Equivariant Quantum Cohomology of the Grassmannian via the Rim Hook Rule

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of k-planes in complex n-space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule then gives an effective algorithm for computing all equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights modulo n, suggesting a direct connection to the Peterson isomorphism relating quantum and affine Schubert calculus.Comment: 24 pages and 4 figures; typos corrected; final version to appear in Algebraic Combinatoric

An equivariant quantum Pieri rule for the Grassmannian on cylindric shapes

The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of Gorbounov and Korff in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of Huang and Li and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety.Comment: 27 pages, 9 figures best viewed in color; updated discussion of several reference

Spectral embedding of weighted graphs

This paper concerns the statistical analysis of a weighted graph through spectral embedding. Under a latent position model in which the expected adjacency matrix has low rank, we prove uniform consistency and a central limit theorem for the embedded nodes, treated as latent position estimates. In the special case of a weighted stochastic block model, this result implies that the embedding follows a Gaussian mixture model with each component representing a community. We exploit this to formally evaluate different weight representations of the graph using Chernoff information. For example, in a network anomaly detection problem where we observe a p-value on each edge, we recommend against directly embedding the matrix of p-values, and instead using threshold or log p-values, depending on network sparsity and signal strength.Comment: 29 pages, 8 figure

An equivariant rim hook rule for quantum cohomology of Grassmannians

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the quantum product in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provide a way to compute quantum products of Schubert classes in the Grassmannian of $k$-planes in complex $n$-space by doing classical multiplication and then applying a combinatorial rimhook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule provides an effective algorithm for computing the equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights that is tantalizingly similar to maps in affine Schubert calculus

Review of code and phase biases in multi-GNSS positioning

A review of the research conducted until present on the subject of Global Navigation Satellite System (GNSS) hardware-induced phase and code biases is here provided. Biases in GNSS positioning occur because of imperfections and/or physical limitations in the GNSS hardware. The biases are a result of small delays between events that ideally should be simultaneous in the transmission of the signal from a satellite or in the reception of the signal in a GNSS receiver. Consequently, these biases will also be present in the GNSS code and phase measurements and may there affect the accuracy of positions and other quantities derived from the observations. For instance, biases affect the ability to resolve the integer ambiguities in Precise Point Positioning (PPP), and in relative carrier phase positioning when measurements from multiple GNSSs are used. In addition, code biases affect ionospheric modeling when the Total Electron Content is estimated from GNSS measurements. The paper illustrates how satellite phase biases inhibit the resolution of the phase ambiguity to an integer in PPP, while receiver phase biases affect multi-GNSS positioning. It is also discussed how biases in the receiver channels affect relative GLONASS positioning with baselines of mixed receiver types. In addition, the importance of code biases between signals modulated onto different carriers as is required for modeling the ionosphere from GNSS measurements is discussed. The origin of biases is discussed along with their effect on GNSS positioning, and descriptions of how biases can be estimated or in other ways handled in the positioning process are provided.QC 20170922</p

The Combinatorics And Geometry Of The Orbits Of The Symplectic Group On Flags In Complex Affine Space

Let F lC2n = B[-] GL2n C be the manifold of flags in C2n . F lC2n has a natural action of S pn by right multiplication. In this thesis we will describe the orbits of S pn on F lC2n . We begin by giving background material in chapter 2 on the combi¨ natorics of S n , the flag manifold, and Grobner bases. In chapter 3 we describe the orbits of B[-] x S pn on full rank 2n x 2n matrices (equivalent to the orbits of S pn on F lC2n ) by mapping those orbits to orbits of B[-] x B+ via M [RIGHTWARDS ARROW] MJM T using [RS90] and then applying the tools available to understand those orbits (see [Ful92]). We recall that the orbits of B[-] x S pn on full rank matrices correspond to fixed-point-free involutions and we explore the combinatorics of the poset of fixed point free involutions to gain insight into the corresponding poset of orbit ¨ closures. We also give a Grobner degeneration of each orbit closure to a union of matrix Schubert varieties. In the chapter 4 we develop understanding of unions of matrix Schubert varieties by finding their equations. In chapter 5 we give the partial results that we have achieved in finding the defining equations for the orbit closures of the orbits of B[-] x S pn