Canonical Height Functions For Monomial Maps

We show that the canonical height function defined by Silverman does not have the Northcott finiteness property in general. We develop a new canonical height function for monomial maps. In certain cases, this new canonical height function has nice properties

On the proper moduli spaces of smoothable K\"ahler-Einstein Fano varieties

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano K\"ahler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit for a punctured flat family of Fano K\"ahler-Einstein manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of \QQ-Gorenstein smoothable, K-semistable Fano varieties, and verify various necessary properties to guarantee that it is a good moduli space.Comment: 41 pages. Final version. Minor change with exposition improved. To appear in Duke Math.

Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds

In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kahler-Einstein Fano manifolds $\mathcal{M}$. As a consequence, we show that the CM line bundle is nef and big on $\bar{\mathcal{M}}$ and its restriction on $\mathcal{M}$ is ample.Comment: 23 pages. Comments are welcom

Evolution of Alfv\'enic fluctuations inside an interplanetary coronal mass ejection and their contributions to local plasma heating: Joint observations from 1.0 AU to 5.4 AU

Directly tracking an interplanetary coronal mass ejection (ICME) by widely separated spacecrafts is a great challenge. However, such an event could provide us a good opportunity to study the evolution of embedded Alfv\'enic fluctuations (AFs) inside ICME and their contributions to local plasma heating directly. In this study, an ICME observed by Wind at 1.0 au on March 4-6 1998 is tracked to the location of Ulysess at 5.4 au. AFs are commonly found inside the ICME at 1.0 au, with an occurrence rate of 21.7% and at broadband frequencies from 4$\times 10^{-4}$ to 5$\times 10^{-2}$ Hz. When the ICME propagates to 5.4 au, the Aflv\'enicity decreases significantly, and AFs are rare and only found at few localized frequencies with the occurrence rate decreasing to 3.0%. At the same time, the magnetic field intensity at the AF-rich region has an extra magnetic dissipation except ICME expansion effect. The energetics of the ICME at different radial distance is also investigated here. Under similar magnetic field intensity situations at 1.0 au, the turbulence cascade rate at the AF-rich region is much larger than the value at the AF-lack region. Moreover, it can maintain as the decrease of magnetic field intensity if there is lack of AFs. However, when there exists many AFs, it reduces significantly as the AFs disappear. The turbulence cascade dissipation rate within the ICME is inferred to be 1622.3 $J\cdot kg^{-1}\cdot s^{-1}$, which satisfies the requirement of local ICME plasma heating rate, 1653.2 $J\cdot kg^{-1}\cdot s^{-1}$. We suggest that AF dissipation is responsible for extra magnetic dissipation and local plasma heating inside ICME.Comment: 15 pages, 3 figures, 1 tabl

Optimal Bounds on Functions of Quantum States under Quantum Channels

Let $\rho_1, \rho_2$ be quantum states and $(\rho_1,\rho_2) \mapsto D(\rho_1, \rho_2)$ be a scalar function such as the trace norm, the fidelity, and the relative entropy, etc. We determine optimal bounds for $D(\rho_1, \Phi(\rho_2))$ for $\Phi \in \mathcal{S}$ for different class of functions $D(\cdot, \cdot)$, where $\mathcal{S}$ is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels

On-Demand Density-Aware UAV Base Station 3D Placement for Arbitrarily Distributed Users with Guaranteed Data Rates

In this letter, we study the on-demand UAV-BS placement problem for arbitrarily distributed users. This UAV-BS placement problem is modeled as a knapsack-like problem, which is NP-complete. We propose a density-aware placement algorithm to maximize the number of covered users subject to the constraint of the minimum required data rates per user. Simulations are conducted to evaluate the performance of the proposed algorithm in a real environment with different user densities. Our numerical results indicate that for various user densities our proposed solution can service more users with guaranteed data rates compared to the existing method, while reducing the transmit power by 29%.Comment: Accepted by IEEE Wireless Communications Letter

Tunneling states in graphene heterostructures consisting of two different graphene superlattices

We have theoretically investigated the properties of electronic transport in graphene heterostructures, which are consisted of two different graphene superlattices with one-dimensional periodic potentials. It is found that such heterostructures possess an unusual tunneling state occurring inside the original forbidden gaps, and the electronic conductance is greatly enhanced and Fano factor is strongly suppressed near the energy of the tunneling state. Finally we present the matching condition of the impedance of the pseudospin wave for occuring the tunneling state by using the Bloch-wave expansion method.Comment: 8 pages, 5 figure

Product of two positive contractions

Several characterizations are given for a square matrix that can be written as the product of two positive (semidefinite) projections. Based on one of these characterizations, and the theory of alternating projections, a Matlab program is written to check the condition and construct the two positive projections whose product equal to the given matrix, if they exist.Comment: 12 page

Numerical Ranges of the product of Operators

We study containment regions of the numerical range of the product of operators $A$ and $B$ such that $W(A)$ and $W(B)$ are line segments. It is shown that the containment region is equal to the convex hull of elliptical disks determined by the spectrum of $AB$, and conditions on $A$ and $B$ for the set equality holding are obtained. The results cover the case when $A$ and $B$ are self-adjoint operators extending the previous results on the numerical range of the product of two orthogonal projections

Modulation of Solar Wind Energy Flux Input on Global Tropical Cyclone Activity

Studies on Sun-climate connection have been carried out for several decades, and almost all of them focused on the effects of solar total irradiation energy. As the second major terrestrial energy source from outer space, the solar wind energy flux exhibits more significant long-term variations. However, its link to the global climate change is rarely concerned and remain a mystery. As a fundamental and important aspect of the Earth's weather and climate system, tropical cyclone activity has been causing more and more attentions. Here we investigate the possible modulation of the total energy flux input from the solar wind into the Earth's magnetosphere on the global tropical cyclone activity during 1963--2012. From a global perspective, the accumulated cyclone energy increases gradually since 1963 and start to decrease after 1994. Compare to the previously frequently used parameters, e,g., the sunspot number, the total solar irradiation, the solar F10.7 irradiation, the tropical sea surface temperature, and the south oscillation index, the total solar wind energy flux input exhibits a better correlation with the global tropical cyclone activity. Furthermore, the tropical cyclones seem to be more intense with higher geomagnetic activities. A plausible modulation mechanism is thus proposed to link the terrestrial weather phenomenon to the seemly-unrelated solar wind energy input.Comment: 18 pages, 4 figure