Some remarkable integrals derived from a simple algebraic identity

2013 ◽  
Vol 97 (539) ◽  
pp. 205-209
Author(s):  
Graham J. O. Jameson ◽  
Timothy P. Jameson
Keyword(s):  
Definite Integral ◽  
Algebraic Identity ◽  
Special Cases ◽  
Long Time ◽  
Comprehensive Survey ◽  
Image Position ◽  
The Way

The identity in question really is simple: it says, for u ≠ −1,We describe two types of definite integral that look quite formidable, but dissolve into a much simpler form by an application of (1) in a way that seems almost magical.Both types, or at least special cases of them, have been mathematical folklore for a long time. For example, case (10) below appears in [1, p. 262], published in 1922 (we are grateful to Donald Kershaw for showing us this example). However, they do not seem to figure in most books on calculus except possibly tucked away as an exercise The comprehensive survey [2] mentions the second type on p. 253, but only as a lemma on the way to an identity the authors call the ‘master formula’ We come back to this formula later, but only after describing a number of other more immediate applications.

Death and Immortality: A Study of the Heraclitus Epigram of Callimachus

Ramus ◽  
1982 ◽  
Vol 11 (1) ◽  
pp. 48-56 ◽  
Author(s):  
J. G. MacQueen
Keyword(s):  
Good Deal ◽  
The Sun ◽  
Classical Literature ◽  
English Version ◽  
Long Time ◽  
Image Position ◽  
The Way

Somebody mentioned your fate, Heraclitus, and he brought me to a tear; and I remembered how often we both made the sun sink in conversation. But you, my guest-friend from Halicarnassus, have, I suppose, been ashes for a very long time. But your nightingales are alive, on which Hades, plunderer of all things, will not lay his hand.This epigram of Callimachus is one of the best known poems in Classical literature, but it suffers more than most from the misfortune of having to live permanently in the shadow of its own translation. It may no longer be the case that every schoolboy knows ‘They told me, Heraclitus, they told me you were dead’, but it is certainly true that Cory's English version is much more widely known, and much more widely quoted, than Callimachus's Greek original. One result of this has been that a good deal of attention has often been given to comparing the two poems, but little time has been spent on examining the Callimachus as a poem in itself in an effort to see what its virtues are. One may occasionally find a few remarks on the restraint or simplicity of the Greek, as opposed to the English, or a note suggesting that Heraclitus of Halicarnassus, the poet to whom the verses are addressed, wrote a volume of verse the title of which was actuallyAēdones(‘Nightingales’) — hence the ‘nightingales’ of the second last line. Occasionally a commentator will go a little further. K. J. Mckay for instance remarks: ‘The high respect in which this epigram is held is fully justified. The way in which the thoughts spill over their barriers in the first four lines, the magic ofkatedusamen(suggestive of a communion of uncommon power), the skilful location of key thoughts (teon moron, katedusamen, aēdones), the pathos of an unknown grave and an abiding grief cannot but move us. Above all, the suggestion of unfathomable sorrow.’

The Logical Square and the Table of Oppositions

2012 ◽  
Vol 15 (1) ◽  
pp. 400-416
Author(s):  
Wolfgang Kienzler
Keyword(s):  
Reduced Form ◽  
Square Of Opposition ◽  
Logical Square ◽  
Long Time ◽  
The Square Of Opposition ◽  
The Way

The way Frege presented the Square of Opposition in a reduced form in 1879 and 1910 can be used to develop two distinct versions of the square: The traditional square that displays inferences and a “Table of Oppositions” displaying variations of negation. This Table of Oppositions can be further simplified and thus be made more symmetrical. A brief survey of versions of the square from Aristotle to the present shows how both aspects of the square have coexisted for a very long time without ever being properly distinguished.

The Beginnings of the Cult of Relics

2018 ◽  
Author(s):  
Robert Wiśniewski
Keyword(s):  
Essential Element ◽  
Fourth Century ◽  
Material Objects ◽  
Long Time ◽  
Hidden Knowledge ◽  
Physical Diseases ◽  
The Dead ◽  
The Way

Christians always admired and venerated martyrs who died for their faith, but for a long time thought that the bodies of martyrs should remain undisturbed in their graves. Initially, the Christian attitude toward the bones of the dead, whether a saint’s or not, was that of respectful distance. This book tells how, in the mid-fourth century, this attitude started to change, swiftly and dramatically. The first chapters show the rise of new beliefs. They study how, when, and why Christians began to believe in the power of relics, first, over demons, then over physical diseases and enemies; how they sought to reveal hidden knowledge at the tombs of saints and why they buried the dead close to them. An essential element of this new belief was a strong conviction that the power of relics was transferred in a physical way and so subsequent chapters study relics as material objects. The book seeks to show what the contact with relics looked like and how close it was. Did people touch, kiss, or look at the very bones, or just at reliquaries which contained them? When did the custom of dividing relics appear? Finally, the book deals with discussions and polemics concerning relics and tries to find out how strong was the opposition which this new phenomenon had to face, both within and outside Christianity on the way to relics becoming an essential element of medieval religiosity.

The Pre-Socratic World-Picture

1952 ◽  
Vol 45 (2) ◽  
pp. 87-104 ◽  
Author(s):  
W. K. C. Guthrie
Keyword(s):  
Religious Thought ◽  
World Picture ◽  
The World ◽  
Long Time ◽  
The Way

I recently became aware that I had for a long time entertained certain preconceptions about the way in which Presocratic thinkers saw the world, without ever having seriously considered the evidence on which my belief was based. This I have now tried to do, with the results which are set forth in this paper. Since in any case it will deal, in a fairly general way, with problems concerning the interaction of philosophical and religious thought in early Greece, I hope it will have a certain interest, whether or not its readers agree with the thesis put forward. The perennial fascination of that topic has been enhanced in recent years by the discussion provoked by Werner Jaeger's book on The Theology of the Early Greek Philosophers, from which I take this sentence as a kind of text for my own reflections: “Though philosophy means death to the old gods, it is itself religion.”

scholarly journals Sistem Informasi Sekolah Berbasis Web

2019 ◽  
Vol 5 (1) ◽  
pp. 87-94
Author(s):  
Eka Puspita Sari ◽  
Asri Wahyuni ◽  
Narti Narti
Keyword(s):  
Information System ◽  
Data Input ◽  
Paper Sheet ◽  
Web Based ◽  
Word Wide Web ◽  
Long Time ◽  
System Information ◽  
Academic Information ◽  
The Way

Abstract: A system of data input and the grade of students who still use manual way namely by filling form of data on student and the grades of students in the form of paper sheet done by administration and teacher. Besides needed a place the immense storage, to find files need a require a long time, because that is the way are considered less effective. Websites is information in word wide web stored in different file as a page web. Academic information system web-based can manage academic information with more effective. The software used for develop information system web is PHP and MySQL for databases. Keywords : Php, MySQL, System, Information, Education

scholarly journals Visual Behavior Based Bio-Inspired Polarization Techniques in Computer Vision and Robotics

2012 ◽  
pp. 243-272 ◽  
Author(s):  
Abd El Rahman Shabayek ◽  
Olivier Morel ◽  
David Fofi
Keyword(s):  
Computer Vision ◽  
Object Recognition ◽  
Visual Perception ◽  
Polarization Vision ◽  
Visual Behavior ◽  
Biologically Inspired ◽  
Long Time ◽  
Comprehensive Survey ◽  
High Level ◽  
And Robotics

For long time, it was thought that the sensing of polarization by animals is invariably related to their behavior, such as navigation and orientation. Recently, it was found that polarization can be part of a high-level visual perception, permitting a wide area of vision applications. Polarization vision can be used for most tasks of color vision including object recognition, contrast enhancement, camouflage breaking, and signal detection and discrimination. The polarization based visual behavior found in the animal kingdom is briefly covered. Then, the authors go in depth with the bio-inspired applications based on polarization in computer vision and robotics. The aim is to have a comprehensive survey highlighting the key principles of polarization based techniques and how they are biologically inspired.

scholarly journals Social Networks, a Populated Picture

2011 ◽  
Vol 10 (02) ◽  
pp. C04
Author(s):  
Fabio Fornasari
Keyword(s):  
Social Networks ◽  
Physical Reality ◽  
Long Time ◽  
Multiple Levels ◽  
The Way

Man, by his very nature, puts things between himself and the environment, turning the latter into a place, a space. He arranges the environment around him on multiple levels, by projecting parts of himself and shaping the frontiers and the horizons that surround, define and represent him. This was learnt a long time ago, but a trace and a memory remain in the way man acts: when mapping reality (both physical reality and the reality explored through digital means), we observe it and find a way through it by adopting behaviours that have always been similar. What has changed in this mapping is the ability to recognise, especially the ability to interpret maps and creatively work them.

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

Connection formulae for spectral functions associated with singular Dirac equations

2004 ◽  
Vol 134 (1) ◽  
pp. 215-223 ◽  
Author(s):  
S. M. Riehl
Keyword(s):  
Dirac Equation ◽  
Spectral Function ◽  
Limit Point ◽  
Initial Condition ◽  
Spectral Functions ◽  
Dirac Equations ◽  
Special Cases ◽  
Limit Point Case ◽  
Image Position ◽  
Connection Formulae

We consider the Dirac equation given by with initial condition y1 (0) cos α + y2(0) sin α = 0, α ε [0; π ) and suppose the equation is in the limit-point case at infinity. Using to denote the derivative of the corresponding spectral function, a formula for is given when is known and positive for three distinct values of α. In general, if is known and positive for only two distinct values of α, then is shown to be one of two possibilities. However, in special cases of the Dirac equation, can be uniquely determined given for only two values of α.

IS ERROR FRAUD?

PEDIATRICS ◽  
1989 ◽  
Vol 84 (4) ◽  
pp. A64-A64
Author(s):  
J. F. L.
Keyword(s):  
Massachusetts Institute Of Technology ◽  
Bad Faith ◽  
Real Science ◽  
Long Time ◽  
The Government ◽  
Institute Of Technology ◽  
Witch Hunts ◽  
First Time ◽  
Published Research ◽  
The Way

Prof. David Baltimore of the Massachusetts Institute of Technology is under attack by Representative John Dingell of Michigan. Why should anyone outside of the Government or basic biomedical research care? Dr. Baltimore's reputation is at stake, but the rest of us will be affected by the outcome of these investigations as well. What has come under a legislative cloud for the first time in a very long time, perhaps ever in this country, is the legitimacy of the scientific method itself. This is an immediate and serious threat to science and medicine. The N.I.H. will have the last word on Dr. Baltimore's published research. But as I understand the Congressman's case, it is that published science must be free of error, and that error itself indicates bad faith and fraudulent intent. This is wrong. Published error is at the heart of any real science. We scientists love to do experiments that show our colleagues to be wrong and, if they are any good, they love to show us to be wrong in turn. By this adversarial process, science reveals the way nature actually works. If we as a country make science a field for only those who enjoy a good lawsuit, we will have shut the door on our future as a technologically serious nation. Clearly Congress cannot wish to do this. I would welcome a Congressional initiative to deal with fraud as such, but I fear that the way Dr. Baltimore is being treated means that witch-hunts are in the offing.

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