Universal Toda brackets of ring spectra

We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.Comment: 38 pages; a few typos corrected, to appear in Trans. Amer. Math. So

Atomic norm denoising with applications to line spectral estimation

Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.Comment: 27 pages, 10 figures. A preliminary version of this work appeared in the Proceedings of the 49th Annual Allerton Conference in September 2011. Numerous numerical experiments added to this version in accordance with suggestions by anonymous reviewer

Algebraic structures for transitive closure

Closed semi-rings and the closure of matrices oven closed semirings are defined and studied. Closed semirings are structures weaker than the structunes studied by Conway [3] and Aho, Hopcnoft and Ullman [1]. Examples of closed semi-rings and closure operations are given, including the case of semirings on which the closure of an element is not always defined. Two algorithms are proved to compute the closure of a matrix oven any closed semiring; the first one based on Gauss-Jordan elimination is a generalization of algorithms by Warshall, Floyd and Kleene; the second one based on Gauss elimination has been studied by Tarjan [11] and [12], from the complexity point of view in a slightly different framework. Simple semirings, where the closure operation for elements is trivial are defined and it is shown that the closure of an n x n matrix over a simple semiring is the sum of its powers of degree less than n. Dijkstna sernirings are defined and it is shown that the rows of the closure of a matrix over a Dijkstra semiring, can he computed by a generalized version of Dijkstra algorithm

High-Performance Small-Scale Solvers for Moving Horizon Estimation

In this paper we present a moving horizon estimation (MHE) formulation suitable to easily describe the quadratic programs (QPs) arising in constrained and nonlinear MHE. We propose algorithms for factorization and solution of the underlying Karush-Kuhn-Tucker (KKT) system, as well as the efficient implementation techniques focusing on small-scale problems. The proposed MHE solver is implemented using custom linear algebra routines and is compared against implementations using BLAS libraries. Additionally, the MHE solver is interfaced to a code generation tool for nonlinear model predictive control (NMPC) and nonlinear MHE (NMHE). On an example problem with 33 states, 6 inputs and 15 estimation intervals execution times below 500 microseconds are reported for the QP underlying the NMHE. 1

Fast and efficient image reconstruction for high density diffuse optical imaging of the human brain

Real-time imaging of human brain has become an important technique within neuroimaging. In this study, a fast and efficient sensitivity map generation based on Finite Element Models (FEM) is developed which utilises a reduced sensitivitys matrix taking advantage of sparsity and parallelisation processes. Time and memory efficiency of these processes are evaluated and compared with conventional method showing that for a range of mesh densities from 50000 to 320000 nodes, the required memory is reduced over tenfold and computational time fourfold allowing for near real-time image recovery

Polystore mathematics of relational algebra

Financial transactions, internet search, and data analysis are all placing increasing demands on databases. SQL, NoSQL, and NewSQL databases have been developed to meet these demands and each offers unique benefits. SQL, NoSQL, and NewSQL databases also rely on different underlying mathematical models. Polystores seek to provide a mechanism to allow applications to transparently achieve the benefits of diverse databases while insulating applications from the details of these databases. Integrating the underlying mathematics of these diverse databases can be an important enabler for polystores as it enables effective reasoning across different databases. Associative arrays provide a common approach for the mathematics of polystores by encompassing the mathematics found in different databases: sets (SQL), graphs (NoSQL), and matrices (NewSQL). Prior work presented the SQL relational model in terms of associative arrays and identified key mathematical properties that are preserved within SQL. This work provides the rigorous mathematical definitions, lemmas, and theorems underlying these properties. Specifically, SQL Relational Algebra deals primarily with relations - multisets of tuples - and operations on and between those relations. These relations can be modeled as associative arrays by treating tuples as non-zero rows in an array. Operations in relational algebra are built as compositions of standard operations on associative arrays which mirror their matrix counterparts. These constructions provide insight into how relational algebra can be handled via array operations. As an example application, the composition of two projection operations is shown to also be a projection, and the projection of a union is shown to be equal to the union of the projections

Development of Scalable Simulator for Spiking Neural Network

A neural network simulator for Spiking Neural Network (SNN) is a useful research tool to model brain functions with a computer. With this tool, different parameters can be explored easily compared to using a real brain. For several decades, researchers have developed many software packages and simulators to accelerate research in computational neuroscience. However, despite their advantages, different neural simulators possess different limitations, such as flexibility of choosing different neuron models and scalability of simulators for large numbers of neurons. This paper demonstrates an efficient and scalable spiking neural simulator that is based on growth transform neurons and runs on a single machine. The growth transform neuron model’s update is based on matrix-vector multiplication, which is optimized using external libraries named BLAS and sparseBLAS. Using sparseBLAS, the scalability of the simulator was optimized with sparse representation of matrix. The optimized tool can simulate up to 1 million neurons and is flexible with neuron model changes behind the simulator. Furthermore, with a simple graphical user interface, a researcher can easily design a variety of network topology with different parameters. He/she can visualize a coupling matrix, simulate a designed network and study the spike train with spike raster plot. This simulator will be made open source so that researchers can benefit from this for large-scale simulations