Academic literature on the topic 'Nonzero elements'

LetRbe an associative ring. An elementa∈Ris said to be dependent on a mappingF:R→Rin caseF(x)a=axholds for allx∈R. In this paper, elements dependent on certain mappings on prime and semiprime rings are investigated. We prove, for example, that in case we have a semiprime ringR, there are no nonzero elements which are dependent on the mappingα+β, whereαandβare automorphisms ofR.

In this short note, we introduce an analogue of Wilson's theorem for all nonzero elements of a finite filed, which reproves Wilson's theorem and yields some Wilson type identities. Finally, we obtain an analogue of Wolstenholme's theorem for nonzero elements of a finite filed.

AbstractUsing properties of${\cal K}$-pairs of sets, we show that every nonzero enumeration degreeabounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump asa. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration degree can be capped by degrees of any possible local jump (i.e., any jump that can be realized by enumeration degrees of the local structure); every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degreeacan be capped belowa.

Algorithms for multiplying several types of sparse n x n-matrices on dynamically reconfigurable n x n-arrays are presented. For some classes of sparse matrices constant time algorithms are given, e.g., when the first matrix has at most kn elements in each column or in each row and the second matrix has at most kn nonzero elements in each row, where k is a constant. Moreover, O(kn ) algorithms are obtained for the case that one matrix is a general sparse matrix with at most kn nonzero elements and the other matrix has at most k nonzero elements in every row or in every column. Also a lower bound of Ω(Kn ) is proved for this and other cases which shows that the algorithms are close to the optimum.

Designing topological and geometrical structures with extended unnatural parameters (negative, near-zero, ultrahigh, or tunable) and counterintuitive properties is a big challenge in the field of metamaterials, especially for relatively unexplored materials with multiphysics coupling effects. For natural piezoelectric ceramics, only five nonzero elements in the piezoelectric matrix exist, which has impeded the design and application of piezoelectric devices for decades. Here, we introduce a methodology, inspired by quasi-symmetry breaking, realizing artificial anisotropy by metamaterial design to excite all the nonzero elements in contrast to zero values in natural materials. By elaborately programming topological structures and geometrical dimensions of the unit elements, we demonstrate, theoretically and experimentally, that tunable nonzero or ultrahigh values of overall effective piezoelectric coefficients can be obtained. While this work focuses on generating piezoelectric parameters of ceramics, the design principle should be inspirational to create unnatural apparent properties of other multiphysics coupling metamaterials.

SUMMARYThere exist several kinds of definitions for complexity in different areas. In mechanical engineering area, some formulations capable of measuring the complexity of robotic architectures have been proposed at the conceptual-design stage. This paper presents the definition of Kinematic Coupling Complexity to measure the kinematic coupling degree of a manipulator based on the input/output velocity matrix. In order to distinguish the differences among many matrixes, three factors will be considered including the proportion of nonzero elements in matrix J, the difference of nonzero elements' numbers in every row among many matrixes, and the difference of nonzero elements' positions among many matrixes. To illustrate the methods introduced in this paper, four types of heavy-payload forging manipulators are analyzed and the type of forging manipulator with the smallest Kinematic Coupling Complexity is chosen as our prototype.

Магістерська дисертація: 100 с., 15 рис., 14 табл., 1 додаток, 83 джерела. Зараз одним з основних етапів при дослідженні об’єктів, явищ і процесів різної природи є математичне моделювання і пов’язаний ним комп’ютерний експеримент. Чисельні експерименти дають можливість, як планувати натурний експеримент, так і отримувати нові знання про ті процеси і явища для яких утруднений, або взагалі неможливий натурний експеримент. Велика кількість математичних моделей після виконання відповідних перетворень можуть бути описанні системами лінійних алгебраїчних рівнянь (СЛАР) з розрідженими матрицями. Основною особливістю таких систем є їхні великі порядки і невелика кількість ненульових елементів. Великі порядки СЛАР виникають за рахунок того, що дослідники хочуть отримати якомога достовірніші результати, через це будуються більш деталізовані моделі. Мала кількість ненульових елементів пояснюється особливостями дискретизації моделі. Зокрема, системи рівнянь з розрідженими матрицями виникають у задачах аналізу міцності конструкцій у цивільному та промисловому будівництві, фільтрації, тепло- та масо переносу, тощо. Область застосування методів розв’язування СЛАР з розрідженими матрицями постійно розширюється. Через це виникає інтерес до проблеми побудови ефективних методів розв’язання таких систем, порядки яких перевищую сотні тисяч. Класичні результати, що стосуються розробки методів розв’язання СЛАР з розрідженими матрицями висвітлюються у ряді монографій американських і вітчизняних авторів: А. Джорджа, Дж. Лю, С. Писанецьки, Дж. Голуба, Р. Тюарсона, І.А. Блатова, М.Е. Ексаревської та інших. Також зростають вимоги до обчислювальної техніки, що використовується для проведення комп’ютерного експерименту. Вона повинна забезпечувати достатню швидкодію і мати необхідну кількість ресурсів, щоб результат експерименту можна було отримати за досить невеликий проміжок часу. Зараз на ринку представлені багато різних архітектур комп’ютерів з паралельною організацією обчислень. Найбільш продуктивними є платформи так званої «гібридної» архітектури. Дані системи поєднують у собі MIMD- (multiple instructions – multiple data) та SIMD-архітектури (single instruction – multiple data), а саме у системі з багатоядерними процесорами обчислення прискорюються за рахунок графічного прискорювача. Отже одним з ефективних підходів до розв’язання СЛАР з розрідженими матрицями є побудова паралельних алгоритмів, що враховують особливості архітектури комп’ютера. Основними проблемами розробки ефективних паралельних алгоритмів є: аналіз структури матриці, або приведення її до відповідного вигляду, застосовуючи відповідні алгоритми перетворення; вибір ефективної декомпозиції даних; визначення ефективної кількості процесорних ядер і графічних прискорювачів, що використовуються для обчислень; визначення топології міжпроцесних зв’язків, яка зменшує кількість комунікацій і синхронізацій. Саме для аналізу структури розрідженої матриці використовується нейрона мережа, яка дозволить виділити групи ненульових елементів, які можуть оброблятись незалежно. За результатами аналізу буде будуватись декомпозиція даних та обиратись кількість обчислювальних ядер, що забезпечить найкоротший час розрахунків для конкретної структури матриці. Мета та завдання дослідження. Метою роботи є розробка та дослідження паралельних методів та комп’ютерних алгоритмів для дослідження та розв’язування СЛАР з розрідженими матрицями нерегулярної структури на комп’ютерах MIMD-архітектури та комбінації MIMD- і SIMD-архітектури, апробація алгоритмів при математичному моделюванні у прикладних задачах. До завдань дослідження належать: • розробка та дослідження ітераційних паралельних алгоритмів для СЛАР з розрідженими матрицями нерегулярної структури з наближеними даними; • розробка алгоритмів та програм дослідження достовірності розв’язків, отриманих прямими та ітераційними методами; • апробація алгоритмів для математичного моделювання в прикладних задачах. Об’єкт дослідження – математичні моделі, що описуються СЛАР з розрідженими матрицями нерегулярної структури. Предмет дослідження – паралельні методи та комп’ютерні алгоритми знаходження розв’язку СЛАР з розрідженими матрицями нерегулярної структури. Методи дослідження. У роботі застосовуються методи теорії матриць, лінійної алгебри, теорії графів, функціонального аналізу, теорії похибок, теорія нейронних мереж.
Now one of the main stages in the study of objects, phenomena and processes of different nature is mathematical modeling and related computer experiment. Numerous experiments give an opportunity to plan a full-scale experiment, as well as to get new knowledge about those processes and phenomena for which it is difficult, or in general, impossible to carry out a full-scale experiment. A large number of mathematical models can be described by systems of linear algebraic equations (SLRs) with soldered matrices after performing the corresponding transformations. The main feature of such systems is their large orders and a small number of non-zero elements. Large orders of SLAR arise due to the fact that researchers want to get the most reliable results, which is why more detailed models are being built. The small number of non-zero elements is due to the discretization of the model. In particular, systems of equations with sparse matrices arise in problems of analysis of the strength of structures in civil and industrial construction, filtration, heat and mass transfer, and others like that. Scope of the methods of solving SLR with sparse matrices is constantly expanding. Because of this, there is an interest in the problem of constructing effective methods for solving such systems, whose orders exceed hundreds of thousands. Classical results concerning the development of methods for solving SLRR with rarefied matrices are covered in a series of monographs of American and domestic authors: A. George, J. Liu, S. Pisanetski, J. Golub, R. Tjurson, I. A. Blatova, ME Ekseryrovskaya and others. Also, the requirements for the computer technology used to conduct a computer experiment are growing. It must provide sufficient speed and have the required amount of resources so that the result of the experiment can be obtained over a relatively short period of time. Now in the market there are many different architectures of computers with parallel computing organization. The most productive are the platforms of the so-called "hybrid" architecture. These systems combine MIMD (multiple instructions - multiple data) and SIMD architecture (single instruction - multiple data), in particular, in a multi-core processor system, computations are accelerated by means of a graphical accelerator. Hence, one of the effective approaches to solving SLR with sparse matrices is the construction of parallel algorithms that take into account the peculiarities of computer architecture. The main problems of developing effective parallel algorithms are: analysis of the structure of the matrix, or bringing it to the corresponding form, using appropriate conversion algorithms; choice of effective data decomposition; determining the effective number of processor cores and graphic accelerators used for calculations; definition of the interprocess communication topology, which reduces the number of communications and synchronizations. It is precisely for analyzing the structure of a sparse matrix that a neural network is used which allows the selection of groups of non-zero elements that can be processed independently. The results of the analysis will be based on the decomposition of data and the number of computing cores to be selected, which will provide the shortest settlement time for a particular matrix structure. The purpose and objectives of the study. The purpose of the work is to develop and research parallel methods and computer algorithms for research and solving SLR with sparse matrices of irregular structure on computers of MIMD architecture and MIMD and SIMD architecture combinations, testing of algorithms in mathematical modeling in applied problems. The research tasks include: • development and research of iterative parallel algorithms for SLR with sparse matrices of irregular structure with approximate data; • development of algorithms and programs for investigating the validity of solutions obtained by direct and iterative methods; • Approbation of algorithms for mathematical modeling in applied problems. The object of the study is the mathematical models described by SLAR with sparse matrices of the irregular structure. The subject of the study is parallel methods and computer algorithms for locating the SLR solution with sparse matrices of the irregular structure. Research methods. The paper uses methods of matrix theory, linear algebra, graph theory, functional analysis, error theory, and the theory of neural networks.

In Chapter 7 I showed how much computational effort could be avoided in a system consisting of a chain of identical equations each coupled just to its neighboring equations. Such systems arise in linear diffusion and heat conduction problems. It is possible to save computational effort because the sleq array that describes the system of simultaneous linear algebraic equations that must be solved has elements different from zero on and immediately adjacent to the diagonal only. This general approach works also for one-dimensional diffusion problems involving several interacting species. In such a system the concentration of a particular species in a particular reservoir is coupled to the concentrations of other species in the same reservoir by reactions between species and is coupled also to adjacent reservoirs by transport between reservoirs. If the differential equations that describe such a system are arranged in appropriate order, with the equations for each species in a given reservoir followed by the equations for each species in the next reservoir and so on, the sleq array still will have elements different from zero close to the diagonal only. All the nonzero elements lie no farther from the diagonal than the number of species. More distant elements are zero. Again, much computation can be eliminated by taking advantage of this pattern. I will show how to solve such a system in this chapter, introducing two new solution subroutines, GAUSSND and SLOPERND, to replace GAUSSD and SLOPERD. I shall apply the new method of solution to a problem of early diagenesis in carbonate sediments. I calculate the properties of the pore fluid in the sediment as a function of depth and time. The different reservoirs are successive layers of sediment at increasing depth. The fluid's composition is affected by diffusion between sedimentary layers and between the top layer and the overlying seawater, the oxidation of organic carbon, and the dissolution or precipitation of calcium carbonate. Because I assume that the rate of oxidation of organic carbon decreases exponentially with increasing depth, there must be more chemical activity at shallow depths in the sediment than at great depths.

One class of important problems involves diffusion in a single spatial dimension, for example, height profiles of reactive constituents in a turbulently mixing atmosphere, profiles of concentration as a function of depth in the ocean or other body of water, diffusion and diagenesis within sediments, and calculation of temperatures as a function of depth or position in a variety of media. The one-dimensional diffusion problem typically yields a chain of interacting reservoirs that exchange the species of interest only with the immediately adjacent reservoirs. In the mathematical formulation of the problem, each differential equation is coupled only to adjacent differential equations and not to more distant ones. Substantial economies of computation can therefore be achieved, making it possible to deal with a larger number of reservoirs and corresponding differential equations. In this chapter I shall explain how to solve a one-dimensional diffusion problem efficiently, performing only the necessary calculations. The example I shall use is the calculation of the zonally averaged temperature of the surface of the Earth (that is, the temperature averaged over all longitudes as a function of latitude). I first present an energy balance climate model that calculates zonally averaged temperatures as a function of latitude in terms of the absorption of solar energy, which is a function of latitude, the emission of long-wave planetary radiation to space, which is a function of temperature, and the transport of heat from one latitude to another. This heat transport is represented as a diffusive process, dependent on the temperature gradient or the difference between temperatures in adjacent latitude bands. I use the energy balance climate model first to calculate annual average temperature as a function of latitude, comparing the calculated results with observed values and tuning the simulation by adjusting the diffusion parameter that describes the transport of energy between latitudes. I then show that most of the elements of the sleq array for this problem are zero. Nonzero elements are present only on the diagonal and immediately adjacent to the diagonal. The array has this property because each differential equation for temperature in a latitude band is coupled only to temperatures in the adjacent latitude bands.

Sorption of organic pollutants by soils and sediments is one of the main chemical processes that controls pollutant migration in the environment. Information about the molecular mechanisms by which an organic pollutant interacts with other solution-phase constituents and with solid-phase sorbents would be invaluable for more accurate prediction of pollutant fate and transport and for optimal design and application of remediation procedures. Many current models and remediation strategies are based upon the “partition theory” of organic compound sorption, which predicts sorption coefficients from properties such as water solubility or octanol-water partition coefficients. Partition theory is well suited for nonpolar hydrocarbons but may not be appropriate for pesticides with electrophilic or weakly acidic or basic substituents, which may interact with soils or organic matter through specific interactions such as hydrogen bonding or charge-transfer complexes. If a pesticide can form hydrogen bonds or a charge-transfer complex with a sorbent, sorption may be greater than in the absence of specific interactions. Nuclear magnetic resonance (NMR) spectroscopy is well suited for the study of pesticide-solution or pesticide-sorbent interactions because NMR is an element-specific method that is extremely sensitive to the electron density (shielding) near the nucleus of interest. Consequently, solution-state NMR can distinguish between closely related functional groups and can provide information about intermolecular interactions. All nuclei with nonzero nuclear spin quantum number can be studied by NMR spectroscopy. Of the more than 100 NMR-active nuclei, 1H and 19F are the easiest to study because both have natural abundances near 100% and greater NMR sensitivity than any other nuclei. In addition, both 1H and 19F have zero quadrupolar moments, which means that sharp, well resolved NMR peaks can be obtained, at least in homogeneous solutions. Proton NMR is well suited for elucidating molecular interactions in solution but cannot be used to study interactions between pesticides and heterogeneous sorbents such as soils, humic acid, or even cell extracts, since protons in the sorbent generally produce broad peaks that mask the NMR peaks from the solute or sorbate of interest. In contrast, 19F NMR can be used to study interactions between fluorine-containing molecules and heterogeneous sorbents because the fluorine concentration in most natural sorbents is negligible.

In order to satisfy the ever-growing demand for high-performance processors for neural networks, the state-of-the-art processing units tend to use application-oriented circuits to replace Processing Engine (PE) on the GPU under circumstances where low-power solutions are required. The application-oriented PE is fully optimized in terms of the circuit architecture and eliminates incorrect data dependency and instructional redundancy. In this paper, we propose a novel encoding approach on a sparse neural network after pruning. We partition the weight matrix into numerous blocks and use a low-rank binary map to represent the validation of these blocks. Furthermore, the elements in each nonzero block are also encoded into two submatrices: one is the binary stream discriminating the zero/nonzero position, while the other is the pure nonzero elements stored in the FIFO. In the experimental part, we implement a well pre-trained sparse neural network on the Xilinx FPGA VC707. Experimental results show that our algorithm outperforms the other benchmarks. Our approach has successfully optimized the throughput and the energy efficiency to deal with a single frame. Accordingly, we contend that Nested Bitmask Neural Network (NBNN), is an efficient neural network structure with only minor accuracy loss on the SoC system.

The graphical lasso is the most popular approach to estimating the inverse covariance matrix of high-dimension data. It iteratively estimates each row and column of the matrix in a round-robin style until convergence. However, the graphical lasso is infeasible due to its high computation cost for large size of datasets. This paper proposes Sting, a fast approach to the graphical lasso. In order to reduce the computation cost, it efficiently identifies blocks in the estimated matrix that have nonzero elements before entering the iterations by exploiting the singular value decomposition of data matrix. In addition, it selectively updates elements of the estimated matrix expected to have nonzero values. Theoretically, it guarantees to converge to the same result as the original algorithm of the graphical lasso. Experiments show that our approach is faster than existing approaches.

The principal coordinates of a non-classically damped linear system are coupled by nonzero off-diagonal element of the modal damping matrix. In the analysis of non-classically damped systems, a common approximation is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is widely accepted that if the modal damping matrix is diagonally dominant, then errors due to the decoupling approximation must be small. In addition, it is intuitively believed that the more diagonal the modal damping matrix, the less will be the errors in the decoupling approximation. Two quantitative measures are proposed in this paper to measure the degree of being diagonal dominant in modal damping matrices. It is demonstrated that, over a finite range, errors in the decoupling approximation can continuously increase while the modal damping matrix becomes more and more diagonal with its off-diagonal elements decreasing in magnitude continuously. An explanation for this unexpected behavior is presented. Within a practical range of engineering applications, diagonal dominance of the modal damping matrix may not be sufficient for neglecting modal coupling in a damped system.

Abstract The problem of synthesizing a planar 4-bar with given pivots such that the coupler curve passes through live precision points is considered. It is shown that the design parameters must satisfy a system of 4 fourth-degree polynomial equations in 4 unknowns which has at most 36 nonzero real solutions. This polynomial system is solved using a continuation method, which thereby generates the collection of all designs that meet the precision-point specification. A computer program that implements this continuation method has been tested on a number of problems. It is reliable and fast enough for the purposes of design. The approach to kinematic design represented by this work is completely general, subject only to computer-time limitations that may arise for problems with many design elements.

For an assembled structure with many bolted joints, predicting its dynamic response with high fidelity is always a difficult problem, because of the nonlinearity introduced by friction contact between jointed interfaces. The friction contact results in nonlinear stiffness and damping to a structure. To realize predictive simulation in structural dynamic design, these nonlinear behaviors must be carefully considered. In this paper, the dynamics of a multi-bolt jointed beam is calculated. A modified IWAN constitutive model, which can consider both tangential micro/macro slip and nonzero residual stiffness at macroslip phase, is developed to model nonlinear contact behaviors due to joint interfaces. A whole interface element integrating the proposed constitutive model is developed. The element is used to model the nonlinear stiffness and damping caused by bolted joints. The interface element is placed between the two contact interfaces. The other part of the beam is modeled by linear beam elements. A Matlab code is developed to realize the proposed nonlinear finite element dynamic analysis method. A hammer impact experiment for the bolt-jointed beam is conducted under different excitation force levels. The calculated nonlinear numerical results are compared with experimental results. It is shown that the effect of joint nonlinearity on structural dynamics can be observed from the response predicted by the proposed method. The numerical results agree well with the experimental results. This work validates the necessary of using nonlinear joint model for dynamic simulation of jointed structures.

Abstract Algorithms have been developed for warping analysis and calculation of the shearing stresses in a general porous cross section of a long rod when it is subjected to twisting torques at its ends. The shape and dimensions of the cross section full of holes are defined from the binary segmented image data with by a micro-CT scanning technique. Finite difference approximation of the Laplace equation governing the cross-sectional warping leading to the matrix solution by a Gauss-Seidel process is discussed. Method of pointer matrix which keeps the locations of the nonzero elements of the coefficient matrix, is employed to expedite the iterative solution. Computer programs are coded in QuickBASIC language to facilitate plotting of the computed distributions of warping and shearing stresses. The classical torsional problem of square and thin-walled cross sections are used to reexamine the accuracy of the developed algorithms and results are found to be in very good agreement.

A method is proposed for identifying a set of reduced order, nonlinear equations which describe the structural behavior of aeroelastic configurations. The strain energy of the system is written as a (polynomial) function of the structures’ modal amplitudes. The unknown coefficients of these polynomials are then computed using the strain energy data calculated from a steady state, high-order, nonlinear finite element model. The resulting strain energy expression can then be be used to develop the modal equations of motion. From these equations zero and nonzero angle of attack flutter and LCO data are computed for a 45 degree delta wing aeroelastic model. The results computed using the reduced order model compare well with those from a high fidelity aeroelastic model and to experiment. A two to three order of magnitude reduction in the number of structural equations and a two order of magnitude reduction in total computational time is accomplished using the current reduced order method.

Abstract Intelligent and adaptive material systems and structures have become very important in engineering applications. The basic characteristic of these systems is the ability to adapt to the environmental conditions. One of the new class of materials with promising applications in structural and mechanical systems are Shape Memory Alloys (SMA). The mechanical behavior of shape memory alloys in particular shows a strong dependence on temperature. This property provides opportunities for the utilization of SMAs in actuators or energy dissipation devices. However, the behavior of systems containing shape memory components under random excitation has not yet been addressed in the literature. Such study is important to verify the feasibility of using SMAs in structural systems. In this work a non-deterministic study of the dynamic behavior of a single degree-of-freedom (SDOF) mechanical system, having a Nitinol spring as a restoring force element is presented. The SMA spring is characterized using a one-dimensional phenomenological constitutive model based on the classical Devonshire theory. Response statistics for zero mean random vibration of the SDOF under a wide range of temperature is obtained. Furthermore, nonzero mean analysis of these systems is carried out.