Locally recoverable J-affine variety codes

A locally recoverable (LRC) code is a code over a finite eld Fq such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. We construct LRC codes correcting more than one erasure, which are sub eld-subcodes of some J-affine variety codes. For these LRC codes, we compute localities (r; )) that determine the minimum size of a set R of positions so that any - 1 erasures in R can be recovered from the remaining r coordinates in this set. We also show that some of these LRC codes with lengths n >> q are ( - 1)-optimal

Locally recoverable codes from the matrix-product construction

Matrix-product constructions giving rise to locally recoverable codes are considered, both the classical $r$ and $(r,\delta)$ localities. We study the recovery advantages offered by the constituent codes and also by the defining matrices of the matrix product codes. Finally, we extend these methods to a particular variation of matrix-product codes and quasi-cyclic codes. Singleton-optimal locally recoverable codes and almost Singleton-optimal codes, with length larger than the finite field size, are obtained, some of them with superlinear length

Introducing locality in some generalized AG codes

In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter $r$ in such codes by using only non-rational places of degrees at most $r$. This is, up to the author's knowledge, a new way to construct locally recoverable codes (LRCs). We give an example of such a code reaching the Singleton-like bound for LRCs, and show the parameters obtained for some longer codes over $\mathbb F_3$. We then investigate similarities with certain concatenated codes. Contrary to previous methods, our construction allows one to obtain directly codes whose dimension is not a multiple of the locality. Finally, we give an asymptotic study using the Garcia-Stichtenoth tower of function fields, for both our construction and a construction of concatenated codes. We give explicit infinite families of LRCs with locality 2 over any finite field of cardinality greater than 3 following our new approach.Comment: 18 page

Ultra-high-speed imaging of bubbles interacting with cells and tissue

Ultrasound contrast microbubbles are exploited in molecular imaging, where bubbles are directed to target cells and where their high-scattering cross section to ultrasound allows for the detection of pathologies at a molecular level. In therapeutic applications vibrating bubbles close to cells may alter the permeability of cell membranes, and these systems are therefore highly interesting for drug and gene delivery applications using ultrasound. In a more extreme regime bubbles are driven through shock waves to sonoporate or kill cells through intense stresses or jets following inertial bubble collapse. Here, we elucidate some of the underlying mechanisms using the 25-Mfps camera Brandaris128, resolving the bubble dynamics and its interactions with cells. We quantify acoustic microstreaming around oscillating bubbles close to rigid walls and evaluate the shear stresses on nonadherent cells. In a study on the fluid dynamical interaction of cavitation bubbles with adherent cells, we find that the nonspherical collapse of bubbles is responsible for cell detachment. We also visualized the dynamics of vibrating microbubbles in contact with endothelial cells followed by fluorescent imaging of the transport of propidium iodide, used as a membrane integrity probe, into these cells showing a direct correlation between cell deformation and cell membrane permeability

Effficient Graph-based Computation and Analytics

With data explosion in many domains, such as social media, big code repository, Internet of Things (IoT), and inertial sensors, only 32% of data available to academic and industry is put to work, and the remaining 68% goes unleveraged. Moreover, people are facing an increasing number of obstacles concerning complex analytics on the sheer size of data, which include 1) how to perform dynamic graph analytics in a parallel and robust manner within a reasonable time? 2) How to conduct performance optimizations on a property graph representing and consisting of the semantics of code, data, and runtime systems for big data applications? 3) How to innovate neural graph approaches (ie, Transformer) to solve realistic research problems, such as automated program repair and inertial navigation? To tackle these problems, I present two efforts along this road: efficient graph-based computation and intelligent graph analytics. Specifically, I firstly propose two theory-based dynamic graph models to characterize temporal trends in large social media networks, then implement and optimize them atop Apache Spark GraphX to improve their performances. In addition, I investigate a semantics-aware optimization framework consisting of offline static analysis and online dynamic analysis on a property graph representing the skeleton of a data-intensive application, to interactively and semi-automatically assist programmers to scrutinize the performance problems camouflaged in the source code. In the design of intelligent graph-based algorithms, I innovate novel neural graph-based approaches with multi-task learning techniques to repair a broad range of programming bugs automatically, and also improve the accuracy of pedestrian navigation systems in only consideration of sensor data of Inertial Measurement Units (IMU, ie accelerometer, gyroscope, and magnetometer). In this dissertation, I elaborate on the definitions of these research problems and leverage the knowledge of graph computation, program analysis, and deep learning techniques to seek solutions to them, followed by comprehensive comparisons with the state-of-the-art baselines and discussions on future research

The Foundations of Infinite-Dimensional Spectral Computations

Spectral computations in infinite dimensions are ubiquitous in the sciences. However, their many applications and theoretical studies depend on computations which are infamously difficult. This thesis, therefore, addresses the broad question, “What is computationally possible within the field of spectral theory of separable Hilbert spaces?” The boundaries of what computers can achieve in computational spectral theory and mathematical physics are unknown, leaving many open questions that have been unsolved for decades. This thesis provides solutions to several such long-standing problems. To determine these boundaries, we use the Solvability Complexity Index (SCI) hierarchy, an idea which has its roots in Smale's comprehensive programme on the foundations of computational mathematics. The Smale programme led to a real-number counterpart of the Turing machine, yet left a substantial gap between theory and practice. The SCI hierarchy encompasses both these models and provides universal bounds on what is computationally possible. What makes spectral problems particularly delicate is that many of the problems can only be computed by using several limits, a phenomenon also shared in the foundations of polynomial root-finding as shown by McMullen. We develop and extend the SCI hierarchy to prove optimality of algorithms and construct a myriad of different methods for infinite-dimensional spectral problems, solving many computational spectral problems for the first time. For arguably almost any operator of applicable interest, we solve the long-standing computational spectral problem and construct algorithms that compute spectra with error control. This is done for partial differential operators with coefficients of locally bounded total variation and also for discrete infinite matrix operators. We also show how to compute spectral measures of normal operators (when the spectrum is a subset of a regular enough Jordan curve), including spectral measures of classes of self-adjoint operators with error control and the construction of high-order rational kernel methods. We classify the problems of computing measures, measure decompositions, types of spectra (pure point, absolutely continuous, singular continuous), functional calculus, and Radon--Nikodym derivatives in the SCI hierarchy. We construct algorithms for and classify; fractal dimensions of spectra, Lebesgue measures of spectra, spectral gaps, discrete spectra, eigenvalue multiplicities, capacity, different spectral radii and the problem of detecting algorithmic failure of previous methods (finite section method). The infinite-dimensional QR algorithm is also analysed, recovering extremal parts of spectra, corresponding eigenvectors, and invariant subspaces, with convergence rates and error control. Finally, we analyse pseudospectra of pseudoergodic operators (a generalisation of random operators) on vector-valued $l^p$ spaces. All of the algorithms developed in this thesis are sharp in the sense of the SCI hierarchy. In other words, we prove that they are optimal, realising the boundaries of what digital computers can achieve. They are also implementable and practical, and the majority are parallelisable. Extensive numerical examples are given throughout, demonstrating efficiency and tackling difficult problems taken from mathematics and also physical applications. In summary, this thesis allows scientists to rigorously and efficiently compute many spectral properties for the first time. The framework provided by this thesis also encompasses a vast number of areas in computational mathematics, including the classical problem of polynomial root-finding, as well as optimisation, neural networks, PDEs and computer-assisted proofs. This framework will be explored in the future work of the author within these settings

Bibliography of Lewis Research Center Technical Publications announced in 1991

This compilation of abstracts describes and indexes the technical reporting that resulted from the scientific engineering work performed and managed by the Lewis Research Center in 1991. All the publications were announced in the 1991 issues of STAR (Scientific and Technical Aerospace Reports) and/or IAA (International Aerospace Abstracts). Included are research reports, journal articles, conference presentations, patents and patent applications, and theses