Elementary Arithmetic Operations Research Articles

Consistent and fast inference in compartmental models of epidemics using Poisson Approximate Likelihoods

AbstractAddressing the challenge of scaling-up epidemiological inference to complex and heterogeneous models, we introduce Poisson approximate likelihood (PAL) methods. In contrast to the popular ordinary differential equation (ODE) approach to compartmental modelling, in which a large population limit is used to motivate a deterministic model, PALs are derived from approximate filtering equations for finite-population, stochastic compartmental models, and the large population limit drives consistency of maximum PAL estimators. Our theoretical results appear to be the first likelihood-based parameter estimation consistency results which apply to a broad class of partially observed stochastic compartmental models and address the large population limit. PALs are simple to implement, involving only elementary arithmetic operations and no tuning parameters, and fast to evaluate, requiring no simulation from the model and having computational cost independent of population size. Through examples we demonstrate how PALs can be used to: fit an age-structured model of influenza, taking advantage of automatic differentiation in Stan; compare over-dispersion mechanisms in a model of rotavirus by embedding PALs within sequential Monte Carlo; and evaluate the role of unit-specific parameters in a meta-population model of measles.

Methods and hardware to accelerate the work of a convolutional neural network

Three main approaches to building computer systems are analyzed and allocated: software, hardware, and problem-oriented. A problem-oriented approach was chosen for the implementation of CNN. This approach uses a processor core with hardware accelerators that implement basic CNN operations. The development of computer systems for the implementation of CNN should be carried out based on an integrated approach. This approach includes a modern element base, existing hardware, and software for the implementation of the CNN; methods and algorithms for the implementation of CNN; methods, algorithms, and VLSI structure for the implementation of basic operations of the CNN; methods and means of computer-aided design of hardware and software focused on the implementation of CNN computer systems. For the development of computer systems for the implementation of CNN chosen approach, which includes: variable composition of equipment; use of the basis of elementary arithmetic operations; organization of the process of calculating the scalar product as execution single operation; pipeline and spatial parallelism; localization and simplification of links between the steps of the conveyor; coordination of the time of formation of input data and weighting coefficients with the duration of the conveyor cycle. It is shown that in order to reduce the processing time of large images, it is most expedient to use parallel-stream VLSI -implementation of basic operations. The modified Booth algorithm for forming partial products in a parallel-threaded computing device is selected, which decreased the number of steps in the pipeline. The method of group summation has been improved, which, with multi-input single-digit adders, combined according to the principle of the Wallace tree, provides a reduction in summation time. The method of parallel-flow calculation of scalar product in a sliding window is developed, which, by coordinating the time of receipt of columns of input data and weighting coefficients with the duration of the conveyor cycle, provides high efficiency of equipment use and calculations in real-time. The main ways regarding coordination of the time of receipt of input data columns and weighting coefficients with the duration of the conveyor stroke of hardware that implement two-dimensional convolution are determined. The hardware structure for the realization of two-dimensional convolution in a sliding window, which is focused on VLSI- implementation with high efficiency of equipment use, has been developed. Programmable logic integrated circuits selected for the implementation of hardware accelerators. Single-bit 7, 15, and 31 input adders were developed and modeled on the basis of FPGA EP3C16F484 of the Cyclone III family of Altera company, and an 8-input 7-bit adder was synthesized on their basis.

Quadrature Rules for the Fm-Transform Polynomial Components

The purpose of this paper is to reduce the complexity of computing the components of the integral Fm-transform, m≥0, whose analytic expressions include definite integrals. We propose to use nontrivial quadrature rules with nonuniformly distributed integration points instead of the widely used Newton–Cotes formulas. As the weight function that determines orthogonality, we choose the generating function of the fuzzy partition associated with the Fm-transform. Taking into account this fact and the fact of exact integration of orthogonal polynomials, we obtain exact analytic expressions for the denominators of the components of the Fm-transformation and their approximate analytic expressions, which include only elementary arithmetic operations. This allows us to effectively estimate the components of the Fm-transformation for 0≤m≤3. As a side result, we obtain a new method of numerical integration, which can be recommended not only for continuous functions, but also for strongly oscillating functions. The advantage of the proposed calculation method is shown by examples.

Algorithms for Stochastically Rounded Elementary Arithmetic Operations in IEEE 754 Floating-Point Arithmetic

We present algorithms for performing the five elementary arithmetic operations ( <inline-formula><tex-math notation="LaTeX">$+$</tex-math></inline-formula> , <inline-formula><tex-math notation="LaTeX">$-$</tex-math></inline-formula> , ×, <inline-formula><tex-math notation="LaTeX">$\div$</tex-math></inline-formula> , and <inline-formula><tex-math notation="LaTeX">$\sqrt{\phantom{x}}$</tex-math></inline-formula> ) in floating point arithmetic with stochastic rounding, and demonstrate the value of these algorithms by discussing various applications where stochastic rounding is beneficial. The algorithms require that the hardware be compliant with the IEEE 754 floating-point standard and that a floating-point pseudorandom number generator be available. The goal of these techniques is to emulate stochastic rounding when the underlying hardware does not support this rounding mode, as is the case for most existing CPUs and GPUs. By simulating stochastic rounding in software, one has the possibility to explore the behavior of this rounding mode and develop new algorithms even without having access to hardware implementing stochastic rounding—once such hardware becomes available, it suffices to replace the proposed algorithms by calls to the corresponding hardware routines. When stochastically rounding double precision operations, the algorithms we propose are between 7.3 and 19 times faster than the implementations that use the GNU MPFR library to simulate extended precision. We test our algorithms on various tasks, including summation algorithms and solvers for ordinary differential equations, where stochastic rounding is expected to bring advantages.

Development of Mobile Facilities of Neuro-like Cryptographic Encryption and Decryption of Data in Real Time

The requirements are formed, the method is chosen and the main stages of development of mobile means of neuro-like cryptographic encryption and real-time data decryption are considered. It is shown that the development of mobile means of neuro-like cryptographic encryption and decryption of real-time data with high efficiency of equipment is reduced to minimize hardware costs while providing a variety of requirements, characteristics and limitations. The tabular-algorithmic method of calculating the scalar product has been improved. Namely, the ability to work with floating-point operands has been added and it is focused on hardware and software implementation. Developed on the basis of a universal processor core, supplemented by specialized modules, mobile means of neuro-like cryptographic encryption and data decryption. Which due to the combination of universal and specialized approaches, software and hardware provides effective implementation of algorithms for cryptographic encryption and decryption of data in real time. It is proposed to use a multioperand approach, tables of macroparticle products and bases of elementary arithmetic operations to achieve high technical and economic indicators in the implementation of specialized modules of neuro-like cryptographic encryption and real-time data decryption. Specialized modules of neuro-like cryptographic encryption and data decryption have been implemented using the VHDL hardware programming language and the Quartus II development environment (version 13.1) on the FPGA. The evaluation of hardware and time parameters of the developed specialized module of neurosimilar cryptographic data decryption is carried out.

Quantum circuit for the fast Fourier transform

We propose an implementation of the algorithm for the fast Fourier transform (FFT) as a quantum circuit consisting of a combination of some quantum gates. In our implementation, a data sequence is expressed by a tensor product of vector spaces. Namely, our FFT is defined as a transformation of the tensor product of quantum states. It is essentially different from the so-called quantum Fourier transform (QFT) defined to be a linear transformation of the amplitudes for the superposition of quantum states. The quantum circuit for the FFT consists of several circuits for elementary arithmetic operations such as a quantum adder, subtractor and shift operations, which are implemented as effectively as possible. Namely, our circuit does not generate any garbage bits. The advantages of our method compared to the QFT are its high versatility, and data storage efficiency in terms, for instance, of the quantum image processing.

Astronomical Constants and Universal Code in Holy Book

At the beginning of 1995, I was looking to produce a new concept of the Astronomical Period (AP) which may be determined by the shortest period of Lunar years which this period includes leap years and common years, just to get a simple formula for calculating the average length of the lunar year where I finally deduced the first formula about this average by using the simple math (the four elementary arithmetic operations).&#x0D; &#x0D; By this rule, I methodically educed what I essentially considered it as an acceptable consequence which, indeed, encouraged me to do more research about the best resources that required to deal with the concept of (AP) where I found something like the hidden signals in Islamic Holy Book (The Great Qur&amp;rsquo;an) which led me by the elicitation method to get the perfect astronomical constants besides of an evolving conclusion about what I considered it as a scientific guide to the universal code.&#x0D; &#x0D; What the exiting in this research is: these (perfect astronomical constants) had successfully passed the test of three physical laws in motion which means that the hypothesis of this research (elicitation method) is not arbitrary, and the conclusions of this research had truly deduced by innovative scientific basis.

Automatic Differentiation in ROOT

In mathematics and computer algebra, automatic differentiation (AD) is a set of techniques to evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.), elementary functions (exp, log, sin, cos, etc.) and control flow statements. AD takes source code of a function as input and produces source code of the derived function. By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program. This paper presents AD techniques available in ROOT, supported by Cling, to produce derivatives of arbitrary C/C++ functions through implementing source code transformation and employing the chain rule of differential calculus in both forward mode and reverse mode. We explain its current integration for gradient computation in TFormula. We demonstrate the correctness and performance improvements in ROOT’s fitting algorithms.

Linear programming using limited-precision oracles

Since the elimination algorithm of Fourier and Motzkin, many different methods have been developed for solving linear programs. When analyzing the time complexity of LP algorithms, it is typically either assumed that calculations are performed exactly and bounds are derived on the number of elementary arithmetic operations necessary, or the cost of all arithmetic operations is considered through a bit-complexity analysis. Yet in practice, implementations typically use limited-precision arithmetic. In this paper we introduce the idea of a limited-precision LP oracle and study how such an oracle could be used within a larger framework to compute exact precision solutions to LPs. Under mild assumptions, it is shown that a polynomial number of calls to such an oracle and a polynomial number of bit operations, is sufficient to compute an exact solution to an LP. This work provides a foundation for understanding and analyzing the behavior of the methods that are currently most effective in practice for solving LPs exactly.

Thick gradual intervals: An alternative interpretation of type-2 fuzzy intervals and its potential use in type-2 fuzzy computations

This paper proposes a new interpretation of type-2 fuzzy intervals (T2FIs) through the joint use of gradual intervals (GIs) and thick intervals (TIs). In this framework, a T2FI is viewed as a thick gradual interval (TGI). This new representation gives an original concept for the manipulation of T2FIs according to the thick gradual representation. Furthermore, this vision allows an extension of the interval arithmetic arsenal and reasoning to the framework of T2FIs. The proposed approach can be regarded as more computationally viable, which will make T2FIs computations more useful in applied scenarios. As an illustration, the proposed concept is used to implement the elementary arithmetic operations on T2FIs. The potentialities of the TGI approach have been validated in the frameworks of T2FI aggregation operators and T2FI regression.

Entropy and Optimal Compression of Some General Plane Trees

We continue developing the information theory of structured data. In this article, we study models generating d -ary trees ( d ≥ 2) and trees with unrestricted degree. We first compute the entropy which gives us the fundamental lower bound on compression of such trees. Then we present efficient compression algorithms based on arithmetic encoding that achieve the entropy within a constant number of bits. A naïve implementation of these algorithms has a prohibitive time complexity of O ( n d ) elementary arithmetic operations (each corresponding to a number f ( n , d ) of bit operations), but our efficient algorithms run in O ( n 2 ) of these operations, where n is the number of nodes. It turns out that extending source coding (i.e., compression) from sequences to advanced data structures such as degree-unconstrained trees is mathematically quite challenging and leads to recurrences that find ample applications in the information theory of general structures (e.g., to analyze the information content of degree-unconstrained non-plane trees).

Fluorescent carbon dots with tunable emission by dopamine for sensing of intracellular pH, elementary arithmetic operations and a living cell imaging based INHIBIT logic gate.

Fluorescent carbon dots (C-dots) have drawn great attention in recent years, and many researchers have focused on developing preparation routes and broadening the applications of C-dots. In this work, novel luminescent C-dots with tunable emission were prepared by the pyrolysis of citric acid (CA) and dopamine (DA). By adjusting the molar ratio of CA and DA in the precursor from 2 : 1 to 2 : 3, the maximum fluorescence emission of the obtained C-dots is at 510 nm in acid solution, while it shifts to 450 nm under neutral conditions, and the fluorescence is quenched in a basic solution. By controlling the unique fluorescence emission at different pH values, a three-state switch is achieved. Half-addition and half-subtraction were performed at 525 nm and 450 nm emissions as outputs. And a living cell imaging based INHIBIT logic gate operation was achieved by imaging the channel ratio of the cells.

Fast Möbius Inversion in Semimodular Lattices and ER-labelable Posets

We consider the problem of fast zeta and Möbius transforms in finite posets, particularly in lattices. It has previously been shown that for a certain family of lattices, zeta and Möbius transforms can be computed in $O(e)$ elementary arithmetic operations, where $e$ denotes the size of the covering relation. We show that this family is exactly that of geometric lattices. We also extend the algorithms so that they work in $e$ operations for all semimodular lattices, including chains and divisor lattices. Finally, for both transforms, we provide a more general algorithm that works in $e$ operations for all ER-labelable posets.

Presentation and investigation of a new two dimensional heterostructure phononic crystal to obtain extended band gap

In this paper, a new heterostructure phononic crystal is introduced. The new heterostructure is composed of square and rhombus phononic crystals. Using finite difference method, a displacement-based algorithm is presented to study elastic wave propagation in the phononic crystal. In contrast with conventional finite difference time domain method, at first by using constitutive equations and strain-displacement relations, elastic wave equations are derived based on displacement. Then, these forms are discretized using finite difference method. By this technique, components of stress tensor can be removed from the updating equations. Since the proposed method needs less elementary arithmetical operations, its computational cost is less than that of the conventional FDTD method. Using the presented displacement-based finite difference time domain algorithm, square phononic crystal, rhombus phononic crystal and the new heterostructure phononic crystal were analyzed. Comparison of transmission spectra of the new heterostructure phononic crystal with those creating lattices, showed that band gap can be extended by using the new structure. Also it was observed that by changing the angular constant of rhombus lattice, a new extended band gap can be achieved.

Matrix Inverse as by-Product of Determinant

The determinant of a given square matrix is obtained as the pr oduct of pivot elements evaluated via the iterative matrix order condensation. It follows as the by-product that the inverse of this matrix is then evaluated via the iterative matrix order expansion. The fast and straightforward basic iterative procedure involves only simple elementary arithmetical operations with out any high mathematical process. Remarkably, the revised optimal iterative process will compute without failing the inverse of any square matrix within minutes, be it real or complex, singular or nons ingular, and amazingly enough even for size as huge as 999x999. The manually extended iteration process is also developed to shorten the iteration process steps.

The First Difference Property of the Present Value Operator

The present value of a future cash flow together with derived risk measures like duration and convexity are some of the most fundamental concepts in financial economics. Technological advances have greatly enhanced methods for numerical calculations of such magnitudes. Consequently, analytical expressions have been pushed into the background to the extent that they are rarely, if ever, found in modern textbooks, despite their widespread use in both the classroom and in the real world. This holds, a fortiori, for the derivations leading to these expressions.Analytical solutions are instructive for both students and practitioners as they help understand the qualitative behavior and outcome of numerical calculations and the driving forces behind.This paper derives closed form solutions for the present value and the derived risk measures for a number of commonly used payment streams. The derivations in the paper only use elementary arithmetic operations and avoid cumbersome differentiation operations as well as complicated summation exercises.

Algorithms, machine intelligence, big data : general considerations

We are experiencing astonishing developments in the areas of big data and artificial intelligence. They follow a pattern that we have now been observing for decades: according to Moore's Law,the performance and efficiency in the area of elementary arithmetic operations increases a thousand-fold every 20 years. Although we have not achieved the status where in the singular sense machines have become as "intelligent" as people, machines are becoming increasingly better. The Internet of Things has again helped to massively increase the efficiency of machines. Big data and suitable analytics do the same. If we let these processes simply continue, our civilization may be endangerd in many instances. If the "containment" of these processes succeeds in the context of a reasonable political global governance, a worldwide eco-social market economy, andan economy of green and inclusive markets, many desirable developments that are advantageous for our future may result. Then, at some point in time, the constant need for more and faster innovation may even stop. However, this is anything but certain. We are facing huge challenges.

Inverse and Determinant of a Square Matrix by Order Expansion and Condensation [EM Programmer's Notebook

A simple and straightforward fast iterative method is presented for computing the inverse and determinant of any square matrix by successively applying order condensation and order expansion in an iterative process. Applying the optimal iteration process, which comprises only some 20 lines of the MATLAB source code (using only simple elementary arithmetical operations), the inverse matrix can be computed within minutes from any given square matrix, even of relatively large size (such as 999), with real or complex entries, and irrespective of whether the matrix is singular or nonsingular.

Open induction in a bounded arithmetic for TC0

The elementary arithmetic operations $${+,\cdot,\le}$$ on integers are well-known to be computable in the weak complexity class TC0, and it is a basic question what properties of these operations can be proved using only TC0-computable objects, i.e., in a theory of bounded arithmetic corresponding to TC0. We will show that the theory VTC 0 extended with an axiom postulating the totality of iterated multiplication (which is computable in TC0) proves induction for quantifier-free formulas in the language $${\langle{+,\cdot,\le}\rangle}$$ (IOpen), and more generally, minimization for $${\Sigma_{0}^{b}}$$ formulas in the language of Buss’s S 2.

A robust neuro-fuzzy classifier for the detection of cardiomegaly in digital chest radiographies

We present a novel procedure that automatically and reliably determines the presence of cardiomegaly in chest image radiographies. The cardiothoracic ratio (CTR) shows the relationship between the size of the heart and the size of the chest. The proposed scheme uses a robust fuzzy classifier to find the correct feature values of chest size, and the right and left heart boundaries to measure the heart enlargement to detect cardiomegaly. The proposed approach uses classical morphology operations to segment the lungs providing low computational complexity and the proposed fuzzy method is robust to find the correct measures of CTR providing a fast computation because the fuzzy rules use elementary arithmetic operations to perform a good detection of cardiomegaly. Finally, we improve the classification results of the proposed fuzzy method using a Radial Basis Function (RBF) neural network in terms of accuracy, sensitivity, and specificity.