Integer Algorithm Research Articles

Influence of Lubricant Traction Coefficient on Cage's Nonlinear Dynamic Behavior in High-Speed Cylindrical Roller Bearing

In this paper, the formulas of elastohydrodynamic traction coefficients of four Chinese aviation lubricating oils, namely, 4109, 4106, 4050, and 4010, were obtained by a great number of elastohydrodynamic traction tests. The nonlinear dynamics differential equations of high-speed cylindrical roller bearing were built on the basis of dynamic theory of rolling bearings and solved by Hilber–Hughes–Taylor (HHT) integer algorithm with variable step. The influence of lubricant traction coefficient on cage's nonlinear dynamic behavior was investigated, and Poincaré map was used to analyze the influence of four types of aviation lubricating oils on the nonlinear dynamic response of cage's mass center. The period of nonlinear dynamic response of cage's mass center was used to assess cage's stability. The results of this paper provide the theoretical basis for selection of aviation lubricating oil.

A branch and price algorithm for a Stackelberg Security Game

Mixed integer optimization formulations are an attractive alternative to solve Stackelberg Game problems thanks to the efficiency of state of the art mixed integer algorithms. In particular, decomposition algorithms, such as branch and price methods, make it possible to tackle instances large enough to represent games inspired in real world domians.In this work we focus on Stackelberg Games that arise from a security application and investigate the use of a new branch and price method to solve its mixed integer optimization formulation. We prove that the algorithm provides upper and lower bounds on the optimal solution at every iteration and investigate the use of stabilization heuristics. Our preliminary computational results compare this solution approach with previous decomposition methods obtained from alternative integer programming formulations of Stackelberg games.

A General Digit-Serial Architecture for Montgomery Modular Multiplication

The Montgomery algorithm is a fast modular multiplication method frequently used in cryptographic applications. This paper investigates the digit-serial implementations of the Montgomery algorithm for large integers. A detailed analysis is given and a tight upper bound is presented for the intermediate results obtained during the digit-serial computation. Based on this analysis, an efficient digit-serial Montgomery modular multiplier architecture using carry save adders is proposed and its complexity is presented. In this architecture, pipelined carry select adders are used to perform two final tasks: adding carry save vectors representing the modular product and subtracting the modulus from this addition, if further reduction is needed. The proposed architecture can be designed for any digit size $\delta $ and modulus $\theta $ . This paper also presents logic formulas for the bits of the precomputation $-\theta ^{-1}\bmod 2^{\delta }$ used in the Montgomery algorithm for $\delta \leq 8$ . Finally, evaluation of the proposed architecture on Virtex 7 FPGAs is presented.

Efficient Hyperparameter Optimization for Deep Learning Algorithms Using Deterministic RBF Surrogates

Automatically searching for optimal hyperparameter configurations is of crucial importance for applying deep learning algorithms in practice. Recently, Bayesian optimization has been proposed for optimizing hyperparameters of various machine learning algorithms. Those methods adopt probabilistic surrogate models like Gaussian processes to approximate and minimize the validation error function of hyperparameter values. However, probabilistic surrogates require accurate estimates of sufficient statistics (e.g., covariance) of the error distribution and thus need many function evaluations with a sizeable number of hyperparameters. This makes them inefficient for optimizing hyperparameters of deep learning algorithms, which are highly expensive to evaluate. In this work, we propose a new deterministic and efficient hyperparameter optimization method that employs radial basis functions as error surrogates. The proposed mixed integer algorithm, called HORD, searches the surrogate for the most promising hyperparameter values through dynamic coordinate search and requires many fewer function evaluations. HORD does well in low dimensions but it is exceptionally better in higher dimensions. Extensive evaluations on MNIST and CIFAR-10 for four deep neural networks demonstrate HORD significantly outperforms the well-established Bayesian optimization methods such as GP, SMAC, and TPE. For instance, on average, HORD is more than 6 times faster than GP-EI in obtaining the best configuration of 19 hyperparameters.

Carrying out entrance control of “Informatics” students’ knowledge with use of specialized computer system

Nowadays the creation and the use of computer testing systems is one of perspective directions in the field of control of knowledge. Computer testing makes it possible to: • minimize time expenses at poll and processing of results and their transfer to a mark; • automatically form databases about the progress, to carry out mathematical processing of results; • estimate the level of knowledge of students, reducing subjectivity of exposure of marks; • avoid «copying off» of answers by the casual choice of tasks from the database. Computer systems of testing have become convenient tools for the teacher, providing quite a high quality of examination by reducing time expenditure on examination of students, in comparison with traditional control methods, and providing him with analytical information that allows him to organize learning process more rationally. The important place in development of such systems is taken by a problem of optimum test formation which consists of searching for set of questions (tasks) allowing to check all elements of knowledge of the considered discipline and is the best from the point of view of certain criteria. Selection of the mentioned tasks usually is carried out empirically based on knowledge and experience of the teacher. At the same time, mathematical models and methods of optimization can be used for the solution of this task. The approach to creating systems of computer testing of students’ knowledge based on the use of tasks and methods of discrete optimization is offered in Omsk branch of Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. The testing EMM-TEST system which is used at Economics Department of Omsk State University for discipline «Economic-Mathematical Methods» is created. The author developed the testing system (a software system) for Economics and Law Departments on disciplines «Economic Informatics» and «Legal Informatics» respectively. The article is devoted to the questions of using the computer control system of students’ knowledge at the beginning of studying of the «Informatics» course. The models of integer linear programming and algorithms of discrete optimization for solving problems of forming optimum tests, offered earlier by the author are used in the system. The results of approbation of system, which have shown efficiency of its use in educational process, are given including those on the stage of entrance control. Further development of the received mathematical models, their application to other sections of the specified course and continuation of a computing experiment is supposed. The introduction of the developed software at the humanitarian departments stated above is planned.

Impact of lubricant traction coefficient on cage’s dynamic characteristics in high-speed angular contact ball bearing

In this paper, the formulas of elasto-hydrodynamic traction coefficients of three Chinese aviation lubricating oils, 4109, 4106 and 4050, were obtained by a great number of elasto-hydrodynamic traction tests. The nonlinear dynamics differential equations of high-speed angular contact ball bearing were built on the basis of dynamic theory of rolling bearings and solved by Gear Stiff (GSTIFF) integer algorithm with variable step. The impact of lubricant traction coefficient on cage’s dynamic characteristics in high-speed angular contact ball bearing was investigated, and Poincaré map was used to analyze the impact of three types of aviation lubricating oils on the dynamic response of cage’s mass center. And then, the period of dynamic response of cage’s mass center and the slip ratio of cage were used to assess the stability of cage under various working conditions. The results of this paper provide the theoretical basis for the selection and application of aviation lubricating oil.

Study on the impact of roller convexity excursion of high-speed cylindrical roller bearing on roller's dynamic characteristics

This paper presents dynamics differential equations of high-speed cylindrical roller bearing considering the impact of roller convexity excursion on mass center of roller, which are solved by GSTIFF (Gear Stiff) integer algorithm with variable step. The impact of roller convexity excursion on roller's dynamic characteristics is investigated theoretically, and the accuracy of dynamic model of high-speed cylindrical roller bearing and the computation results are demonstrated via Caba3D. The findings are as following: 1) Roller convexity excursion has tiny impact on tilting angle of roller but great impact on skewing angle, and the bigger convexity excursion the bigger skewing angle; 2) For PCRC(partially crowned roller with the center of crowned profile locating at the center line of roller), PCRE(partially crowned roller with the center of crowned profile locating at the ends of roller), fully crowned roller and logarithmic crowned roller, the amounts of roller convexity excursion are less than 0.6mm, 0.4mm, 0.4mm and 0.4mm, respectively; 3) The sensitivity to roller convexity excursion for four types of rollers is different, and the sensitivity level of roller from small to large is logarithmic crowned roller, PCRE, PCRC, fully crowned roller. 4) Among four types of crowned rollers, anti-speed capability and anti-load capability of logarithmic crowned roller are the most excellent, followed by PCRE and PCRC, and fully crowned roller is the worst one in both aspects.

An algorithm for counting smooth integers

An integer number n > 0 is called y-smooth for y > 0 if any prime factor p of n satisfies p ≤ y. Let ψ(x, y) be the number of all y-smooth integers less or equal to x. In this paper we elaborate a new algorithm for approximate calculation of ψ(x, y) at large x and relatively small y < log x.

Exploiting Binary Floating-Point Representations for Constraint Propagation

Floating-point computations are quickly finding their way in the design of safety- and mission-critical systems, despite the fact that designing floating-point algorithms is significantly more difficult than designing integer algorithms. For this reason, verification and validation of floating-point computations are hot research topics. An important verification technique, especially in some industrial sectors, is testing. However, generating test data for floating-point intensive programs proved to be a challenging problem. Existing approaches usually resort to random or search-based test data generation, but without symbolic reasoning it is almost impossible to generate test inputs that execute complex paths controlled by floating-point computations. Moreover, because constraint solvers over the reals or the rationals do not natively support the handling of rounding errors, the need arises for efficient constraint solvers over floating-point domains. In this paper, we present and fully justify improved algorithms for the propagation of arithmetic IEEE 754 binary floating-point constraints. The key point of these algorithms is a generalization of an idea by B. Marre and C. Michel that exploits a property of the representation of floating-point numbers.

On the number of binary signed digit representations of a given weight

Binary signed digit representations (BSDR's) of integers have been studied since the 1950's. Their study was originally motivated by multiplication and division algorithms for integers and later by arithmetics on elliptic curves. Our paper is motivated by differential cryptanalysis of hash functions. We give an upper bound for the number of BSDR's of a given weight. Our result improves the upper bound on the number of BSDR's with minimal weight stated by Grabner and Heuberger in On the number of optimal base $2$ representations, Des. Codes Cryptogr. 40 (2006), 25--39, and introduce a new recursive upper bound for the number of BSDR's of any given weight.

Algorithm design and theoretical analysis of a novel CMM modular exponentiation algorithm for large integers

Modular exponentiation is an important operation in public-key cryptography. The Common-Multiplicand-Multiplication (CMM) modular exponentiation is an efficient exponentiation algorithm. This paper presents a novel method for speeding up the CMM modular exponentiation algorithm based on a Modified Montgomery Modular Multiplication (M4) algorithm. The M4 algorithm uses a new multi bit scan-multi bit shift technique by employing a modified encoding algorithm. In the M4 algorithm, three operations (the zero chain multiplication, the required additions and the nonzero digit multiplication) are relaxed to a multi bit shift and one binary addition in only one clock cycle. Our computational complexity analysis shows that the average number of required multiplication steps (clock cycles) is considerably reduced in comparison with other CMM modular exponentiation algorithms.

A chronological and mathematical overview of digital circle generation algorithms – introducing efficient 4- and 8-connected circles

Circles are one of the basic drawing primitives for computers and while the naive way of setting up an equation for drawing circles is simple, implementing it in an efficient way using integer arithmetic has resulted in quite a few different algorithms. We present a short chronological overview of the most important publications of such digital circle generation algorithms. Bresenham is often assumed to have invented the first all integer circle algorithm. However, there were other algorithms published before his first official publication, which did not use floating point operations. Furthermore, we present both a 4- and an 8-connected all integer algorithm. Both of them proceed without any multiplication, using just one addition per iteration to compute the decision variable, which makes them more efficient than previously published algorithms.

Fraction-Free Unit Circle Stability Tests

The paper considers five fraction-free (FF) tests to determine whether a complex or a real polynomial has all its zeros inside the unit-circle. A FF test for complex polynomial is applicable to both complex and real polynomials where the FF property means that when it is applied to Gaussian or real integer coefficients, it can be completed over the respective ring of integers, i.e., it is Gaussian integers preserving (GIP) for Gaussian integer coefficient polynomials and integer preserving (IP) for integer coefficient polynomials. Three of the FF tests are for both complex and real polynomials. Two of the FF tests are specific for real polynomials and are IP for a polynomial with integer coefficients. Two GIP tests and two corresponding IP tests are immittance-type (stem from the Bistritz test). The third GIP test, a modified test proposed by Jury, is scattering-type (stems from the Schur---Cohn test). The two real immittance-type IP tests are significantly more efficient as integer algorithms than using any of the GIP tests for a real integer polynomial. The focus of the paper is on stability conditions. However, assuming normal conditions, stability conditions are always embedded in rules for counting also zeros outside the unit-circle.

A Non-restoring Division Algorithm

Non-restoring division method originally defined by Robertson in 1958. Restoring and non-restoring division processes are the algorithms conventionally used to program division method on microprocessors to minimize the hardware cost. A new hardware algorithm is to be proposed for non-restoring division algorithm for nonnegative integers. Restoring division algorithm maximizes the hardware cost where as non-restoring division algorithm minimizes the hardware cost.

Optimal Fraction-Free Routh Tests for Complex and Real Integer Polynomials

The Routh test is the simplest and most efficient algorithm to determine whether all the zeros of a polynomial have negative real parts. However, the test involves divisions that may decrease its numerical accuracy and are a drawback in its use for various generalized applications. The paper presents fraction-free forms for this classical test that enhance it with the property that the testing of a polynomial with Gaussian or real integer coefficients can be completed over the respective ring of integers. Two types of algorithms are considered one, named the G-sequence, which is most efficient (as an integer algorithm) for Gaussian integers, and another, named the R-sequence, which is most efficient for real integers. The G-sequence can be used also for the real case, but the R-sequence is by far more efficient for real integer polynomials. The count of zeros with positive real parts for normal polynomials is also presented for each algorithm.

Differential Evolution Nonlinear Integer Algorithm based on Chaotic Local Search Strategy

Differential Evolution Nonlinear Integer Algorithm based on Chaotic Local Search Strategy

Spending Scarce Funds More Efficiently—Including the Pattern of Interdependence in Cost-Benefit Analysis

The financial crisis has substantially aggravated shortages of public funds all over the world, particularly in many developed countries. After the expiration of investments made within the framework of public economic recovery plans, the availability of public funds for infrastructure investments will be subject to severe constraints. These conditions require that scarce available funds be spent in an efficient and effective manner. Evaluation methods to select and prioritize infrastructure projects of an investment package often presuppose that infrastructure projects are independent of one another. However, it is inherent in the nature of transport networks that links are interdependent and that changes in one link will affect other parts of the network. Some infrastructure projects might be characterized by substitutive interdependence, while others might interact in a synergistic context. Thus, the cost and benefits of a project are strongly dependent on the realization of other projects. This paper confronts this issue by comprehensively addressing the nature of the problem to derive an optimal set of investment projects. The approach utilizes a dynamic mixed integer algorithm, and the complexity of the combinatorial problem is reduced by elaborating on a heuristic method. Under application of these requisites, this paper develops a framework for integrating the interdependence between infrastructure projects in cost-benefit analysis. Results obtained from an example application related to investments made in the Trans-European Network (TEN-T) emphasize the importance and relevance of considering interdependence for evaluating the investment potential of infrastructure projects.

Successive Approximation Algorithm for Binary Divisionof Unsigned Integers

Successive Approximation Algorithm for Binary Divisionof Unsigned Integers

Approximate shortest paths in weighted graphs

We present an approximation algorithm for the all pairs shortest paths (APSP) problem in weighed graphs. Our algorithm solves the APSP problem for weighted directed graphs, with real (positive or negative) weights, up to an additive error of ϵ. For any pair of vertices u , v , the algorithm finds a path whose length is at most δ ( u , v ) + ϵ . The algorithm is randomized and runs in O ˜ ( n ( ω + 3 ) / 2 ) < O ( n 2.688 ) time, where n is the number of vertices and ω is the matrix multiplication exponent. The absolute weights are allowed to mildly depend upon n, being at most n o ( 1 ) (we note that even if the weights are constants, δ ( u , v ) can be linear in n, while the error requirement ϵ is a small constant independent of n). Clearly, ϵ-additive approximations generalize exact algorithms for integer weighted instances. Hence, if ω = 2 + o ( 1 ) , this algorithm is as fast as any algorithm known for integer APSP in directed graphs, and is more general.

Fast Algoritms for Resource Allocation in Radio-over-Fiber Access Networks

This article presents algorithms for the optimization of radio resources that can be implemented in dynamic mobile networks based on RoF. The algorithms are based on linear relaxation techniques for integer linear programming (ILP) problems. The formulation models a multi-tier structure of antennas with increasing radius. Considering this antenna structure, the optimizer performs dynamic cell merging and cell splitting to save resources and improve network availability. Both the integer and the relaxation formulations produced similar results for all the experiments, but the time of processing required by the relaxed algorithms was much shorter than those required by the integer algorithms for large instance of the problem, which highlights the advantages of relaxation techniques under time constraints.