}$, $S_n=E_n^{(2)}$ and $φ(n)$ is Euler's function.
We generalize our previous work on ππscattering to Kπscattering, and re-analyze the experiment data of Kπscattering below 1.6 GeV. Without any free parameter, we explain KπI=3/2 S-wave phase shift very well by using t-channel rho and u-channel K^* meson exchange. With the t-channel and u-channel meson exchange fixed as the background term, we fit the KπI=1/2 S-wave data of the LASS experiment quite well by introducing one or two s-channel resonances. It is found that there is only one s-channel resonance between Kπthreshold and 1.6 GeV, i.e., K_0^*(1430) with a mass around 1438~1486 MeV and a width about 346 MeV, while the t-channel rho exchange gives a pole at (450-480i) MeV for the amplitude.
The experimental isoscalar $ππ\toππ$ S-wave amplitude squared below 1.75 GeV is characterized by a very broad structure $f_0(400-1200)$ with two narrow dips due to its destructive interference with the $f_0(980)$ and $f_0(1500)$. The $f_0(1370)$ and $f_0(1710)$ do not show up clearly in the $ππ\toππ$ S-wave amplitude due to their weak coupling to $ππ$. This paper is about the controversial nature of the broad $f_0(400-1200)$. We decompose it into two parts, i.e., t-channel $ρ$ meson exchange plus an additional s-channel resonance $f_0(X)$. With the t-channel $ρ$ meson exchange fixed by the isotensor $ππ\toππ$ S-wave scattering, we re-fit the CERN-Munich (CM) data on $ππ$ scattering to get parameters for the $f_0(X)$. We find that the $f_0(X)$ is very broad with a nearby pole at (1.67-0.26i) GeV, while the t-channel $ρ$ meson exchange part produces a pole at (0.36-0.53i) GeV. The implication of our results for the study of $σ$ and glueball is discussed.