On the asymptotic S_n-structure of invariant differential operators on symplectic manifolds

We consider the space of polydifferential operators on n functions on symplectic manifolds invariant under symplectic automorphisms, whose study was initiated by Mathieu in 1995. Permutations of inputs yield an action of S_n, which extends to an action of S_{n+1}. We study this structure viewing n as a parameter, in the sense of Deligne's category. For manifolds of dimension 2d, we show that the isotypic part of this space of <= 2d+1-th tensor powers of the reflection representation h=C^n of S_{n+1} is spanned by Poisson polynomials. We also prove a partial converse, and compute explicitly the isotypic part of <= 4-th tensor powers of the reflection representation. We give generating functions for the isotypic parts corresponding to Young diagrams which only differ in the length of the top row, and prove that they are rational functions whose denominators are related to hook lengths of the diagrams obtained by removing the top row. This also gives such a formula for the same isotypic parts of induced representations from Z/(n+1) to S_{n+1} where n is viewed as a parameter. We apply this to the Poisson and Hochschild homology associated to the singularity C^{2dn}/S_{n+1}. Namely, the Brylinski spectral sequence from the zeroth Poisson homology of the S_{n+1}-invariants of the n-th Weyl algebra of C^{2d} with coefficients in the whole Weyl algebra degenerates in the 2d+1-th tensor power of h, as well as its fourth tensor power. Furthermore, the kernel of this spectral sequence has dimension on the order of 1/n^3 times the dimension of the homology group.Comment: v2: 47 pages; removed what was part (ii) of Theorem 1.3.45 since its proof was invalid. Nothing else was affected. v3: Several corrections; final version to be published in J. Algebr

Algorithms for Mumford curves

Mumford showed that Schottky subgroups of $PGL(2,K)$ give rise to certain curves, now called Mumford curves, over a non-Archimedean field K. Such curves are foundational to subjects dealing with non-Archimedean varieties, including Berkovich theory and tropical geometry. We develop and implement numerical algorithms for Mumford curves over the field of p-adic numbers. A crucial and difficult step is finding a good set of generators for a Schottky group, a problem solved in this paper. This result allows us to design and implement algorithms for tasks such as: approximating the period matrices of the Jacobians of Mumford curves; computing the Berkovich skeleta of their analytifications; and approximating points in canonical embeddings. We also discuss specific methods and future work for hyperelliptic Mumford curves.Comment: 32 pages, 4 figure

Cayley-Bacharach Formulas

The Cayley-Bacharach Theorem states that all cubic curves through eight given points in the plane also pass through a unique ninth point. We write that point as an explicit rational function in the other eight.Comment: 13 pages, 4 figure

Tropicalization of classical moduli spaces

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.Comment: 33 page

Latency-Optimized and Energy-Efficient MAC Protocol for Underwater Acoustic Sensor Networks: A Cross-Layer Approach

Considering the energy constraint for fixed sensor nodes and the unacceptable long propagation delay, especially for latency sensitive applications of underwater acoustic sensor networks, we propose a MAC protocol that is latency-optimized and energy-efficient scheme and combines the physical layer and the MAC layer to shorten transmission delay. On physical layer, we apply convolution coding and interleaver for transmitted information. Moreover, dynamic code rate is exploited at the receiver side to accelerate data reception rate. On MAC layer, unfixed frame length scheme is applied to reduce transmission delay, and to ensure the data successful transmission rate at the same time. Furthermore, we propose a network topology: an underwater acoustic sensor network with mobile agent. Through fully utilizing the supper capabilities on computation and mobility of autonomous underwater vehicles, the energy consumption for fixed sensor nodes can be extremely reduced, so that the lifetime of networks is extended

Bitangents of tropical plane quartic curves

We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.Comment: 13 pages, 9 figures. Minor revisions; accepted for publication in Mathematische Zeitschrif

Tropicalization of del Pezzo surfaces

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, respectively. There are 27 trees on each tropical cubic surface, attached to a bounded complex with up to 73 polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic types of such surfaces

EBウイルス膜蛋白質LMP1によるIL-8誘導と上咽頭癌血管新生に関する研究

取得学位 : 博士（医学）, 学位授与番号 : 医博甲第1532号 , 学位授与年月日 : 平成14年3月22日, 学位授与大学 : 金沢大

Computational approaches to Poisson traces associated to finite subgroups of Sp(2n,C)

We reduce the computation of Poisson traces on quotients of symplectic vector spaces by finite subgroups of symplectic automorphisms to a finite one, by proving several results which bound the degrees of such traces as well as the dimension in each degree. This applies more generally to traces on all polynomial functions which are invariant under invariant Hamiltonian flow. We implement these approaches by computer together with direct computation for infinite families of groups, focusing on complex reflection and abelian subgroups of GL(2,C) < Sp(4,C), Coxeter groups of rank <= 3 and A_4, B_4=C_4, and D_4, and subgroups of SL(2,C).Comment: 37 pages, 6 figure