Vector Operations Research Articles

An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators

The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators (A_1,ldots ,A_n). A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.

Output-Feedback Robust Control of Uncertain Systems via Online Data-Driven Learning.

Although robust control has been studied for decades, the output-feedback robust control design is still challenging in the control field. This article proposes a new approach to address the output-feedback robust control for continuous-time uncertain systems. First, we transform the robust control problem into an optimal control problem of the nominal linear system with a constructive cost function, which allows simplifying the control design. Then, a modified algebraic Riccati equation (MARE) is constructed by further investigating the corresponding relationship with the state-feedback optimal control. To solve the derived MARE online, the vectorization operation and Kronecker's product are applied to reformulate the output Lyapunov function, and then, a new online data-driven learning method is suggested to learn its solution. Consequently, only the measurable system input and output are used to derive the solution of the MARE. In this case, the output-feedback robust control gain can be obtained without using the unknown system states. The control system stability and convergence of the derived solution are rigorously proved. Two simulation examples are provided to demonstrate the efficacy of the suggested methods.

About the B-spline wavelet discrete-continual finite element method of the local plate analysis

Introduction. This distinctive paper addresses the local semi-analytical solution to the problem of plate analysis. Isotropic plates featuring the regularity (constancy) of physical and geometric parameters (modulus of elasticity of the plate material, Poisson’s ratio of the plate material, dimensions of the cross section of the plate) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction.
 Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial operational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov.
 Results. Some relevant issues of construction of normalized basis functions of the B-spline are considered; the technique of approximation of corresponding vector functions and operators within DCFEM is described. The problem remains continual if analyzed along the basic direction, and its exact analytical solution can be obtained, whereas the finite element approximation is used in combination with a wavelet analysis apparatus in respect of the non-basic direction. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method of solving such problems was developed, described and verified in the numerous papers of the co-authors. In particular, we consider the simplest sample analysis of a plate (rectangular in plan) fixed along the side faces exposed to the influence of the load concentrated in the center of the plate.
 Conclusions. The solution to the verification problem obtained using the proposed version of wavelet-based DCFEM was in good agreement with the solution obtained using the conventional finite element method (the corresponding solutions were constructed with and without localization; these solutions almost completely coincided, while the advantages of the numerical-analytical approach were quite obvious). It is shown that the use of B-splines of various degrees within wavelet-based DCFEM leads to a significant reduction in the number of unknowns.

Novasep turns the corner

When Thermo Fisher Scientific purchased a viral vector business in Belgium earlier this year, all eyes were on the drug services giant as it added a European arm to the US viral vector operation it acquired for $1.7 billion in 2019. But the real action was arguably taking place at the midsize French firm that sold the business for $875 million. That company, Novasep, emerged from the sale with a clean balance sheet after spending nearly a decade working down debt accrued through mergers and acquisitions—a burden that nearly put it out of business. The deal also completed the process of remaking Novasep. It is now a contract development and manufacturing organization (CDMO) emphasizing small-molecule active pharmaceutical ingredient (API) production and tightly focused on customer molecules that are in late-stage clinical trials or on the market. Novasep managers credit that focus with delivering several months of phenomenal sales growth, resulting

AERO: A 1.28 MOP/s/LUT Reconfigurable Inference Processor for Recurrent Neural Networks in a Resource-Limited FPGA

This study presents a resource-efficient reconfigurable inference processor for recurrent neural networks (RNN), named AERO. AERO is programmable to perform inference on RNN models of various types. This was designed based on the instruction-set architecture specializing in processing primitive vector operations that compose the dataflows of RNN models. A versatile vector-processing unit (VPU) was incorporated to perform every vector operation and achieve a high resource efficiency. Aiming at a low resource usage, the multiplication in VPU is carried out on the basis of an approximation scheme. In addition, the activation functions are realized with the reduced tables. We developed a prototype inference system based on AERO using a resource-limited field-programmable gate array, under which the functionality of AERO was verified extensively for inference tasks based on several RNN models of different types. The resource efficiency of AERO was found to be as high as 1.28 MOP/s/LUT, which is 1.3-times higher than the previous state-of-the-art result.

New physics effects in leptonic and semileptonic decays

We discuss the possibilities of extracting the constraints on New Physics by using the current data on the leptonic and semileptonic decays of pseudoscalar mesons. In doing so we use a general low energy Lagrangian that besides the vector and axial operators also includes the (pseudo-)scalar and tensor ones. In obtaining constraints on New Physics couplings, we combine the experimental information concerning several decay modes with the accurate and precise lattice QCD results for the hadronic matrix elements. We propose to study new observables that can be extracted from the angular analysis of the semileptonic decays and discuss their values both in the Standard Model and in some specific scenarios of physics beyond the Standard Model.

A general definition of $JT_a$ -- deformed QFTs

We propose a general path-integral definition of two-dimensional quantum field theories deformed by an integrable, irrelevant vector operator constructed from the components of the stress tensor and those of a U(1) current. The deformed theory is obtained by coupling the original QFT to a flat dynamical gauge field and ``half'' a flat dynamical vielbein. The resulting partition function is shown to satisfy a geometric flow equation, which perfectly reproduces the flow equations for the deformed energy levels that were previously derived in the literature. The S-matrix of the deformed QFT differs from the original S-matrix only by an overall phase factor that depends on the charges and momenta of the external particles, thus supporting the conjecture that such QFTs are UV complete, although intrinsically non-local. For the special case of an integrable QFT, we check that this phase factor precisely reproduces the change in the finite-size spectrum via the Thermodynamic Bethe Ansatz equations.

Matrix iteration algorithms for solving the generalized Lyapunov matrix equation

In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. Then we introduce the matrix versions of biconjugate residual (BICR), biconjugate gradients stabilized (Bi-CGSTAB), and conjugate residual squared (CRS) algorithms. This study’s primary motivation is to avoid the increase of computational complexity by using the Kronecker product and vectorization operation. Finally, we offer several numerical examples to show the effectiveness of the derived algorithms.

Fuzzy granular convolutional classifiers

Convolutional operations extracting effective features have been widely used in the field of deep learning. For the convolution is difficult to process set data, we propose two convolutional operators on fuzzy sets, and build a fuzzy granular classifier. Firstly, a fuzzy granulation is performed on single-atom features of classification systems to form fuzzy conditional granules and fuzzy decision granules. Then, a fuzzy conditional granular vector is constructed from the fuzzy conditional granules, and a convolutional operation is carried out on the granular vector. After that, a new fuzzy feature granule is obtained. The fuzzy feature granule is compared with its corresponding fuzzy decision granule. The result of comparison is back-propagated to the fuzzy granular vector. Simultaneously, weights of the fuzzy granular vector are modified. Thus, a fuzzy granular convolutional classifier is formed by iterating and optimizing the weights of fuzzy granular vectors several times. Furthermore, we prove the difference and derivative of fuzzy granular convolution, which provide a theoretical basis for the back-propagation of the fuzzy granular convolutional classifier. Finally, the convergence effects of the fuzzy granular convolutional operations and the classification performance of the proposed classifier are tested on some UCI datasets. The theoretical analysis and experimental results show that the convolutional operations of fuzzy granular vectors have the characteristics of fast convergence, and the fuzzy granular convolutional classifier obtains a better classification performance.

Two-meson form factors in unitarized chiral perturbation theory

We present a comprehensive analysis of form factors for two light pseudoscalar mesons induced by scalar, vector, and tensor quark operators. The theoretical framework is based on a combination of unitarized chiral perturbation theory and dispersion relations. The low-energy constants in chiral perturbation theory are fixed by a global fit to the available data of the two-meson scattering phase shifts. Each form factor derived from unitarized chiral perturbation theory is improved by iteratively applying a dispersion relation. This study updates the existing results in the literature and explores those that have not been systematically studied previously, in particular the two-meson tensor form factors within unitarized chiral perturbation theory. We also discuss the applications of these form factors as mandatory inputs for low-energy phenomena, such as the semi-leptonic decays Bs→ π+π−ℓ+ℓ− and the τ lepton decay τ → π−π0ντ, in searches for physics beyond the Standard Model.

Electromagnetic Vector Sparse Nested Array: Array Structure Design, Off-Grid Parameter Estimation Algorithm

In this paper, a new array structure of sparse nested array (SNA) for electromagnetic vector sensor is designed. An electromagnetic vector sensor is composed of six spatially colocated, orthogonally oriented, diversely polarized antennas, which can measure three-dimensional electric and magnetic field components. By introducing sparse factor (SF) between every adjacent sensor, the proposed SNA has flexibility of extending the array aperture and reducing the mutual coupling effect. Meanwhile, a low-complexity multiparameter estimation algorithm is proposed for SNA. First, the vectorization operation for array manifold ensures the large degrees of freedom for multiparameter estimation, where the initial coarse estimates decrease search range. In addition, the improved off-grid orthogonal matching pursuit method obtains joint direction of arrival (DOA) and polarization estimates with a relatively small overcomplete dictionary because this off-grid method achieves high performance even if the estimates do not fall on the grid of the dictionary. Theoretical analysis and simulation results verify the superiority of the proposed array structure and the algorithm.

Orthogonality Measurement of Three-Axis Motion Trajectories for Micromanipulation Robot Systems.

In robotic micromanipulation systems, the orthogonality of the three-axis motion trajectories of the motion control systems influences the accuracy of micromanipulation. A method of measuring and evaluating the orthogonality of three-axis motion trajectories is proposed in this paper. Firstly, a system for three-axis motion trajectory measurement is developed and an orthogonal reference coordinate system is designed. The influence of the assembly error of laser displacement sensors on the reference coordinate system is analyzed using simulation. An approach to estimating the orthogonality of three-axis motion trajectories and to compensating for its error is presented using spatial line fitting and vector operation. The simulation results show that when the assembly angle of the laser displacement sensors is limited within a range of 10°, the relative angle deviation of the coordinate axes of the reference coordinate frame is approximately 0.09%. The experiment results show that precision of spatial line fitting is approximately 0.02 mm and relative error of the orthogonality measurement is approximately 0.3%.

PathologyGAN: Learning deep representations of cancer tissue

Histopathological images of tumours contain abundant information about how tumours grow and how they interact with their micro-environment. Better understanding of tissue phenotypes in these images could reveal novel determinants of pathological processes underlying cancer, and in turn improve diagnosis and treatment options. Advances of Deep learning makes it ideal to achieve those goals, however, its application is limited by the cost of high quality labels from patients data. Unsupervised learning, in particular, deep generative models with representation learning properties provides an alternative path to further understand cancer tissue phenotypes, capturing tissue morphologies. In this paper, we develop a framework which allows Generative Adversarial Networks (GANs) to capture key tissue features and uses these characteristics to give structure to its latent space. To this end, we trained our model on two different datasets, an H&E colorectal cancer tissue from the National Center for Tumor diseases (NCT, Germany) and an H&E breast cancer tissue from the Netherlands Cancer Institute (NKI, Netherlands) and Vancouver General Hospital (VGH, Canada). Composed of 86 slide images and 576 tissue micro-arrays (TMAs) respectively. We show that our model generates high quality images, with a Frechet Inception Distance (FID) of 16.65 (breast cancer) and 32.05 (colorectal cancer). We further assess the quality of the images with cancer tissue characteristics (e.g. count of cancer, lymphocytes, or stromal cells), using quantitative information to calculate the FID and showing consistent performance of 9.86. Additionally, the latent space of our model shows an interpretable structure and allows semantic vector operations that translate into tissue feature transformations. Furthermore, ratings from two expert pathologists found no significant difference between our generated tissue images from real ones. The code, generated images, and pretrained model are available at https://github.com/AdalbertoCq/Pathology-GAN

Analysis method and algorithm design of biological sequence problem based on generalized k-mer vector

K-mer can be used for the description of biological sequences and k-mer distribution is a tool for solving sequences analysis problems in bioinformatics. We can use k-mer vector as a representation method of the k-mer distribution of the biological sequence. Problems, such as similarity calculations or sequence assembly, can be described in the k-mer vector space. It helps us to identify new features of an old sequence-based problem in bioinformatics and develop new algorithms using the concepts and methods from linear space theory. In this study, we defined the k-mer vector space for the generalized biological sequences. The meaning of corresponding vector operations is explained in the biological context. We presented the vector/matrix form of several widely seen sequence-based problems, including read quantification, sequence assembly, and pattern detection problem. Its advantages and disadvantages are discussed. Also, we implement a tool for the sequence assembly problem based on the concepts of k-mer vector methods. It shows the practicability and convenience of this algorithm design strategy.

Constraints on quasinormal modes and bounds for critical points from pole-skipping

We consider a holographic thermal state and perturb it by a scalar operator whose associated real-time Green’s function has only gapped poles. These gapped poles correspond to the non-hydrodynamic quasinormal modes of a massive scalar perturbation around a Schwarzschild black brane. Relations between pole-skipping points, critical points and quasinormal modes in general emerge when the mass of the scalar and hence the dual operator dimension is varied. First, this novel analysis reveals a relation between the location of a mode in the infinite tower of quasinormal modes and the number of pole-skipping points constraining its dispersion relation at imaginary momenta. Second, for the first time, we consider the radii of convergence of the derivative expansions about the gapped quasinormal modes. These convergence radii turn out to be bounded from above by the set of all pole-skipping points. Furthermore, a transition between two distinct classes of critical points occurs at a particular value for the conformal dimension, implying close relations between critical points and pole-skipping points in one of those two classes. We show numerically that all of our results are also true for gapped modes of vector and tensor operators.

RcdMathLib: An Open Source Software Library for Computing on Resource-Limited Devices.

We developped an open source library called RcdMathLib for solving multivariate linear and nonlinear systems. RcdMathLib supports on-the-fly computing on low-cost and resource-constrained devices, e.g., microcontrollers. The decentralized processing is a step towards ubiquitous computing enabling the implementation of Internet of Things (IoT) applications. RcdMathLib is modular- and layer-based, whereby different modules allow for algebraic operations such as vector and matrix operations or decompositions. RcdMathLib also comprises a utilities-module providing sorting and filtering algorithms as well as methods generating random variables. It enables solving linear and nonlinear equations based on efficient decomposition approaches such as the Singular Value Decomposition (SVD) algorithm. The open source library also provides optimization methods such as Gauss–Newton and Levenberg–Marquardt algorithms for solving problems of regression smoothing and curve fitting. Furthermore, a positioning module permits computing positions of IoT devices using algorithms for instance trilateration. This module also enables the optimization of the position by performing a method to reduce multipath errors on the mobile device. The library is implemented and tested on resource-limited IoT as well as on full-fledged operating systems. The open source software library is hosted on a GitLab repository.

Thermal aspects of interacting quantum gases in Lorentz-violating scenarios

In this work, we study the interaction of quantum gases in Lorentz-violating scenarios considering both boson and fermion sectors. In the latter case, we investigate the consequences of a system governed by scalar, vector, pseudovector and tensor operators. Besides, we examine the implications of $\left( \hat{k}_{a}\right) ^{\kappa }$ and $\left( \hat{k}_{c}\right) ^{\kappa \xi }$ operators for the boson case as well. For doing so, we regard the grand canonical ensemble seeking the so-called partition function, which suffices to provide analytically the calculations of interest, i.e., mean particle number, entropy, mean total energy and pressure. Furthermore, in low temperature regime, such quantities converge until reaching a similar behavior being in contrast with what is shown in high temperature regime, which brings out the differentiation of their effects. In addition, particle number, entropy and energy exhibit an extensive characteristic even in the presence of Lorentz violation. Finally, for peseudovector and tensor operators, we notice a remarkable feature due to the breaking process of spin degeneracy: the system turns out to have greater energy and particle number for the spin-down particles in comparison with spin-up ones.

Research of Three Phase PWM Rectifier based on Direct Vector Control Method

In oilfield worldwide, beam pumping units are wildly used to produce oil. the working mechanism of beam pumping unit determines that its driving motor frequently enters into the state of power generation, so PWM reversible rectifier is an ideal choice for recycle the generated energy, therefore it is necessary to research the control theory of PWM reversible rectifier. The paper presents a digital algorithm for the PWM rectifier, on the basis of coordinate analyzing theory and SVPWM method. The method that directly calculates the basic vector operation time and position is adopted, so the arc-tangent calculation is avoided, calculation process is simplified. Based on the vector coordinate transformation of three-phase reversible PWM rectifier, the control equation of direct current control method is derived in this paper. The rectifier control algorithm is realized with high performance DSP TMS320F28335. The experimental results prove that the method proposed in the paper is practical and correct.

Developing a Multi-GPU-Enabled Preconditioned GMRES with Inexact Triangular Solves for Block Sparse Matrices

Solving triangular systems is the building block for preconditioned GMRES algorithm. Inexact preconditioning becomes attractive because of the feature of high parallelism on accelerators. In this paper, we propose and implement an iterative, inexact block triangular solve on multi-GPUs based on PETSc’s framework. In addition, by developing a distributed block sparse matrix-vector multiplication procedure and investigating the optimized vector operations, we form the multi-GPU-enabled preconditioned GMRES with the block Jacobi preconditioner. In the implementation, the GPU-Direct technique is employed to avoid host-device memory copies. The preconditioning step used by PETSc’s structure and the cuSPARSE library are also investigated for performance comparisons. The experiments show that the developed GMRES with inexact preconditioning on 8 GPUs can achieve up to 4.4x speedup over the CPU-only implementation with exact preconditioning using 8 MPI processes.

Cambricon-G: A Polyvalent Energy-Efficient Accelerator for Dynamic Graph Neural Networks

Graph neural networks (GNNs), which extend traditional neural networks for processing graph-structured data, have been widely used in many fields. The GNN computation mainly consists of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">edge processing</i> to generate messages by combining the edge/vertex features and the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vertex processing</i> to update the vertex features with aggregated messages. In addition to nontrivial vector operations in the edge processing, huge random accesses and neural network operations in the vertex processing, the graph topology of GNNs may also vary during the computation (i.e., dynamic GNNs). The above characteristics pose significant challenges on existing architectures. In this article, we propose a novel accelerator named CAMBRICON-G for efficient processing of both dynamic and static GNNs. The key of CAMBRICON-G is to abstract the irregular computation of a broad range of GNN variants to the process of regularly tiled <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">adjacent cuboid</i> (which extends the traditional adjacent matrix of graph by adding the dimension of vertex features). The intuition is that the adjacent cuboid facilitates exploitation of both data locality and parallelism by offering <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multidimensional multilevel tiling</i> (including spatial and temporal tiling) opportunities. To perform the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multidimensional spatial tiling</i> , the CAMBRICON-G architecture mainly consists of the cuboid engine (CE) and hybrid on-chip memory. The CE has multiple vertex processing units (VPUs) working in a coordinated manner to efficiently process the sparse data and dynamically update the graph topology with dedicated instructions. The hybrid on-chip memory contains the topology-aware cache and multiple scratchpad memory to reduce off-chip memory access. To perform the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multidimensional temporal tiling</i> , an easy-to-use programming model is provided to flexibly explore different tiling options for large graphs. Experimental results show that compared against Nvidia P100 GPU, the performance and energy efficiency can be improved by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$7.14\times $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$20.18\times $ </tex-math></inline-formula> , respectively, on various GNNs, which validates both the versatility and energy efficiency of CAMBRICON-G.