Researches on the integral calculus. Part I

1837 ◽  
Vol 3 ◽  
pp. 380-381
Keyword(s):  
Differential Equations ◽  
Finite Number ◽  
Great Improvement ◽  
Integral Calculus ◽  
Exact Integration ◽  
Fourth Degree ◽  
Euler’S Theorem ◽  
Straight Line ◽  
The Difference ◽  
Historical Sketch

The author premises a brief historical sketch of the progress of discovery in this branch of analytical science. He observes that the first inventors of the integral calculus obtained the exact integration of a certain number of formulæ only ; resolving them into a finite number of terms, involving algebraic, circular, or logarithmic quantities, and developing the integrals of others into in finite series. The first great improvement in this department of analysis was made by Fagnani, about the year 1714, by the discovery of a method of rectifying the differences of two arcs of a given biquadratic parabola, whose equation is x 4 = y . He published, subsequently, a variety of important theorems respecting the division into equal parts of the arcs of the lemniscate, and respecting the ellipse and hyperbola ; in both of which he showed how two arcs may be determined, of which the difference is a known straight line. Further discoveries in the algebraic integration of differential equations of the fourth degree were made by Euler ; and the inquiry was greatly extended by Legendre, who examined and classified the properties of elliptic integrals, and presented the results of his researches in a luminous and well-arranged theory. In the year 1828, Mr. Abel, of Christiana, in Norway, published a remarkable theorem, which gives the sum of a series of integrals of a more general form, and extending to higher powers than those in Euler’s theorem ; and furnishes a multitude of solutions for each particular case of the problem. Legendre, though at an advanced age, devoted a large portion of time to the verification of this important theorem, the truth of which he established upon the basis of the most rigorous demonstration. M. Poisson has, in a recent memoir, considered various forms of integrals which are not comprehended in Abel’s formula.

XIV. Researches in the integral calculus.—Part I

1836 ◽  
Vol 126 ◽  
pp. 177-215
Keyword(s):  
Infinite Series ◽  
Point Of View ◽  
Integral Calculus ◽  
Exact Relation ◽  
Exact Integration ◽  
Italian Journal ◽  
Scientific World ◽  
Analytical Point ◽  
The Subject ◽  
Historical Sketch

1. Brief historical sketch of the subject. The first inventors of the integral calculus observed that only a certain number of formulæ were susceptible of exact integration, or could be reduced to a finite number of terms involving algebraic, circular, or logarithmic quantities. When this result could not be attained, they were accustomed to develop the integral in an infinite series. But this method, although useful when numerical values are to be computed, is entirely inadequate, in an analytical point of view, to supply the place of the exact integral; for the progress of analysis has shown many instances of exact relation between different integrals which cannot by any means be inferred from the infinite series in which they are developed. The first great improvement beyond this was made by Fagnani about the year 1714. This most acute and ingenious mathematician proposed the following question to the scientific world in an Italian journal: “Given a biquadratic parabola whose equation is x 4 = y , and an arc of it, to find another arc, so that their difference may be rectifiable.”

Automatic Tread Gaging Machine

1979 ◽  
Vol 7 (1) ◽  
pp. 31-39
Author(s):  
G. S. Ludwig ◽  
F. C. Brenner
Keyword(s):  
Experimental Tests ◽  
Automatic Machine ◽  
Wear Rates ◽  
Handling Device ◽  
Straight Line ◽  
Component Systems ◽  
Before And After ◽  
Good Repeatability ◽  
The Difference ◽  
Straight Lines

Abstract An automatic tread gaging machine has been developed. It consists of three component systems: (1) a laser gaging head, (2) a tire handling device, and (3) a computer that controls the movement of the tire handling machine, processes the data, and computes the least-squares straight line from which a wear rate may be estimated. Experimental tests show that the machine has good repeatability. In comparisons with measurements obtained by a hand gage, the automatic machine gives smaller average groove depths. The difference before and after a period of wear for both methods of measurement are the same. Wear rates estimated from the slopes of straight lines fitted to both sets of data are not significantly different.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

scholarly journals Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

2021 ◽  
Vol 58 (2) ◽  
pp. 469-483
Author(s):  
Jesper Møller ◽  
Eliza O’Reilly
Keyword(s):  
Finite Number ◽  
Point Process ◽  
Point Processes ◽  
Parametric Models ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
The Difference ◽  
Weaker Conditions ◽  
Palm Distributions

AbstractFor a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

scholarly journals XXIII.—On the Disruptive Discharge of Electricity: An Experimental Thesis for the Degree of Doctor of Science, Department A

1878 ◽  
Vol 28 (2) ◽  
pp. 633-671 ◽  
Author(s):  
Alexander Macfarlane
Keyword(s):  
Copper Wire ◽  
Late Summer ◽  
The Other ◽  
Straight Line ◽  
Summer Session ◽  
The Difference ◽  
A Chain ◽  
The One ◽  
Metallic Parts ◽  
The University

The experiments to which I shall refer were carried out in the physical laboratory of the University during the late summer session. I was ably assisted in conducting the experiments by three students of the laboratory,—Messrs H. A. Salvesen, G. M. Connor, and D. E. Stewart. The method which was used of measuring the difference of potential required to produce a disruptive discharge of electricity under given conditions, is that described in a paper communicated to the Royal Society of Edinburgh in 1876 in the names of Mr J. A. Paton, M. A., and myself, and was suggested to me by Professor Tait as a means of attacking the experimental problems mentioned below.The above sketch which I took of the apparatus in situ may facilitate tha description of the method. The receiver of an air-pump, having a rod capable of being moved air-tight up and down through the neck, was attached to one of the conductors of a Holtz machine in such a manner that the conductor of the machine and the rod formed one conducting system. Projecting from the bottom of the receiver was a short metallic rod, forming one conductor with the metallic parts of the air-pump, and by means of a chain with the uninsulated conductor of the Holtz machine. Brass balls and discs of various sizes were made to order, capable of being screwed on to the ends of the rods. On the table, and at a distance of about six feet from the receiver, was a stand supporting two insulated brass balls, the one fixed, the other having one degree of freedom, viz., of moving in a straight line in the plane of the table. The fixed insulated ball A was made one conductor with the insulated conductor of the Holtz and the rod of the receiver, by means of a copper wire insulated with gutta percha, having one end stuck firmly into a hole in the collar of the receiver, and having the other fitted in between the glass stem and the hollow in the ball, by which it fitted on to the stem tightly. A thin wire similarly fitted in between the ball B and its insulating stem connected the ball with the insulated half ring of a divided ring reflecting electrometer.

A reduction theory of second order meromorphic differential equations, I

1987 ◽  
Vol 101 (2) ◽  
pp. 323-342
Author(s):  
W. B. Jurkat ◽  
H. J. Zwiesler
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Standard Form ◽  
Reduction Theory ◽  
Fundamental Solution Matrix ◽  
Equivalent Equation ◽  
The Difference ◽  
Solution Matrix ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

In this article we investigate the meromorphic differential equation X′(z) = A(z) X(z), often abbreviated by [A], where A(z) is a matrix (all matrices we consider have dimensions 2 × 2) meromorphic at infinity, i.e. holomorphic in a punctured neighbourhood of infinity with at most a pole there. Moreover, X(z) denotes a fundamental solution matrix. Given a matrix T(z) which together with its inverse is meromorphic at infinity (a meromorphic transformation), then the function Y(z) = T−1(z) X(z) solves the differential equation [B] with B = T−1AT − T−1T [1,5]. This introduces an equivalence relation among meromorphic differential equations and leads to the question of finding a simple representative for each equivalence class, which, for example, is of importance for further function-theoretic examinations of the solutions. The first major achievement in this direction is marked by Birkhoff's reduction which shows that it is always possible to obtain an equivalent equation [B] where B(z) is holomorphic in ℂ ¬ {0} (throughout this article A ¬ B denotes the difference of these sets) with at most a singularity of the first kind at 0 [1, 2, 5, 6]. We call this the standard form. The question of how many further simplifications can be made will be answered in the framework of our reduction theory. For this purpose we introduce the notion of a normalized standard equation [A] (NSE) which is defined by the following conditions:(i) , where r ∈ ℕ and Ak are constant matrices, (notation: )(ii) A(z) has trace tr for some c ∈ ℂ,(iii) Ar−1 has different eigenvalues,(iv) the eigenvalues of A−1 are either incongruent modulo 1 or equal,(v) if A−1 = μI, then Ar−1 is diagonal,(vi) Ar−1 and A−1 are triangular in opposite ways,(vii) a12(z) is monic (leading coefficient equals 1) unless a12 ≡ 0; furthermore a21(z) is monic in case that a12 ≡ 0 but a21 ≢ 0.

scholarly journals The Level of Specific Endurance and Technical and Tactical Efficiency of Young Football Players of Different Level of Competition / Nivo specifične izdržljivosti i tehničko-taktičke efikasnosti mladih fudbalera različitog stepena takmičenja

2016 ◽  
Vol 10 (2) ◽  
Author(s):  
Toplica Stojanović ◽  
Slobodan Goranović ◽  
Aleksandar Šakanović ◽  
Darko Stojanović
Keyword(s):  
The Other ◽  
Measuring Instruments ◽  
Football Players ◽  
Five Factors ◽  
Straight Line ◽  
Specific Performance ◽  
The Difference

In order to determine at which level is the specific performance and technical and tactical efficiency of young players of different level of competition, and whether the level of competition can be an indicator of level differences of these abilities, a research was conducted on the sample of young football players aged 14 to 16 from the eight clubs, half of them competing in the higher and the other half in the lower level of competition. A sample of measuring instruments consisted of 13 tests for evaluation of five factors of specific endurance: starting endurance, stamina in maintaining the shallow formation, endurance during fast dribbling, ball pressing endurance, and evaluation of technical and tactical efficiency of football players. The results of the research showed that the young players of higher level of competition had significantly greater technical and tactical efficiency, as well as specific performance in tests which included curvilinear movement and dribbling, as well as control and passing the ball in motion, but the difference is not recorded with straight-line movements and sprints.

Regularities of changes in microhardness in the volume of EDT-69N binder cured at different temperatures

2020 ◽  
Vol 4 ◽  
pp. 65-71
Author(s):  
E.A. Veshkin ◽  
◽  
V.I. Postnov ◽  
V.V. Semenychev ◽  
E.V. Krasheninnikova ◽  
...  
Keyword(s):  
Cross Section ◽  
High Sensitivity ◽  
Sample Volume ◽  
Curing Temperature ◽  
Molding Temperature ◽  
Parabolic Law ◽  
Dimensionless Coefficient ◽  
Straight Line ◽  
Different Temperatures ◽  
The Difference

The change in the microhardness over the thickness of samples made of EDT-69N binder cured in vacuum and at atmospheric pressure at temperatures from 130 to 170°C was investigated. It was found that the change in microhardness along the thickness of the samples occurs according to the parabolic law, with the maximum values being achieved in the middle of the sample cross-section along the thickness. With an increase in the molding temperature, the microhardness in the middle section of the sample increases from 222 MPa at a molding temperature of 130°C to 410 MPa during molding at 170°C. At the critical molding temperature (170°C), the microhardness in all zones of the specimen cross section (subsurface, semi-average, and core) levels off, while the parabolic dependence degenerates into a straight line. It is shown that the method of scratching (sclerometry) demonstrated a sufficiently high sensitivity to the state of samples cured at different temperatures. With an increase in the molding temperature, the width of the sclerometric grooves decreases. At a critical molding temperature of 170°C, the groove width is stabilized and becomes constant throughout the sample thickness. To characterize the difference in the values of the microhardness of the cured binder in the sample volume, it is proposed to use a dimensionless “coefficient of volume anisotropy,” which can take a positive, negative or zero value. With an increase in the curing temperature of the binder and, accordingly, with an increase in the microhardness of the sample, the coefficient of volume anisotropy decreases, and when the samples are molded at the critical temperature, it turns to zero, which indicates the absence of anisotropy.

A Matrix-Vector Operation-Based Numerical Solution Method for Linear m-th Order Ordinary Differential Equations: Application to Engineering Problems

2013 ◽  
Vol 5 (03) ◽  
pp. 269-308 ◽  
Author(s):  
M. Aminbaghai ◽  
M. Dorn ◽  
J. Eberhardsteiner ◽  
B. Pichler
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Solution Method ◽  
Variable Coefficients ◽  
Exact Integration ◽  
Robust Solution ◽  
Vector Operation ◽  
Engineering Sciences ◽  
Engineering Problems ◽  
Matrix Vector

AbstractMany problems in engineering sciences can be described by linear, inhomogeneous,m-th order ordinary differential equations (ODEs) with variable coefficients. For this wide class of problems, we here present a new, simple, flexible, and robust solution method, based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals. The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus. Based on cubic approximation polynomials, the presented method can be expected to perform (i) similar to the Runge-Kutta method, when applied to stiff initial value problems, and (ii) significantly better than the finite difference method, when applied to boundary value problems. Therefore, we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum, steady-state heat transfer through a cooling web, and the structural analysis of a slender tower based on second-order beam theory. Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.

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