scholarly journals Inequalities which include q-integrals

2006 ◽  
Vol 133 (31) ◽  
pp. 137-146 ◽  
Author(s):  
M.S. Stankovic ◽  
P.M. Rajkovic ◽  
Sladjana Marinkovic
Keyword(s):  
Subject Classification ◽  
Integral Calculus ◽  
The Difference ◽  
Infinite Sums

The main problem in analyzing inequalities which include q-integrals is the fact that q-integral of a function over an interval [a; b] (0 < a < b) is defined by the difference of two infinite sums. Thus defined q- integral properties must include the points outside of interval of integration. In this paper, we will signify to some directions for solving this problem and derive some inequalities which are analogues to well-known ones in standard integral calculus. AMS Mathematics Subject Classification (2000): 33D60, 26D15.

scholarly journals XX. On the theory of the elliptic transcendents

1831 ◽  
Vol 121 ◽  
pp. 349-377 ◽  
Keyword(s):  
Algebraic Equation ◽  
Elliptic Function ◽  
Elliptic Integral ◽  
Elliptic Functions ◽  
The Other ◽  
Algebraic Equations ◽  
Integral Calculus ◽  
Circular Arc ◽  
Regular Theory ◽  
The Difference

The branch of the integral calculus which treats of elliptic transcendents originated in the researches of Fagnani, an Italian geometer of eminence. He discovered that two arcs of the periphery of a given ellipse may be determined in many ways, so that their difference shall be equal to an assignable straight line; and he proved that any arc of the lemniscata, like that of a circle, may be multiplied any number of times, or may be subdivided into any number of equal parts, by finite algebraic equations. These are particular results; and it was the discoveries of Euler that enabled geometers to advance to the investigation of the general properties of the elliptic functions. An integral in finite terms deduced by that geometer from an equation between the differentials of two similar transcendent quantities not separately integrable, led immediately to an algebraic equation between the amplitudes of three elliptic functions, of which one is the sum, or the difference, of the other two. This sort of integrals, therefore, could now be added or subtracted in a manner analogous to circular arcs, or logarithms; the amplitude of the sum, or of the difference, being expressed algebraically by means of the amplitudes of the quantities added or subtracted. What Fagnani had accomplished with respect to the arcs of the lemniscata, which are expressed by a particular elliptic integral, Euler extended to all transcendents of the same class. To multiply a function of this kind, or to subdivide it into equal parts, was reduced to solving an algebraic equation. In general, all the properties of the elliptic transcendents, in which the modulus remains unchanged, are deducible from the discoveries of Euler. Landen enlarged our knowledge of this kind of functions, and made a useful addition to analysis, by showing that the arcs of the hyperbola may be reduced, by a proper transformation, to those of the ellipse. Every part of analysis is indebted to Lagrange, who enriched this particular branch with a general method for changing an elliptic function into another having a different modulus, a process which greatly facilitates the numerical calculation of this class of integrals. An elliptic function lies between an arc of the circle on one hand, and a logarithm on the other, approaching indefinitely to the first when the modulus is diminished to zero, and to the second when the modulus is augmented to unit, its other limit. By repeatedly applying the transformation of Lagrange, we may compute either a scale of decreasing moduli reducing the integral to a circular arc, or a scale of increasing moduli bringing it continually nearer to a logarithm. The approximation is very elegant and simple, and attains the end proposed with great rapidity. The discoveries that have been mentioned occurred in the general cultivation of analysis; but Legendre has bestowed much of his attention and study upon this particular branch of the integral calculus. He distributed the elliptic functions in distinct classes, and reduced them to a regular theory. In a Mémoire sur les Transcendantes Elliptiques, published in 1793, and in his Exercices de Calcul Intégral, which appeared in 1817 he has developed many of their properties entirely new; investigated the easiest methods of approximating to their values; computed numerical tables to facilitate their application; and exemplified their use in some interesting problems of geometry and mechanics. In a publication so late as 1825, the author, returning to the same subject, has rendered his theory still more perfect, and made many additions to it which further researches had suggested. In particular we find a new method of making an elliptic function approach as near as we please to a circular arc, or to a logarithm, by a scale of reduction very different from that of which Lagrange is the author, the only one before known. This step in advance would unavoidably have conducted to a more extensive theory of this kind of integrals, which, nearly about the same time, was being discovered by the researches of other geometers.

scholarly journals PENGEMBANGAN BAHAN AJAR KALKULUS INTEGRAL DENGAN PENDEKATAN APTITUDE TREATMENT INTERACTION (ATI) BERBANTU MACROMEDIA FLASH

Eksponen ◽  
2020 ◽  
Vol 10 (1) ◽  
pp. 31-39
Author(s):  
Venty Meilasari
Keyword(s):  
Learning Outcomes ◽  
Student Learning Outcomes ◽  
Teaching Material ◽  
Development Research ◽  
Integral Calculus ◽  
Optimal Learning ◽  
Treatment Interaction ◽  
Aptitude Treatment Interaction ◽  
The Difference ◽  
Calculus Teaching

ABSTRACT The difficulty of students in understanding the material in the integral calculus print book is one factor that makes student learning outcomes low. The difference in students' abilities is one of the considerations in developing process. This study aims to develop integral calculus teaching material using the aptitude treatment interaction (ATI) approach assisted by Macromedia flash. This research belongs to the development research. The instruments used in this study were questionnaires and tests. Indicators of the results of this study are teaching materials declared worthy of use and students who get optimal learning outcomes> 75%. The stages in the study consisted of ten stages that began with discovering potentials and problems to mass production. The results of the study showed that the integral calculus teaching material made was feasible to use. As many as 79% of students achieve optimal learning outcomes.

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.

scholarly journals Perturbation analysis of a nonlinear equation arising in the Schaefer-Schwartz model of interest rates

2018 ◽  
Vol 68 (3) ◽  
pp. 617-624
Author(s):  
Beáta Stehlíková
Keyword(s):  
Nonlinear Equation ◽  
Interest Rates ◽  
Closed Form ◽  
Special Functions ◽  
Analytical Formula ◽  
Elementary Functions ◽  
Bond Prices ◽  
The Difference ◽  
The Interest Rate ◽  
Infinite Sums

Abstract We deal with the interest rate model proposed by Schaefer and Schwartz, which models the long rate and the spread, defined as the difference between the short and the long rates. The approximate analytical formula for the bond prices suggested by the authors requires a computation of a certain constant, defined via a nonlinear equation and an integral of a solution to a system of ordinary differential equations. A quantity entering the nonlinear equation is expressed in a closed form, but it contains infinite sums and evaluations of special functions. In this paper we use perturbation methods to compute the constant of interest as an asymptotic serie with coefficients given in closed form and expressed using elementary functions. A quick computation of the bond prices, which our approach allows, is essential for example in calibration of the model by means of fitting the observed yields, where the theoretical bond prices need to be recalculated for every observed date and maturity, as well as every combination of parameters considered. The first step of our derivation is identification of a small parameter in the problem, since it is not immediately clear. We verify our choice by numerical experiments using the values of parameters from the literature.

The Origin of the Moon

1962 ◽  
Vol 14 ◽  
pp. 149-155 ◽  
Author(s):  
E. L. Ruskol
Keyword(s):  
Chemical Composition ◽  
Solar System ◽  
The Moon ◽  
The Earth ◽  
The Difference ◽  
Origin Of The Moon

The difference between average densities of the Moon and Earth was interpreted in the preceding report by Professor H. Urey as indicating a difference in their chemical composition. Therefore, Urey assumes the Moon's formation to have taken place far away from the Earth, under conditions differing substantially from the conditions of Earth's formation. In such a case, the Earth should have captured the Moon. As is admitted by Professor Urey himself, such a capture is a very improbable event. In addition, an assumption that the “lunar” dimensions were representative of protoplanetary bodies in the entire solar system encounters great difficulties.

Influence of Cell Wall Composition on the Fossilisation of Bacteria and the Implications for the Search for Early Life Forms

1997 ◽  
Vol 161 ◽  
pp. 491-504 ◽  
Author(s):  
Frances Westall
Keyword(s):  
Cell Wall ◽  
Life Forms ◽  
Cation Binding ◽  
Positive Bacterium ◽  
Gram Positive ◽  
Gram Positive Bacteria ◽  
Transmission Electron ◽  
Hydrothermal Sediments ◽  
Identification Of Bacteria ◽  
The Difference

AbstractThe oldest cell-like structures on Earth are preserved in silicified lagoonal, shallow sea or hydrothermal sediments, such as some Archean formations in Western Australia and South Africa. Previous studies concentrated on the search for organic fossils in Archean rocks. Observations of silicified bacteria (as silica minerals) are scarce for both the Precambrian and the Phanerozoic, but reports of mineral bacteria finds, in general, are increasing. The problems associated with the identification of authentic fossil bacteria and, if possible, closer identification of bacteria type can, in part, be overcome by experimental fossilisation studies. These have shown that not all bacteria fossilise in the same way and, indeed, some seem to be very resistent to fossilisation. This paper deals with a transmission electron microscope investigation of the silicification of four species of bacteria commonly found in the environment. The Gram positiveBacillus laterosporusand its spore produced a robust, durable crust upon silicification, whereas the Gram negativePseudomonas fluorescens, Ps. vesicularis, andPs. acidovoranspresented delicately preserved walls. The greater amount of peptidoglycan, containing abundant metal cation binding sites, in the cell wall of the Gram positive bacterium, probably accounts for the difference in the mode of fossilisation. The Gram positive bacteria are, therefore, probably most likely to be preserved in the terrestrial and extraterrestrial rock record.

Structuring, Motions and Shapes of Corona Emission Line Profiles

1994 ◽  
Vol 144 ◽  
pp. 421-426
Author(s):  
N. F. Tyagun
Keyword(s):  
Emission Line ◽  
Filling Factor ◽  
Direct Relationship ◽  
Coronal Plasma ◽  
Line Of Sight ◽  
Line Profiles ◽  
The Difference ◽  
Yellow Line ◽  
Red Line

AbstractThe interrelationship of half-widths and intensities for the red, green and yellow lines is considered. This is a direct relationship for the green and yellow line and an inverse one for the red line. The difference in the relationships of half-widths and intensities for different lines appears to be due to substantially dissimilar structuring and to a set of line-of-sight motions in ”hot“ and ”cold“ corona regions.When diagnosing the coronal plasma, one cannot neglect the filling factor - each line has such a factor of its own.

Difference Fourier Analysis of Glucose Embedded and Frozen Hydrated Purple Membrane

1982 ◽  
Vol 40 ◽  
pp. 74-75
Author(s):  
Jules S. Jaffe ◽  
Robert M. Glaeser
Keyword(s):  
Purple Membrane ◽  
Data Set ◽  
High Resolution Data ◽  
X Ray ◽  
X Ray Crystallography ◽  
Fourier Techniques ◽  
Versus Protein ◽  
The Difference ◽  
Difference Fourier ◽  
Ideal Method

Although difference Fourier techniques are standard in X-ray crystallography it has only been very recently that electron crystallographers have been able to take advantage of this method. We have combined a high resolution data set for frozen glucose embedded Purple Membrane (PM) with a data set collected from PM prepared in the frozen hydrated state in order to visualize any differences in structure due to the different methods of preparation. The increased contrast between protein-ice versus protein-glucose may prove to be an advantage of the frozen hydrated technique for visualizing those parts of bacteriorhodopsin that are embedded in glucose. In addition, surface groups of the protein may be disordered in glucose and ordered in the frozen state. The sensitivity of the difference Fourier technique to small changes in structure provides an ideal method for testing this hypothesis.

Destruction of Actin Filaments by Osmium Tetroxide

1977 ◽  
Vol 35 ◽  
pp. 466-467
Author(s):  
P. Maupin-Szamier ◽  
T. D. Pollard
Keyword(s):  
Actin Filaments ◽  
Time Course ◽  
Osmium Tetroxide ◽  
Rabbit Muscle ◽  
Muscle Actin ◽  
Sodium Phosphate Buffer ◽  
Transmission Electron ◽  
The Difference ◽  
Rates Of Decay ◽  
Short Time

We have studied the destruction of rabbit muscle actin filaments by osmium tetroxide (OSO4) to develop methods which will preserve the structure of actin filaments during preparation for transmission electron microscopy.Negatively stained F-actin, which appears as smooth, gently curved filaments in control samples (Fig. 1a), acquire an angular, distorted profile and break into progressively shorter pieces after exposure to OSO4 (Fig. 1b,c). We followed the time course of the reaction with viscometry since it is a simple, quantitative method to assess filament integrity. The difference in rates of decay in viscosity of polymerized actin solutions after the addition of four concentrations of OSO4 is illustrated in Fig. 2. Viscometry indicated that the rate of actin filament destruction is also dependent upon temperature, buffer type, buffer concentration, and pH, and requires the continued presence of OSO4. The conditions most favorable to filament preservation are fixation in a low concentration of OSO4 for a short time at 0°C in 100mM sodium phosphate buffer, pH 6.0.

Antiphase Boundaries in Ordered Ni3FE Crystals

1969 ◽  
Vol 27 ◽  
pp. 62-63
Author(s):  
Y. H. Liu
Keyword(s):  
Domain Size ◽  
Antiphase Domain ◽  
X Ray Diffraction ◽  
X Ray ◽  
Hardening Mechanism ◽  
Superlattice Structure ◽  
Disordered Phase ◽  
Domain Boundaries ◽  
Atomic Scattering ◽  
The Difference

Ordered Ni3Fe crystals possess a LI2 type superlattice similar to the Cu3Au structure. The difference in slip behavior of the superlattice as compared with that of a disordered phase has been well established. Cottrell first postulated that the increase in resistance for slip in the superlattice structure is attributed to the presence of antiphase domain boundaries. Following Cottrell's domain hardening mechanism, numerous workers have proposed other refined models also involving the presence of domain boundaries. Using the anomalous X-ray diffraction technique, Davies and Stoloff have shown that the hardness of the Ni3Fe superlattice varies with the domain size. So far, no direct observation of antiphase domain boundaries in Ni3Fe has been reported. Because the atomic scattering factors of the elements in NijFe are so close, the superlattice reflections are not easily detected. Furthermore, the domain configurations in NioFe are thought to be independent of the crystallographic orientations.

Sign in / Sign up

Export Citation Format

Share Document