scholarly journals Note on Dr Muir's Paper on a Peculiar Set of Linear Equations

1902 ◽  
Vol 23 ◽  
pp. 261-263
Author(s):  
Charles Tweedie
Keyword(s):  
Symmetric Functions ◽  
Elementary Theory ◽  
Linear Equations ◽  
Simple Method

In Dr Muir's Paper on a Peculiar Set of Linear Equations (communicated December 3, 1900) there occur two Determinants of the nth order, the expansions of which are given by Dr Muir, As the paper in question has so much to do with Symmetric Functions, the following simple method of obtaining their expansions may not prove uninteresting, based, as it is, upon the elementary theory of Symmetric Functions and the so-called Principle of Indeterminate Coefficients.

scholarly journals A NOVEL SIMPLE-BASED PRESSURE-ENTHALPY COUPLING SCHEME FOR ENGINE FLOW PROBLEMS

2012 ◽  
Vol 17 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Maximilian Emans ◽  
Zoran Žunič ◽  
Branislav Basara ◽  
Sergey Frolov
Keyword(s):  
Linear Equations ◽  
Computing Time ◽  
Simple Algorithm ◽  
Variable Density ◽  
Coupling Scheme ◽  
The Novel ◽  
Simple Method ◽  
Flow Problems ◽  
Novel Method ◽  
Coupling Terms

A novel method in CFD derived from the SIMPLE algorithm is presented. Instead of solving the linear equations for each variable and the pressurecorrection equation separately in a so-called segregated manner, it relies on the solution of a linear system that comprises the discretisation of enthalpy and pressurecorrection equation which are linked through physical coupling terms. These coupling terms reflect a more accurate approximation of the density update with respect to thermodynamics (compared to standard SIMPLE method). We show that the novel method is a reasonable extension of existing CFD techniques for variable density flows based on SIMPLE. The novel method leads to a reduction of the number of iterations of SIMPLE which translates in many – but not in all – cases to a reduction in computing time. We will therefore demonstrate practical advantages and restrictions in terms of computational efficiency for industrial CFD applications in the field of piston engine simulations.

Equally Inclined Spheres

1962 ◽  
Vol 58 (2) ◽  
pp. 420-421 ◽  
Author(s):  
J. G. Mauldon
Keyword(s):  
Symmetric Functions ◽  
Linear Equations ◽  
Quadratic Equation ◽  
Cartesian Coordinates ◽  
Elementary Symmetric Functions ◽  
Equation Solving ◽  
Image Position

If five spheres in 3-space are such that each pair is inclined at the same non-zero angle θ, then where b1, …, b5 (the ‘bends’ (1) of the spheres) are the reciprocals of their radii. To prove this result, establish a system of rectangular cartesian coordinates (x, y, z) and let the spheres have centres (xi, yi, zi) and radii , where i = 1,…, 5. Then for x5, y5, z5, r5 we have the equations which, on subtraction, yield three linear equations and one quadratic equation. Solving the three linear equations for x5, y5, z5 and substituting, we see that the required relation is algebraic (indeed quadratic) in r5 and hence in b5. Since it is also symmetric in b1,…, b5, it follows that it can be expressed as a polynomial relation in the elementary symmetric functions p1, …, p5 in b5, …, b5.

EFFICIENT INFERENCE OF HAPLOTYPES FROM GENOTYPES ON A PEDIGREE

2003 ◽  
Vol 01 (01) ◽  
pp. 41-69 ◽  
Author(s):  
JING LI ◽  
TAO JIANG
Keyword(s):  
Large Scale ◽  
Gaussian Elimination ◽  
Linear Equations ◽  
Simulated Data ◽  
Exact Algorithm ◽  
Real Data ◽  
Haplotype Reconstruction ◽  
Pedigree Data ◽  
Simple Method ◽  
Complexity Result

We study haplotype reconstruction under the Mendelian law of inheritance and the minimum recombination principle on pedigree data. We prove that the problem of finding a minimum-recombinant haplotype configuration (MRHC) is in general NP-hard. This is the first complexity result concerning the problem to our knowledge. An iterative algorithm based on blocks of consecutive resolved marker loci (called block-extension) is proposed. It is very efficient and can be used for large pedigrees with a large number of markers, especially for those data sets requiring few recombinants (or recombination events). A polynomial-time exact algorithm for haplotype reconstruction without recombinants is also presented. This algorithm first identifies all the necessary constraints based on the Mendelian law and the zero recombinant assumption, and represents them using a system of linear equations over the cyclic group Z2. By using a simple method based on Gaussian elimination, we could obtain all possible feasible haplotype configurations. A C++ implementation of the block-extension algorithm, called PedPhase, has been tested on both simulated data and real data. The results show that the program performs very well on both types of data and will be useful for large scale haplotype inference projects.

scholarly journals On a simple method of determing the homology and the cohomology of finitely presented groups

1970 ◽  
Vol 34 (2) ◽  
pp. 103-114
Author(s):  
Subrata Majumdar ◽  
Nasima Akhter
Keyword(s):  
Linear Equations ◽  
Free Resolution ◽  
Nilpotent Groups ◽  
General Situation ◽  
Group Rings ◽  
Integral Group ◽  
Group Presentation ◽  
Simple Method ◽  
Finitely Presented Groups ◽  
Finitely Presented

In this paper the authors obtained a method of constructing free resolutions of Z for finitely presented groups directly from their presentations by extending Lyndon’s 3-term partial resolution to a full-length resolution. Authors resolutions and the method of their construction are such that free generators of the modules and the boundary homomorphisms are directly and explicitly obtained by solving of linear equations over the corresponding integral group rings, and hence these are immediately applicable for computing homology and cohomology of the groups for arbitrary coefficient modules. Authors have also described a general situation where their method is valid. The method has been used for a number of classes of group including Fuchsian groups, a few Euclidean crystallographic groups, NEC groups, the fundamental groups of a few interesting manifolds, groups of isometries of the hyperbolic plane and a few nilpotent groups of class 2. Key words: Group presentation; Free resolution; Homology; Cohomology DOI: 10.3329/jbas.v34i2.6854Journal of Bangladesh Academy of Sciences, Vol. 34, No. 2, 103-114, 2010

An elementary theory of equations

1971 ◽  
Vol 18 (7) ◽  
pp. 457-462
Author(s):  
L. Clark Lay
Keyword(s):  
Middle Grades ◽  
Elementary Mathematics ◽  
Elementary Theory ◽  
Linear Equations ◽  
First Grade ◽  
Lower Grade ◽  
Curriculum Changes ◽  
Beginning Algebra ◽  
The Past ◽  
Grade Levels

Curriculum changes in elementary mathematics during the past fifteen years have featured the introduction of algebraic content at lower grade levels than formerly. For example, there has been a drop from the ninth grade to the first grade for the first mention and use of parentheses and inequalities. But some ideas from beginning algebra have been more resistant to introduction in the primary and middle grades. One such topic that has lacked an elementary introduction is truly fundamental. It is the study of the solutions, and the transformations to equivalent forms, of simple linear equations.

Evaluation of Mechanical Behavior of Taper Pipe Threads in the Tightening Process by Finite Element Analysis and Elementary Theory of Solid Mechanics

2017 ◽  
Author(s):  
Toshimichi Fukuoka ◽  
Yuki Hirai
Keyword(s):  
Finite Element ◽  
Contact Force ◽  
Solid Mechanics ◽  
Rotation Angle ◽  
Elementary Theory ◽  
Element Analysis ◽  
Simple Method ◽  
Male And Female ◽  
Sealing Performance ◽  
The Relationship

There are two types of pipe threads, i.e., parallel and tapered ones. The former is used for mechanically connecting hollow cylinder-shaped structures, and the latter is usually employed for connecting thin pipes and tubes. The primary function required for taper pipe threads is to prevent the leakage of contained fluids. In order to ensure the sealing performance, target taper pipe threads need to be tightened with proper conditions. However, it seems that a standard tightening guideline with sufficient mechanical background has not been established. In this paper, using helical thread models, the relationship between assembly torque and rotation angle of threaded pipe is studied by FEA. The relationship between rotation angle and radial contact force between male and female threads, which is regarded as an index of the sealing performance, is also evaluated in like manner. In the numerical calculations, finite element analyses are performed as elastic and elastic-plastic problems, in which nominal dimeter of threads, pipe wall thickness and coefficient of friction on the thread contact surface are changed systematically, aiming at the establishment of a practical tightening guideline. Additionally, a simple method is proposed to evaluate the contact force between male and female threads, using elementary theory of solid mechanics. It is shown that the simple method can predict the contact force with sufficient accuracy, comparing to the calculation results by FEA.

On μ-symmetric polynomials

2021 ◽  
pp. 2250233
Author(s):  
Jing Yang ◽  
Chee K. Yap
Keyword(s):  
Clustering Algorithm ◽  
Symmetric Functions ◽  
Complexity Analysis ◽  
Root Function ◽  
Linear Equations ◽  
Symmetric Polynomials ◽  
Root Functions ◽  
Multiplicity Formula ◽  
Polynomial Formula ◽  
Integer Polynomial

We study functions of the roots of an integer polynomial [Formula: see text] with [Formula: see text] distinct roots [Formula: see text] of multiplicity [Formula: see text], [Formula: see text]. Traditionally, root functions are studied via the theory of symmetric polynomials; we generalize this theory to [Formula: see text]-symmetric polynomials. We initiate the study of the vector space of [Formula: see text]-symmetric polynomials of a given degree [Formula: see text] via the concepts of [Formula: see text]-gist and [Formula: see text]-ideal. In particular, we are interested in the root function [Formula: see text]. The D-plus discriminant of [Formula: see text] is [Formula: see text]. This quantity appears in the complexity analysis of the root clustering algorithm of Becker et al. (ISSAC 2016). We conjecture that [Formula: see text] is [Formula: see text]-symmetric, which implies [Formula: see text] is rational. To explore this conjecture experimentally, we introduce algorithms for checking if a given root function is [Formula: see text]-symmetric. We design three such algorithms: one based on Gröbner bases, another based on canonical bases and reduction, and the third based on solving linear equations. Each of these algorithms has variants that depend on the choice of a basis for the [Formula: see text]-symmetric functions. We implement these algorithms (and their variants) in Maple and experiments show that the latter two algorithms are significantly faster than the first.

Determinant Theory, Symmetric Functions, and Dyadic

2014 ◽  
pp. 224-246
Author(s):  
Eberhard Knobloch
Keyword(s):  
Symmetric Functions ◽  
Linear Equations ◽  
Number System ◽  
Work Related ◽  
Inhomogeneous Systems ◽  
System P ◽  
Systems Of Linear Equations ◽  
Ars Inveniendi ◽  
Determinant Theory ◽  
Common Variable

This chapter examines Leibniz’s determinant theory and analyzes the contribution of ars characteristica, ars combinatoria, and ars inveniendi to this theory. It explains that the art of inventing suitable characters led to numerical double indices while the combinatorial art helped to represent a determinant as a sum. Moreover, the chapter discusses inhomogeneous systems of linear equations and the elimination of a common variable in the determinant theory. It also explores Leibniz’s work related to symmetric functions, dyadic, and duodecimal number system.

Finite Element Analysis of the Mechanical Behavior of Multibolted Joints Subjected to Shear Loads

2018 ◽  
Vol 140 (5) ◽  
Author(s):  
Toshimichi Fukuoka
Keyword(s):  
Mechanical Behavior ◽  
Load Distribution ◽  
Solid Mechanics ◽  
Elementary Theory ◽  
Load Capacity ◽  
Shear Load ◽  
3D Fem ◽  
Simple Method ◽  
Friction Forces ◽  
Shear Loads

When subjected to external forces, bolted joints behave in a complex manner especially in the case of the joints being clamped with multiple bolts. Friction type joints are widely used for the joints subjected to shear loads. Bearing type joints, which support the shear loads on the bolt cylindrical surface, are used less frequently, since its mechanical behavior is too complicated to accurately estimate the load capacity. In this study, mechanical behavior of the bearing type multibolted joints subjected to shear loads is analyzed by three-dimensional (3D) FEM. As a result of comprehensive calculations, it has been found that the shear load applied to bearing type joints distributes with a concave shape along the load direction, and a fair amount of the shear load is supported by friction forces as in the case of friction type joints. In addition, a simple method that calculates the shear load distribution using elementary theory of solid mechanics is proposed, which can estimate the shear load distribution with sufficient accuracy especially for the case of small friction coefficient.

Finite Element Analysis of the Mechanical Behavior of Multi-Bolted Joints Subjected to Shear Loads

2010 ◽  
Author(s):  
Toshimichi Fukuoka ◽  
Masataka Nomura ◽  
Takahiro Kamihira
Keyword(s):  
Mechanical Behavior ◽  
Load Distribution ◽  
Solid Mechanics ◽  
Elementary Theory ◽  
Load Capacity ◽  
Bolted Joints ◽  
Shear Load ◽  
Element Analysis ◽  
Simple Method ◽  
Shear Loads

When subjected to external forces, bolted joints behave in a complex manner especially in the case of the joints being clamped with multiple bolts. Friction type joints are widely used for the joints subjected to shear loads. Bearing type joints, which support the shear load on the surface of bolt body, are used less frequently, since its mechanical behavior is too complicated to accurately estimate the load capacity. In this study, mechanical behavior of multi-bolted joints subjected to shear loads is comprehensively analyzed by three-dimensional FEM. Load distribution patterns of the bearing type joint are compared to those of the friction type joint. It has been found that the shear load applied to a bearing type joint distributes with a concave shape along the load direction as clamping bolts are increased, and a fair amount of the shear load is also supported by friction. Additionally, a simple method that calculates the shear load distribution is proposed using elementary theory of solid mechanics, which can estimate the shear load distribution with sufficient accuracy for the case of small friction coefficient.

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