Equations of parametric surfaces with base points via syzygies

Suppose $S$ is a parametrized surface in complex projective 3-space $mathbf{P}^3$ given as the image of $phi: mathbf{P}^1 imes mathbf{P}^1 o mathbf{P}^3$. The implicitization problem is to compute an implicit equation $F=0$ of $S$ using the parametrization $phi$. An algorithm using syzygies exists for computing $F$ if $phi$ has no base points, i.e. $phi$ is everywhere defined. This work is an extension of this algorithm to the case of a surface with multiple base points of total multiplicity k. We accomplish this in three chapters. In Chapter 2, we develop the concept and properties of Castelnuovo-Mumford regularity in biprojective spaces. In Chapter 3, we give a criterion for regularity in biprojective spaces. These results are applied to the implicitization problem in Chapter 4

Multigraded regularity: coarsenings and resolutions

Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for each i and suppose that M is a finitely generated Z^r-graded S-module. In this paper we describe how to find finite subsets of Z^r containing the multidegrees of the minimal multigraded syzygies of M. To find such a set, we first coarsen the grading of M so that we can view M as a Z-graded S-module. We use a generalized notion of Castelnuovo-Mumford regularity, which was introduced by D. Maclagan and G. Smith, to associate to M a number which we call the regularity number of M. The minimal degrees of the multigraded minimal syzygies are bounded in terms of this invariant.Comment: 20 pages, 1 figure; small corrections made; final version; to appear in J. of Algebr

Ferroelectric metals in 1T/1T'-phase transition metal dichalcogenides bilayers

Ferroelectricity and metallicity cannot coexist due to the screening effect of conducting electrons, and a large number of stable monolayers with 1T/1T$^{\prime}$ phase lack spontaneous polarization due to inversion symmetry (IS). In this work, we have constructed the $\pi$-bilayer structures for transition metal dichalcogenides (TMD; $M$Te$_2,M =$ Pt, Pd, and Ni) with van der Waals stacking, where two monolayers are related by $C_{2z}$ rotation, and have demonstrated that these $\pi$-bilayers are typical ferroelectric metals (FEMs). The $\pi$-bilayer structure widely exist in nature, such as 1T$^{\prime}$/T$_d$-TMD, $\alpha$-Bi$_4$Br$_4$. The computed vertical polarization of PtTe$_2$ and MoTe$_2$ $\pi$-bilayers are 0.46 and 0.25 pC/m, respectively. We show that the switching of polarization can be realized through interlayer sliding, which only requires crossing a low energy barrier. The interlayer vdW charge transfer is the source of both vertical polarization and metallicity, and these properties are closely related to the spatially extended Te-$p_z$ orbital. Finally, we reveal that electron doping can significantly adjust the vertical polarization of these FEMs in both magnitude and direction. Our findings introduce a new class of FEMs, which have potential applications in functional nanodevices such as ferroelectric tunneling junction and nonvolatile ferroelectric memory

Stress Monitoring for Anchor Rods System in Subway Tunnel Using FBG Technology

This paper presents a model test, used on the tunnels on Xi’an Metro Line 2, as the prototype for evaluating the reinforcing effect of the anchor rod in tunnel construction in loess areas. An independently designed fiber Bragg grating (FBG) sensor was used to monitor the seven strain conditions of the rock bolts during the construction. The result shows that the axial stress of the rock bolt changes after the excavation and increases steadily with the growing pressure in the wall rock. Results additionally show that the anchor rods at the tunnel vault are subjected to a compressive stress that remains relatively constant after the primary and the secondary lining, while those at the spandrel and the corner of the tunnel are subjected to increased tensile stress. This paper demonstrates the feasibility and the superiority of FBG technology for tunnel model tests