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from The American Heritage® Dictionary of the English Language, 4th Edition
• n. A unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius of the circle, approximately 57°17ʹ44.6〞. See Table at measurement.
from Wiktionary, Creative Commons Attribution/Share-Alike License
• n. In the International System of Units, the derived unit of plane angular measure of angle equal to the angle subtended at the centre of a circle by an arc of its circumference equal in length
to the radius of the circle. Symbol: rad
from the GNU version of the Collaborative International Dictionary of English
• n. An arc of a circle which is equal to the radius, or the angle measured by such an arc.
from The Century Dictionary and Cyclopedia
• n. The angle subtended at the center of a circle by an are equal in length to the radius. Also called the unit angle in circular measure. It is equal to 57° 17′ 44″. 80625 nearly.
• n. A unit of angular velocity equivalent, approximately, to 0.15916 revolutions per second, or, strictly, to revolutions per second: in full, radian per second.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
• n. the unit of plane angle adopted under the Systeme International d'Unites; equal to the angle at the center of a circle subtended by an arc equal in length to the radius (approximately 57.295
radi(us) + -an^1.
(American Heritage® Dictionary of the English Language, Fourth Edition)
From radi(us) + -an (Wiktionary)
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Tucker, GA
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trimming a cylinder from PI to 3*PI (Geom_RectangularTrimmedSurface)
Mauro Mariotti 2012/06/28 15:54
It seems to me there is an inconsistency in the Geom_RectangularTrimmedSurface class.
It allows to trim the basis surface along both U and V parameters, or just along one of them.
Suppose I want to trim a Geom_CylindricalSurface with these parameters:
PI, 3*PI, 0, 100.
It is a closed cylinder, trimmed just along V from 0 to 100, but the U interval starts from PI instead of 0.
Which constructor should I use?
If I trim along both U and V, the internal flag isutrimmed is set to true.
Then the method IsUClosed returns Standard_False!
If I trim along V only, isutrimmed is set to false.
Then the method VIso creates a Geom_Circle which goes from 0 to 2*PI, not from PI to 3*PI
In my opinion the first option is correct (otherwise no information is kept about PI and
3*PI), but Geom_RectangularTrimmedSurface::IsUClosed, when isutrimmed is true, should
check if utrim1 and utrim2 are coincident and return Standard_True.
I wonder if this could give troubles to some OCC algorithms.
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Mathematics in medieval Islam
From Wikipedia, the free encyclopedia
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics, is the mathematics developed in the Islamic world between 622 and 1600, during what is known as the
Islamic Golden Age, in that part of the world where Islam was the dominant religion. Islamic science and mathematics flourished under the Islamic caliphate (also known as the Islamic Empire)
established across the Middle East, Central Asia, North Africa, Southern Italy, the Iberian Peninsula, and, at its peak, parts of France and India as well. Islamic activity in mathematics was largely
centered around modern-day Iraq and Persia, but at its greatest extent stretched from North Africa and Spain in the west to India in the east.^[1]
While most scientists in this period were Muslims and wrote in Arabic,^[2] many of the best known contributors were Persians^[3]^[4] as well as Arabs,^[4] in addition to Berber, Moorish and Turkic
contributors, as well as some from other religions (Christians, Jews, Sabians, Zoroastrians, and the irreligious).^[2] Arabic was the dominant language—much like Latin in Medieval Europe, Arabic was
the written lingua franca of most scholars throughout the Islamic world.
Use of the term "Islam"
Bernard Lewis writes the following on the historical usage of the term "Islam" in What Went Wrong? Western Impact and Middle Eastern Response:^[5]
"There have been many civilizations in human history, almost all of which were local, in the sense that they were defined by a region and an ethnic group. This applied to all the ancient
civilizations of the Middle East—
; to the great civilizations of Asia—
; and to the civilizations of
Pre-Columbian America
. There are two exceptions:
and Islam. These are two civilizations defined by religion, in which religion is the primary defining force, not, as in India or China, a secondary aspect among others of an essentially regional
and ethnically defined civilization. Here, again, another word of explanation is necessary."
"In English we use the word “Islam” with two distinct meanings, and the distinction is often blurred and lost and gives rise to considerable confusion. In the one sense, Islam is the counterpart
; that is to say, a
in the strict sense of the word: a system of belief and worship. In the other sense, Islam is the counterpart of Christendom; that is to say, a civilization shaped and defined by a religion, but
containing many elements apart from and even hostile to that religion, yet arising within that civilization."
In this article, "Islam" and the adjective "Islamic" is used in the meaning described above; that is, of a civilization.
Origins and influences
The first century of the Islamic Arab Empire saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and
research in other parts of the world had faded. In the second half of the eighth century Islam had a cultural awakening, and research in mathematics and the sciences increased.^[6] The Muslim Abbasid
caliph al-Mamun (809-833) is said to have had a dream where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translation be made of as many Greek works as possible,
including Ptolemy's Almagest and Euclid's Elements. Greek works would be given to the Muslims by the Byzantine Empire in exchange for treaties, as the two empires held an uneasy peace.^[6] Many of
these Greek works were translated by Thabit ibn Qurra (826-901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.^[7] Historians are in debt to many Islamic
translators, for it is through their work that many ancient Greek texts have survived only through Arabic translations.
Greek, Indian and Babylonian all played an important role in the development of early Islamic mathematics. The works of mathematicians such as Euclid, Apollonius, Archimedes, Diophantus, Aryabhata
and Brahmagupta were all acquired by the Islamic world and incorporated into their mathematics. Perhaps the most influential mathematical contribution from India was the decimal place-value
Indo-Arabic numeral system, also known as the Hindu numerals.^[8] The Persian historian al-Biruni (c. 1050) in his book Tariq al-Hind states that al-Ma'mun had an embassy in India from which was
brought a book to Baghdad that was translated into Arabic as Sindhind. It is generally assumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.^[9] The earliest translations
from Sanskrit inspired several astronomical and astrological Arabic works, now mostly lost, some of which were even composed in verse.^[10] Biruni described Indian mathematics as a "mix of common
pebbles and costly crystals".^[11]
Indian influences were later overwhelmed by Greek mathematical and astronomical texts. It is not clear why this occurred but it may have been due to the greater availability of Greek texts in the
region, the larger number of practitioners of Greek mathematics in the region, or because Islamic mathematicians favored the deductive exposition of the Greeks over the elliptic Sanskrit verse of the
Indians. Regardless of the reason, Indian mathematics soon became mostly eclipsed by or merged with the "Graeco-Islamic" science founded on Hellenistic treatises.^[10] Another likely reason for the
declining Indian influence in later periods was due to Sindh achieving independence from the Caliphate, thus limiting access to Indian works. Nevertheless, Indian methods continued to play an
important role in algebra, arithmetic and trigonometry.^[12]
Besides the Greek and Indian tradition, a third tradition which had a significant influence on mathematics in medieval Islam was the "mathematics of practitioners", which included the applied
mathematics of "surveyors, builders, artisans, in geometric design, tax and treasury officials, and some merchants." This applied form of mathematics transcended ethnic divisions and was a common
heritage of the lands incorporated into the Islamic world.^[8] This tradition also includes the religious observances specific to Islam, which served as a major impetus for the development of
mathematics as well as astronomy.^[13]
Islam and mathematics
A major impetus for the flowering of mathematics as well as astronomy in medieval Islam came from religious observances, which presented an assortment of problems in astronomy and mathematics,
specifically in trigonometry, spherical geometry,^[13] algebra^[14] and arithmetic.^[15]
The Islamic law of inheritance served as an impetus behind the development of algebra (derived from the Arabic al-jabr) by Muhammad ibn Mūsā al-Khwārizmī and other medieval Islamic mathematicians.
Al-Khwārizmī's Hisab al-jabr w’al-muqabala devoted a chapter on the solution to the Islamic law of inheritance using algebra. He formulated the rules of inheritance as linear equations, hence his
knowledge of quadratic equations was not required.^[14] Later mathematicians who specialized in the Islamic law of inheritance included Al-Hassār, who developed the modern symbolic mathematical
notation for fractions in the 12th century,^[15] and Abū al-Hasan ibn Alī al-Qalasādī, who developed an algebraic notation which took "the first steps toward the introduction of algebraic symbolism"
in the 15th century.^[16]
In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Ptolemy's method to calculate the place of the moon and stars.
The method Ptolemy used to solve spherical triangles, however, was a clumsy one devised late in the first century by Menelaus of Alexandria. It involved setting up two intersecting right triangles;
by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications
of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.^[13]
Regarding the issue of moon sighting, Islamic months do not begin at the astronomical new moon, defined as the time when the moon has the same celestial longitude as the sun and is therefore
invisible; instead they begin when the thin crescent moon is first sighted in the western evening sky.^[13] The Qur'an says: "They ask you about the waxing and waning phases of the crescent moons,
say they are to mark fixed times for mankind and Hajj."^[17]^[18] This led Muslims to find the phases of the moon in the sky, and their efforts led to new mathematical calculations.^[19]
Predicting just when the crescent moon would become visible is a special challenge to Islamic mathematical astronomers. Although Ptolemy's theory of the complex lunar motion was tolerably accurate
near the time of the new moon, it specified the moon's path only with respect to the ecliptic. To predict the first visibility of the moon, it was necessary to describe its motion with respect to the
horizon, and this problem demands fairly sophisticated spherical geometry. Finding the direction of Mecca and the time of Salah are the reasons which led to Muslims developing spherical geometry.
Solving any of these problems involves finding the unknown sides or angles of a triangle on the celestial sphere from the known sides and angles. A way of finding the time of day, for example, is to
construct a triangle whose vertices are the zenith, the north celestial pole, and the sun's position. The observer must know the altitude of the sun and that of the pole; the former can be observed,
and the latter is equal to the observer's latitude. The time is then given by the angle at the intersection of the meridian (the arc through the zenith and the pole) and the sun's hour circle (the
arc through the sun and the pole).^[13]^[20]
Muslims are also expected to pray towards the Kaaba in Mecca and orient their mosques in that direction. Thus they need to determine the direction of Mecca (Qibla) from a given location.^[21]^[22]
Another problem is the time of Salah. Muslims need to determine from celestial bodies the proper times for the prayers at sunrise, at midday, in the afternoon, at sunset, and in the evening.^[13]^[20
J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:
"Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to
mathematicians of the 16th, 17th, and 18th centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied
today is far closer in style to that of Islamic mathematics than to that of
Greek mathematics
R. Rashed wrote in The development of Arabic mathematics: between arithmetic and algebra:
's successors undertook a systematic application of
, algebra to arithmetic, both to
, algebra to the
Euclidean theory of numbers
, algebra to
, and geometry to algebra. This was how the creation of
polynomial algebra
combinatorial analysis
numerical analysis
, the numerical solution of
, the new elementary theory of numbers, and the geometric construction of equations arose."
Al-Ḥajjāj ibn Yūsuf ibn Maṭar (786 – 833)
Al-Ḥajjāj translated Euclid's Elements into Arabic.
Muḥammad ibn Mūsā al-Khwārizmī (c. 780 Khwarezm/Baghdad – c. 850 Baghdad)
Al-Khwārizmī was a Persian mathematician, astronomer, astrologer and geographer. He worked most of his life as a scholar in the House of Wisdom in Baghdad. His Algebra was the first book on the
systematic solution of linear and quadratic equations. Latin translations of his Arithmetic, on the Indian numerals, introduced the decimal positional number system to the Western world in the
12th century. He revised and updated Ptolemy's Geography as well as writing several works on astronomy and astrology.
Al-ʿAbbās ibn Saʿid al-Jawharī (c. 800 Baghdad? – c. 860 Baghdad?)
Al-Jawharī was a mathematician who worked at the House of Wisdom in Baghdad. His most important work was his Commentary on Euclid's Elements which contained nearly 50 additional propositions and
an attempted proof of the parallel postulate.
ʿAbd al-Hamīd ibn Turk (fl. 830 Baghdad)
Ibn Turk wrote a work on algebra of which only a chapter on the solution of quadratic equations has survived.
Yaʿqūb ibn Isḥāq al-Kindī (c. 801 Kufa – 873 Baghdad)
Al-Kindī (or Alkindus) was a philosopher and scientist who worked as the House of Wisdom in Baghdad where he wrote commentaries on many Greek works. His contributions to mathematics include many
works on arithmetic and geometry.
Hunayn ibn Ishaq (808 Al-Hirah – 873 Baghdad)
Hunayn (or Johannitus) was a translator who worked at the House of Wisdom in Baghdad. Translated many Greek works including those by Plato, Aristotle, Galen, Hippocrates, and the Neoplatonists.
Banū Mūsā (c. 800 Baghdad – 873+ Baghdad)
The Banū Mūsā were three brothers who worked at the House of Wisdom in Baghdad. Their most famous mathematical treatise is The Book of the Measurement of Plane and Spherical Figures, which
considered similar problems as Archimedes did in his On the Measurement of the Circle and On the sphere and the cylinder. They contributed individually as well. The eldest, Jaʿfar Muḥammad (c.
800) specialised in geometry and astronomy. He wrote a critical revision on Apollonius' Conics called Premises of the book of conics. Aḥmad (c. 805) specialised in mechanics and wrote a work on
pneumatic devices called On mechanics. The youngest, al-Ḥasan (c. 810) specialised in geometry and wrote a work on the ellipse called The elongated circular figure.
Thabit ibn Qurra (Syria-Iraq, 835-901)
Al-Hashimi (Iraq? ca. 850-900)
Muḥammad ibn Jābir al-Ḥarrānī al-Battānī (c. 853 Harran – 929 Qasr al-Jiss near Samarra)
Abu Kamil (Egypt? ca. 900)
Sinan ibn Tabit (ca. 880 - 943)
Ibrahim ibn Sinan (Iraq, 909-946)
Al-Khazin (Iraq-Iran, ca. 920-980)
Al-Karabisi (Iraq? 10th century?)
Ikhwan al-Safa' (Iraq, first half of 10th century)
The Ikhwan al-Safa' ("brethren of purity") were a (mystical?) group in the city of Basra in Irak. The group authored a series of more than 50 letters on science, philosophy and theology. The
first letter is on arithmetic and number theory, the second letter on geometry.
Al-Uqlidisi (Iraq-Iran, 10th century)
Al-Saghani (Iraq-Iran, ca. 940-1000)
Abū Sahl al-Qūhī (Iraq-Iran, ca. 940-1000)
Abū al-Wafāʾ al-Būzjānī (Iraq-Iran, ca. 940-998)
Ibn Sahl (Iraq-Iran, ca. 940-1000)
Al-Sijzi (Iran, ca. 940-1000)
Labana of Cordoba (Spain, ca. 10th century)
One of the few Islamic female mathematicians known by name, and the secretary of the Umayyad Caliph al-Hakem II. She was well-versed in the exact sciences, and could solve the most complex
geometrical and algebraic problems known in her time.^[23]
Ibn Yunus (Egypt, ca. 950-1010)
Abu Nasr ibn `Iraq (Iraq-Iran, ca. 950-1030)
Kushyar ibn Labban (Iran, ca. 960-1010)
Al-Karaji (Iran, ca. 970-1030)
Ibn al-Haytham (Iraq-Egypt, ca. 965-1040)
Abū al-Rayḥān al-Bīrūnī (September 15, 973 in Kath, Khwarezm – December 13, 1048 in Gazna)
Ibn Sina (Avicenna)
Al-Jayyani (Spain, ca. 1030-1090)
Ibn al-Zarqalluh (Azarquiel, al-Zarqali) (Spain, ca. 1030-1090)
Al-Mu'taman ibn Hud (Spain, ca. 1080)
al-Khayyam (Iran, ca. 1050-1130)
Ibn Yaḥyā al-Maghribī al-Samawʾal (ca. 1130, Baghdad – c. 1180, Maragha)
Al-Hassār (ca. 1100s, Maghreb)
Developed the modern mathematical notation for fractions and the digits he uses for the ghubar numerals also cloesly resembles modern Western Arabic numerals.
Ibn al-Yāsamīn (ca. 1100s, Maghreb)
The son of a Berber father and black African mother, he was the first to develop a mathematical notation for algebra since the time of Brahmagupta.
Sharaf al-Dīn al-Ṭūsī (Iran, ca. 1150-1215)
Ibn Mun`im (Maghreb, ca. 1210)
al-Marrakushi (Morocco, 13th century)
Naṣīr al-Dīn al-Ṭūsī (18 February 1201 in Tus, Khorasan – 26 June 1274 in Kadhimain near Baghdad)
Muḥyi al-Dīn al-Maghribī (c. 1220 Spain – c. 1283 Maragha)
Shams al-Dīn al-Samarqandī (c. 1250 Samarqand – c. 1310)
Ibn Baso (Spain, ca. 1250-1320)
Ibn al-Banna' (Maghreb, ca. 1300)
Kamal al-Din Al-Farisi (Iran, ca. 1300)
Al-Khalili (Syria, ca. 1350-1400)
Ibn al-Shatir (1306–1375)
Qāḍī Zāda al-Rūmī (1364 Bursa – 1436 Samarkand)
Jamshīd al-Kāshī (Iran, Uzbekistan, ca. 1420)
Ulugh Beg (Iran, Uzbekistan, 1394–1449)
Abū al-Hasan ibn Alī al-Qalasādī (Maghreb, 1412–1482)
Last major medieval Arab mathematician. Pioneer of symbolic algebra.
The term algebra is derived from the Arabic term al-jabr in the title of Al-Khwarizmi's Al-jabr wa'l muqabalah. He originally used the term al-jabr to describe the method of "reduction" and
"balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.^[24]
There are three theories about the origins of Islamic algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence, and the third emphasizes Greek
influence. Many scholars believe that it is the result of a combination of all three sources.^[25]
Throughout their time in power, before the fall of Islamic civilization, the Arabs used a fully rhetorical algebra, where sometimes even the numbers were spelled out in words. The Arabs would
eventually replace spelled out numbers (eg. twenty-two) with Arabic numerals (eg. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra,^[7] until the work of Ibn al-Banna
al-Marrakushi in the 13th century and Abū al-Hasan ibn Alī al-Qalasādī in the 15th century.^[16]
There were four conceptual stages in the development of algebra, three of which either began in, or were significantly advanced in, the Islamic world. These four stages were as follows:^[26]
Static equation-solving algebra
Al-Khwarizmi and Al-jabr wa'l muqabalah
The Muslim^[27] Persian mathematician Muhammad ibn Mūsā al-Khwārizmī (c. 780-850) was a faculty member of the "House of Wisdom" (Bait al-hikma) in Baghdad, which was established by Al-Mamun.
Al-Khwarizmi, who died around 850 A.D., wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian Sindhind.^[6] One of al-Khwarizmi's most famous books
is entitled Al-jabr wa'l muqabalah or The Compendious Book on Calculation by Completion and Balancing, and it gives an exhaustive account of solving polynomials up to the second degree.^[28] The book
also introduced the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on
opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as al-jabr.^[24]
Al-Jabr is divided into six chapters, each of which deals with a different type of formula. The first chapter of Al-Jabr deals with equations whose squares equal its roots (ax² = bx), the second
chapter deals with squares equal to number (ax² = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax² + bx = c), the
fifth chapter deals with squares and number equal roots (ax² + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax²).^[29]
J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:
"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how
significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed
rational numbers
irrational numbers
, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and
provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which
had not happened before."
The Hellenistic mathematician Diophantus was traditionally known as "the father of algebra"^[30]^[31] but debate now exists as to whether or not Al-Khwarizmi deserves this title instead.^[30] Those
who support Diophantus point to the fact that the algebra found in Al-Jabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully
rhetorical.^[30] Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,^[32] introduced the
fundamental methods of reduction and balancing,^[24] and was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of
numbers.^[33] In addition, R. Rashed and Angela Armstrong write:
"Al-Khwarizmi's text can be seen to be distinct not only from the
Babylonian tablets
, but also from Diophantus'
. It no longer concerns a series of
to be resolved, but an
which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand,
the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is
specifically called on to define an infinite class of problems."
Ibn Turk and Logical Necessities in Mixed Equations
'Abd al-Hamīd ibn Turk (fl. 830) authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as,
or even possibly earlier than, Al-Jabr.^[35] The manuscript gives the exact same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes
beyond Al-Jabr by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution.^[35] The similarity between these two works has led some historians to
conclude that Islamic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.^[35]
Abū Kāmil and al-Karkhi
Arabic mathematicians were also the first to treat irrational numbers as algebraic objects.^[36] The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850-930) was the first to accept irrational
numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.^[37] He was also the first to solve three non-linear
simultaneous equations with three unknown variables.^[38]
Al-Karkhi (953-1029), also known as Al-Karaji, was the successor of Abū al-Wafā' al-Būzjānī (940-998) and he was the first to discover the solution to equations of the form ax^2n + bx^n = c.^[39]
Al-Karkhi only considered positive roots.^[39] Al-Karkhi is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which
are at the core of algebra today. His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics F. Woepcke, in Extrait du
Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this,
Al-Karaji investigated binomial coefficients and Pascal's triangle.^[40]
Linear algebra
In linear algebra and recreational mathematics, magic squares were known to Arab mathematicians, possibly as early as the 7th century, when the Arabs got into contact with Indian or South Asian
culture, and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics. It has also been suggested that the idea came via China. The first magic squares of order
5 and 6 appear in an encyclopedia from Baghdad circa 983 AD, the Rasa'il Ihkwan al-Safa (Encyclopedia of the Brethren of Purity); simpler magic squares were known to several earlier Arab
The Arab mathematician Ahmad al-Buni, who worked on magic squares around 1200 AD, attributed mystical properties to them, although no details of these supposed properties are known. There are also
references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs.^[41]
Geometric algebra
Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond Al-Jabr to include equations of the third degree.^[42] Omar Khayyám provided both arithmetic and geometric solutions for quadratic
equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible.^[42] His method of solving cubic equations by using
intersecting conics had been used by Menaechmus, Archimedes, and Alhazen, but Omar Khayyám generalized the method to cover all cubic equations with positive roots.^[42] He only considered positive
roots and he did not go past the third degree.^[42] He also saw a strong relationship between Geometry and Algebra.^[42]
Dynamic functional algebra
In the 12th century, Sharaf al-Dīn al-Tūsī found algebraic and numerical solutions to cubic equations and was the first to discover the derivative of cubic polynomials.^[43] His Treatise on Equations
dealt with equations up to the third degree. The treatise does not follow Al-Karaji's school of algebra, but instead represents "an essential contribution to another algebra which aimed to study
curves by means of equations, thus inaugurating the beginning of algebraic geometry." The treatise dealt with 25 types of equations, including twelve types of linear equations and quadratic equations
, eight types of cubic equations with positive solutions, and five types of cubic equations which may not have positive solutions.^[44] He understood the importance of the discriminant of the cubic
equation and used an early version of Cardano's formula^[45] to find algebraic solutions to certain types of cubic equations.^[43]
Sharaf al-Din also developed the concept of a function. In his analysis of the equation $\ x^3 + d = bx^2$ for example, he begins by changing the equation's form to $\ x^2 (b - x) = d$. He then
states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value $\ d$. To determine this, he finds a maximum value for the
function. He proves that the maximum value occurs when $x = \frac{2b}{3}$, which gives the functional value $\frac{4b^3}{27}$. Sharaf al-Din then states that if this value is less than $\ d$, there
are no positive solutions; if it is equal to $\ d$, then there is one solution at $x = \frac{2b}{3}$; and if it is greater than $\ d$, then there are two solutions, one between $\ 0$ and $\frac{2b}
{3}$ and one between $\frac{2b}{3}$ and $\ b$. This was the earliest form of dynamic functional algebra.^[46]
Numerical analysis
In numerical analysis, the essence of Viète's method was known to Sharaf al-Dīn al-Tūsī in the 12th century, and it is possible that the algebraic tradition of Sharaf al-Dīn, as well as his
predecessor Omar Khayyám and successor Jamshīd al-Kāshī, was known to 16th century European algebraists, or whom François Viète was the most important.^[47]
A method algebraically equivalent to Newton's method was also known to Sharaf al-Dīn. In the 15th century, his successor al-Kashi later used a form of Newton's method to numerically solve $\ x^P - N
= 0$ to find roots of $\ N$. In western Europe, a similar method was later described by Henry Biggs in his Trigonometria Britannica, published in 1633.^[48]
Symbolic algebra
Al-Hassār, a mathematician from the Maghreb (North Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions
, where the numerator and denominator are separated by a horizontal bar. This same fractional notation appeared soon after in the work of Fibonacci in the 13th century.^[15]
Abū al-Hasan ibn Alī al-Qalasādī (1412–1482) was the last major medieval Arab algebraist, who improved on the algebraic notation earlier used in the Maghreb by Ibn al-Banna in the 13th century^[16]
and by Ibn al-Yāsamīn in the 12th century.^[15] In contrast to the syncopated notations of their predecessors, Diophantus and Brahmagupta, which lacked symbols for mathematical operations,^[49]
al-Qalasadi's algebraic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols
using characters from the Arabic alphabet.^[16]
The symbol x now commonly denotes an unknown variable. Even though any letter can be used, x is the most common choice. This usage can be traced back to the Arabic word šay' شيء = “thing,” used in
Arabic algebra texts such as the Al-Jabr, and was taken into Old Spanish with the pronunciation “šei,” which was written xei, and was soon habitually abbreviated to x. (The Spanish pronunciation of
“x” has changed since). Some sources say that this x is an abbreviation of Latin causa, which was a translation of Arabic شيء. This started the habit of using letters to represent quantities in
algebra. In mathematics, an “italicized x” ($x\!$) is often used to avoid potential confusion with the multiplication symbol.
Arabic numerals
The Indian numeral system came to be known to both the Persian mathematician Al-Khwarizmi, whose book On the Calculation with Hindu Numerals written circa 825, and the Arab mathematician Al-Kindi,
who wrote four volumes, On the Use of the Indian Numerals (Ketab fi Isti'mal al-'Adad al-Hindi) circa 830, are principally responsible for the diffusion of the Indian system of numeration in the
Middle-East and the West [3]. In the 10th century, Middle-Eastern mathematicians extended the decimal numeral system to include fractions using decimal point notation, as recorded in a treatise by
Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952-953.
In the Arab world—until early modern times—the Arabic numeral system was often only used by mathematicians. Muslim astronomers mostly used the Babylonian numeral system, and merchants mostly used the
Abjad numerals. A distinctive "Western Arabic" variant of the symbols begins to emerge in ca. the 10th century in the Maghreb and Al-Andalus, called the ghubar ("sand-table" or "dust-table")
numerals, which is the direct ancestor to the modern Western Arabic numerals now used throughout the world.^[50]
The first mentions of the numerals in the West are found in the Codex Vigilanus of 976 [4]. From the 980s, Gerbert of Aurillac (later, Pope Silvester II) began to spread knowledge of the numerals in
Europe. Gerbert studied in Barcelona in his youth, and he is known to have requested mathematical treatises concerning the astrolabe from Lupitus of Barcelona after he had returned to France.
Al-Khwārizmī, the Persian scientist, wrote in 825 a treatise On the Calculation with Hindu Numerals, which was translated into Latin in the 12th century, as Algoritmi de numero Indorum, where
"Algoritmi", the translator's rendition of the author's name gave rise to the word algorithm (Latin algorithmus) with a meaning "calculation method".
Al-Hassār, a mathematician from the Maghreb (North Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for
fractions, where the numerator and denominator are separated by a horizontal bar. The "dust ciphers he used are also nearly identical to the digits used in the current Western Arabic numerals. These
same digits and fractional notation appear soon after in the work of Fibonacci in the 13th century.^[15]
Decimal fractions
In discussing the origins of decimal fractions, Dirk Jan Struik states that (p. 7):^[51]
"The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphelet De Thiende, published at Leyden in 1585, together with a French translation,
La Disme, by the Flemish mathematician Simon Stevin (1548-1620), then settled in the Northern Netherlands. It is true that decimal fractions were used by the Chinese many centuries before Stevin
and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).^[52]"
While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were
first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.^[38]
Real numbers
The Middle Ages saw the acceptance of zero, negative, integral and fractional numbers, first by Indian mathematicians and Chinese mathematicians, and then by Arabic mathematicians, who were also the
first to treat irrational numbers as algebraic objects,^[36] which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more
general idea of real numbers, and they criticized Euclid's idea of ratios, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude.^[53] In his
commentary on Book 10 of the Elements, the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and
irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows:^[54]
"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is
impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc."
In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also
introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes:^[54]
"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."
The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form
of square roots, cube roots and fourth roots.^[37] In the 10th century, the Iraqi mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he
considered multiplication, division, and other arithmetical functions.^[55] Abū Ja'far al-Khāzin (900-971) provides a definition of rational and irrational magnitudes, stating that if a definite
quantity is:^[56]
"contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to a rational number. . . . Each time when this (latter) magnitude comprises a half, or a
third, or a quarter of the given magnitude (of the unit), or, compared with (the unit), comprises three, five, or three fifths, it is a rational magnitude. And, in general, each magnitude that
corresponds to this magnitude (i.e. to the unit), as one number to another, is rational. If, however, a magnitude cannot be represented as a multiple, a part (l/n), or parts (m/n) of a given
magnitude, it is irrational, i.e. it cannot be expressed other than by means of roots."
Many of these concepts were eventually accepted by European mathematicians some time after the Latin translations of the 12th century. Al-Hassār, an Arabic mathematician from the Maghreb (North
Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated
by a horizontal bar. This same fractional notation appears soon after in the work of Fibonacci in the 13th century.^[15]
Number theory
In number theory, Ibn al-Haytham solved problems involving congruences using what is now called Wilson's theorem. In his Opuscula, Ibn al-Haytham considers the solution of a system of congruences,
and gives two general methods of solution. His first method, the canonical method, involved Wilson's theorem, while his second method involved a version of the Chinese remainder theorem. Another
contribution to number theory is his work on perfect numbers. In his Analysis and synthesis, Ibn al-Haytham was the first to discover that every even perfect number is of the form 2^n−1(2^n − 1)
where 2^n − 1 is prime, but he was not able to prove this result successfully (Euler later proved it in the 18th century).^[57]
In the early 14th century, Kamāl al-Dīn al-Fārisī made a number of important contributions to number theory. His most impressive work in number theory is on amicable numbers. In Tadhkira al-ahbab fi
bayan al-tahabb ("Memorandum for friends on the proof of amicability") introduced a major new approach to a whole area of number theory, introducing ideas concerning factorization and combinatorial
methods. In fact, al-Farisi's approach is based on the unique factorization of an integer into powers of prime numbers.
The successors of Muhammad ibn Mūsā al-Khwārizmī (born 780) undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory
of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new
elementary theory of numbers, and the geometric construction of equations arose.
Al-Mahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji (born 953) completely freed algebra from geometrical operations
and replaced them with the arithmetical type of operations which are at the core of algebra today.
Early Islamic geometry
See also Applied mathematics
Thabit ibn Qurra (known as Thebit in Latin) (born 836) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical
discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. An important
geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had
dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities
previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalization of the number concept. Another important contribution Thabit made to geometry was
his generalization of the Pythagorean theorem, which he extended from special right triangles to all right triangles in general, along with a general proof.^[58]
In some respects, Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning
motion in his geometrical arguments.
Ibrahim ibn Sinan ibn Thabit (born 908), who introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of
Greek higher geometry in the Islamic world. These mathematicians, and in particular Ibn al-Haytham (Alhazen), studied optics and investigated the optical properties of mirrors made from conic
sections (see Mathematical physics).
Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. For example Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves
required in the construction of sundials. Abu'l-Wafa and Abu Nasr Mansur pioneered spherical geometry in order to solve difficult problems in Islamic astronomy. For example, to predict the first
visibility of the moon, it was necessary to describe its motion with respect to the horizon, and this problem demands fairly sophisticated spherical geometry. Finding the direction of Mecca (Qibla)
and the time for Salah prayers and Ramadan are what led to Muslims developing spherical geometry.^[13]^[20]
Algebraic and analytic geometry
In the early 11th century, Ibn al-Haytham (Alhazen) was able to solve by purely algebraic means certain cubic equations, and then to interpret the results geometrically.^[59] Subsequently, Omar
Khayyám discovered the general method of solving cubic equations by intersecting a parabola with a circle.^[60]
Omar Khayyám (1048–1122) was a Persian mathematician, as well as a poet. Along with his fame as a poet, he was also famous during his lifetime as a mathematician, well known for inventing the general
method of solving cubic equations by intersecting a parabola with a circle. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made
their way to England, where they contributed to the eventual development of non-Euclidean geometry. Omar Khayyam also combined the use of trigonometry and approximation theory to provide methods of
solving algebraic equations by geometrical means. His work marked the beginnings of algebraic geometry^[36]^[61] and analytic geometry.^[62]
In a paper written by Khayyam before his famous algebra text Treatise on Demonstration of Problems of Algebra, he considers the problem: Find a point on a quadrant of a circle in such manner that
when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal.
Khayyam shows that this problem is equivalent to solving a second problem: Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse.
This problem in turn led Khayyam to solve the cubic equation x^3 + 200x = 20x^2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a
circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic
requires the use of conic sections and that it cannot be solved by compass and straightedge, a result which would not be proved for another 750 years.
His Treatise on Demonstration of Problems of Algebra contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. In fact Khayyam
gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations. Indeed, as Khayyam writes, the contributions by earlier writers such as
al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of Muḥammad ibn Mūsā al-Ḵwārizmī). However, Khayyam
himself seems to have been the first to conceive a general theory of cubic equations.
Omar Khayyám saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra^[42] with his
geometric solution of the general cubic equations,^[62] but the decisive step in analytic geometry came later with René Descartes.^[42]
Persian mathematician Sharafeddin Tusi (born 1135) did not follow the general development that came through al-Karaji's school of algebra but rather followed Khayyam's application of algebra to
geometry. He wrote a treatise on cubic equations, entitled Treatise on Equations, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus
inaugurating the study of algebraic geometry.^[44]
Non-Euclidean geometry
In the early 11th century, Ibn al-Haytham (Alhazen) made the first attempt at proving the Euclidean parallel postulate, the fifth postulate in Euclid's Elements, using a proof by contradiction,^[63]
where he introduced the concept of motion and transformation into geometry.^[64] He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert
quadrilateral",^[65] and his attempted proof also shows similarities to Playfair's axiom.^[66]
In the late 11th century, Omar Khayyám made the first attempt at formulating a non-Euclidean postulate as an alternative to the Euclidean parallel postulate,^[67] and he was the first to consider the
cases of elliptical geometry and hyperbolic geometry, though he excluded the latter.^[68]
In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-Euclidean geometry, although this was not his intention. In trying to prove the parallel postulate he
accidentally proved properties of figures in non-Euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios. The
importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios (which was that first proposed by Eudoxus ) and the definition of equality of ratios as
proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions. Khayyam proved that the two definitions are equivalent. He also posed the question of whether a
ratio can be regarded as a number but leaves the question unanswered.
The Khayyam-Saccheri quadrilateral was first considered by Omar Khayyam in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.^[65] Unlike many
commentators on Euclid before and after him (including of course Saccheri), Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated
from "the principles of the Philosopher" (Aristotle):
Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.^[69]
Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted
the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid. It wasn't until 600 years later that Giordano Vitale made an advance on the understanding of this
quadrilateral in his book Euclide restituo (1680, 1686), when he used it to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant.
Saccheri himself based the whole of his long, heroic and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties
along the way.
In 1250, Nasīr al-Dīn al-Tūsī, in his Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion Which Removes Doubt about Parallel Lines), wrote detailed critiques of the Euclidean
parallel postulate and on Omar Khayyám's attempted proof a century earlier. Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate.^[70] He was one of the first to
consider the cases of elliptical geometry and hyperbolic geometry, though he ruled out both of them.^[68]
His son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented one of the earliest arguments for a non-Euclidean
hypothesis equivalent to the parallel postulate.^[70]^[71] Sadr al-Din's work was published in Rome in 1594 and was studied by European geometers. This work marked the starting point for Giovanni
Girolamo Saccheri's work on the subject, and eventually the development of modern non-Euclidean geometry.^[70] A proof from Sadr al-Din's work was quoted by John Wallis and Saccheri in the 17th and
18th centuries. They both derived their proofs of the parallel postulate from Sadr al-Din's work, while Saccheri also derived his Saccheri quadrilateral from Sadr al-Din, who himself based it on his
father's work.^[72]
The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were the first theorems on
elliptical geometry and hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works marked the beginning of non-Euclidean geometry and had a considerable
influence on its development among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis, and Giovanni Girolamo Saccheri.^[73]
The early Indian works on trigonometry were translated and expanded in the Muslim world by Arab and Persian mathematicians, who enunciated a large number of theorems which freed the subject of
trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due to the application of Menelaus' theorem. According to E. S. Kennedy, it was after this
development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the spherical or plane triangle, its sides and angles."^[74]
Another important development was the subject's separation from astronomy. All works on trigonometry up until the 12th century treated it mainly as an adjunct to astronomy; the first treatment of
trigonometry as a subject in its own right was by Nasīr al-Dīn al-Tūsī in the 13th century.^[75]
Trigonometric functions
In the early 9th century, Muhammad ibn Mūsā al-Khwārizmī (c. 780-850) produced tables for the trigonometric functions of sines and cosine,^[76] and the first tables for tangents.^[77] In 830, Habash
al-Hasib al-Marwazi produced the first tables of cotangents as well as tangents.^[75]^[78] Muhammad ibn Jābir al-Harrānī al-Battānī (853-929) discovered the reciprocal functions of secant and
cosecant, and produced the first table of cosecants, which he referred to as a "table of shadows" (in reference to the shadow of a gnomon), for each degree from 1° to 90°.^[75] By the 10th century,
in the work of Abū al-Wafā' al-Būzjānī (959-998), Muslim mathematicians were using all six trigonometric functions, and had sine tables in 0.25° increments, to 8 decimal places of accuracy, as well
as accurate tables of tangent values.
Jamshīd al-Kāshī (1393-1449) gives trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added
for each 1/60 of 1°.^[79] In one of his numerical approximations of π, he correctly computed 2π to 9 sexagesimal digits.^[80] In order to determine sin 1°, al-Kashi discovered the following
triple-angle formula often attributed to François Viète in the 16th century:^[81]
$\sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\,.$
Al-Kashi, alongside his colleague Ulugh Beg (1394-1449), gave accurate tables of sines and tangents correct to 8 decimal places. Taqi al-Din (1526-1585) contributed to trigonometry in his Sidrat
al-Muntaha, in which he was the first mathematician to compute a highly accurate numeric value for sin 1°. He discusses the values given by his predecessors, explaining how Ptolemy (ca. 150) used an
approximate method to obtain his value of sin 1° and how Abū al-Wafā, Ibn Yunus (ca. 1000), al-Kashi, Qāḍī Zāda al-Rūmī (1337-1412), Ulugh Beg and Mirim Chelebi improved on the value. Taqi al-Din
then solves the problem to obtain the value of sin 1° to a precision of 8 sexagesimals (the equivalent of 14 decimals):^[82]
$\sin 1^\circ = 1^P 2' 49'' 43''' 11'''' 14''''' 44''''''16''''''' \ (= 1/60 + 2/60^2 + 49/60^3 + \cdots)\,.$
Laws and identities
Muhammad ibn Jābir al-Harrānī al-Battānī (853-929) formulated a number of important trigonometrical relationships such as:
$\tan a = \frac{\sin a}{\cos a}$
$\sec a = \sqrt{1 + \tan^2 a }$
In the 10th century, Abū al-Wafā' al-Būzjānī discovered the law of sines for spherical trigonometry:^[78]
$\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}.$
Abū al-Wafā' also developed the following trigonometric formula:
$\sin 2x = 2 \sin x \cos x \$
Abū al-Wafā also established the angle addition identities, e.g. sin (a + b).^[78] Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful
trigonometric calculations and demonstrated the following formula:
$\cos a \cos b = \frac{\cos(a+b) + \cos(a-b)}{2}$
Also in the 11th century, Al-Jayyani's The book of unknown arcs of a sphere introduced the general law of sines.^[83] In the 13th century, Nasīr al-Dīn al-Tūsī, in his On the Sector Figure, stated
the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for these laws.^[38] Jamshīd al-Kāshī (1393-1449) provided the first
explicit statement of the law of cosines in a form suitable for triangulation.^[79] As such, the law of cosines is known the théorème d'Al-Kashi in France.
Spherical trigonometry
Hellenistic methods dealing with spherical triangles were known, particularly the method of Menelaus of Alexandria, who developed Menelaus' theorem to deal with spherical problems.^[84]^[85] However,
E. S. Kennedy points out that while it was possible in pre-lslamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the
application of the theorem to spherical problems was very difficult in practice.^[86] In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon,
astronomers initially used Menelaus' method to calculate the place of the moon and stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles
; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated
applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.^[13]
In the early 9th century, Muhammad ibn Mūsā al-Khwārizmī was an early pioneer in spherical trigonometry and wrote a treatise on the subject.^[77] In the 10th century, Abū al-Wafā' al-Būzjānī
discovered the law of sines for spherical trigonometry.^[78] In the 11th century, Al-Jayyani (989–1079) of Al-Andalus wrote The book of unknown arcs of a sphere, which is considered "the first
treatise on spherical trigonometry" in its modern form.^[83] It "contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar
triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are
likely to have influenced Regiomontanus.^[83] In the 13th century, Nasīr al-Dīn al-Tūsī developed spherical trigonometry into its present form,^[75] and listed the six distinct cases of a
right-angled triangle in spherical trigonometry.^[38] In his On the Sector Figure, he also stated the law of sines for plane and spherical triangles, and discovered the law of tangents for spherical
Other advances
The method of triangulation, which was unknown in the Greco-Roman world, was also first developed by Muslim mathematicians, who applied it to practical uses such as surveying^[87] and Islamic
geography, as described by Abu Rayhan Biruni in the early 11th century. Biruni employed triangulation techniques to measure the size of the Earth and the distances between places (see Mathematical
geography and geodesy section).^[88]
In the late 11th century, Omar Khayyám (1048-1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables (see Geometric algebra and Algebraic and
analytic geometry sections). Jamshīd al-Kāshī (1393-1449) provided the first explicit statement of the law of cosines in a form suitable for triangulation.^[79]
Integral calculus
Around 1000 AD, Al-Karaji, using mathematical induction, found a proof for the sum of integral cubes.^[89] The historian of mathematics, F. Woepcke,^[90] praised Al-Karaji for being "the first who
introduced the theory of algebraic calculus." Shortly afterwards, Ibn al-Haytham (known as Alhazen in the West), an Iraqi mathematician working in Egypt, was the first mathematician to derive the
formula for the sum of the fourth powers, and using an early proof by mathematical induction, he developed a method for determining the general formula for the sum of any integral powers. He used his
result on sums of integral powers to perform an integration, in order to find the volume of a paraboloid. He was thus able to find the integrals for polynomials up to the fourth degree, and came
close to finding a general formula for the integrals of any polynomials. This was fundamental to the development of infinitesimal and integral calculus. His results were repeated by the Moroccan
mathematicians Abu-l-Hasan ibn Haydur (d. 1413) and Abu Abdallah ibn Ghazi (1437-1514), by Jamshīd al-Kāshī (c. 1380-1429) in The Calculator's Key, and by the Indian mathematicians of the Kerala
school of astronomy and mathematics in the 15th-16th centuries.^[70]
Differential calculus
In the 12th century, the Persian mathematician Sharaf al-Dīn al-Tūsī was the first to discover the derivative of cubic polynomials, an important result in differential calculus.^[43] His Treatise on
Equations developed concepts related to differential calculus, such as the derivative function and the maxima and minima of curves, in order to solve cubic equations which may not have positive
solutions. For example, in order to solve the equation $\ x^3 + a = bx$, al-Tusi finds the maximum point of the curve $\ bx - x^3 = a$. He uses the derivative of the function to find that the maximum
point occurs at $x = \sqrt{\frac{b}{3}}$, and then finds the maximum value for y at $2(\frac{b}{3})^\frac{3}{2}$ by substituting $x = \sqrt{\frac{b}{3}}$ back into $\ y = bx - x^3$. He finds that the
equation $\ bx - x^3 = a$ has a solution if $a \le 2(\frac{b}{3})^\frac{3}{2}$, and al-Tusi thus deduces that the equation has a positive root if $D = \frac{b^3}{27} - \frac{a^2}{4} \ge 0$, where D
is the discriminant of the equation.^[44]
Applied mathematics
Geometric art and architecture
Geometric artwork in the form of the Arabesque was not widely used in the Middle East or Mediterranean Basin until the golden age of Islam came into full bloom, when Arabesque became a common feature
of Islamic art. Euclidean geometry as expounded on by Al-Abbās ibn Said al-Jawharī (ca. 800-860) in his Commentary on Euclid's Elements, the trigonometry of Aryabhata and Brahmagupta as elaborated on
by Muhammad ibn Mūsā al-Khwārizmī (ca. 780-850), and the development of spherical geometry^[13] by Abū al-Wafā' al-Būzjānī (940–998) and spherical trigonometry by Al-Jayyani (989-1079)^[83] for
determining the Qibla and times of Salah and Ramadan,^[13] all served as an impetus for the art form that was to become the Arabesque.
Recent discoveries have shown that geometrical quasicrystal patterns were first employed in the girih tiles found in medieval Islamic architecture dating back over five centuries ago. In 2007,
Professor Peter Lu of Harvard University and Professor Paul Steinhardt of Princeton University published a paper in the journal Science suggesting that girih tilings possessed properties consistent
with self-similar fractal quasicrystalline tilings such as the Penrose tilings, predating them by five centuries.^[91]^[92]
Mathematical astronomy
Main articles:
Islamic astronomy
An impetus behind mathematical astronomy came from Islamic religious observances, which presented a host of problems in mathematical astronomy, particularly in spherical geometry. In solving these
religious problems the Islamic scholars went far beyond the Greek mathematical methods.^[13] For example, predicting just when the crescent moon would become visible is a special challenge to Islamic
mathematical astronomers. Although Ptolemy's theory of the complex lunar motion was tolerably accurate near the time of the new moon, it specified the moon's path only with respect to the ecliptic.
To predict the first visibility of the moon, it was necessary to describe its motion with respect to the horizon, and this problem demands fairly sophisticated spherical geometry. Finding the
direction of Mecca and the time of Salah are the reasons which led to Muslims developing spherical geometry. Solving any of these problems involves finding the unknown sides or angles of a triangle
on the celestial sphere from the known sides and angles. A way of finding the time of day, for example, is to construct a triangle whose vertices are the zenith, the north celestial pole, and the
sun's position. The observer must know the altitude of the sun and that of the pole; the former can be observed, and the latter is equal to the observer's latitude. The time is then given by the
angle at the intersection of the meridian (the arc through the zenith and the pole) and the sun's hour circle (the arc through the sun and the pole).^[13]^[20]
The Zij treatises were astronomical books that tabulated the parameters used for astronomical calculations of the positions of the Sun, Moon, stars, and planets. Their principal contributions to
mathematical astronomy reflected improved trigonometrical, computational and observational techniques.^[93]^[94] The Zij books were extensive, and typically included materials on chronology,
geographical latitudes and longitudes, star tables, trigonometrical functions, functions in spherical astronomy, the equation of time, planetary motions, computation of eclipses, tables for first
visibility of the lunar crescent, astronomical and/or astrological computations, and instructions for astronomical calculations using epicyclic geocentric models.^[95] Some zījes go beyond this
traditional content to explain or prove the theory or report the observations from which the tables were computed.^[96]
In observational astronomy, Muhammad ibn Mūsā al-Khwārizmī's Zij al-Sindh (830) contains trigonometric tables for the movements of the sun, the moon and the five planets known at the time.^[97]
Al-Farghani's A compendium of the science of stars (850) corrected Ptolemy's Almagest and gave revised values for the obliquity of the ecliptic, the precessional movement of the apogees of the sun
and the moon, and the circumference of the earth.^[98] Muhammad ibn Jābir al-Harrānī al-Battānī (853-929) discovered that the direction of the Sun's eccentric was changing,^[99] and studied the times
of the new moon, lengths for the solar year and sidereal year, prediction of eclipses, and the phenomenon of parallax.^[100] Around the same time, Yahya Ibn Abi Mansour wrote the Al-Zij al-Mumtahan,
in which he completely revised the Almagest values.^[101] In the 10th century, Abd al-Rahman al-Sufi (Azophi) carried out observations on the stars and described their positions, magnitudes,
brightness, and colour and drawings for each constellation in his Book of Fixed Stars (964). Ibn Yunus observed more than 10,000 entries for the sun's position for many years using a large astrolabe
with a diameter of nearly 1.4 meters. His observations on eclipses were still used centuries later in Simon Newcomb's investigations on the motion of the moon, while his other observations inspired
Laplace's Obliquity of the Ecliptic and Inequalities of Jupiter and Saturn's.^[102]
In the late 10th century, Abu-Mahmud al-Khujandi accurately computed the axial tilt to be 23°32'19" (23.53°),^[103] which was a significant improvement over the Greek and Indian estimates of
23°51'20" (23.86°) and 24°,^[104] and still very close to the modern measurement of 23°26' (23.44°). In 1006, the Egyptian astronomer Ali ibn Ridwan observed SN 1006, the brightest supernova in
recorded history, and left a detailed description of the temporary star. He says that the object was two to three times as large as the disc of Venus and about one-quarter the brightness of the Moon,
and that the star was low on the southern horizon. In 1031, al-Biruni's Canon Mas’udicus introduced the mathematical technique of analysing the acceleration of the planets, and first states that the
motions of the solar apogee and the precession are not identical. Al-Biruni also discovered that the distance between the Earth and the Sun is larger than Ptolemy's estimate, on the basis that
Ptolemy disregarded annular eclipses.^[105]^[106]
During the "Maragha Revolution" of the 13th and 14th centuries, Muslim astronomers realized that astronomy should aim to describe the behavior of physical bodies in mathematical language, and should
not remain a mathematical hypothesis, which would only save the phenomena. The Maragha astronomers also realized that the Aristotelian view of motion in the universe being only circular or linear was
not true, as the Tusi-couple showed that linear motion could also be produced by applying circular motions only.^[107] Unlike the ancient Greek and Hellenistic astronomers who were not concerned with
the coherence between the mathematical and physical principles of a planetary theory, Islamic astronomers insisted on the need to match the mathematics with the real world surrounding them,^[108]
which gradually evolved from a reality based on Aristotelian physics to one based on an empirical and mathematical physics after the work of Ibn al-Shatir. The Maragha Revolution was thus
characterized by a shift away from the philosophical foundations of Aristotelian cosmology and Ptolemaic astronomy and towards a greater emphasis on the empirical observation and mathematization of
astronomy and of nature in general, as exemplified in the works of Ibn al-Shatir, Ali Qushji, al-Birjandi and al-Khafri.^[109]^[110]^[111] In particular, Ibn al-Shatir's geocentric model was
mathematically identical to the later heliocentric Copernical model.^[112]
Mathematical geography and geodesy
The Muslim scholars, who held to the spherical Earth theory, used it in an impeccably Islamic manner, to calculate the distance and direction from any given point on the earth to Mecca. This
determined the Qibla, or Muslim direction of prayer. Muslim mathematicians developed spherical trigonometry which was used in these calculations.^[113]
Around 830, Caliph al-Ma'mun commissioned a group of astronomers to measure the distance from Tadmur (Palmyra) to al-Raqqah, in modern Syria. They found the cities to be separated by one degree of
latitude and the distance between them to be 66 2/3 miles and thus calculated the Earth's circumference to be 24,000 miles.^[114] Another estimate given by Al-Farghānī was 56 2/3 Arabic miles per
degree, which corresponds to 111.8 km per degree and a circumference of 40,248 km, very close to the currently modern values of 111.3 km per degree and 40,068 km circumference, respectively.^[115]
In mathematical geography, Abū Rayhān al-Bīrūnī, around 1025, was the first to describe a polar equi-azimuthal equidistant projection of the celestial sphere.^[116] He was also regarded as the most
skilled when it came to mapping cities and measuring the distances between them, which he did for many cities in the Middle East and western Indian subcontinent. He often combined astronomical
readings and mathematical equations, in order to develop methods of pin-pointing locations by recording degrees of latitude and longitude. He also developed similar techniques when it came to
measuring the heights of mountains, depths of valleys, and expanse of the horizon, in The Chronology of the Ancient Nations. He also discussed human geography and the planetary habitability of the
Earth. He hypothesized that roughly a quarter of the Earth's surface is habitable by humans, and also argued that the shores of Asia and Europe were "separated by a vast sea, too dark and dense to
navigate and too risky to try" in reference to the Atlantic Ocean and Pacific Ocean.^[117]
Abū Rayhān al-Bīrūnī is considered the father of geodesy for his important contributions to the field,^[118]^[119] along with his significant contributions to geography and geology. At the age of 17,
al-Biruni calculated the latitude of Kath, Khwarazm, using the maximum altitude of the Sun. Al-Biruni also solved a complex geodesic equation in order to accurately compute the Earth's circumference,
which were close to modern values of the Earth's circumference.^[105]^[120] His estimate of 6,339.9 km for the Earth radius was only 16.8 km less than the modern value of 6,356.7 km. In contrast to
his predecessors who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-Biruni developed a new method of using trigonometric calculations based on
the angle between a plain and mountain top which yielded more accurate measurements of the Earth's circumference and made it possible for it to be measured by a single person from a single location.^
Mathematical physics
Ibn al-Haytham's work on geometric optics, particularly catoptrics, in "Book V" of the Book of Optics (1021) contains the important mathematical problem known as "Alhazen's problem" (Alhazen is the
Latinized name of Ibn al-Haytham). It comprises drawing lines from two points in the plane of a circle meeting at a point on the circumference and making equal angles with the normal at that point.
This leads to an equation of the fourth degree. This eventually led Ibn al-Haytham to derive the earliest formula for the sum of the fourth powers, and using an early proof by mathematical induction,
he developed a method for determining the general formula for the sum of any integral powers, which was fundamental to the development of infinitesimal and integral calculus.^[70] Ibn al-Haytham
eventually solved "Alhazen's problem" using conic sections and a geometric proof, but Alhazen's problem remained influential in Europe, when later mathematicians such as Christiaan Huygens, James
Gregory, Guillaume de l'Hôpital, Isaac Barrow, and many others, attempted to find an algebraic solution to the problem, using various methods, including analytic methods of geometry and derivation by
complex numbers.^[66] Mathematicians were not able to find an algebraic solution to the problem until the end of the 20th century.^[124]
Ibn al-Haytham also produced tables of corresponding angles of incidence and refraction of light passing from one medium to another show how closely he had approached discovering the law of constancy
of ratio of sines, later attributed to Snell. He also correctly accounted for twilight being due to atmospheric refraction, estimating the Sun's depression to be 19 degrees below the horizon during
the commencement of the phenomenon in the mornings or at its termination in the evenings.^[125]
Ibn al-Haytham systematically endeavoured to mathematize physics in the context of his experimental research and controlled testing, which was oriented by geometric models of the structural
mathematical principles that governed physical phenomena, particularly in relation to the explication of the behaviour and nature of vision and light.^[126] Ibn al-Haytham also advanced in his
Discourse on Place (Qawl fi al-makan) a geometrical understanding of place as mathematical space that is akin to the 17th century conceptions of extensio by Descartes and analysis situs by Leibniz.
Ibn al-Haytham established his geometrical thesis about place as space in the context of his mathematical refutation of the Aristotelian physical definition of topos as a boundary surface of a
containing body (as argued in Book delta [IV] of Aristotle's Physics).^[127]
Abū Rayhān al-Bīrūnī (973-1048), and later al-Khazini (fl. 1115-1130), were the first to apply experimental scientific methods to the statics and dynamics fields of mechanics, particularly for
determining specific weights, such as those based on the theory of balances and weighing. Muslim physicists applied the mathematical theories of ratios and infinitesimal techniques, and introduced
algebraic and fine calculation techniques into the field of statics.^[128]
Abu 'Abd Allah Muhammad ibn Ma'udh, who lived in Al-Andalus during the second half of the 11th century, wrote a work on optics later translated into Latin as Liber de crepisculis, which was
mistakenly attributed to Alhazen. This was a "short work containing an estimation of the angle of depression of the sun at the beginning of the morning twilight and at the end of the evening
twilight, and an attempt to calculate on the basis of this and other data the height of the atmospheric moisture responsible for the refraction of the sun's rays." Through his experiments, he
obtained the accurate value of 18°, which comes close to the modern value.^[129]
In 1574, Taqi al-Din estimated that the stars are millions of kilometres away from the Earth and that the speed of light is constant, that if light had come from the eye, it would take too long for
light "to travel to the star and come back to the eye. But this is not the case, since we see the star as soon as we open our eyes. Therefore the light must emerge from the object not from the eyes."
Other fields
In the 9th century, al-Kindi was a pioneer in cryptanalysis and cryptology. He gave the first known recorded explanation of cryptanalysis in A Manuscript on Deciphering Cryptographic Messages. In
particular, he is credited with developing the frequency analysis method whereby variations in the frequency of the occurrence of letters could be analyzed and exploited to break ciphers (i.e.
crypanalysis by frequency analysis).^[131] This was detailed in a text recently rediscovered in the Ottoman archives in Istanbul, A Manuscript on Deciphering Cryptographic Messages, which also covers
methods of cryptanalysis, encipherments, cryptanalysis of certain encipherments, and statistical analysis of letters and letter combinations in Arabic.^[132] Al-Kindi also had knowledge of
polyalphabetic ciphers centuries before Leon Battista Alberti. Al-Kindi's book also introduced the classification of ciphers, developed Arabic phonetics and syntax, and described the use of several
statistical techniques for cryptoanalysis. This book apparently antedates other cryptology references by several centuries, and it also predates writings on probability and statistics by Pascal and
Fermat by nearly eight centuries.^[133]
Ahmad al-Qalqashandi (1355-1418) wrote the Subh al-a 'sha, a 14-volume encyclopedia which included a section on cryptology. This information was attributed to Taj ad-Din Ali ibn ad-Duraihim ben
Muhammad ath-Tha 'alibi al-Mausili who lived from 1312 to 1361, but whose writings on cryptology have been lost. The list of ciphers in this work included both substitution and transposition, and for
the first time, a cipher with multiple substitutions for each plaintext letter. Also traced to Ibn al-Duraihim is an exposition on and worked example of cryptanalysis, including the use of tables of
letter frequencies and sets of letters which can not occur together in one word.
Mathematical induction
The first known proof by mathematical induction was introduced in the al-Fakhri written by Al-Karaji around 1000 AD, who used it to prove arithmetic sequences such as the binomial theorem, Pascal's
triangle, and the sum formula for integral cubes.^[134]^[135] His proof was the first to make use of the two basic components of an inductive proof, "namely the truth of the statement for n = 1 (1 =
1^3) and the deriving of the truth for n = k from that of n = k - 1."^[136]
Shortly afterwards, Ibn al-Haytham (Alhazen) used the inductive method to prove the sum of fourth powers, and by extension, the sum of any integral powers, which was an important result in integral
calculus. He only stated it for particular integers, but his proof for those integers was by induction and generalizable.^[137]^[138]
Ibn Yahyā al-Maghribī al-Samaw'al came closest to a modern proof by mathematical induction in pre-modern times, which he used to extend the proof of the binomial theorem and Pascal's triangle
previously given by al-Karaji. Al-Samaw'al's inductive argument was only a short step from the full inductive proof of the general binomial theorem.^[139]
The astrolabe is a mathematical tool that could be used to solve all the standard problems of spherical astronomy in five different ways.
See also
1. ^ O'Connor 1999
2. ^ ^a ^b Hogendijk 1999
3. ^ Joseph A. Schumpeter, Historian of Economics: Selected Papers from the History of Economics Society Conference, 1994, y Laurence S. Moss, Joseph Alois Schumpeter, History of Economics Society.
Conference, Published by Routledge, 1996, ISBN 041513353X, p.64. Excerpt: A great portion (and most of the best) of medieval Muslim philosophers, physicians, ethicists, scientists, Islamic
jurists, historians, and geographers were Persian-speaking Iranians
4. ^ ^a ^b Ibn Khaldun, Franz Rosenthal, N. J. Dawood (1967), The Muqaddimah: An Introduction to History, p. x, Princeton University Press, ISBN 0691017549. page 430: "Only the Persians engaged in
the task of preserving knowledge and writing systematic scholarly works. Thus, the truth of the following statement by the Prophet becomes apparent:"If scholarship hung suspended in the highest
parts of heaven, the Persians would attain it. [...] This situation continued in the cities as long as the Persians and the Persian countries, the 'Iraq, Khurasan, and Transoxania, retained their
sedentary culture. But when those cities fell into ruins, sedentary culture, which God has devised for the attainment of sciences and crafts, disappeared from them. Along with it, scholarship
altogether disappeared from among the non-Arabs (Persians), who were (now) engulfed by the desert attitude. Scholarship was restricted to cities with an abundant sedentary culture. Today, no
(city) has a more abundant sedentary culture than Cairo (Egypt). It is the mother of the world, the great center (Iwan) of Islam, and the mainspring of the sciences and the crafts. Some sedentary
culture has also survived in Transoxania, because the dynasty there provides some sedentary culture. Therefore, they have there a certain number of the sciences and the crafts, which cannot be
denied. Our attention was called to this fact by the contents of the writings of a (Transoxanian) scholar, which have reached us in this country. He is Sa'd-ad-din at-Taftazani. As far as the
other non-Arabs (Persians) are concerned, we have not seen, since the imam Ibn al-Khatib and Nasir-ad-din at-Tusi, any discussions that could be referred to as indicating their ultimate
5. ^ Bernard Lewis in What Went Wrong? Western Impact and Middle Eastern Response
6. ^ ^a ^b ^c Boyer (1991). "The Arabic Hegemony". p. 227. "The first century of the Muslim empire had been devoid of scientific achievement. This period (from about 650 to 750) had been, in fact,
perhaps the nadir in the development of mathematics, for the Arabs had not yet achieved intellectual drive, and concern for learning in other parts of the world had faded. Had it not been for the
sudden cultural awakening in Islam during the second half of the eighth century, considerably more of ancient science and mathematics would have been lost. [...] It was during the caliphate of
al-Mamun (809-833), however, that the Arabs fully indulged their passion for translation. The caliph is said to have had a dream in which Aristotle appeared, and in the new world peolpe nee
fjjdhew r as a consequence al-Mamun determined to have Arabic versions made of all the Greek works that he could lay his hands on, including Ptolemy's Almagest and a complete version of Euclid's
Elements. From the Byzantine Empire, with which the Arabs maintained an uneasy peace, Greek manuscripts were obtained through peace treaties. Al-Mamun established at Baghdad a "House of Wisdom"
(Bait al-hikma) comparable to the ancient Museum at Alexandria. Among the faculty members was a mathematician and astronomer, Mohammed ibn-Musa al-Khwarizmi, whose name, like that of Euclid,
later was to become a household word in Western Europe. The scholar, who died sometime before 850, wrote more than half a dozen astronomical and mathematical works, of which the earliest were
probably based on the Sindhad derived from India."
7. ^ ^a ^b Boyer (1991). "The Arabic Hegemony". p. 234. "but al-Khwarizmi's work had a serious deficiency that had to be removed before it could serve its purpose effectively in the modern world: a
symbolic notation had to be developed to replace the rhetorical form. This step the Arabs never took, except for the replacement of number words by number signs. [...] Thabit was the founder of a
school of translators, especially from Greek and Syriac, and to him we owe an immense debt for translations into Arabic of works by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius."
8. ^ ^a ^b Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 516. ISBN
9780691114859. "The mathematics, to speak only of the subject of interest here, came principally from three traditions. The first was Greek mathematics, from the great geometrical classics of
Euclid, Apollonius, and Archimedes, through the numerical solutions of indeterminate problems in Diophantus's Arithmatica, to the practical manuals of Heron. But, as Bishop Severus Sebokht
pointed out in the mid-seventh century, "there are also others who know something." Sebokht was referring to the Hindus, with their in genius arithmetic system based on only nine signs and a dot
for an empty place. But they also contributed algebraic methods, a nascent trigonometry, and methods from solid geometry to solve problems in astronomy. The third tradition was what one may call
the mathematics of practitioners. Their numbers included surveyors, builders, artisans, in geometric design, tax and treasury officials, and some merchants. Part of an oral tradition, this
mathematics transcended ethnic divisions and was common heritage of many of the lands incorporated into the Islamic world."
9. ^ Boyer (1991). "The Arabic Hegemony". p. 226. "By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally
thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not
long afterwards (ca. 780) that Ptolemy's astrological Tetrabiblos was translated into Arabic from the Greek."
10. ^ ^a ^b Plofker, Kim (2007). p. 434. "The early translations from Sanskrit inspired several other astronomical/astrological works in Arabic; some even imitated the Sanskrit practice of composing
technical treatises in verse. Unfortunately, the earliest texts in this genre have now mostly been lost, and are known only from scattered fragments and allusions in later works. They reveal that
the emergent Arabic astronomy adopted many Indian parameters, cosmological models, and computational techniques, including the use of sines.
These Indian influences were soon overwhelmed - although it is not completely clear why - by those of the Greek mathematical and astronomical texts that were translated into Arabic under the
Abbasid caliphs. Perhaps the greater availability of Greek works in the region, and of practitioners who understood them, favored the adoption of the Greek tradition. Perhaps its prosaic and
deductive expositions seemed easier for foreign readers to grasp than elliptic Sanskrit verse. Whatever the reasons, Sanskrit inspired astronomy was soon mostly eclipsed by or merged with the
"Graeco-Islamic" science founded on Hellenistic treatises."
11. ^ (Boyer 1991, "China and India" p. 211)
12. ^ Haq, Syed Nomanul, The Indian and Persian background, pp. 60–3 , in Seyyed Hossein Nasr, Oliver Leaman (1996), History of Islamic Philosophy, Routledge, pp. 52–70, ISBN 0415131596
13. ^ ^a ^b ^c ^d ^e ^f ^g ^h ^i ^j ^k ^l Gingerich, Owen (April 1986), "Islamic astronomy", Scientific American 254 (10): 74, http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm,
retrieved 2008-05-18
14. ^ ^a ^b Gandz, Solomon (1938), "The Algebra of Inheritance: A Rehabilitation of Al-Khuwarizmi", Osiris (University of Chicago Press) 5: 319–91, doi:10.1086/368492
15. ^ ^a ^b ^c ^d ^e ^f Prof. Ahmed Djebbar (June 2008). "Mathematics in the Medieval Maghrib: General Survey on Mathematical Activities in North Africa". FSTC Limited. http://muslimheritage.com/
topics/default.cfm?ArticleID=952. Retrieved 2008-07-19.
16. ^ ^a ^b ^c ^d O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics archive, University of St Andrews, http://
www-history.mcs.st-andrews.ac.uk/Biographies/Al-Qalasadi.html .
17. ^ Qur'an 2:189
18. ^ Syed Mohammad Hussain Tabatabai, "Volume 3: Surah Baqarah, Verse 189", Tafsir al-Mizan, http://www.almizan.org/Tafseer/Volume3/Baqarah47.asp, retrieved 2008-01-24
19. ^ Khalid Shaukat (September 23, 1997). "The Science of Moon Sighting". http://www.chowk.com/site/articles/index.php?id=4026. Retrieved 2008-01-24.
20. ^ ^a ^b ^c ^d Syed Mohammad Hussain Tabatabai, Volume 2: Surah Baqarah, Verses 142-151, , Tafsir al-Mizan, http://www.almizan.org/Tafseer/Volume2/Baqarah32.asp, retrieved 2008-01-24
21. ^ Qur'an 2:144
22. ^ Qur'an 2:150
23. ^ Samuel Parsons Scott (1904). "xxix: Moorish art in southern Europe". History of the Moorish Empire in Europe. 3 (1 ed.). Philadelphia & London: J.B. Lippincott Company. p. 447. ISBN
978-1402144851 (published in 2004). OCLC 3522061. http://www.archive.org/stream/historyofmoorish03scotuoft#page/447/mode/1up. Retrieved 2010-01-15.
24. ^ ^a ^b ^c (Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation
above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word
muqabalah is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation."
25. ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. p. 230. ISBN 0471543977.
"Al-Khwarizmi continued: "We have said enough so far as numbers are concerned, about the six types of equations. Now, however, it is necessary that we should demonsrate geometrically the
truth of the same problems which we have explained in numbers." The ring of this passage is obviously Greek rather than Babylonian or Indian. There are, therefore, three main schools of
thought on the origin of Arabic algebra: one emphasizes Hindu influence, another stresses the Mesopotamian, or Syriac-Persian, tradition, and the third points to Greek inspiration. The truth
is probably approached if we combine the three theories."
26. ^ Victor J. Katz, Bill Barton (October 2007), "Stages in the History of Algebra with Implications for Teaching", Educational Studies in Mathematics (Springer Netherlands) 66 (2): 185–201, doi:
27. ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. pp. 228–229. ISBN 0471543977.
"the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to al-Mamun, "the Commander of the Faithful"."
28. ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. p. 228. ISBN 0471543977.
"The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization - respects in which neither Diophantus nor the Hindus excelled."
29. ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. p. 229. ISBN 0471543977. "in six short chapters, of the six types of
equations made up from the three kinds of quantities: roots, squares, and numbers (that is x, x^2, and numbers). Chapter I, in three short paragraphs, covers the case of squares equal to roots,
expressed in modern notation as x^2 = 5x, x^2/3 = 4x, and 5x^2 = 10x, giving the answers x = 5, x = 12, and x = 2 respectively. (The root x = 0 was not recognized.) Chapter II covers the case of
squares equal to numbers, and Chapter III solves the cases of roots equal to numbers, again with three illustrations per chapter to cover the cases in which the coefficient of the variable term
is equal to, more than, or less than one. Chapters IV, V, and VI are mor interesting, for they cover in turn the three classical cases of three-term quadratic equations: (1) squares and roots
equal to numbers, (2) squares and numbers equal to roots, and (3) roots and numbers equal to squares."
30. ^ ^a ^b ^c (Boyer 1991, "The Arabic Hegemony" p. 228) "Diophantus sometimes is called "the father of algebra," but this title more appropriately belongs to al-Khwarizmi. It is true that in two
respects the work of al-Khwarizmi represented a retrogression from that of Diophantus. First, it is on a far more elementary level than that found in the Diophantine problems and, second, the
algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek Arithmetica or in Brahmagupta's work. Even numbers were written out in words rather than symbols!
It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither
al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers."
31. ^ (Derbyshire 2006, "The Father of Algebra" p. 31) "Diophantus, the father of algebra, in whose honor I have named this chapter, lived in Alexandria, in Roman Egypt, in either the 1st, the 2nd,
or the 3rd century CE."
32. ^ (Boyer 1991, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and
exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."
33. ^ Gandz and Saloman (1936), The sources of al-Khwarizmi's algebra, Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because
Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
34. ^ Rashed, R.; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 11–2, ISBN 0792325656, OCLC 29181926
35. ^ ^a ^b ^c Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. p. 234. ISBN 0471543977. "The Algebra of al-Khwarizmi usually is
regarded as the first work on the subject, but a recent publication in Turkey raises some questions about this. A manuscript of a work by 'Abd-al-Hamid ibn-Turk, entitled "Logical Necessities in
Mixed Equations," was part of a book on Al-jabr wa'l muqabalah which was evidently very much the same as that by al-Khwarizmi and was published at about the same time - possibly even earlier. The
surviving chapters on "Logical Necessities" give precisely the same type of geometric demonstration as al-Khwarizmi's Algebra and in one case the same illustrative example x^2 + 21 = 10x. In one
respect 'Abd-al-Hamad's exposition is more thorough than that of al-Khwarizmi for he gives geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution.
Similarities in the works of the two men and the systematic organization found in them seem to indicate that algebra in their day was not so recent a development as has usually been assumed. When
textbooks with a conventional and well-ordered exposition appear simultaneously, a subject is likely to be considerably beyond the formative stage. ... Note the omission of Diophantus and Pappus,
authors who evidently were not at first known in Arabia, although the Diophantine Arithmetica became familiar before the end of the tenth century."
36. ^ ^a ^b ^c O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews, http://
www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html .
37. ^ ^a ^b Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN
38. ^ ^a ^b ^c ^d ^e Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518.
ISBN 9780691114859.
39. ^ ^a ^b Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. p. 239. ISBN 0471543977. "Abu'l Wefa was a capable algebraist as well
as a trigonometer. ... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus - but without Diophantine analysis! ... In particular, to al-Karkhi is
attributed the first numerical solution of equations of the form ax^2n + bx^n = c (only equations with positive roots were considered),"
40. ^ O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji", MacTutor History of Mathematics archive, University of St Andrews, http://
www-history.mcs.st-andrews.ac.uk/Biographies/Al-Karaji.html .
41. ^ ^a ^b Swaney, Mark. History of Magic Squares.
42. ^ ^a ^b ^c ^d ^e ^f ^g Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. pp. 241–242. ISBN 0471543977. :
Omar Khayyam (ca. 1050-1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided
for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were
impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took
the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not
envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap
between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks
algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts
which are proved."
43. ^ ^a ^b ^c J. L. Berggren (1990), "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2): 304-9
44. ^ ^a ^b ^c O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk
/Biographies/Al-Tusi_Sharaf.html .
45. ^ Rashed, Roshdi; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 342–3, ISBN 0792325656
46. ^ Victor J. Katz, Bill Barton (October 2007), "Stages in the History of Algebra with Implications for Teaching", Educational Studies in Mathematics (Springer Netherlands) 66 (2): 185–201 [192],
47. ^ Ypma, Tjalling J. (December 1995), "Historical Development of the Newton-Raphson Method", SIAM Review (Society for Industrial and Applied Mathematics) 37 (4): 531–551 [534], doi:10.1137/1037125
48. ^ Ypma, Tjalling J. (December 1995), "Historical Development of the Newton-Raphson Method", SIAM Review (Society for Industrial and Applied Mathematics) 37 (4): 531–551 [539], doi:10.1137/1037125
49. ^ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for
operations and relations, as well as of the exponential notation."
50. ^ Gandz, Solomon (November 1931), "The Origin of the Ghubār Numerals, or the Arabian Abacus and the Articuli", Isis 16 (2): 393–424, doi:10.1086/346615
51. ^ D.J. Struik, A Source Book in Mathematics 1200-1800 (Princeton University Press, New Jersey, 1986). ISBN 0-691-02397-2
52. ^ P. Luckey, Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī (Steiner, Wiesbaden, 1951).
53. ^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences 500: 253–277 [254], doi:10.1111/
54. ^ ^a ^b Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences 500: 253–277 [259], doi:10.1111/
55. ^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences 500: 253–277 [260], doi:10.1111/
56. ^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences 500: 253–277 [261], doi:10.1111/
57. ^ O'Connor, John J.; Robertson, Edmund F., "Abu Ali al-Hasan ibn al-Haytham", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/
Biographies/Al-Haytham.html .
58. ^ Aydin Sayili (1960), "Thabit ibn Qurra's Generalization of the Pythagorean Theorem", Isis 51 (1): 35-37
59. ^ Kline, M. (1972), Mathematical Thought from Ancient to Modern Times, Volume 1, p. 193, Oxford University Press
60. ^ Kline, M. (1972), Mathematical Thought from Ancient to Modern Times, Volume 1, pp. 193-5, Oxford University Press
61. ^ R. Rashed (1994). The development of Arabic mathematics: between arithmetic and algebra. London.
62. ^ ^a ^b Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", The Journal of the American Oriental Society 123.
63. ^ (Katz 1998, p. 269):
In effect, this method characterized parallel lines as lines always equidisant from one another and also introduced the concept of motion into geometry.
64. ^ ^a ^b (Rozenfeld 1988, p. 65)
65. ^ ^a ^b (Smith 1992)
66. ^ Victor J. Katz (1998), History of Mathematics: An Introduction, p. 270, Addison-Wesley, ISBN 0321016181:
"In some sense, his treatment was better than ibn al-Haytham's because he explicitly formulated a new postulate to replace Euclid's rather than have the latter hidden in a new definition."
67. ^ ^a ^b Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [469], Routledge, London and New
"Khayyam's postulate had excluded the case of the hyperbolic geometry whereas al-Tusi's postulate ruled out both the hyperbolic and elliptic geometries."
68. ^ Boris A Rosenfeld and Adolf P Youschkevitch (1996), Geometry, p.467 in Roshdi Rashed, Régis Morelon (1996), Encyclopedia of the history of Arabic science, Routledge, ISBN 0415124115.
69. ^ ^a ^b ^c ^d ^e Victor J. Katz (1998), History of Mathematics: An Introduction, p. 270-271, Addison-Wesley, ISBN 0321016181:
"But in a manuscript probably written by his son Sadr al-Din in 1298, based on Nasir al-Din's later thoughts on the subject, there is a new argument based on another hypothesis, also
equivalent to Euclid's, [...] The importance of this latter work is that it was published in Rome in 1594 and was studied by European geometers. In particular, it became the starting point
for the work of Saccheri and ultimately for the discovery of non-Euclidean geometry."
70. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [469], Routledge, London and New York:
"In Pseudo-Tusi's Exposition of Euclid, [...] another statement is used instead of a postulate. It was independent of the Euclidean postulate V and easy to prove. [...] He essentially revised
both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements."
71. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [469], Routledge, London and New York:
"His book published in Rome considerably influenced the subsequent development of the theory of parallel lines. Indeed, J. Wallis (1616-1703) included a Latin translation of the proof of
postulate V from this book in his own writing On the Fifth Postulate and the Fifth Definition from Euclid's Book 6 (De Postulato Quinto et Definitione Quinta lib. 6 Euclidis, 1663). Saccheri
quited this proof in his Euclid Cleared of all Stains (Euclides ab omni naevo vindicatus, 1733). It seems possible that he borrowed the idea of considering the three hypotheses about the
upper angles of the 'Saccheri quadrangle' from Pseudo-Tusi. The latter inserted the exposition of this subject into his work, taking it from the writings of al-Tusi and Khayyam."
72. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [470], Routledge, London and New York:
"Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the
ninteenth century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse,
embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It
is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of
parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European couterparts. The first European attempt to prove the postulate on parallel lines - made by
Witelo, the Polish scientists of the thirteenth century, while revising Ibn al-Haytham's
Book of Optics
Kitab al-Manazir
) - was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the
above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that
Pseudo-Tusi's Exposition of Euclid
had stimulated borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
73. ^ Kennedy, E. S. (1969), "The History of Trigonometry", 31st Yearbook (National Council of Teachers of Mathematics, Washington DC) (cf. Haq, Syed Nomanul, The Indian and Persian background, pp.
60–3 , in Seyyed Hossein Nasr, Oliver Leaman (1996), History of Islamic Philosophy, Routledge, pp. 52–70, ISBN 0415131596 )
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75. ^ Kennedy, E.S. (1956), A Survey of Islamic Astronomical Tables; Transactions of the American Philosophical Society, 46, Philadelphia: American Philosophical Society, pp. 26–9
76. ^ ^a ^b O'Connor, John J.; Robertson, Edmund F., "Mathematics in medieval Islam", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/
Biographies/Al-Khwarizmi.html .
77. ^ ^a ^b ^c ^d Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN
78. ^ ^a ^b ^c O'Connor, John J.; Robertson, Edmund F., "Ghiyath al-Din Jamshid Mas'ud al-Kashi", MacTutor History of Mathematics archive, University of St Andrews, http://
www-history.mcs.st-andrews.ac.uk/Biographies/Al-Kashi.html .
79. ^ Al-Kashi, author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256
80. ^ Marlow Anderson, Victor J. Katz, Robin J. Wilson (2004), Sherlock Holmes in Babylon and Other Tales of Mathematical History, Mathematical Association of America, p. 139, ISBN 0883855461
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82. ^ ^a ^b ^c ^d O'Connor, John J.; Robertson, Edmund F., "Abu Abd Allah Muhammad ibn Muadh Al-Jayyani", MacTutor History of Mathematics archive, University of St Andrews, http://
www-history.mcs.st-andrews.ac.uk/Biographies/Al-Jayyani.html .
83. ^ O'Connor, John J.; Robertson, Edmund F., "Menelaus of Alexandria", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/
Menelaus.html . "Book 3 deals with spherical trigonometry and includes Menelaus's theorem."
84. ^ Boyer (1991). "Greek Trigonometry and Mensuration". p. 163. "In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles.
Included is a theorem without Euclidean analogue - that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric
spherical triangles); and the theorem A + B + C > 180° is established. The second book of the Sphaerica describes the application of spherical geometry to astronomical phenomena and is of little
mathematical interest. Book III, the last, contains the well known "theorem of Menelaus" as part of what is essentially spherical trigonometry in the typical Greek form - a geometry or
trigonometry of chords in a circle. In the circle in Fig. 10.4 we should write that chord AB is twice the sine of half the central angle AOB (multiplied by the radius of the circle). Menelaus and
his Greek successors instead referred to AB simply as the chord corresponding to the arc AB. If BOB' is a diameter of the circle, then chord A' is twice the cosine of half the angle AOB
(multiplied by the radius of the circle)."
85. ^ Kennedy, E. S. (1969), "The History of Trigonometry", 31st Yearbook (National Council of Teachers of Mathematics, Washington DC): 337 (cf. Haq, Syed Nomanul, The Indian and Persian background,
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88. ^ Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255-259. Addison-Wesley. ISBN 0321016181.
89. ^ F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
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91. ^ Kennedy, Islamic Astronomical Tables, p. 51
92. ^ Benno van Dalen, PARAMS (Database of parameter values occurring in Islamic astronomical sources), "General background of the parameter database"
93. ^ Kennedy, Islamic Astronomical Tables, pp. 17-23
94. ^ Kennedy, Islamic Astronomical Tables, p. 1
95. ^ (Dallal 1999, p. 163)
96. ^ (Dallal 1999, p. 164)
97. ^ (Singer 1959, p. 151) (cf. (Zaimeche 2002))
98. ^ (Wickens 1976) (cf. (Zaimeche 2002))
99. ^ 23rd Annual Conference on the History of Arabic Science, Aleppo, Syria, October 2001 (cf. (Zaimeche 2002))
100. ^ Aulie, Richard P. (March 1994), "Al-Ghazali Contra Aristotle: An Unforeseen Overture to Science In Eleventh-Century Baghdad", Perspectives on Science and Christian Faith 45: 26–46 (cf.
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101. ^ (Saliba 2007)
102. ^ ^a ^b "Khwarizm". Foundation for Science Technology and Civilisation. http://muslimheritage.com/topics/default.cfm?ArticleID=482. Retrieved 2008-01-22.
103. ^ Saliba, George (1980), "Al-Biruni", in Strayer, Joseph, Dictionary of the Middle Ages, 2, Charles Scribner's Sons, New York, p. 249
104. ^ (Saliba 1994b, pp. 245, 250, 256-257)
105. ^ Saliba, George (Autumn 1999), "Seeking the Origins of Modern Science?", BRIIFS 1 (2), http://www.riifs.org/review_articles/review_v1no2_sliba.htm, retrieved 2008-01-25
106. ^ (Saliba 1994b, pp. 42 & 80)
107. ^ Dallal, Ahmad (2001-2002), The Interplay of Science and Theology in the Fourteenth-century Kalam, From Medieval to Modern in the Islamic World, Sawyer Seminar at the University of Chicago,
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108. ^ (Huff 2003, pp. 217-8)
109. ^ (Saliba 1994b, pp. 254 & 256-257)
110. ^ David A. King, Astronomy in the Service of Islam, (Aldershot (U.K.): Variorum), 1993.
111. ^ Gharā'ib al-funūn wa-mulah al-`uyūn (The Book of Curiosities of the Sciences and Marvels for the Eyes), 2.1 "On the mensuration of the Earth and its division into seven climes, as related by
Ptolemy and others," (ff. 22b-23a)[1]
112. ^ Edward S. Kennedy, Mathematical Geography, pp. 187-8, in (Rashed & Morelon 1996, pp. 185-201)
113. ^ David A. King (1996), "Astronomy and Islamic society: Qibla, gnomics and timekeeping", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 1, p. 128-184 [153]. Routledge
, London and New York.
114. ^ Scheppler, Bill (2006), Al-Biruni: Master Astronomer and Muslim Scholar of the Eleventh Century, The Rosen Publishing Group, ISBN 1404205128
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118. ^ Lenn Evan Goodman (1992), Avicenna, p. 31, Routledge, ISBN 041501929X.
119. ^ Behnaz Savizi (2007), "Applicable Problems in History of Mathematics: Practical Examples for the Classroom", Teaching Mathematics And Its Applications (Oxford University Press) 26 (1): 45–50,
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124. ^ El-Bizri, Nader (2007), "In Defence of the Sovereignty of Philosophy: al-Baghdadi’s Critique of Ibn al-Haytham’s Geometrisation of Place", Arabic Sciences and Philosophy: A Historical Journal
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125. ^ Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642, in (Morelon & Rashed 1996, pp. 614-642)
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131. ^ Victor J. Katz (1998), History of Mathematics: An Introduction, p. 255-259, Addison-Wesley, ISBN 0321016181:
"Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used
such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary n. He stated his theorem
for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer.
132. ^ O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji", MacTutor History of Mathematics archive, University of St Andrews, http://
www-history.mcs.st-andrews.ac.uk/Biographies/Al-Karaji.html .
"Al-Karaji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle."
133. ^ Katz (1998), p. 255:
"Al-Karaji's argument includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 1^3) and the deriving of the truth for n
= k from that of n = k - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather
than proceeding upward. Nevertheless, his argument in al-Fakhri is the earliest extant proof of the sum formula for integral cubes."
134. ^ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), p. 163-174:
"The central idea in
ibn al-Haytham
's proof of the sum formulas was the derivation of the equation [...] Naturally, he did not state this result in general form. He only stated it for particular integers, [...] but his proof
for each of those
is by induction on
and is immediately generalizable to any value of
135. ^ Katz (1998), p. 255-259.
136. ^ Katz (1998), p. 259:
"Like the proofs of al-Karaji and ibn al-Haytham, al-Samaw'al's argument contains the two basic components of an inductive proof. He begins with a value for which the result is known, here n
= 2, and then uses the result for a given integer to derive the result for the next. Although al-Samaw'al did not have any way of stating, and therefore proving, the general binomial theorem,
to modern readers there is only a short step from al-Samaw'al's argument to a full inductive proof of the binomial theorem."
Further reading
• Berggren, J. Lennart (1986). Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag. ISBN 0-387-96318-9. (Reviewed: Toomer, Gerald J. (1988), "Episodes in the Mathematics of
Medieval Islam", American Mathematical Monthly 95 (6): 567, doi:10.2307/2322777, http://links.jstor.org/sici?sici=0002-9890%28198806%2F07%2995%3A6%3C567%3AEITMOM%3E2.0.CO%3B2-3 ; Hogendijk, Jan
P. (1989), "Episodes in the Mathematics of Medieval Islam by J. Lennart Berggren", Journal of the American Oriental Society 109 (4): 697–698, doi:10.2307/604119 )
• Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". in Victor J. Katz. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. ISBN
• Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. ISBN 0471543977.
• Cooke, Roger (1997). "Islamic Mathematics". The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0471180823.
• Daffa', Ali Abdullah al- (1977). The Muslim contribution to mathematics. London: Croom Helm. ISBN 0-85664-464-1.
• Daffa, Ali Abdullah al-; Stroyls, J.J. (1984). Studies in the exact sciences in medieval Islam. New York: Wiley. ISBN 0471903205.
• Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, Rutgers University, http://www.math.rutgers.edu/~cherlin/History/Papers2000/eder.html,
retrieved 2008-01-23
• Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd Edition ed.). Princeton University Press. ISBN 0691006598. (Reviewed: Katz, Victor J. (1992), "
The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph", The College Mathematics Journal 23 (1): 82–84, doi:10.2307/2686206 )
• Katz, Victor J. (1998), History of Mathematics: An Introduction, Addison-Wesley, ISBN 0321016181, OCLC 38199387
• Kennedy, E. S. (1984). Studies in the Islamic Exact Sciences. Syracuse Univ Press. ISBN 0815660677.
• O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/
HistTopics/Arabic_mathematics.html .
• Rashed, Roshdi (2001). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Transl. by A. F. W. Armstrong. Springer. ISBN 0792325656.
• Rashed, Roshdi (2009). Al-Khwarizmi:The Beginnings of Algebra. Transl. by Judith Field with revision of trans. by Nader El-Bizri. Saqi Books. ISBN 0863564305.
• Rozenfeld, Boris A. (1988), A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, Springer Science+Business Media, ISBN 0387964584, OCLC 15550634
• Sánchez Pérez, José A (1921). Biografías de Matemáticos Árabes que florecieron en España. Madrid: Estanislao Maestre.
• Sezgin, Fuat (1997) (in German). Geschichte Des Arabischen Schrifttums. Brill Academic Publishers. ISBN 9004020071.
• Smith, John D. (1992), "The Remarkable Ibn al-Haytham", The Mathematical Gazette (Mathematical Association) 76 (475): 189–198, doi:10.2307/3620392
• Suter, Heinrich (1900). Die Mathematiker und Astronomen der Araber und ihre Werke. Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft. Leipzig.
• Youschkevitch, Adolf P.; Boris A. Rozenfeld (1960). Die Mathematik der Länder des Ostens im Mittelalter. Berlin. Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62–160.
• Youschkevitch, Adolf P. (1976). Les mathématiques arabes: VIII^e-XV^e siècles. translated by M. Cazenave and K. Jaouiche. Paris: Vrin. ISBN 978-2-7116-0734-1.
External links
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Road to Solovay's Land.
up vote 6 down vote favorite
In the first semester of 2012 I took a course in General Topology and Set Theory, at undergraduate level. For topology, I was instructed to use Engelking's General Topology; albeit I had a great
difficult to approach it, I got used to the text and did (and I'm still doing) some exercises (but none of the problems until now). For the Set Theory course we used Jech & Hrbacek's Introduction to
Set Theory, which I think was suitable for my level back there. In these courses, I heard about Boolean Algebras, Forcing, Independence Proofs, Models, Topological Games, Kunen's book (which I just
bought a copy), and others interesting things that caused me to change my favorite mathematical area (in fact I was a physics undergrad student when this year began).
In this semester, I enrolled myself in Measure and Integration course, also at undergraduate level, where I discovered about Solovay's model, which completely drove me to madness.
I'm looking for advice about my background and the path that I have to follow to reach these mentioned topics; is too early to begin? do my background is sufficiently enough to start? And where to
begin with ? what books do I have to read?
P.S.: I had no background in mathematical logic, the only thing I can do is some proofs with truth-tables.
soft-question reference-request set-theory lo.logic
add comment
2 Answers
active oldest votes
With no background in logic, it's a bit of a long road to Solovay's model; the good news is that every step of it is incredibly interesting!
(Some of this you may already know - I'm just listing a complete roadmap to Solovay.)
To start with, you need a good understanding of what models of ZFC look like. The last couple chapters of Hrbacek/Jech cover this; alternatively, it's at the beginning of Kunen's book.
Then comes forcing. Forcing is basically a way of building models of ZFC "to specification." This is a big deal, since ZFC is a really complicated theory, unlike, say, the theory of
rings: while it's very easy to build lots and lots of (models of the theory of) rings, it's incredibly hard to build models of ZFC, and forcing accomplishes this.
The picture of forcing in ZFC is reasonably straightforward (although the details, of course, take a lot of work): you take a model V of ZFC to start with, look at some poset P in V, and
the machinery of forcing gives you a* model V[G] containing V with properties that can be discovered reasonably easily by looking at P, and conversely, there are natural strategies for
building a P such that the resulting V[G] will have properties you want it to. Playing around with Martin's Axiom might make forcing make a lot more sense; it certainly did for me!
(*OK, actually forcing gives you lots of different models, one for each "generic filter" G of P over V, but for almost all intents and purposes the precise generic filter doesn't matter,
and all the information is contained in the poset P alone.)
up vote 12
down vote Now we can prove lots of nice properties about forcing over models of ZFC, including one which for our purposes is actually a bad property: any V[G] is also a model of ZFC. The reason
accepted this is bad for us is that Solovay's model is definitely not a model of Choice, so we have to add another layer of complexity: the symmetric submodel construction. By doing some
complicated shenanigans** with automorphisms of P, we can build intermediate models W of ZF set theory, containing V and contained in V[G]. Solovay's model is built in this fashion.
(**Specifically, elements of the extension V[G] have "names" in V; the symmetric submodel construction is a way of defining "hereditarily symmetric" names, which are basically names
fixed by "a lot" of automorphisms of P (the precise choice of definition of "a lot" determines the properties of the symmetric submodel), and models W consist of the elements of V[G]
with hereditarily symmetric names.)
So there's really four different steps in getting to Solovay's model: understanding the ZFC picture of the universe (Hrbacek/Jech's final chapters, or Kunen's intro chapter, do this
well); understanding forcing over models of ZFC (Kunen covers this well, as does Jech's gigantic set theory tome); understanding symmetric submodels (this is covered in Jech's big tome,
but not Kunen; so it might be a good idea to use Jech throughout); and finally, understanding the details behind Solovay's particular construction (covered in a bunch of sources,
including Jech's book). Basically, Jech's giant tome of set theory - "Set Theory," Third Millennium Edition - has everything you need. It's pretty expensive, though.
Good luck!
Having some recursion theory knowledge would also be very helpful to understand Solovay's construction. – Liang Yu Jan 13 '13 at 0:33
To elaborate on Yu's point: recursion theory (AKA computability theory) is definitely not a prerequisite for Solovay's construction. The relevance is that the intuition behind the
symmetric submodel construction is (at least for me) a "limited knowledge" sort of thing: I imagine the model of set theory I'm building as a character who is only aware of some
(specific) portion of the larger model $V[G]$. Recursion theory is all about this sort of picture, so a familiarity with recursion theory can really make Solovay's construction
"click." – Noah S Jan 13 '13 at 21:33
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Let me give an alternative ending to Noah's road map. The splitting point is at symmetric models.
After you've understood the basics of forcing well, you can switch to Kanamori's The Higher Infinite. In the chapter about the real numbers and forcing he again reviews forcing (and if
you're new to this - such review is always good) and constructs Solovay's model in a very clear approach.
He avoids [1] talking about symmetric models (which can be a rather complicated tool) by using the "external" construction: we add some sort of generic set to $V$ then we consider an
inner model of $V[G]$ which is $HOD(\mathbb R)$ or $L(\mathbb R)$, the latter being thrown around a lot in discussions about models of set theory without choice.
In Kanamori you can find a good introduction to large cardinals (if you haven't run into them in previous steps) which also play a role in Solovay's construction, although that appears in
up vote 10 another chapter of the book.
down vote
I want to add that studying the construction of symmetric extensions is a good idea. This is an extremely illuminating construction which sheds a lot of light on how set theory works, at
least this is how I felt in the past year. However for this particular case I think that using the approach of relative constructibility is better.
1. This is not entirely true that Kanamori avoids the symmetric models because as it turns out all symmetric models are $HOD(A)$ (whatever that means) of some generic set $A$. In the
case of Solovay's model it is just much simpler to use this sort of construction rather going through the complication of symmetric forcing.
1 Let me second Asaf's statement that, in this case, inner models are clearer than symmetric submodels. My preference for inner models comes purely from the fact that I really like other
things they do - constructing amorphous sets, for example - where they are very clear, so I like them for their own sake. – Noah S Aug 11 '12 at 17:32
Noah, I suppose you meant your preference for symmetric models and not inner models. Do note that Grigorieff proved that every symmetric extension is actually this sort of an inner
model. In particular those that have amorphous sets etc.; however I agree completely that for general negations of AC symmetric models are preferable because they allow you better
control, in some sense. – Asaf Karagila Aug 11 '12 at 17:55
Yeah, that was a typo. Re: Grigorieff, I'm aware of that; I just meant (as you said) that many results make more sense to me when phrased in terms of symmetric submodel constructions
than as inner models. – Noah S Aug 12 '12 at 2:56
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Not the answer you're looking for? Browse other questions tagged soft-question reference-request set-theory lo.logic or ask your own question.
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problems with RLC bandpass filter simulation in LTspice - All About Circuits Forum
Originally Posted by
Is this correct? and that it should be more around 100Hz.
fo =100Hz
Originally Posted by
In this case, my question is now a little different. Everywhere else I've looked it said that the natural frequency should be 1/sqrt(LC) so could you please explain to me why f=sqrt(fh*fl) is valid?.
for LC circuit is equal 1/sqrt(LC) becaues only for this frequency
Xc = XL
is located "halfway" between F2, F1.
But for mother nature "halfway" in not in the "linear" axis (arithmetic mean) but in geometric mean.
And that why Fo = √(F1*F2) it's true.
So for you circuit
Fo = 100Hz and for R = 1KΩ ---> Q = 0.101
L = 0.160762H
C = 15.7563μF
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Looking for constructive criticism
05-24-2008 #1
Registered User
Join Date
May 2008
Looking for constructive criticism
I am finished with my first assignment in a C prog 2 class. I am going to post it here and hopefully get some constructive criticism. This program is nothing really to shout about...adds and
subtracts arbitrarily large numbers, not designed to handle negative numbers. I know that there is nothing earthshattering in here, but I have definitely learned a lot in the past two weeks. If
some of you programming masters can take a look and provide me with some constructive criticism it would be much appreciated.
I will also attach this as a .c file
I am trying to improve my knowledge of C and programming in general, feel free to be brutal.
// Wes Kendrick
// BIGINT.C
// THIS PROGRAMS ALLOWS FOR SIMPLE MANIPULATION OF LARGE INTEGERS.
// THE USER CAN ADD, OR SUBTRACT TWO LARGE INTEGERS INPUT FROM THE KEYBOARD
// THIS PROGRAM DOES NOT ALLOW FOR A NEGATIVE RESULT. THE SMALLER NUMBER WILL
// ALWAYS BE SUBTRACTED FROM THE LARGER NUMBER WHEN SUBTRACTION IS THE OPERATION
// DESIRED
// This program was written in Dev-C++ 4.9.9.2, and is written strictly for use
// on a PC. Not all functions used are assumed to be ANSI C
#include <stdio.h>
#include <stdlib.h>
#define TRUE 1
#define FALSE 0
#define ADD 1
#define SUBTRACT 2
#define QUIT 3
#define MAX 200
//Preconditions: the first parameter is string that stores
// only contains digits, doesn't start with
// 0, and is 200 or fewer characters long.
//Postconditions: The function will read the digits of the
// large integer character by character,
// convert them into integers and place them in
// nodes of a linked list. The pointer to the
// head of the list is returned.
struct integer* read_integer();
//Preconditions: p is a pointer to a big integer, stored in
// reverse order, least to most significant
// digit, with no leading zeros.
//Postconditions: The big integer pointed to by p is
// printed out.
void print_bigint(struct integer *p);
//Preconditions: p and q are pointers to big integers,
// stored in reverse order, least to most
// significant digit, with no leading zeros.
//Postconditions: A new big integer is created that stores
// the sum of the integers pointed to by p
// and q and a pointer to it is returned.
struct integer* add(struct integer *p, struct integer *q);
//Preconditions: p and q are pointers to big integers,
// stored in reverse order, least to most
// significant digit, with no leading zeros.
//Postconditions: A new big integer is created that stores
// the absolute value of the difference
// between the two and a pointer to this is
// returned.
struct integer* subtract(struct integer *p, struct integer *q);
//Preconditions: Both parameters of the function are
// pointers to struct integer.
//Postconditions: The function compares the digits of two
// numbers recursively and returns:
// -1 if the first number is smaller than the second,
// 0 if the first number is equal to the second number,
// 1 if the first number is greater than the second.
int compare(struct integer *p, struct integer *q);
//Preconditions: Both parameters point to struct integer
//Postconditions: Prints spaces so that the numbers are
// aligned to the right
//Preconditions: p, q, and answer point to struct integer, operation is
// the operation to be performed
//Postconditions: Formats the output
void format_output(struct integer *p, struct integer *q, struct integer *answer, int operation);
//Preconditions: choice, low, and high are all int...low is low bound of
// acceptable input, high is high bound
//Postconditions: Returns TRUE if input is valid, otherwise returns FALSE
int valid_choice(int choice, int low, int high);
//Preconditions: None
//Postconditions: Retrieves the user choice from the keyboard and returns it
int get_user_choice();
//Preconditions: how_many is an int, character is a char
//Postconditions: prints out character how_many times
void print_characters(int how_many, char character);
//Preconditions: answer is of type struct integer, it has at least one
// leading zero (because method of complements is used for
// subtraction.)
//Postconditions: All leading zeroes are removed, and the size
// of answer is adjusted accordingly
void remove_zeroes(struct integer *answer);
struct integer
int *digits;
int size;
int main()
int choice, operation;
struct integer *bigint1,*bigint2,*answer;
// Run until user chooses to quit
if(choice !=QUIT)
printf("Please enter the first number: ");
printf("\nPlease enter the second number: ");
// Executes users choice
case ADD:
answer = add(bigint1,bigint2);
case SUBTRACT:
answer = subtract(bigint1,bigint2);
} // End switch
// Frees memory
// End program
printf("\n\nThank you for using the Big Int calculator!\n\n\n");
return 0;
int get_user_choice()
int choice;
printf("1. Add two large integers\n2. Subtract two large integers\n3. Quit\n");
printf("\n\nEnter the number corresponding to your choice: ");
// Checks for valid input in the while loop
return choice;
int valid_choice(int choice, int low, int high)
// Checks entry against bounds
if(choice<low || choice > high)
printf("\n\nThat is invalid input.\n\n\n");
return FALSE;
return TRUE;
struct integer* read_integer()
// Temporary string is created to read in the number from the user
char string_num[MAX];
// Allocates memory for the struct integer, the address of this will be returned
struct integer *temp= (struct integer *)malloc(sizeof(struct integer));
int counter;
// Reads string
// Sets the size of the "integer"
// Dynamically allocates memory for the array of int
temp->digits=(int *)(malloc((temp->size*sizeof(int))));
// Subtracts '0' from the character to get the digit...numbers are stored
// with the least signifigant digit in index 0...this makes the code
// slightly simpler
return temp;
struct integer* add(struct integer *p, struct integer *q)
// Allocates integer
struct integer *answer=malloc(sizeof(struct integer));
int counter, carry=0;
// If p is bigger than q, size the array in answer to the size of p
// To simplify the code, we reallocate memory to the smaller number and set its
// size to the size of the bigger one (if they are the same number of digits, nothing
// happens
for(counter=q->size; counter < p-> size; counter++)
// Q is bigger, repeat process as above, but switch p and q
if(q->size != p->size)
for(counter=p->size;counter< q->size;counter++)
// Allocates enough memory for the array of int inside answer
answer->digits=(int *) malloc(answer->size*sizeof(int));
// Runs counter for entire length of answer
for(counter=0;counter< (answer->size);counter++)
// Sets answer->digits[counter] to the sum of the digit in the same index position
// of q and p, as well as "carry" which is initialized to zero
answer->digits[counter]=p->digits[counter]+q->digits[counter] + carry;
// Number is greater than 9...must be reduced and reset carry
// Carry is zero, answer->digits[counter] is less than 10
// End of the for-loop, if carry is still 1, reallocate the size of answer->digits
// to answer->size+1, and increment answer->size
// turns digit from > 10
// Will never be greater than one...(worst case example: 999+999 = 1998 (4 digits, leading 1)
// Reallocate p and q to their original sizes, only the added zeroes will be removed
// address of answre is returned
return answer;
struct integer* subtract(struct integer *p, struct integer *q)
// COMPLEMENTS stores the nines complement of a digit 0-9
int COMPLEMENTS[10], counter, difference;//, not_a_leading_zero, how_big_answer;
// Required for complement method of subtraction, to add one to the result
// This segment sets the attributes of one
struct integer *one= (struct integer *)(malloc(sizeof(struct integer)));
one->digits=(int *)(malloc(one->size*sizeof(int)));
// Nines complement will hold the nines complement of the smaller number
struct integer *nines_complement=(struct integer *)(malloc(sizeof(struct integer)));
struct integer *answer;
// Initializes the nines complement of digits 0-9
// If p > q
// Difference will represent the number of additional digits that must
// be added to make both numbers the same size
difference=p->size - q->size;
// sets nines_complement->size to the bigger numbers size
// Allocates memory for the int array inside nines_complement
nines_complement->digits=(int *)(malloc(nines_complement->size*sizeof(int)));
// Sets all "unaccounted for digits" to 9...counter stops when it equals the difference in
// size between nines_complement and the smaller number
// sets the rest of the "accounted for digits" to the respective digit in the smaller of
// the two numbers (in this case q)
// This is the method of complements, the intermediate step answer is set to
// the sum of the larger number and the nines complement of the smaller number
// In this case, p is smaller (or equal to) q...same logic as above except the
// p's and q's have been switched
difference=q->size - p->size;
nines_complement->digits=(int *)(malloc(nines_complement->size*sizeof(int)));
// Second to last step of complements method, one is added to answer
// Finished with nines_complement and one
// We know that the size of answer has been incremented by one, and this new
// digit holds a 1...final step of this algorithm is to remove the one.
// Removes leading zeroes from answer
return answer;
void remove_zeroes(struct integer *answer)
// First digit will be a zero
int not_a_leading_zero=0, counter;
// Runs until counter > 0 because if answer is 0 we want it left that way
if(answer->digits[counter]==0 && not_a_leading_zero==0)
// Size of answer is smaller
// Reallocate memory for answer, only the leading zeroes will be removed
void print_bigint(struct integer *p)
int counter;
// Since the number is stored in reverse order, we print it out in reverse order
// to get the number as it should appear
int compare(struct integer *p, struct integer *q)
int counter;
// If p has more digits then it is larger
return 1;
// if p has fewer digits then p is smaller
return -1;
// If any digit from p is greater than the digit in the same place of q
// is larger, then p is larger
return 1;
// If any digit from q is greater than the digit in the same place of p
// then q is larger
return -1;
// Digits are equal
return 0;
void format_output(struct integer *p, struct integer *q, struct integer *answer,
int operation)
int counter, difference=0;
case ADD:
// If p is greater than q, then print p and print spaces equal to the
// difference in size of p and q...this is strictly aesthetic
print_characters(p->size-q->size,' ');
// If q is larger, then we print spaces, then print p, then print q
print_characters(q->size-p->size, ' ');
// The specs for the program do not allow for negative numbers...this uses
// the same logic as above, only prints the larger number on top of the smaller
// number. Since the result of subtraction can have a variety of sizes, we set
// difference to the larger numbers size minus the size of the answer
case SUBTRACT:
print_characters(p->size-q->size,' ');
print_characters(q->size-p->size, ' ');
} // End Switch
// Prints spaces to format the answer, this will always be zero
// in the case of addition
print_characters(difference,' ');
void print_characters(int how_many, char character)
int counter;
// Prints character n times
You're doing this is an extremely complicated way, way more than necessary.
C has built-in types for integers. You can use int or long long for numbers instead and it would be easier.
Note also that some structs have pointers as members where you allocate memory to, but you never free those. Freeing the struct does not free the memory associated with its pointers.
You also read strings with scanf in the wrong way. See http://cpwiki.sourceforge.net/Buffer_overrun
Also take a look at http://www.cse.scu.edu/~tschwarz/COE...es/Strings.ppt if you can, as it highlights the pitfalls and security issues of C.
Last edited by Elysia; 05-24-2008 at 03:04 PM.
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
You are right about the scanf, I was just reading someone elses post about that....thanks for pointing that out.
In the assignment we are to create a the type "integer" that is meant to handle arbitrarily large integers, say hundreds of digits long. Although the output may not be pretty with a 199 digit
number, the answer should still be correct.
Could you elaborate a little further about how to free the memory associated with the structs pointers?
Thanks for the help
Data type
Typical academia type stuff, do something the hard way just to make you learn something
I mean if you do
struct s
char* p;
s* s1 = malloc(sizeof(*s1));
s1.p = malloc(10);
/* You never free p; it is not automatically freed */
And btw, never cast the return of malloc.
And I still think you can simplify it. Just treat the whole number as a string and convert it to a number when doing archimethic operations.
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
in your example
Is there a way to free "p" ?
say would
Last edited by wd_kendrick; 05-24-2008 at 03:24 PM.
Yes, that would be the way to do it.
Each allocated pointer must be separately freed.
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
As said before, you making things more complicated than it should be and perhaps most of the thing can be done more easily. With that i could that you have learnt quite a lot of logics and stuff
which you could impalement them in C. But which are not really necessary at all here.
I can also see few thing which, like casting malloc and realloc which you don't have to do them, since the return type is of void * type for both two function. And you read integer as a string
and then you convert them using a standard procedure, as said before you could use int data type or you could still read them as a string and them convert them using atoi or using sscanf function
as well.
And keeping things simple will give your code more clarity and more marks indeed
Right on
I have made the changes recommended above...Most of the function prototypes were provided by the instructor, and we had to implement them using his defined pre and post conditions. Some of the
code may seem overly complex because of this.
As far as atoi goes...I could not get it to work...
so I decided to subtract a '0' and get the same result.
This code was quite a leap from the topics covered in Programming 1 (where we barely even touched on pointers, and covered what a struct was in the last day of class)
Maybe some of the code can help someone who i having trouble passing structs to functions, returning pointers to structs from functions, using malloc, or realloc.
It would have certainly saved me a lot of time!
Thanks again, any other advice is welcome.
atoi usage
#include <stdio.h>
#include <stdlib.h>
int main()
char *numinstr = "1234";
int numinint;
numinint = atoi( numinstr);
if( numinint != 0 )
printf("Converted number - %d\n", numinint );
printf("Conversion failed\n");
return 0;
/* my output
Converted number - 1234
Atoi or strtol takes a string, and not a single char.
The fact that you have troubles with pointers suggests that you should make the program less complex.
Btw, ssharish2005, don't forget that atoi returns 0 if the string is "0"!
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
Hello Wes, i am a Novice in C programming and eager to learn more.
Do you have any advice how to understand C better since i am having a hard time understanding the whole concept.
This is my understanding so far:
1 Declare variable so compiler will have variables to use
2. Write the main function
3. Larger programs have to have multiple functions that will be called by main
Any other suggestions??
kirksson, you need to google search for C tutorials. Go through them on-line, and you will learn a lot, free.
You can't beat that!
They will cover a lot more info than we can cover in a forum, and the information is already there, so no need to re-do it, here.
Kendrick - I'm tempted to show you how easily this job can be done - but I don't want to send you into terminal depression.
In RL, don't ever do a program in such a Rube Goldberg, way. I know you had a lot of conditions from your instructor, that had to be met. Just letting you know.
Last edited by Adak; 05-27-2008 at 06:30 PM.
Ah, but books are usually more in-depths than tutorials, so one of those may be suggested. See the recommended books thread.
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
I think "implement an arbitrarily big number data type along with its basic operators" is part of virtually every programming course in universities. I think it's pretty useful too.
Computer Programming: An Introduction for the Scientifically Inclined
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FastC++: Coding Cpp Efficiently
A common operation for two 3D vectors is the
cross product
|a.x| |b.x| | a.y * b.z - a.z * b.y |
|a.y| X |b.y| = | a.z * b.x - a.x * b.z |
|a.z| |b.z| | a.x * b.y - a.y * b.x |
Executing this operation using scalar instructions requires 6 multiplications and three subtractions. When using vectorized SSE code, the same operation can be performed using 2 multiplications, one
subtraction and 4 shuffle operations:
inline __m128 CrossProduct(__m128 a, __m128 b)
return _mm_sub_ps(
_mm_mul_ps(_mm_shuffle_ps(a, a, _MM_SHUFFLE(3, 0, 2, 1)), _mm_shuffle_ps(b, b, _MM_SHUFFLE(3, 1, 0, 2))),
_mm_mul_ps(_mm_shuffle_ps(a, a, _MM_SHUFFLE(3, 1, 0, 2)), _mm_shuffle_ps(b, b, _MM_SHUFFLE(3, 0, 2, 1)))
Both registers
contain three floats (x, y and z) where the highest float of the 128-bit register is unused. The values can be loaded using the
function or SSE set methods such as
_mm_setr_ps(x, y, z, 0)
3 comments:
1. Was looking for a simd cross product implementation and found your nice post. I eventually discovered that you can do it with only 3 shuffle instructions:
inline __m128 CrossProduct( __m128 a, __m128 b )
__m128 result = _mm_sub_ps(
_mm_mul_ps(b, _mm_shuffle_ps(a, a, _MM_SHUFFLE(3, 0, 2, 1))),
_mm_mul_ps(a, _mm_shuffle_ps(b, b, _MM_SHUFFLE(3, 0, 2, 1)))
return _mm_shuffle_ps(result, result, _MM_SHUFFLE(3, 0, 2, 1 ));
1. I don't know what the other anonymous guy who came on here was smoking, but his code does not produce a proper 3D cross product... like at all... Here's what the original code would have done
with two vectors, A, and B:
(A3, A0, A2, A1) * (B3, B1, B0, B2) - (A3, A1, A0, A2) * (B3, B0, B2, B1)
Since the multiplications go straight through, slot by slot, the resulting two vector is:
(0, (A0B1 - A1B0), (A2B0 - A0B2), (A1B2 - A2B1))
Here's what the code the anonymous person posted does:
(B0, B1, B2, B3) * (A3, A0, A2, A1) - (A0, A1, A2, A3) * (B3, B0, B2, B1)
Which results in the vector:
((B2A2 - A2B2), (B0A3 - A0B3), (B3A1 - A3B1), (B1A0 - A1B0))
These are clearly not the same (to anyone who understands anything about... well... numbers... or variables... so anyone who would ever be interested in this sort of thing, anyway :P)
2. This code produces (1, 0, 0) X (0, 1, 0) = (0, 0, -1) so we don't have a right-handed coordinate system.
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MathGroup Archive: May 2005 [00185]
[Date Index] [Thread Index] [Author Index]
Mathematica Notebook Organiztion
• To: mathgroup at smc.vnet.net
• Subject: [mg56816] Mathematica Notebook Organiztion
• From: "David Park" <djmp at earthlink.net>
• Date: Fri, 6 May 2005 03:01:35 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com
I've made this a new topic because we have rather drifted off from the
subject of writing packages to the subject of using notebooks in the best
It is my view that Mathematica notebooks (and similar such entities) are an
entirely new publishing form. They FAR SURPASS printed books and articles
because of the ability to interactively meld text, calculations, graphics
and animations in one document. Theodore Gray deserves a lot of credit for
his work on this concept. We are still learning how to use this media. But
things are not perfect yet and Professor Siegman has touched on some issues.
There is no reason that the Initialization and Routines Sections couldn't be
at the end of the notebook. The Input cells in these Sections should be made
into Initialization cells (and choose NOT to save as an AutoSave package).
That way one doesn't have to necessarily evaluate a notebook from the top.
The initializations are automatically performed when the first statement,
anywhere, is evaluated. I like to make my notebooks such that a reader can
start at any Section and begin evaluating. If this is not possible because
of a rigid progression in the sections then the reader should be so
Often I will select the Initialization and Routine section headings and
change the FontColor to Gray. I also often add "Automatically Initialized".
This subdues the sections and tells the reader he can generally ignore them.
Sections are not automatically opened when Initialization cells are
evaluated. My experience is that the sections remain closed. Also you can
select a Section and completely evaluate it without ever opening it, or
seeing the results. (I've had super geniuses complain that they evaluated my
notebook but got no results, simply because they didn't know how to open
Graphics code can be put in closed cells in the running sections. It doesn't
necessarily have to be put in the Routines section. That way you can
intermix text, calculations and graphics in a smooth manner. The only
problem is getting the reader to evaluate the closed cells, even if it has
been carefully explained in an Introduction. They are so thin and small new
readers often overlook them. It might be nice if one had the option of
having a closed cell display a cell tag. It would also be nice if closed
cells could be opened and closed in the same way as Sections.
It is also possible to generate proofs, derivations or step by step
calculations by interspersing Print statements with %% referenced
statements. These can also be put in closed cells so that the main code is
For printing (It will take some time for people to give up the security
blanket of printed documents - inferior as they are!) there is no reason why
some Sections can't be open and others closed.
Professor Siegman's case:
Text (a few paragraphs introducing the section)
is a good point. I don't see any direct way around it other than making the
Text an Introductory Subsection, which may be objectionable because it is so
short, or manually closing these Text cells, but this is too difficult for
the reader to work with. Perhaps there might be a FrontEnd command that
gives the "outline view".
Another approach would be to make a Table of Contents Section. The various
items in the Table of Contents could actually be links to the corresponding
sections of the notebook. This is like pdf documents where there is often a
table of contents with links in the side bar. It requires extra work to
write the sections, but then it also requires extra work in a pdf document.
It would also be nice to have the following construction:
Text and Input cells
Text and Input cells
End of BoxSection marker
Text and Input cells
where the BoxSection could be closed or terminated, and subsequent Text and
Input cells would NOT be part of the BoxSection, but part of the containing
section. The BoxSections would be like boxes in textbooks which contain a
side discussion without interrupting the main flow of material. (Possibly
there could be a way to have manual grouping only in some subsection of a
notebook, but I would much prefer a more versatile automatic grouping
because manual grouping is too subject to abuse.)
I have only looked a little at the Author's Tools application. It does give
information about constructing Help documentation, which I omitted to
mention in an earlier posting. But otherwise I haven't figured out just what
Author's Tools does for one in the way of constructing better notebooks for
readers. I wish WRI had provided a short elegant example with the
It might also be nice to have the ability to construct stand alone browsers.
Then the categories in the browser would be like the table of contents. In
essence, authors would write Mathematica browsers, in which Mathematica
notebooks formed the various chapters and sections.
I wish that there were better standard notebook styles supplied with
Mathematica. I find many of the standard ones useless. WRI needs to hire
Edward Tufte, or someone equivalent, to design some notebook styles. It
certainly is preferable to use a standard style because then one can count
on readers having it.
I would like to see one more Section level in notebooks. I would like to see
the default to have GroupOpenCloseIcons on all the Section levels - but NOT
on anything else. (Especially not on Input/Output groups.) The triangular
open/close icons are intuitive to new readers - the cell brackets are not. I
would like to see a better balance, actually a smaller range, of font sizes.
In the Default style, for instance, I think the Title font size is much too
large, and the Text font size is too small. The Text, Input and Output font
sizes should be reasonably close in size. After all, text cells and
Input/Output cells are of equal importance (IMHO) and should better blend.
Look at any technical article or book and you will see that the equations
and text have roughly comparable font sizes.
David Park
djmp at earthlink.net
From: AES [mailto:siegman at stanford.edu]
To: mathgroup at smc.vnet.net
Agreed, this is the sensible way to [include routines in notebooks], and how
I generally do it.
But two gripes about the result:
1) In PhD dissertations, journal papers, books, reports, the (sometimes
lengthy) "Routines" are most commonly are sent to the end, e.g. are
stuck in Appendices, and the Initialization (or Introduction) section is
immediately followed by the important (to the reader) sections such as
Calculations and Results. Among other things that lets you easily
select and print the Introduction, the Calculations and the Results to
toss in a file folder or (three-hole-type) notebook, leaving off the
lengthy Routines stuff.
Mathematica doesn't make it easy to organize its notebooks that way.
2) In my (limited) experience if I use Automatic Grouping and try to
close groups to see only the section headings (to get an overview of the
notebook structure and faster scrolling to , this doesn't work right
(i.e., the way I want it!) unless the cell structure is strictly
hierarchical. E.g., if I have repeated cell sequences in the form
Text (a few paragraphs introducing the section)
closing these groups so I'll see just the Section headings does not
close the Text cells, although it does close the Subsections (maybe I'm
not doing things right?).
Also, closing the Routines section, then running the notebook from the
top (to get a fresh start) opens the Routines section, doesn't it?
• Follow-Ups:
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[Numpy-discussion] fixed_pt some progress and a question
Neal Becker ndbecker2@gmail....
Tue Sep 29 10:10:09 CDT 2009
josef.pktd@gmail.com wrote:
> On Tue, Sep 29, 2009 at 10:22 AM, Neal Becker <ndbecker2@gmail.com> wrote:
>> This doesn't work either:
>> def as_double (self):
>> import math
>> def _as_double_1 (x):
>> return math.ldexp (x, -self.frac_bits)
>> vecfunc = np.vectorize (_as_double_1, otypes=[np.float])
>> return vecfunc (self)
>> In [49]: obj.as_double()
>> Out[49]: fixed_pt_array([ 0., 1., 2., 3., 4.])
>> The values are correct, but I wanted a float array, not fixed_pt_array
>> for output.
>> _______________________________________________
>> NumPy-Discussion mailing list
>> NumPy-Discussion@scipy.org
>> http://mail.scipy.org/mailman/listinfo/numpy-discussion
> I don't understand much, but if you just want to convert to a regular
> float array, you can just create a new array with np.array in
> as_double, or not?
I could force an additional conversion using np.array (xxx, dtype=float).
Seems wasteful.
The bigger question I have is, if I've subclassed an array, how can I get at
the underlying array type? In this example, fixed_pt is really an 'int64'
array. But I can't find any way to get that. Any function such as 'view'
just silently does nothing. It always returns the fixed_pt_array class.
Consider the fixed_pt_array method as_base(). It should return the
underlying int array. How could I do this?
More information about the NumPy-Discussion mailing list
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• March 8 — Prof. Colleen Mitchell, Dept. of Mathematics
Electro-physiology of cardiac arrhythmia
• March 1 — Prof. Thomas L. Casavant, Dept. of Biomedical Engineering
Highly Collaborative Bioinformatics in Partnership with Post-Genome Medicine
• February 22 — Prof. Rodica Curtu, Dept. of Mathematics
Modeling the unfolded protein response under stress in mammalian cells
• February 15 — Prof. Qihe Tang, dept. of Statistics and Actuarial Science
Reducing risk by merging counter-monotonic risks
• February 8 — Prof. Gregory R. Carmichael, Dept. of Chemical and Biochemical Engineering
Aerosol Feedbacks on Chemistry and Climate at Urban and Regional Scales
• February 1 — Prof. J. Robert Manak, Departments of Biology & Pediatrics
Creating epileptic flies and (mathematically) modeling ataxia
• May 4 — Prof. David S Bates, Dept. of Finance
On estimating stock market volatility and crash risk
• April 27 — Prof. Sharif Rahman, Dept of Mechanical and Industrial Engineering
Approximation Errors in High-Dimensional Uncertainty Quantification
• April 20 — Prof. John P. Spencer, Dept of Psychology & Delta Center
Dynamic Thinking--Capturing Cognition and Development with Dynamic Neural Fields
• April 13 — Dr. Scott Small, IIHR Hydroscience Eng
Real-Time Flood Forecasting in Iowa
• April 6 — Prof. Barrett Thomas, Dept. of Management Sciences
A Rollout Policy Framework for Dynamic Programming Approximations to the Vehicle Routing Problem with Stochastic Demand and Duration Limits
• March 30 — Prof. Jia Lu, Dept of Mechanical and Industrial Engineering
Cloth simulation by isogeometric modeling
• March 23 — Prof. Jianfeng Cai, Department of Mathematics
Image Restoration: Total Variation; Wavelet Frames; and Beyond
• March 9 — Prof. Palle Jorgensen, Dept of Math
Trends in Mathematics Inspired by Financial Mathematics
• March 2 — Prof. Victor Camillo, Dept of Math
Regulated Functions and Average Variation
• February 24 — Prof. Joe Eichholz, Rose-Hulman Institute of Technology
Introduction to the mathematics of bioluminescence tomography
• February 17 — Prof. George R. Neumann, Dept of Economics
Why do prediction markets work?
• February 10 — Mr. Nathan Ellingwood, AMCS
Q:What do Graduate Research, Particle Tracking Simulations and VideoGames have in common? A: Taiwan!
• February 3 — Prof. Yannick Meurice, Dept of Physics and Astronomy
Mathematical Problems in Lattice Field Theory
• January 27 — Dr. Frederick Qiu, IMA, University of Minnesota
An analysis of the practical DPG method
• May 1 — Professor Glenn Luecke, Iowa State University
Numerical Analysis and Current Trends in High Performance Computing
• April 29 — Mr. Hongbo Dong, AMCS
Symmetric Tensor Approximation Hierarchies for the Completely Positive Cone
• April 22 — Prof. Bruce Ayati, Department of Mathematics
Mathematical Descriptions of Bone Remodeling Dynamics in Myeloma Bone Disease
• April 15 — Prof. Suely Oliveira, Dept. of Computer Science
Clustering Applications in Biology
• April 1 — Prof. Karim Abdel-Malek, Departments of Biomedical and Mechanical Engineering, Center for Computer Aided Design
Santos: A human simulation environment
• March 31 — Mr. Joe Eichholz, AMCS
Dicontinuous Galerkin methods for solving the radiative transfer equation and its approximations
• March 25 — Prof. Keith Stroyan, Department of Mathematics
Visual Depth Perception from Motion
• February 28 — Prof. David Stewart, Department of Mathematics
Understanding bouncing balls
• February 25 — Prof. Victor Camillo, Department of Mathematics
Uniform Limits of Step Functions
• February 18 — Prof. Palle Jorgensen, Dept. of Mathematics
Interaction between ideas from engineering of signals and approximation in mathematics (wavelets and more)
• February 11 — Prof. Johna Leddy, Department of Chemistry
• January 31 — Prof. Wayne Polyzou, Department of Physics and Astronomy
The Relativistic Quantum Mechanical Three-Body Problem
• January 28 — Prof. Y. Cheng, University of Texas, Austin
Discontinuous Galerkin Schemes for Boltzmann Equations in Semiconductor Device Simulation
• May 1 — Mr. Jon Van Laarhoven, AMCS
Exact and Heuristic Algorithms for the Euclidean Steiner Tree Problem
• April 30 — Prof. Kurt M Anstreicher, Dept of Management Sciences
Comparing Convex Relaxations for Quadratically Constrained Quadratic Programming
• April 16 — Prof. Goran Lesaja, Georgia Southern University
Kernel-Based Interior-Point Methods for Monotone Linear Complementarity Problems over Symmetric Cones
• April 1 — Prof. Pavlo Krokhmal, Dept. of Mechanical and Industrial Engineering
Combinatorial Optimization Problems on Hypergraph Matchings: A Probabilistic Analysis
• March 31 — Prof. Jia Lu, Dept. of Mechanical and Industrial Engineering
Isogeometric Analysis: A Brief Introduction
• March 26 — Mr. Joe Eichholz, AMCS
A numerical method for the Radiative Transfer Equation
• March 12 — Mr. Tim Kreutzmann, Karlsruhe Institute of Technology (KIT)
The Domain Derivative and its Application to Bioluminescence Tomography
• February 28 — Prof. Kai Tan, Department of Internal Medicine and Biomedical Engineering
Computational tools for reading epigenomes
• February 26 — Prof. Alberto Segre, Department of Computer Science
UI Computational Epidemiology Group Research
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Calculus with Real Infinitesimals
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Views of the subnucleonic world
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Reproducible Statistical Analysis or Literate Programming via StatWeave
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Challenges of Flood Forecasting and Prediction
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• April 24 — Prof. Jia Lu, Dept. of Mechanical and Industrial Engineering
Discrete Method in Solid Mechanics
• April 17 — Prof. Jinhu Xiong, Dept. of Radiology
Mapping Functional Organizations of the Human Brain Using MRI
• April 10 — Dr. Alan Huebner, ACT
Multi-dimensional a-stratification for Computerized Adaptive Testing
• March 31 — Ms. Fengrong Wei, AMCS
Group Selection in High-dimensional Regression
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Effective resistance in infinite networks
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Poincare invariant quantum mechanics
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Asymptotic Aspects of the Expected Discounted Penalty Function in the Renewal Risk Model Using Wiener-Hopf Factorization and Convolution Equivalence
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Microarray-based integrative genomics: From genome annotation to mapping mutations to genes
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The Query Complexity of Estimating Weighted Averages
Google Tech Talk
February 4, 2010
Presented by Tony Wirth.
The query complexity of estimating the mean of some [0, 1] variables is well known to the theory community. Inspired by some work by Carterette et al. [SIGIR 2006, pp 26875] on evaluating retrieval
systems, and by Moffat and Zobel's new proposal for such evaluation [under review], we decided to examine the query complexity of weighted average calculation. In general, the problem requires the
same number of queries as estimating the mean, as the latter is a special case. In fact, there is a matching upper bound for the weighted mean. This result remains true for any set of weights that is
the normalized prefix of a divergent series. However, if the weights follow a geometric sequence, a much smaller sample is sufficient. Finally, we investigate power-law sequences of weights and show
matching lower and upper bounds. This is joint work with Amit Chakrabarti and Venkatesan Guruswami and Andrew Wirth.
Tony Wirth joined the faculty of the University of Melbourne's Computer Science department in 2005. Prior to that, he completed a PhD, as a Gordon Wu Fellow, at Princeton University in 2004 on
approximation algorithms for clustering problems. Tony completed his undergraduate degree at the University of Melbourne, majoring in statistics. His research interests also include sequence problems
in bioinformatics and adaptive sampling.
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First Grade Algebra - Printables, Worksheets, and Lessons
'); } var S; S=topJS(); SLoad(S); //-->
Math First Grade Algebra
First Grade Printables, Worksheets, and Lessons
First Grade Algebra Mixed Review Book
First Grade Algebra Mixed Review Book
Skip Counting
Easy Skip Counting with Numberline
Count by 2's, 3's, or 4's - addition
Count by 5's - addition
Count by 2's, 3's, or 4's - subtraction
Count by 5's - subtraction
Circle Growing or Decreasing
Circle growing or decreasing
Circle growing, decreasing, or repeating
Mystery Numbers: Greater and Less Than
One Mystery Number with Number Line
Numbers 1 to 9: Missing Number - with Number Line - Color In (greater/less than signs always same order)
Numbers 1 to 9: Missing Number - with Number Line - Color In (greater/less than signs in different orders)
Numbers 1 to 15: Missing Number - with Number Line - Color In (greater/less than signs always same order)
Numbers 1 to 15: Missing Number - with Number Line - Color In (greater/less than signs in different orders)
One Mystery Number Fill-in
Numbers 1 to 9: Missing Number - Fill-in - Color In (greater/less than signs always same order)
Numbers 1 to 9: Missing Number - Fill-in - Color In (greater/less than signs in different orders)
Numbers 1 to 19: Missing Number - Fill-in - Color In (greater/less than signs always same order)
Numbers 1 to 19: Missing Number - Fill-in - Color In (greater/less than signs in different orders)
More Patterns
Patterns - Worksheets and Printables Unit
Change in Numbers: How Did it Change?
Numbers 1-9
Numbers 1 to 9 - the change will be from 1 to 3 - equation is given
Ordered Pairs
Make the Scale Balance
Easier: Balance is Equal - Add the Missing Blocks
Add the missing blocks
Add the missing blocks and write the number of blocks on each side
Add the missing blocks - one problem does not need any blocks
More problems per page
More problems per page - one problem does not need any blocks
One Side of Balance is Heavier
Add the missing blocks
Add the missing blocks and write the number of blocks on each side
Add the missing blocks - one problem does not need any blocks
More problems per page
More problems per page - one problem does not need any blocks
Some Balances are Equal - Others have Heavier Sides
Add the missing blocks
Add the missing blocks and write the number of blocks on each side
Add the missing blocks - one problem does not need any blocks
More problems per page
More problems per page - one problem does not need any blocks
First Grade Geometry
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Leave your suggestions or comments about edHelper!
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Precision loss
Author Precision loss
Ranch Hand
Dec 24, Hi Ranchers, I'm trying to find the result of 2^1000. With "double x = Math.pow(2,1000)". The result is deviating, i mean its rounding it off at a particular limit and I need the full
2010 values. Can someone please help?
Posts: 70
A double does not guarantee precision. There are limits to what you can do with only so many bits...
Oct 02, Further, why would you use something designed to store floating point decimal numbers when you are computing what is clearly an integer value?
Posts: If you need something more accurate, look at something like the BitInteger class.
There are only two hard things in computer science: cache invalidation, naming things, and off-by-one errors
I like...
fred rosenberger wrote:If you need something more accurate, look at something like the BitInteger class.
Mar 17,
2011 I think that's BigInteger, Fred.
7064 @Shamsudeen - More specifically, something like:
private static final BigInteger TWO = BigInteger.valueOf(2L); (I don't know why BigInteger doesn't define this constant itself)
16 and then elsewhere in your code:
I like... Winston
Isn't it funny how there's always time and money enough to do it WRONG?
Artlicles by Winston can be found here
Shamsudeen Akanbi wrote: . . . With "double x = Math.pow(2,1000)". The result is deviating, i mean its rounding it off at a particular limit . . .
Oct 13, All the primitives have limits of range, and the floating-point types have limits to their precision, too. The limits are clearly stated in the Java Language Specification and Java
2005 Tutorials, and you can find more by Googling for IEEE754. 2 to the 1000 is actually inside the range of a double, so you won’t get infinitiy, but it will tell you it is only precise to 15
Posts: or 16 Significant figures. Floating-point arithmetic is designed for engineers to use.
When I was an undergraduate:
What’t an engineer?
He’s somebody you ask “What’s 2×2?”, and he gets his slide‑rule and says, “ . . . two by two . . . three point nine nine . . . that’s four, near as makes no difference.”
Floating-point arithmetic is intended for people who don’t mind such imprecision. You have already been given the correct solution, twice, with and without spelling errors
Mar 17,
2011 Campbell Ritchie wrote:When I was an undergraduate: What's an engineer?...
Nice one.
I like...
Winston Gutkowski wrote:TWO.pow(1000);Winston
Mar 17,
2011 BTW: If you fancy it (and you have the memory space and - possibly more importantly - time
Posts: try:
7064 BigInteger FOUR = new BigInteger.valueOf(4L);
I've never got it to work; probably because my heap memory is too small (possibilities run to anywhere around 3Gb, depending on efficiency) and I can't be bothered to fix it; but I
suspect very strongly that it will display a negative number..SILENTLY. And this after repeated queries about it to Oracle and Sun.
I like...
Oct 13,
Posts: You already know its bit length. I can tell you it free. 2147483649.
Mar 17,
2011 Campbell Ritchie wrote:You already know its bit length. I can tell you it free. 2147483649.
And BigInteger.bitLength() returns what?
I like...
Oct 13,
2005 2147483649. Presumably rounded with the ROUND_PEG_SQUARE_HOLE algorithm, to the nearest int.
36508 . . . And how long does it take to run? (Or crash?)
Joined: Campbell Ritchie wrote:2147483649. Presumably rounded with the ROUND_PEG_SQUARE_HOLE algorithm, to the nearest int.
Mar 17, . . . And how long does it take to run? (Or crash?)
7064 Actually, nanoseconds; unless the number is negative and an exact power of 2; so, on average...
16 My objection is theoretical: bitLength() returns an int, which has a limit of 23^31 -1. The magnitude of a BigInteger is held in an int[], which can hold up to ≈2^36 bits. I've just never
been able to prove that it does what I think it will.
I like... Winston
Oct 13, Winston Gutkowski wrote: . . . unless the number is negative and an exact power of 2; so, on average...
Posts: How can a number be negative and an exact power of 2?
I did work out that something would go wrong, so I said it would use the ROUND_PEG_SQUARE_HOLE algorithm. I have had the d*mn thing running for several hours with no sign of any output
16 yet.
Campbell Ritchie wrote:
Mar 17, Winston Gutkowski wrote: . . . unless the number is negative and an exact power of 2; so, on average...
Posts: How can a number be negative and an exact power of 2?
16 I guess I should have said -(2^n).
BigIntegers hold their values in sign/magnitude form, and bitLength() is defined as returning:
I like... "the number of bits in the minimal two's-complement representation of this BigInteger, excluding [the] sign bit."
I take that to mean "the number of bits from the first one in the 2's-C form that differs from the sign bit"; otherwise the check doesn't make much sense.
Personally, I think they could've saved themselves a lot of bother by simply returning the bit length of the absolute value, but maybe they had their reasons.
BTW: How did your attempt work out?
Oct 13,
Posts: I gave up after it had a day, on and off, and still hadn’t completed. I had to close the shell to terminate it; even ctrl-C didn’t seem to stop it
subject: Precision loss
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Representation of all pass transfer functions/inner functions as Blaschke product.
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What is the proof for: 'An all pass transfer function/inner function can be represented by a Blaschke Product' ?
add comment
Atkinson, Discrete and continuous boundary problems, page 8.
Thanks Alexandre.
Mohan Apr 19 '13 at 5:22
add comment
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Mandelbrot in fragment shaders? [Archive] - OpenGL Discussion and Help Forums
08-30-2002, 06:41 PM
I'm just idly wondering which implementation of the Mandelbrot fractal set renderer would run faster:
- do the actual exponentiation and test (with KIL) in a big unrolled loop
- do the successive exponentiation using a exponented-value look-up texture
I suppose the first solution would be more precise, as the exponentiation look-up texture by necessity needs to quantize the input values to 1/1024 resolution or so.
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Parkandbush, NJ Statistics Tutor
Find a Parkandbush, NJ Statistics Tutor
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unless you have already submitted a credit card number to Wyzant, I will not be able to see your phone nu...
25 Subjects: including statistics, chemistry, physics, calculus
...I have been helping high school students prepare for the SAT and ACT for over two years.Algebra II is often difficult for students because they have trouble seeing the link between the
equations (which are abstract) and the real world. I try to bridge the gap to show how the math represents the ...
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Hypertext Help with LaTeX
Ellipses (three dots) can be produced by the following commands
• \ldots - horizontally at bottom of line
• \cdots - horizontally center of line (math mode only)
• \ddots - diagonal (math mode only)
• \vdots - vertical (math mode only)
See also Math Formulas, Math Miscellany.
Back to the Table of Contents
Revised 25 April 1995.
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Gamma function & integral
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. It has a scale parameter θ and a shape parameter k. If k is an integer
then the distribution represents the sum of k exponentially distributed random variables, each of which has parameter .
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Chapter 3: Two-Dimensional Motion and Vectors
Welcome to Holt Physics. Chapter 3 describes scalar and vector quantities and projectile motion. Use the links below to find learning tools that will help you review the chapter and extend your
study of two-dimensional motion and vectors.
Review and Practice
Reading Skills: K-W-L
This worksheet will help you organize your thoughts before and after you read the chapter or an article.
Glossary of Terms: Chapter 3
Use this handy glossary to enhance your proficiency with the terms found in this chapter.
Homework Help: Chapter 3 Study Guide
Select one of the sections listed below to review key skills and concepts from the chapter.
Section 1 || Section 2 || Section 3 || Section 4 || Mixed Review
Practice Problems: Chapter 3
Select one of the topics listed below to practice using the concepts found in this chapter.
Finding Resultant Magnitude and Direction
Resolving Vectors
Adding Vectors Algebraically
Projectiles Launched Horizontally
Projectiles Launched at an Angle
Relative Velocity
Quiz Yourself: Two-Dimensional Motion and Vectors
Click here to test your knowledge of two-dimensional motion and vectors.
Teacher Notes
Enrichment and Extension
Online Research: SciLinks
Visit NSTA's SciLinks Web site, which links you to some of the most up-to-date science information on the Internet. When you get to the site, log in, and then enter a keyword for the topic you
want to research. The topics and keywords for Chapter 3 are listed below.
TOPIC: Vectors
SciLinks CODE: HF2031
TOPIC: Projectile motion
SciLinks CODE: HF2032
TOPIC: Speed of Light
SciLinks CODE: HF2033
Integrating Mathematics: Jesse Owens in the 100 Meter Dash
Make a graph and analyze the speed of a sprinter in the 100 m dash.
Integrating Mathematics: Using Quantitative Statements to Solve Problems
Translate between words and symbols to solve problems in physics.
Integrating Mathematics: Using Comparisons to Understand Space Statistics
Compare other planets to Earth to find out how big they really are.
Real World Applications: Hiking in Yellowstone
Use Naismith’s rule to determine how changing altitude affects the travel time of hikers.
Career: Experimental Physicist
Learn about the life of an experimental physicist, and get tips on how to find out whether a career in physics is right for you!
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How can I tell if y is a function of x in a random sample?
up vote 3 down vote favorite
I have some data and believe that a given metric is a function of another metric. I have the values of both metrics and many different sets of these values. Can I tell if one is a function of the
other through some simple exercise like a regression? I'm not sure if the function is linear. I'm not a math expert so apologies if this is a trival question.
Edit: Here's my (Anton's) interpretation of the question. If I misunderstood, I hope gitkin corrects it.
Given a bunch of data points $\{(x_i,y_i)\}$ in the plane, I can find the line best fitting the data. Then I can compute the coefficient of determination $R^2$ to see how good the fit is. More
generally, given a model $y=f(x)$ (where $f$ may not be linear), I can do various things to determine how well the model fits the data.
Is there some way to determine if there exists a model $y=f(x)$ fitting the data well? In other words, is there a way to measure your confidence that the $x$ values completely determine the $y$
values (in some reasonable way) in the system you've sampled? Intuitively, you should somehow vary over all possible functions $f$, measure how much the model $y=f(x)$ fails to explain the data, add
some penalty depending on the complexity of $f$ relative to the size of the sample,^† and return the lowest value you get. Is there a precise, theoretically justified way to do this?
^† e.g. the penalty should be very high if $f$ is a polynomial of degree comparable to the number of data points.
1 poor use of terminology on my part. It's just a value. "property was quantitatively measured" is exactly what I meant. – gitkin Jan 17 '10 at 22:38
1 I think a version of this question, rephrased in the language of statistics, could be very good. As it is currently written, I suspect there will be votes to close, although I hope it is not, in
that Douglas's answer below sufficiently interprets the question mathematically (although not deeply so). – Theo Johnson-Freyd Jan 17 '10 at 23:58
1 I disagree. This is not a real question. It is asking for a way to fit a curve to data points on a 2d-graph. Of course, you can construct a function that hits almost every point, but having a
perfect or near-perfect fit curve doesn't help you prove anything. The whole point of modeling data like this is to see a clear trend and compute how effective your estimate is. If the function is
too complicated, all that you get is a really strange looking interpolated graph. – Harry Gindi Jan 18 '10 at 4:15
I'm going to close this as "not a real question" because it's too ambiguous. We've resolved the confusion about "metric," but now I don't see what you mean by "function" or "the values of both
1 metrics." If you have all the values, then x is a function of y if no value of y has two different values of x, but I suspect this isn't what you're looking for. Since you used the stats tag, you
probably have samples. Are you asking if there's a dependence between the two? If you edit the question to ask a precise question, I'm happy to consider reopening it. – Anton Geraschenko Jan 18
'10 at 7:41
1 Although you can interpolate when there is no relation, nonmathematicians including physicists are likely to look at that mess and say, "That's not a function!" Perhaps they want the function to
be $K$-Lipschitz for some $K$ which isn't too large, or $\alpha$-Holder continuous for some $\alpha$ which isn't too small. – Douglas Zare Jan 18 '10 at 12:07
show 4 more comments
4 Answers
active oldest votes
If you are only interested in correlation between the two feature values, then there are a lot of ways to compute it (simple correlation, rank correlation, linear or nonlinear
regression, etc.).
up vote 5 down If you are interested in causality, a few places to look at are: Granger causality
and NIPS workshops on causality: 2008, 2009
1 I'm not sure why no one is upvoting this. To my limited knowledge, these look like very useful links. – David Speyer Jan 19 '10 at 13:22
I wish I understood that notion of causality well enough to say whether it fit the question. – Douglas Zare Feb 2 '10 at 6:27
add comment
Metric is a technical term in mathematics, but I'll ignore the usual technical meaning.
In practice, I would plot the points $(metric_1,metric_2)$. Decide whether you would call the graph a function, whether you can predict the value of one from the other.
A linear regression will only detect linear functions perfectly since the linear correlation will be +1 or -1. You can detect any increasing or decreasing function with a Spearman rank
up vote 4 down correlation coefficient, or just sort by one metric and see if that sorts the other.
This will not detect a relationship which is not monotone like $metric_1 = \sin(metric_2)$. If you have a good guess that something like this is the case, you might try testing the
rank correlation of $metric_1$ and $\sin(metric_2)$.
I believe there is another technique which may be useful, which is to consider the topology of set of points within r of a data point as you increase r. It's too bad the question is
closed. – Douglas Zare Jan 30 '10 at 10:47
add comment
I do have a little to add at a much lower level. The first step is to plot lots of points and see if you still believe one quantity is determined by the other. Next, rewrite the pairs as $
(x_i, y_i)$ where you believe the $x$ value may determine the $y$ value (you might need to switch the order of every pair). One necessary condition is that there be no repeated $x_i.$
Finally and hardest, you really need to GUESS a functional relationship. As long as your function is determined by a (small) finite number of quantities the method of least squares can be
applied. If you think you have a sine wave, you define the general curve by constants $A,B,C$ in the function $f(x) = A \sin (B x + C).$ Least squares says you minimize $ \sum_{i} ( y_i - f
up vote (x_i) )^2, $ which is a process involving your data pairs and something called partial derivatives. You should get individual help with this process, it is commonly taught just for lines
3 down (regression). If the best curve matches the data points very well perhaps you have it.
Finally, the reason you are absolutely required to guess a function eventually is that, under the assumption that there is a dependence (no repeated $x_i$) there are infinitely many
mathematical functions $g(x)$ that satisfy all $ y_i = g(x_i)$ exactly, for example $g(x)$ can be a polynomial of high degree. What you really want is a function that will be deemed
reasonable in your line of work.
add comment
Strictly a function is a mapping that assigns a unique value in a set B to every point in a set A. So the only way (in this strict sense) in which your second "metric" will not be a
function of the first metric is if there are two datapoints for which the second metric gives different values but your first metric gives the same value.
up vote 0 However I suspect you have in mind the looser non-mathematical sense in which you can write the second "metric" (the quotation marks are because metric has a particular technical meaning
down vote in mathematics) as a closed-form function of the first metric plus some reasonably well-behaved error term. In this case a suitable linear regression may help you with your problem but
only once you make some assumptions about the form of the functional relationship.
4 I also suspect this question will ultimately be closed as insufficiently mathematical/suitable for MathOverflow. – Tom Smith Jan 17 '10 at 22:35
add comment
Not the answer you're looking for? Browse other questions tagged st.statistics or ask your own question.
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Homework Help
Posted by James on Monday, November 21, 2011 at 2:11pm.
Fan and Dan took part in a 1000 meter competition. When Fan was at the 800 meter mark, Dan was 300 meters away from the finish line. How many meters away from the finish line was Dan when Fan crossed
the finish line?
• Math - Ms. Sue, Monday, November 21, 2011 at 2:18pm
100 m
• Math - Reiny, Monday, November 21, 2011 at 2:33pm
let the time at which was at the 800 m mark be t
then Fran's rate = 800/t
Dan's rate = 300/t
Let the time it took Fran to run the whole 1000 m be k
1000 = k(800/t)
k = 1000t/800
in that same time of k, distance covered by Dan was
D = k(300/t) , using Distance = rate x time
= (1000t/800)(300/t) = 375
so Dan is still 625 m from the finish line when Fran crosses.
• correction - Math - Reiny, Monday, November 21, 2011 at 3:28pm
I think I misread the question.
Dan was 300 m from the finish line, so he had gone 700 m, I took it that he went 300 m
so change all my 300's to 700
D = (1000t/800)(700/t) = 875
so he was 125 m from the finish line.
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Alhambra, CA Trigonometry Tutor
Find an Alhambra, CA Trigonometry Tutor
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Orthogonal complements
April 26th 2008, 12:49 PM
Please Help!!!
Let V be an inner product space, and let U and W be subspaces of V. Show that
$(U \cap W)* = U*+W*$
$(U+W)=U* \cap W*$
here, U* denotes the orthogonal complement of U.
having trouble even finding where to start this one.
April 27th 2008, 02:49 PM
i know i need to prove subspaces both ways.
x $\in$ the perp of the intersection of U and W what does that tell me?
i'm stuck.
April 27th 2008, 07:15 PM
hey have you been able to figure out this problem. I also have the same problem just wondering if u found a way to do it or maybe we can but our ideas together to solve it.
April 27th 2008, 11:16 PM
April 28th 2008, 02:06 AM
See here.
April 28th 2008, 08:45 PM
the link sends me to a login for live journal? thank you though
April 29th 2008, 12:02 AM
Sorry, I hadn't noticed that LiveJournal page was "friends only". Here's a copy of the relevant comment. The "perp" symbol ⊥ (denoting an orthogonal complement) has come out on the line instead
of as a superscript, which makes it a bit hard to read.
You need to know that U⊥⊥ = U. Also, if G and H are subspaces with G⊆H, then H⊥⊆G⊥.
If x = y+z with y∈U⊥ and z∈W⊥ then it is easy to see that y and z both lie in (U∩W)⊥, hence so does x. Therefore U⊥+W⊥⊆(U∩W)⊥.
For the converse inclusion, if x∈(U⊥+W⊥)⊥ then x∈U⊥⊥ = U, and similarly x∈W⊥⊥ = W. Thus (U⊥+W⊥)⊥⊆U∩W. Take the perp of each side to see that (U∩W)⊥⊆U⊥+W⊥.
Thus (U∩W)⊥ = U⊥+W⊥. You get the other identity by taking the perp of both sides and replacing U with U⊥ and W with W⊥.
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667pages on
this wiki
Zero is a number denoting the lack of any quantity or measure. In the Hindu-Arabic numeration system, it is denoted by the numeral 0. In algebraic structures, zero is also a name given to an element
in an additive monoid, group, ring, or field that acts as the additive identity, such as zero itself (in the context of real numbers), zero vector, zero matrix, and zero function. It is neither a
negative or a positive number.
Zero's properties
Property one: Neither negative nor positive.
Property two: Anything multiplied by zero is zero, including zero multiplied by zero.
Property three: Zero divided by anything is zero, except zero divided by zero is undefined.
Property four: Anything divided by zero is undefined.
Property five: Zero has the lowest possible absolute value.
Property six: Anything to the power of zero is one, except zero to the power of zero is undefined.
Property seven: Zero to any power but itself is zero.
Property eight: Anything added to zero is itself, including zero added to zero.
Property nine: Anything subtracted from zero is itself with changed sign, except zero (which has no sign).
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Aligned Resources
Shodor > Interactivate > Standards > Alaska Performance Standards: Grade 6 > Aligned Resources
Alaska Performance Standards
Grade 6
Process Skills:
The student demonstrates an ability to use logic and reason.
Lesson • Activity • Show All
Lesson (...)
Lesson: Students discover algorithms as they sort shapes into Venn diagrams. Then students compare the efficiency of their algorithms using box plots.
Lesson: In this lesson, students explore sets, elements, and Venn diagrams.
Venn Diagrams
Lesson: Help students learn about classifying numbers into various categories through answering questions about Venn Diagrams.
Activity (...)
Activity: Sort colored shapes into a three circle Venn Diagram.
Activity: Sort colored shapes into a Venn diagram based on various characteristics. Venn Diagram Shape Sorter is one of the Interactivate assessment explorers.
Venn Diagrams
Activity: Classify various objects into categories in a Venn Diagram. Learn how categories in Venn Diagrams work. Venn Diagrams is one of the Interactivate assessment explorers.
No Results Found
©1994-2014 Shodor Website Feedback
Alaska Performance Standards
Grade 6
Process Skills:
The student demonstrates an ability to use logic and reason.
Lesson • Activity • Show All
Lesson (...)
Lesson: Students discover algorithms as they sort shapes into Venn diagrams. Then students compare the efficiency of their algorithms using box plots.
Lesson: In this lesson, students explore sets, elements, and Venn diagrams.
Venn Diagrams
Lesson: Help students learn about classifying numbers into various categories through answering questions about Venn Diagrams.
Activity (...)
Activity: Sort colored shapes into a three circle Venn Diagram.
Activity: Sort colored shapes into a Venn diagram based on various characteristics. Venn Diagram Shape Sorter is one of the Interactivate assessment explorers.
Venn Diagrams
Activity: Classify various objects into categories in a Venn Diagram. Learn how categories in Venn Diagrams work. Venn Diagrams is one of the Interactivate assessment explorers.
No Results Found
The student demonstrates an ability to use logic and reason.
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Asymptotic behavior of coefficients
Given the difference equation
The asymptotic behavior of the coefficients is given by
a_n^{(1)}\sim 2^{-n}n^{-1-\lambda/2}\sum_{s=0}^{\infty}\frac{c_s^{(1)}}{n^s}
a_n^{(2)}\sim n^{-3}\sum_{s=0}^{\infty}\frac{c_s^{(2)}}{n^s}
I have to do the a similar calculation in my research project but I couldn't find out the procedure used. Please, someone can show me the steps to such a solution?
I tried to helplessly follow the text by Saber Elaydi, An Introduction to Difference Equations.
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Rewriting expressions with logic laws (4)
April 12th 2010, 01:23 AM #1
Super Member
Oct 2007
Rewriting expressions with logic laws (4)
Rewrite: $q \wedge ( \sim (((p \vee (p \wedge q)) \wedge (p \vee q)) \vee q))$ and determine if its tautology, contradiction or neither. Using truth table I got contradiction.
My working out:
$q \wedge ( \sim (((p \vee (p \wedge q)) \wedge (p \vee q)) \vee q))$
$\equiv$$q \wedge ( \sim ((p \wedge (p \vee q)) \vee q))$
$\equiv$$q \wedge ( \sim (p \vee q))$
$\equiv$$q \wedge ( \sim p \wedge \sim q)$
$\equiv$$q \wedge ( \sim q \wedge \sim p)$
$\equiv$$(q \wedge \sim q) \wedge \sim p$
$\equiv$$p \wedge \sim p$
So I finished off with a contradiction. Is this correct? Any help would be much appreciated.
Can anyone confirm? thanks
Rewrite: $q \wedge ( \sim (((p \vee (p \wedge q)) \wedge (p \vee q)) \vee q))$ and determine if its tautology, contradiction or neither. Using truth table I got contradiction.
My working out:
$q \wedge ( \sim (((p \vee (p \wedge q)) \wedge (p \vee q)) \vee q))$
$\equiv$$q \wedge ( \sim ((p \wedge (p \vee q)) \vee q))$
$\equiv$$q \wedge ( \sim (p \vee q))$
$\equiv$$q \wedge ( \sim p \wedge \sim q)$
$\equiv$$q \wedge ( \sim q \wedge \sim p)$
$\equiv$${\color{red}(q \wedge \sim q)} \wedge \sim p$
So far so good. But now things get slightly off the rails:
$\equiv$${\color{red}p} \wedge \sim p$
This should, imho, be
$\equiv {\color{blue}F}\wedge \sim p$
$\equiv F$
So I finished off with a contradiction. Is this correct?
The result is correct, in my opinion, but see the above glitch in the derivation.
Right, but to be honest, I personally use a much more general rule: whenever there is a "false" anywhere in no matter how large a conjunction ( $\wedge$), the whole conjunction collapses to the
value "false".
It's akin to having a 0 in a product of numbers: the whole product collapses to the value 0.
April 13th 2010, 02:26 AM #2
Super Member
Oct 2007
April 13th 2010, 02:47 AM #3
April 13th 2010, 03:03 AM #4
Super Member
Oct 2007
April 13th 2010, 03:14 AM #5
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Statistics 5 Probability and Z-Scores
1.Similar to t-test Ð except IV has 2 or more levels
n1 IV analyzed and 1 DV measured
nH0: population means are all equal (H0: µ1 = µ2 = µ3)
nH1: population means are NOT equal (µ1 µ2 µ3)
nTreatment Ð may have more than 2 levels
3.Hypothesis testing
nH0 (null) and H1 (alternate) hypotheses are stated (Assume H0 is true)
nSampling distribution Ð t or F
nGather data and calculate F value
nCompare calculated F to table value and Interpret results
4.F distribution
nF is a sampling distribution (just as t distribution is)
nRatio Ð treatment/between groups variation to within group variation
¤B - between groups Ð do groups differ due to treatment
¤W or E within (error) Ð unbiased estimate of variability in same group
¤B > W reject Ho (treatment had effect)
¤Small variability (F=1.00) Ð keep H0, large var (F > 1.00) Ð reject H0
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Cars Guessing Game NetLogo Model
Produced for the book series "Artificial Intelligence";
Author: W. J. Teahan; Publisher: Ventus Publishing Aps, Denmark.
powered by NetLogo
view/download model file: Cars-Guessing-Game.nlogo
WHAT IS IT?
This model plays a simple game trying to guess the colour of cars as they drive past. Various agents maintain probability distributions that they use to predict the arrival of the cars. The source
distribution is a fixed distribution that is used to generate the cars, so is the most accurate, and therefore its entropy is the source entropy and corresponding code length is the lowest (the code
length is the optimal cost of encoding the sequence of cars given its probability distribution).
Agents 01 to 03 maintain fixed distributions which the user can adjust as they see fit by changing the slider values to the left of the Interface. An adaptive agent also maintains a dynamic
distribution by updating counts of cars that have previously been seen. These counts are shown in the Interface by monitors under the heading "Adaptive Agent's distribution". The entropy and code
length calculations are shown by monitors at the middle top of the Interface. The adaptive entropy and code length is the one that usually gets close to the source entropy and code length, whereas
the other agents' entropy and code lengths reflect how different the slider settings are from the source distribution.
The purpose of this model is show how entropy and code length calculations are made given a probability distribution.
A car turtle agent is used to represent the cars in the environment. These are created with colours distributed according to the source distribution. The arrival of the next car to the left of the
environment is determined by a random number generated according to the slider next-car-random-tick-interval.
Turtle agents called "agents" are also present, but are not shown in the animation. These can be considered to be observers watching the cars go past. Each owns a distribution which is a list of
counts that are used to calculate the probabilities in order to guess the upcoming cars.
HOW TO USE IT
Setting the values on the sliders for the Source Distribution will determine the distribution for the cars that appear in the animation. Setting the next-car-random-tick-interval will control how
often the cars appear. The user can then set the slider values for the three Agent distributions on the left to see how this affects the entropy and code lengths for these distributions.
The model's Interface buttons are defined as follows:
- setup: This resets the animation and initialises the counts, entropy and code length values.
- animate-cars: This starts the animation in the environment (the light blue rectangle shown middle bottom of the Interface).
The model's Interface sliders have the following naming convention:
This sets the count for the specified colour in the list of counts maintained for the agent whose name is <agent-name>. These counts are used to determine the probability for a specific colour using
the following formula:
P(colour) = C(colour) / C_total
where C(colour) is the count for the colour and C_total is the sum of all counts for all colours.
The model's Interface monitors are defined as follows:
- adaptive-<colour>: This is the count of the number of cars observed in the animation of the respective colour that is adaptively updated as the animation proceeds.
- <agent-name> Entropy : This is the entropy of the probability distribution maintained by the agent whose name is <agent-name>.
- <agent-name> Code Length : This is the cost of optimally encoding the sequence of observed cars given the agent's probability distribution.
The model's plot is defined as follows:
- The distributions are converted to a number that uniquely represents the values of the counts in the distribution. (For example, the number 1234 can be thought of as representing four separate
counts - 1, 2, 3 and 4 - that combine to produce a unique number. In this case, the separate counts do not range from 0 to 9; they can range from 0 to 1000). The number that uniquely represents the
distribution is then plotted versus ticks to show how the distributions evolve over the simulation.
Notice how well the adaptive distribution does compared to the source distribution. The source distribution will have the lowest code length total compared to the others, but usually the nearest
distribution will be the adaptive one (unless the counts for one or more of the Agent's distribution exactly match the source distribution counts).
If the source distribution has equal counts, then the adaptive counts will rise at a similar rate, and as a consequence, the red line in the plot will rise diagonally from bottom-left to top-right.
A horizontal line in the plot indicates a fixed distribution. Modifying the distribution counts in the middle of the animation will result in the lines in the plot changing to reflect this.
Try changing the distribution counts in the sliders to see what affect this has on the entropy and code length calculations, and on what happens in the plot. Note that the lower colours (blues and
pinks) will cause the greatest shifts in the line plots. (Why?)
Can you achieve a situation where the code length of one of the three non-adaptive Agents is better than the Adaptive Agent? (i.e. the code length is smaller and closer to the source agent's code
length total).
This model was created by William John Teahan.
To refer to this model in publications, please use:
Cars Guessing Game NetLogo model.
Teahan, W. J. (2010). Artificial Intelligence. Ventus Publishing Aps
; Cars Guessing Game model.
; Three agents try to guess the probabilities of cars passing by.
; Copyright 2010 William John Teahan. All Rights Reserved.
extensions [ array ]
breed [cars car] ;; represents cars that the agents observe going past
breed [agents agent] ;; represents the agents guessing what car will come next
;; the process generating the arrival of the cars is also considered as an 'agent' here
[ agent-id ;; identification number associated with the agent
distribution ;; probability distribution (represented using frequency counts) that
;; the agent is using to predict or generate the colour of the cars
dist-total ;; the total of the distribution's counts
dist-entropy ;; the entropy of the distribution
codelength-total ;; total code length for the sequence
[ next-car-tick ;; tick when the next car arrives
source-agent ;; source 'agent' that is generating the cars
adaptive-agent ;; adaptive agent that adapts distribution to the cars as they appear by counting them
car-colours ;; colours of the upcoming cars
car-colours-index ;; position in car-colours array
cars-dist-total ;; source distribution's total used to generate the cars
to setup
ca ;; clear everything
ask patches with [ pycor > -9 ] [ set pcolor 88 ] ; make everything light blue for the "sky"
ask patches with [ pycor = -9 ] [ set pcolor black ] ; draw the road surface
let id 0
create-agents 5
set agent-id id
set id id + 1
set codelength-total 0
set dist-total 0
set dist-entropy 0
set source-agent one-of agents with [agent-id = 0]
set adaptive-agent one-of agents with [agent-id = 4]
set car-colours-index 0
set-current-plot "Distributions versus tick"
to setup-distributions
;; sets the distributions for each agent
if (ticks = 0)
[ ask agents with [agent-id = 4] ; initialise the adaptive distribution
[ set distribution (list 1 1 1 1 1) ]] ; all colours are equiprobable at the beginning for the adaptive agent
;; the adaptive agent's distribution counts are updated susequently elsewhere in update-adaptive-count
ask agents
[ set dist-total 0
set dist-entropy 0
foreach distribution [ set dist-total dist-total + ?] ; calculate total first
foreach distribution [ set dist-entropy dist-entropy + (? / dist-total) * neg-log-prob ? dist-total]]
to reset-distributions
;; resets the distribution counts (the user may have altered the slider values mid-stream)
ask agents with [agent-id = 0]
[ set distribution (list source-white source-black source-red source-blue source-pink) ]
ask agents with [agent-id = 1]
[ set distribution (list agent-01-white agent-01-black agent-01-red agent-01-blue agent-01-pink) ]
ask agents with [agent-id = 2]
[ set distribution (list agent-02-white agent-02-black agent-02-red agent-02-blue agent-02-pink) ]
ask agents with [agent-id = 3]
[ set distribution (list agent-03-white agent-03-black agent-03-red agent-03-blue agent-03-pink) ]
to-report neg-log-prob [p q]
;; returns the negative of the log to base 2 of the probability p/q.
report (- log (p / q) 2)
to-report distribution-as-a-point [agent]
; return the agent's distribution represented as a single point
report ;; note that maximum count for non-adaptive distributions is 1000
;; so make sure that max-count does not exceed 1000 for adaptive distribution
(remainder (item 0 [distribution] of agent) 1000) +
(remainder (item 1 [distribution] of agent) 1000) * 1000 +
(remainder (item 2 [distribution] of agent) 1000) * 1000 * 1000 +
(remainder (item 3 [distribution] of agent) 1000) * 1000 * 1000 * 1000 +
(remainder (item 4 [distribution] of agent) 1000) * 1000 * 1000 * 1000 * 1000
to plot-distributions
;; Plots the distributions as points to show how they vary with time.
set-current-plot-pen "Agent 01"
plot distribution-as-a-point one-of agents with [agent-id = 1]
; type "Agent 01 point = " print distribution-as-a-point one-of agents with [agent-id = 1]
set-current-plot-pen "Agent 02"
plot distribution-as-a-point one-of agents with [agent-id = 2]
; type "Agent 02 point = " print distribution-as-a-point one-of agents with [agent-id = 2]
set-current-plot-pen "Agent 03"
plot distribution-as-a-point one-of agents with [agent-id = 3]
; type "Agent 03 point = " print distribution-as-a-point one-of agents with [agent-id = 3]
set-current-plot-pen "Adaptive"
plot distribution-as-a-point one-of agents with [agent-id = 4]
; type "Adaptive point = " print distribution-as-a-point one-of agents with [agent-id = 4]
set-current-plot-pen "Source"
plot distribution-as-a-point one-of agents with [agent-id = 0]
; type "Source point = " print distribution-as-a-point one-of agents with [agent-id = 0]
to create-car-colours
;; creates the colours of the upcoming cars
let d 0 let i 0 let k 0
let total ([dist-total] of source-agent)
let colour black
set car-colours array:from-list n-values total [ 0 ] ; initialise array to 'black'
set d 0
set cars-dist-total [dist-total] of source-agent
foreach [distribution] of source-agent ; generate car colours according to current source distribution
[ ; create ? amount of colors
ifelse d = 0
[set colour white]
[ifelse d = 1
[set colour 1] ; almost black to make doors and windows of the car visible
[ifelse d = 2
[set colour red]
[ifelse d = 3
[set colour blue]
[set colour pink]]]]
set k 0
set i 0
while [k < ?]
[ ; insert new colour in a random location
set i i + random-poisson (round (total / ?)) + 1
if (i >= total) [set i (remainder i total)]
while [array:item car-colours i != black]
[ set i i + 1
if (i >= total) [set i 0]
array:set car-colours i colour
set k k + 1
set d d + 1
to-report next-car-colour
;; Returns the colour of the next car. Creates the colours of the upcoming cars
;; in advance according to the source distribution until it runs out, then creates a
;; whole new bunch again and again.
let total [dist-total] of source-agent
if (car-colours-index >= total) or (total != cars-dist-total)
[ ; we've run out of cars or user has changed source distributions counts
create-car-colours ; create next lot of upcoming cars in advance
set car-colours-index 0
set car-colours-index car-colours-index + 1 ; move onto next colour for next time
report array:item car-colours (car-colours-index - 1)
to-report source-entropy
report [dist-entropy] of source-agent
to-report adaptive-entropy
report [dist-entropy] of adaptive-agent
to-report agent-01-entropy
report [dist-entropy] of one-of agents with [agent-id = 1]
to-report agent-02-entropy
report [dist-entropy] of one-of agents with [agent-id = 2]
to-report agent-03-entropy
report [dist-entropy] of one-of agents with [agent-id = 3]
to-report source-codelength
report [codelength-total] of source-agent
to-report adaptive-codelength
report [codelength-total] of adaptive-agent
to-report agent-01-codelength
report [codelength-total] of one-of agents with [agent-id = 1]
to-report agent-02-codelength
report [codelength-total] of one-of agents with [agent-id = 2]
to-report agent-03-codelength
report [codelength-total] of one-of agents with [agent-id = 3]
to-report adaptive-white
report item 0 ([distribution] of adaptive-agent)
to-report adaptive-black
report item 1 ([distribution] of adaptive-agent)
to-report adaptive-red
report item 2 ([distribution] of adaptive-agent)
to-report adaptive-blue
report item 3 ([distribution] of adaptive-agent)
to-report adaptive-pink
report item 4 ([distribution] of adaptive-agent)
to-report create-new-car
;; creates a new car that enters screen from the left and
;; reports its colour
let this-colour black
create-cars 1
[ set color next-car-colour
set shape "car"
set heading 90
set size 10
set xcor (- max-pxcor)
set ycor -5
set this-colour color
report this-colour
to update-adaptive-count [colour]
;; updates the colour's count for the adaptive agent
let index 0
let this-count 0
ask adaptive-agent
ifelse colour = white
[set index 0]
[ifelse colour = 1 ; almost black to make doors and windows of the car visible
[set index 1]
[ifelse colour = red
[set index 2]
[ifelse colour = blue
[set index 3]
[set index 4]]]]
set this-count (item index distribution)
set distribution replace-item index distribution (this-count + 1)
set dist-total dist-total + 1
set dist-entropy 0
foreach distribution [ set dist-entropy dist-entropy + neg-log-prob ? dist-total]
to encode-this-car [colour]
;; returns the cost of encoding this car's colour according to the agent's distribution
let codelength 0
ask agents
set codelength 0
ifelse colour = white
[set codelength neg-log-prob (item 0 distribution) dist-total]
[ifelse colour = 1 ; almost black to make doors and windows of the car visible
[set codelength neg-log-prob (item 1 distribution) dist-total]
[ifelse colour = red
[set codelength neg-log-prob (item 2 distribution) dist-total]
[ifelse colour = blue
[set codelength neg-log-prob (item 3 distribution) dist-total]
[set codelength neg-log-prob (item 4 distribution) dist-total]]]]
set codelength-total codelength-total + codelength
to animate-cars
; move the cars across the screen from left to right
let colour black
setup-distributions ; reset the distributions (in some cases user might have changed one of the counts)
if (ticks = next-car-tick)
[ set colour create-new-car
encode-this-car colour
update-adaptive-count colour
set next-car-tick ticks + random next-car-random-tick-interval + 12] ; make sure cars are apart
ask cars
[ if (xcor + 1 > max-pxcor)
[ die ] ; move this car off the screen
set xcor xcor + 1 ; move this car to the right by 1
; Copyright 2010 by William John Teahan. All rights reserved.
; Permission to use, modify or redistribute this model is hereby granted,
; provided that both of the following requirements are followed:
; a) this copyright notice is included.
; b) this model will not be redistributed for profit without permission
; from William John Teahan.
; Contact William John Teahan for appropriate licenses for redistribution for
; profit.
; To refer to this model in publications, please use:
; Teahan, W. J. (2010). Cars Guessing Game NetLogo model.
; Artificial Intelligence. Ventus Publishing Aps.
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Find the minimal polynomial for a = π^2 over Q(π^3)
March 26th 2012, 10:11 AM #1
Mar 2011
Find the minimal polynomial for a = π^2 over Q(π^3)
If it were in Q then the polynomial would be p(x) = x - π^2, but I don't know if being over Q(π^3) changes that. I think that when you compare the fields you get degree 1, but again, I'm not
completely sure.
Re: Find the minimal polynomial for a=π^2 over Q(π^3)
If the extension is $\mathbb Q(\pi)/\mathbb Q\left(\pi^3\right)$ the minimal polynomial is $x^3-\pi^6.$
Re: Find the minimal polynomial for a = π^2 over Q(π^3)
Is this because the minimal polynomial needs to be degree three?
Re: Find the minimal polynomial for a=π^2 over Q(π^3)
The minimal polynomial here is the monic polynomial $f(x)$of minimal degree with coefficients in $\mathbb Q\left(\pi^3\right)$ such that $f\left(\pi^2\right)=0.$ If $f(x)$ has degree $1$ or $2,$
$f\left(\pi^2\right)$ is of the form $a_0+\pi^2$ or $a_0+a_1\pi^2+\pi^4,$ where $a_0,a_1\in\mathbb Q\left(\pi^3\right);$ clearly these cannot be $0.$
March 26th 2012, 10:33 AM #2
March 26th 2012, 11:54 AM #3
Mar 2011
March 26th 2012, 01:40 PM #4
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Posts by
Posts by VIVI
Total # Posts: 21
What are two important unifying forces in central asia? HELP ME PLEASE!!!
A 1.6 N loop of wire is hung in a magnetic field as shown. If the spring scale reads 0.60 N when a current of 1.0 A passes through the loop what is the magnitude and direction of the magnetic field?
Holly is flipping a coin and pulling a marble from a bag. There are 4 white marbles, 2 blue marbles, and 5 green marbles, all of the same size, in the bag. What is the probability that the coin lands
on heads and she pulls a green marble from the bag?
A road worker holds a flag in position R with an out strech arm. He swings the flag downward and it forms angle of 30degrees with the ground P. His shoe Q is 2,5 away from P. His head is 24cm above
his shoulder. calculate the height of the man
Car A traveling southward towards Point P at 45 mi/hr. Car B is traveling east away from P at 30mi/hr. AT the instant when the distance AP is 60 mi and PB is 80 mi, what is the rate of change of
distance AB?
Water is draining at a rate of 2 cubic feet per minute from the bottom of a conically shaped storage tank of overall height 6 feet and radius 2 feet . How fast is the height of water in the tank
changing when 8 cubic feet of water remain the the tank? Include appropraite units...
Consider a box with a square base of side length s and height h. If the sum of the length, width, and height of the the box is 120 inches, what is the maximum possible volume? Justify your answer.
The concentration of a drug in a patient's bloodstream t hours after it is taken after it is taken is given by C(t) = 0.016t/(t+2)^2 mg/cm^3 Find the maximum concentration of the drug and the time at
which it occurs.
The concentration of a drug in a patient's bloodstream t hours after it is taken after it is taken is given by C(t) = 0.016t/(t+2)^2 mg/cm^3 Find the maximum concentration of the drug and the time at
which it occurs.
us history
I need to do a thesis on a late-present American history about anything. I can't think of something that will be easy and easily researched. Any ideas?
Roberto ate 3 pieces of a pizza and them felt that he should pay 1/4 of the cost because that's the fraction he ate. How many pieces was the pizza cut into?
Roberto ate 3 pieces of a pizza and them felt that he should pay 1/4 of the cost because that's the fraction he ate. How many pieces was the pizza cut into?
Thomas is playing Tic-Tac-Toe with a computer. It is the computer's turn to place an "x" on the board. If the computer makes its moves at random in the open spaces, what is the chance it will win on
this move?
4th grade
MY DAUGTHER WILL BE 10 YEARS NEXT WEEK AND THE HOMEWORK SAID: A NORMAL PERSON BLINKS ABOUT 25 TIMES PER MINUTE WHEN AWAKE. A) HOW OLD WILL YOU BE ON YOUR NEXT BIRTHDAY; B) TO THE NEAREST MILLION, HOW
MANY TIMES WILL YOU HAVE BLINKED ON YOUR NEXT BIRTHDAY?. ASSUME YOU SLEEP 8 H...
4th grade
On the average your heart beats about 72 times per minute. At this rate, about how many times will it beat: a) in a 30-day month; b) in a year; c)in your lifetime, if you live to 72 years of age?
4th grade
IN A TUG OF WAR, 5 DONKEYS ARE EXACTLY EQUAL TO 2 ELEPHANTS. IN ANOTHER TUG OF WAR, 3 ELEPHANTS ARE EQUAL TO 1 CAR. WHICH TEAM SHOULD WIN IF A CAR AND 3 DONKEYS ARE MATCHED AGAINST 4 ELEPHANTS?
The edge of a cube is increasing at a rate of .05 cm/s. In terms of the side of the cube, s, what is the rate of change of the volume of the cube?
This is a quote from an essay on Liberty by John Stuart Mills. The peculiar evil of silencing the expression of opinion is that it is robbing the human race. If the opinion is rights, they are
deprived of the opportunity for exchanging error for truth. If wro...
Today "freedom of speech" is limited in interest of public safety by the clear and present danger test. do you feel this is a good standard?
The lions ate up all the prophets
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Bolyai (bōˈlyoi) [key], family of Hungarian mathematicians. The father, Farkas, or Wolfgang, Bolyai, 1775–1856, b. Bolya, Transylvania, was educated in Nagyszeben from 1781 to 1796 and studied in
Germany during the next three years at Jena and Göttingen, where he began a lifelong friendship with Carl F. Gauss. From 1804 to 1853 he was professor of mathematics at Maros Vásárhely. His primary
interest was in the Euclidean parallel postulate. His principal work, the Tentamen (1832–33), inspired by his mathematically gifted son János, is an attempt at a rigorous and systematic foundation of
geometry (Vol. I) and of arithmetic, algebra, and analysis (Vol. II).
János, or Johann, Bolyai, 1802–60, b. Koloszvár, Transylvania, was educated by his father in Maros Vásárhely and from 1818 to 1822 in Vienna, where he received military training at the imperial
engineering academy. In 1820 he began to work in a direction that ultimately led him to a non-Euclidean geometry. In 1823, after vain attempts to prove the Euclidean parallel postulate, he developed
his system by assuming that a geometry could be constructed without the parallel postulate. His theory of absolute space was published as an appendix to his father's Tentamen and constituted the sole
work published in his lifetime.
The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.
More on Bolyai from Fact Monster:
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Teachers and Homeschoolers: Teaching ideas to make probability easy
Dear Visitor,
Let's review what we want to do to make math fun both for you and for your students.
FIRST: spend a short time each class period (perhaps 5 minutes) on something mathematical which is puzzling or fascinating.
SECOND: Let your students help you teach. Don't force proofs or methods down their throats. Give them a lead, and then let them make suggestions.
THIRD: Teach how to solve problems. Try Prof. George Polya's steps:
• Understand the problem: (what's unknown? What's given?)
• Devise a plan (Have you seen this problem, or a similar one, before? Think of a similar problem with the same unknown, and see if you can apply it.)
• Carry out the plan.
• Check the result (And see if you can find another way of finding it.)
FOURTH: Think of teaching as an art. Develop your own style just as an actor, dancer, painter, or writer does.
EXAMPLE 1
[Start as follows at the end of class.
When you see the ... that means you should wait for an answer.]
"You're playing Monopoly with your buddies, and you own Park Place and want to buy Boardwalk. When you last rolled the dice, you found yourself just four squares away from Boardwalk.
"What are your chances of rolling a four the next time it's your turn?"
[Let the kids guess, and write down their answers on the blackboard.]
"Tomorrow we'll start the class by figuring just how lucky you have to be to buy your heart's desire."
[Start as follows at the beginning of the next class.]
"Yesterday we were playing Monopoly, and were four squares from Boardwalk. We wanted to know what our chances were of rolling a four with the dice when it was our turn again.
"How do we solve this problem? Our plan should be to [write on board] Understand the problem, Plan how to solve it, Carry out the plan, Check our answer,
"So first -- Understand the problem. Here the problem is, we have two dice and want to know our chances of getting a four. How can we get a four with two dice? ... Roll two two's? Very good. [Write
that on the board]. Any other way to get a four? ... Roll a one and a three? Very good. [Write on board]. Any other way? ... When we said 'Roll a one and a three', we meant 'Get a one on die 1 and a
three on die 2'. How else can we get a four? ... How about if we get a three on die 1 and a one on die 2? [Write on board.]
"Now we think we understand the problem. There are three ways to get a four when we roll two dice, and we have to figure the chances of one of those three possibilities occurring. Now we must plan
how to get an answer. How do we figure our chances of rolling one of those three combinations? ...
"Have we seen a problem like this one before? ... No? One scheme for solving a problem we haven't seen before is to try to think of a similar problem that's easier. Anyone have an idea of a similar
easier problem? ... How many sides does a die have? ... Six, that's right. What could we use that has less than six sides? ... How about a coin? How many sides does a coin have? ... Two.
"Before we tackle the dice problem, let's look a similar problem with coins. If we flip one coin, what's the chance we'll get a head?
"Now let's make the problem a little more complicated. Let's flip a second coin, and ask: what's the chance of our getting a Head on one coin and a Tail on the other? Can anyone guess the answer? ...
[Looking at the drawing on the board] "Let's use the same scheme we used when we flipped one coin. When we flip the second coin, we 'start' after we've flipped the first -- either here where we had
got a Head, or below where we got a Tail. Suppose we got a Head. What are the possibilities of the second coin? ... Right. Either a Head or a Tail. [Draws the upper two branches]. And what are our
chances of getting a Head with this second toss? ... One-half. That's right. Every time we flip a coin, no matter what the last flip was, our chance on the next flip is 1/2 for a Head, and 1/2 for a
Tail. The coin has no idea what happened last time it was flipped. So even if the last 100 flips were Heads, the chance of a Head or Tail on the next flip is 1/2. [Put 1/2 on upper branch.]
"What's the chance of getting a Tail on this second toss? ... Right. One-half again. [Put 1/2 on lower branch.]
"Now suppose our first flip got us a Tail. What's the chance that the next flip will give us a Head? ... Right. 1/2. And a Tail? ... Right. 1/2 again. [Draw the two 1/2's on the lower part of the
diagram.] So with two coins, there are four possible results. Two Heads, a Head then a Tail, a Tail then a Head, and two Tails.
"What's the chance of getting two Heads? [Point to the top branch. ... Right! One in four, or 1/4th. How did you get that? ... Well, our chance was 1/2 of getting the first Head, and 1/2 of getting
the second. The chance of getting the first, then the second is 1/2 times 1/2, or 1/4. What's the chance of getting each of the other three results? ... Right! 1/4. For each possibility, we multiply
the possibilities along the path.
"Another way of looking at the problem is to realize that we're equally likely to get any of the four results. So the four chances must all be equal. But what must the add up to? What must be the
total of the four chances? ... The chance we'll get two Heads, or a Head and a Tail, or a Tail and a Head, or two Tails, is 'for sure'. What probability number goes with 'for sure'? ... That's right,
one. So the four chances must be equal and must add to one. So each of them must be 1/4.
"Now we have a plan. We'll treat the dice just like we did the coins. Let's carry out the plan. How many branches should I have from the starting point? ... How many results can I get from the first
roll of the die? ... That's right, six. [Draws the six branches and their results.] What's the chance of my rolling a one? ... That's right. One time in 6 I'll get a one, so the chance is 1/6. What's
the chance of rolling a two? ... Right 1/6. What's the chance of getting a three or a four or a five or a six? ... Right. 1/6 for each. [Writes the 6 1/6ths].
"Now for the second roll of the dice. How many branches will we have from each of these six branches of the first roll? ... If we roll a one on the first roll, how many possible results will there be
from the second roll? ... Right, six. So if we have six results from each of the six first-roll results, how many results will there bealtogether? ... Right. 36. Six time six.
"Do we have to look at all 36 results? ... What are we interested in? ... What total do we want, for the two rolls? ... That's right, 4. We're trying to roll a 4. So: do we have to look at all 36
results? ... If the first roll gives us a 6, can we have a total of 4 for two rolls? ... Of course not. So we don't have to look at what happens after we roll a six. What first-rolls DO we have to
look at? ... Right. Just a 1 or a 2 or a 3. If we roll a 4, 5, or 6 on the first roll, we can't get a total of 4.
"Suppose we rolled a one on the first roll. What must we roll the second time, to get a total of 4? ... That's right, 3. And what's the chance of getting a 3 on that second roll? ... One in six, or 1
/6. (Remember, the second die doesn't 'know' what the first one did. No matter how the first tie turned, the chance of getting a 3 with the second is 1/6.) What's the chance, then of our getting
first a 1 and then a 3? ... How did we calculate the chance of getting first a head, then another head? ... We multiplied the chances, didn't we? ... So what's the chance of getting first a 1 and
then a 3? ... That's right, one in 36 or 1/36 -- 1/6 times 1/6. [Draw the branch from the first one, and mark one-three with 1/36.]
"What's the chance of rolling first a two, and then another two? ... Right. 1 in 36, or 1/36. How about first a three, and then a one? ... Right. 1/36 again. [Draw these last results]. In fact, how
many total results can we have from the two rolls? ... Right, 36. And are all of the 36 results equally likely? ... Yes, they are. And what must all the chances add up to? ... Right, 1. So if they're
all equal, and they add to 1, what must each of them be? ... Right. 1/36.
"Now we're almost finished. What's the chance of rolling a 4? ... What's the chance of rolling either a 1 and 3, or 2 and 2, or 3 and 1? ... How did we find the chance of getting a Head-Tail or a
Tail- Head? ... Right we added the chances of each of them together. So what's the chance of rolling a 4? ... Right. We add 1/36 + 1/36 + 1/36 and get 3/36, or 1/12. One time in twelve we'll get a
four when we roll two dice.
"How can we check our result? Any ideas? ... Here's one sort-of-check. Suppose we're 10 squares from Boardwalk and want to know our chances of getting there in one roll of two dice. What rolls will
give us a 10? ... A five and a five. Right. Any others? ... A six then a four. Any others? ... A four, then a six. Any others? ... No. No more. So there are three ways of getting a 10, just like
there are 3 ways of getting a 4. How do we compute the chance of getting a 10? ... What's the chance of getting two fives? ... Right. 1 in 36, or 1/36. So what's the chance of getting a 10? ...
Right. 1/36 + 1/36 + 1/36 = 3/36 = 1/12. The same as for getting a 4. That's a sort of check, anyway."
So there's a first example. How can you find more examples -- enough so you have a new one every day in the week for a whole year?
There are a lot of books full of ideas, and many math links on the Web full of ideas. Click on the Book and Link buttons down below to find them. A particularly good link is Cut-the Knot.com --an
award-winning site full of good ideas. And a particularly useful book is Teaching Mathematics--a Sourcebook by M.A. Sobel and E.A. Kaletsky. Finally, if you subscribe to our hilarious free math
newsletter, the Gnarly Gnews, you'll find new ideas in every issue. Just click the Newsletter button down below to subscribe.
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limit questions
April 11th 2011, 09:48 AM #1
Mar 2011
limit questions
Find the radius of convergence of the power series: $((n!)^2/(2n)!)x^n$
My answer; 0 using $a(n+1)/a(n)$
Find the limits, if they exists, of
i) $100(ln(x))/x^3$ as n tends to infinity
ii) $(sinx-x)/(cosx -x^2-1)$ as n tends to 0
My working i) pretty certain its infinity but not sure how to show it
ii) no idea on this one.
$a(n+1) = \dfrac{((n+1)!)^2}{(2(n+1))!}.$ But $(n+1)! = (n+1)n!$, and $(2(n+1))! = (2n+2)! = (2n+2)(2n+1)(2n)!$. Now take another look at $a(n+1)/a(n)$ and see if you still get the same result.
L'Hôpital's rule should do these (apply it repeatedly until you get an answer). For i), you can guess what the answer should be (and it's not infinity) if you remember the general principle that
ln(x) tends to infinity more slowly than any positive power of x.
ok I got 0.5, 0 and non exsistent respectively. Can I use l'hopital's rule for sequences or just functions? Thank for the help.
I am doing an analysis exam next month on limits of sequences, series and functions. I don't suppose you have any tips as I often feel like i'm stumbling in the darkness trying to figure out
which method to use.
April 11th 2011, 11:02 AM #2
April 12th 2011, 02:25 AM #3
Mar 2011
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London 1889 - Reading 1961
Brief scientific biography
Eric Harold Neville was born in London on 1 January 1889. Attending the William Ellis School, his mathematical abilities were recognised and encouraged by his mathematics teacher, T. P. Nunn. In
1907, he entered Trinity College, Cambridge, graduating as second wrangler two years later, and subsequently winning a Trinity fellowship. While there he became acquainted with other Cambridge
fellows, most notably Bertrand Russell and G. H. Hardy.
The story of Hardy and his Indian protégé Srinivasa Ramanujan is one of the most famous in the history of modern mathematics. But Neville's role in it is less well known. In 1914, as a visiting
lecturer, he travelled to India, where, in response to a request from Hardy, he managed to persuade the cautious Ramanujan to accompany him back to England, thus playing a vital role in the
initiation of one of the most celebrated mathematical collaborations of the last hundred years.
Despite being eligible to serve during the First World War, Neville did not join the army when hostilities erupted in the summer of 1914. Although poor eyesight would have prevented him from active
service, he openly declared his opposition to the conflict and refused to fight. While no reason was ever given, it was probably this pacifist declaration that resulted in the non-renewal of his
Trinity fellowship in 1919. On leaving Cambridge, he was appointed to the chair of mathematics at the small University College, Reading. In a few years, he had built up the mathematics department
there, working vigorously to enable the institution to receive a university charter and award its own degrees from 1926.
Neville had a wide variety of mathematical interests, but his principal areas of expertise were geometrical, with differential geometry dominating much of his early work. Early on in his Trinity
fellowship, in a dissertation on moving axes, he extended Darboux's method of the moving triad and coefficients of spin by removing the restriction of the orthogonal frame. He later wrote an
introductory tract on how to generalise concepts and operations of 3-space into the four-dimensional arena. But his ambition to write a comprehensive treatise on differential geometry was never
During his time in Cambridge, he had been greatly influenced by Bertrand Russell's work on the logical foundations of mathematics and in 1922 he published his
Prolegomena to analytical geometry
prompted by the absence, in his view, of adequate foundational treatments of the topic. Highly influenced by Russell's work, it is a detailed and logical investigation of the foundations of
analytical geometry, providing an axiomatic development of the subject.
Neville had long held a keen interest in elliptic functions, having taught the subject to postgraduate students at Reading since the 1920s. He believed that the subject's recent decline in popularity
was due to its dependence on a mass of complicated formulae, a variety of differing and confusing notations, and an artificial definition relying on a familiarity with theta functions. A period of
recuperation from an illness in 1940 gave him the opportunity to put several years of lecture notes into publishable form. The result was his best-known, and perhaps most original, work:
Jacobian elliptic functions
By starting with the Weierstrass
-function and associating with it a group of doubly-periodic functions with two simple poles, he was able to give a simple derivation of the Jacobian elliptic functions, as well as modifying the
existing notation to provide a more systematic approach to the subject. Like all of Neville's books,
Jacobian elliptic functions
, while intricate and not easy to read, is expertly crafted and painstakingly thorough. Unfortunately, it failed to achieve its author's stated intention "to restore the Jacobian functions to the
elementary curriculum" (NEVILLE 1951, vi) and its appearance came too late to have any real effect on the dominance of the classical approach to elliptic functions.
Neville was an active member of several mathematical and scientific bodies. Elected to membership of the London Mathematical Society in 1913, he served on its council from 1926 to 1931. He regularly
attended meetings of the British Association for the Advancement of Science, being President of Section A (Mathematics and Physics) in 1950. He also chaired its Mathematical Tables Committee from
1931 to 1947 and, when it came under the auspices of the Royal Society, he contributed two sets of tables, on
Farey series of order 1025
(1950) and
Rectangular-polar conversion tables
Neville published many papers, but the vast majority were short items, focusing on concise and succinctly-solved problems, often in the
Mathematical Gazette
, to which he was a frequent contributor. As with all of his writings, they were focused and highly polished, yet, as one obituary says with regret, "so brilliant and versatile a talent could have
been harnessed to some major mathematical investigation" (BROADBENT 1962, 482). Indeed, a former student at Reading could "never understand why his published work of substance was so small in
quantity" (LANGFORD 1964, 133).
Neville retired from the University of Reading in 1954, after which he continued to publish papers in the
Mathematical Gazette
. He was working on a sequel to his book on elliptic functions when he died on 22 August 1961.
Neville's work in mathematical education
Neville's work in mathematical education manifested itself in three main forums: his teaching at the University of Reading; his work for the Mathematical Association; and his membership of the
Executive Committee of ICMI.
By all accounts, in his lectures at Reading, Neville was a skillful and considerate teacher, possessing a "unique gift of handling problems with superb technical skill and economy" (LANGFORD 1964,
132). From the perspective of one of his students from the 1920s, "as a teacher he was an inspiring guide (though sometimes so far ahead as to be almost out of sight) but with the small classes of
those days-there were never more than three of us in the honours group-a lecture could always become a seminar if
wished, and he delighted in the arguments which could develop" (LANGFORD 1964, 134). However, according to one of his colleagues at Reading, the sharpness of his mind and the depth of his knowledge
could often leave the less able students feeling rather baffled:
Honours students were inspired by the brilliance of his lectures and the immensity of his erudition; and if the pass degree pupils sometimes found him above their heads, this was never from any
failure of his sympathy, but because he could often modestly forget how fast his own mind worked (BROADBENT 1962, 479)
The challenges of teaching mathematics at Reading prompted the beginning of Neville's active involvement in issues concerning mathematical education, and his membership of the chief organisation in
Britain devoted to mathematical pedagogy, the Mathematical Association. We are told: "In the steady growth of the Association from 1920 onwards, in the widening of its interests, in the spreading of
its influence, Neville played a part second to none" (BROADBENT 1964, 139).
Introduced to its activities by his former teacher, T. P. Nunn, in 1922 he chaired a sub-committee of the Association's General Teaching Committee charged with reporting on the teaching of geometry
in British schools. The resulting report recommended dividing school geometry into stages: experimental, deductive, systematising, and advanced. Later described as "revolutionary", one committee
member later wrote: "It is perhaps not giving away a secret to say that T.P.N. and E.H.N. were the two principally responsible for the Report, which has been a best-seller ever since" (BROADBENT
1964, 137).
His involvement with the Association's chief publication,
The Mathematical Gazette
, began while he was still at Cambridge. For over four decades, Neville contributed a multitude of articles, classroom notes and book reviews on a wide variety of mathematical topics. He also briefly
edited the journal in the late 1920s, following the illness and death of the then editor, until a replacement was appointed. He was the Association's Librarian for over thirty years, from 1923 to
1954, and served as its President in 1934.
His presidential address, "The Food of the Gods", drew attention to the widening gap between school and university mathematics. He began by noting that, in the quarter-century since he took his
degree, British university mathematics courses had grown considerably, featuring subjects (such as matrices, vectors, Lebesgue integration, tensor calculus, statistics, relativity and wave mechanics)
that were all but unknown to British undergraduates 25 years before. His central argument was that "the university builds a different mathematical structure, but is content to build it on foundations
which have not changed since the beginning of the century" (NEVILLE 1935, 7). He continued:
The burden of my plea this afternoon is that changes in emphasis in creative mathematics, which have now a direct influence on teaching at the university, ought to have a greater and a far more
rapid influence on teaching at the school than they seem to have. (NEVILLE 1935, 16)
It was his opinion that, while the more experienced teachers have greater authority and influence, it is younger teachers who are more in touch with what will best equip a student for university
Twenty years ago we did know what were the best current methods of presentation, where emphasis had to be placed to serve most efficiently the needs of those who were soon to be undergraduate
students of mathematics. How many of us who are engaged in teaching rather than research can make the same boast to-day? (NEVILLE 1935, 16)
Urging his colleagues to consider changes to the secondary school curriculum, he asked:
Is it absolutely certain that the [school] curriculum is perfect, that there is nothing which could be postponed in favour of some subject now acquired at a later stage? Is it quite indisputable
that none of the teaching is wasteful, that nowhere would better methods enable us to explain in one hour a principle over which we have got into the habit of spending two? (NEVILLE 1935, 7)
The theme is as pertinent today as when it was delivered in 1935.
Neville's work for the Mathematical Association and British mathematical education generally brought him recognition from overseas. The high regard in which he was held by the British pedagogical
community was reflected in his election in 1932 as a member of the Central Committee of ICMI, on which he served with
as President and
as Secretary General. He was re-elected in 1936.
Essential bibliography
E.H. NEVILLE 1921,
Multilinear functions of direction, and their uses in differential geometry
, Cambridge, Cambridge University Press
E.H. NEVILLE 1921,
The fourth dimension
, Cambridge, Cambridge University Press
E.H. NEVILLE 1922,
Prolegomena to analytical geometry in anisotropic Euclidean space of three dimensions
, Cambridge, Cambridge University Press
E.H. NEVILLE 1944,
Jacobian elliptic functions
, Oxford, Clarendon Press, Second edition, 1951
E.H. NEVILLE 1950,
The Farey series of order 1025. Displaying solutions of the Diophantine equation
by - ax = 1
, Cambridge, Cambridge University Press
E.H. NEVILLE 1956,
Rectangular-polar conversion tables
, Cambridge, Cambridge University Press
T. A. A. BROADBENT 1962,
Eric Harold Neville
, Journal of the London Mathematical Society, 37, 479-482
T. A. A. BROADBENT 1964,
On the Teaching Committee
, The Mathematical Gazette, 48, 136-139
W. J. LANGFORD 1964,
Professor Eric Harold Neville, M.A., B.Sc.: The man
, The Mathematical Gazette, 48, 131-136
Articles on the teaching of mathematics
E.H. NEVILLE 1919,
Notes for lessons introductory to differential geometry
, The Mathematical Gazette, 9, 369-371
E.H. NEVILLE 1930,
Higher trigonometry for schools
, The Mathematical Gazette, 15, 180
E.H. NEVILLE 1933,
The teaching of geometry
, The Mathematical Gazette, 17, 307-312
E.H. NEVILLE 1935,
The food of the gods
, The Mathematical Gazette, 19, 5-17
E.H. NEVILLE 1937,
The influence of the university on school geometry
, The Mathematical Gazette, 21, 339-343
E.H. NEVILLE 1964,
Mathematical notation
, The Mathematical Gazette, 48, 145-163
Adrian Rice
Randolph-Macon College, Ashland, Virginia
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Finding a solution given initial value problem
y'-y=7te^2t , y(0)=1
I would like to know how to solve this Differential Equation. I think I have to find the integrating factor.
It does look like it follows the the standard form So the integrating factor is e^∫px
Would the integrating factor be e^∫-y ?
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The Art of the Subnet Cheat Sheet And Other Subnetting Tips
As I've grown older, I've grown less tolerant of memorizing useless information. After hearing people talk of memorizing a subnetting "cheat sheet," to help on Cisco exams, my thought was I would
just memorize the formulas and do the math instead of memorizing yet more useless information. However, that mistake became painfully obvious as I failed the Cisco INTRO exam because I ran out of
time before I was finished with the test. And because I scored over 700 points on what I did complete, I'm reasonably sure I would have passed if I had been able to complete all of the questions.
When I first started doing subnetting, I was actually doing boolean AND operations, and to figure out the proper subnet mask, I was writing out the netmask in binary and converting it to decimal.
While I'm sure it is good to know all of that, Cisco let a nugget of truth slip out in its CCNA Self-Study ICND Exam Certification Guide (emphasis added):
Using the binary math required to find the subnet number and broadcast address really does help you understand subnetting to some degree. To get the correct answers faster on the exam, you
might want to avoid all the conversion and binary math.
The following subnet cheat sheet will not solve all of the subnetting questions on the exam. What it will do, however, is gain valuable time. Since you can't take anything into the exam, the
trick is to write the following chart out on your dry-erase board before you start the exam:
And if you examine the chart very closely, you can actually reproduce it with very little memorization. Here is the way to do it. First, duplicate column one of the table, which is fairly easy,
and then fill in the second column, which is nothing but multiples of two, starting out at four. (if you aren't good with multiplication, when you get to the higher numbers, you can actually just
write the numbers out to the side twice and add them together to get the number for the next row).
Next, fill in the netmask for the /24 network and the /16 network, which should also be easy to remember (if you are about to take the exam and can't remember /24 and /16 netmasks, you might as
well hang it up). The netmask for the /30 network is also fairly easy to remember, but if you forget any of the netmasks, all that needs to be done is subtract the number of hosts directly to the
left of it to get the next netmask. For example, 255.255.255.252 provides for four hosts (two useable, because zero is the network and .4 is the broadcast address). If you take 252 and subtract 4
from it, you get the netmask for the next row, 255.255.255.248. If you take 255.255.255.248 and subtract its 8 hosts, you get 255.255.255.240, which is the netmask for the next row. This works
all the way down to the /24 network.
The only odd netmask to memorize is the /23 netmask, which is 255.255.254.0. That is also not hard to remember because it is just one off from the .255 directly above it. Note that after the /23
network, all of the network masks are identical to the /25 - /30 networks, just move them over one octet to the left. So from /22 to /17, you already have the needed information, just fill it in.
For the fourth column, just put a 1 in the /24 network and put in multiples of two up the chart from the /24 network, as well as down the chart, to the /16 row.
As you can see, once you understand the table, it can be reproduced with very little memorization.
How to use the chart
Now that you have the chart, if a simulation question calls for a /27 netmask, instead of writing it out in binary and converting it to decimal, you can just refer to the chart and plug in the
netmask 255.255.255.224.
If the exam question asks for the network and broadcast address of a host, for example, 192.168.1.68 /27, simply look at the hosts provided by the /27 network mask, which is 32. Now simply add by
32 until you get to a subnet higher than the .68 host (be sure to add 32 each time and not get into using multiples of 2, which is easy to do here. If you get the numbers, 32, 64, 128, 256, you
doing it wrong and will miss questions on the exam). What you should come up with is this:
Since the address is 192.168.1.68, it must fall in the subnet between 64 and 128. And since the first address of the subnet is the network address and the last address of the subnet is the
broadcast, the network address is 192.168.1.64 and the broadcast address is 192.168.1.95.
If the exam asks to find a network that will allow for 4 subnets and at least 48 hosts per subnet, just look at the table and pick the row that matches:
┃ │Hosts│ Netmask │Number of Subnets ┃
┃/26│64 │255.255.255.192 │4 ┃
If a question involves two IP addresses, for example 172.145.1.85 /28 and 172.145.1.92 /28 and the questions asks if they are in the same network, just look at the hosts provided by the /28
netmask, which is 16. Count by 16 until you pass the networks involved:
Since you know that the network address is 172.145.1.80 and the broadcast address is 172.145.1.95, then you know both IP addresses are in the same network.
The Cisco exams are geared toward people who are sharp at math. If you aren't a math wizard, you will be at a disadvantage to those who are, because a good portion of the INTRO and ICND exams
deal with subnetting. If you aren't good with math, the subnetting cheat sheet will only help so much (and it won't help with converting binary or hex at all), as some of the questions are asked
in a manner designed specifically to confuse a person as much as possible. If a subnetting question is especially confusing, don't waste a large amount of time on it.
If you know your stuff, you can miss some subnetting questions and still pass the exam. It would be better to miss some subnetting questions and answer all of the questions on the exam, then to
run out of time on the exam because you spent too much time on a particularly confusing subnetting question. Remember, if you pass the exam by one point, you are just as Cisco certified as the
math wizard who passed the exam with a score of 978.
If you aren't sharp at math, your goal should be to know your stuff well enough that you can miss some of the trickier subnetting questions on the exam and still pass comfortably. It can't be
said enough times, you have to know your stuff.
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"Day Off, a Boss's Perspective"
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"Day Off, a Boss's Perspective"
So, you want the day off?
Let's take a look at what you are asking for. There are 365 days per year available for work. There are 52 weeks a year in which you already get 2 days off per week, leaving 261 days available for
work. Since you spend 16 hours each day away from work, you have used up 170 days, leaving 1 days available. You spend 30 minutes a day on a coffee break. That accounts for 23 days each year, leaving
only 68 days available. With one Hour for lunch period each day you use up another 46 days, leaving only 22 days available to work. You normally spend 2 days a year for sick leave. This leaves you
only 20 days available for work. We are off for 5 holidays per year, so your available working time is down to 15 days. We generously give you 2 weeks off for vacation per year. This only leaves 1
day available for work.
And I'll be darned if you're going to take that day off!!
Next Joke --> Return to Jokes Back to Jokes - Work
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
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Functional Dependencies
Dirk Reckmann reckmann at cs.tu-berlin.de
Thu Jul 21 05:30:03 EDT 2005
Hello everybody!
I wanted to have some fun with functional dependencies (see
http://www.cs.chalmers.se/~hallgren/Papers/wm01.html), and tried some
examples from this paper as well as some own experiments. The idea is to use
the type checker for computations by "abuse" of type classes with functional
The example in the attached file is taken from the above paper. Due to the
functional dependencies, I expected the type of seven to be uniquely
determined to be (Succ (Succ (Succ ...))), i. e. seven, but ghc (version 6.4)
gives me following error message:
Couldn't match the rigid variable `a' against `Succ s'
`a' is bound by the type signature for `seven'
Expected type: Succ s
Inferred type: a
When using functional dependencies to combine
Add (Succ n) m (Succ s), arising from the instance declaration at
Add (Succ (Succ (Succ Zero))) (Succ (Succ (Succ (Succ Zero)))) a,
arising from the type signature for `seven' at Add.hs:13:0-77
When generalising the type(s) for `seven'
However, using the definition of Add to define Fibonacci numbers does work,
and a similar function declaration can be used to compute numbers by the type
The same definition of Add works in Hugs...
So, is this a bug in ghc, or am I doing something wrong?
Thanks in advance,
Dirk Reckmann
-------------- next part --------------
{-# OPTIONS -fglasgow-exts #-}
module Add where
data Zero
data Succ n
class Add n m s | n m -> s
instance Add Zero m m
instance Add n m s => Add (Succ n) m (Succ s)
seven :: Add (Succ (Succ (Succ Zero))) (Succ (Succ (Succ (Succ Zero)))) a => a
seven = undefined
More information about the Glasgow-haskell-users mailing list
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Commerce, GA Algebra 2 Tutor
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Finding Applied Minimum of a Poster (Calculus Word Problem)
November 2nd 2008, 05:20 PM #1
Sep 2008
Finding Applied Minimum of a Poster (Calculus Word Problem)
I've been doing pretty good on these Min/Max problems so far, but this one is kind of getting to me. It goes like this:
Determine the minimum area of a poster that will contain 50 square inches of printed material and have 4 inch margins on the top and bottom and 2 inch margins on the left and right. Prove your
I was able to draw out a rectangles and define the x and y sides while filling in the appropriate labels. It looks like this:
It's very messy, and I apologize for that, but it's the best I could do and it fits the description I gave. How do I go about setting this one up so I can find the derivative, factor it, get the
points (min and max), and find the restrictions?
EDIT: And also, this one is kind of stupid, but I'm pretty burnt out and I just can't factor this problem for some reason.
$12x^2 - 36x^3$
Once I get that, I can handle the rest of the problem from there. It's due tomorrow morning, so I greatly appreciate the help.
$12x^2 ( 1 - 3x )$
Does that help?
November 2nd 2008, 05:28 PM #2
Junior Member
Oct 2008
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Oblique Parallel Projection
A selection of articles related to oblique parallel projection.
Original articles from our library related to the Oblique Parallel Projection. See Table of Contents for further available material (downloadable resources) on Oblique Parallel Projection.
Oblique Parallel Projection is described in multiple online sources, as addition to our editors' articles, see section below for printable documents, Oblique Parallel Projection books and related
Suggested Pdf Resources
Suggested News Resources
In any case, an oblique parallel projection should be seen as just a special case of a dimetric (or sometimes isometric) projection.
Suggested Web Resources
In both oblique projection and orthographic projection, parallel lines of the source object produce parallel lines in the projected image.
oblique views.
Projection Normalization for Oblique Parallel. Projections. • Orthogonal parallel projection can be seen as just a special case of an oblique parallel projection.
Oblique Projections. H&B. DOP not perpendicular to view plane.
Jun 13, 2010 A Cartographic 3D View in Oblique Parallel Projection.
Great care has been taken to prepare the information on this page. Elements of the content come from factual and lexical knowledge databases, realmagick.com library and third-party sources. We
appreciate your suggestions and comments on further improvements of the site.
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San Ramon Trigonometry Tutor
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Whither "intuitionistic"?
Whither "intuitionistic"?
The terms >intuitionism< and -- derived from the latter -- >intuitionistic<
are due to Luitzen Egbertus Jan Brouwer, a dutch mathematician who in 1912
(in his inaugural lecture at Amsterdam) introduced the term for a position
opposed to what he called formalism: the axiomatic approach to the
foundation of mathematics favoured by Hilbert, Zermelo et al. Brouwer did
not initiate intuitionism -- before him, others like Poincar\'e and
Kronecker had voiced similar objections against formalism -- but he coined
the term and became, during the 20s and 30s the most fervent proponent of
the related views on the foundation of mathematics.
Brouwer suggested to base Mathematics and all science on simple intuitions:
unity, difference and counting. He was strictly opposed to the the
employment of the law of the excluded middle in the reasoning about
infinite sets. A proof of existence should be given by constructive means,
not ex negativo by the proof of the impossibility of nonexistence.
In the context of his work on natural deduction, Gentzen considered --
besides, what, when opposed to >intuitionistic<, has come to be called
>classical< logic, including the law of the excluded middle -- the
formalization of intuitionistic reasoning. Thereby he draw on earlier work
done by Arend Heyting.
A good source to enhance the knowledge about intuitionism ist still this one:
A. HEYTING:
Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie.
Berlin: Springer, 1934 (Reprint 1974).
Rainer Fischbach
Rainer Fischbach \_\_\_\_\_\_\_\_\_\_\_ Fon +49 7474 91366
Information Technology \_\_\_\_\_\_\_\_ Fax +49 7474 91369
Consultant & Writer \_\_\_\_\_\_\_ Mobil +49 171 4141570
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1. Addition of vectors
Two or more vectors may be added together to produce their ADDITION. If two vectors have the same direction, their resultant has a magnitude equal to the sum of their magnitudes and will also have
the same direction.
Similarly orientated vectors can be subtracted the same manner.
It follows that vectors can also be multiplied by a scalar, so for example if the vector A were multiplied by the number m, the magnitude of the vector, |A|, would increase to m|A|, but its direction
would not change.
In general, since vectors may have any direction, we must use one of three methods for adding vectors. These are, the POLYGON METHOD, PARALLELOGRAM METHOD and the METHOD OF COMPONENTS.
2. Polygon method
Two vectors A and B are added by drawing the arrows which represent the vectors in such a way that the initial point of B is on the terminal point of A. The resultant C = A + B, is the vector from
the initial point of A to the terminal point of B.
Many vectors can be added together in this way by drawing the successive vectors in a head-to-tail fashion, as shown here on the left.
If the polygon is closed, the resultant is a vector of zero magnitude and has no direction. This is called the NULL VECTOR, or 0 (see above on the right).
3. Parallelogram method
In the parallelogram method for vector addition, the vectors are translated, (i.e., moved) to a common origin and the parallelogram constructed as follows:
The resultant R is the diagonal of the parallelogram drawn from the common origin.
4. Method of components
The components of a vector are those vectors which, when added together, give the original vector.
The sum of the components of two vectors is equal to the sum of these two vectors.
If components are appropriately chosen, this theorem can be used as a convenient method for adding vectors.
The direction of vectors is always defined relative to a system of axes. For example, in discussing displacement on the surface of the earth, it is convenient to use axes directed from South to North
and from West to East:
In such a situation, an arbitrary displacement A can be thought of as being made up of two components A1 and A2 directed along these axes, such that A = A1 + A2.
The components could be determined by constructing a scale diagram, but they are easily calculated as follows (Note: It is convenient to specify Θ clockwise from North when referring to displacements
on the earth:
A1, the component in an easterly direction, will have a magnitude |A1| = |A| cosΘ.
A2, the component in a northerly direction, will have a magnitude |A2| = |A|sinΘ
5. Rectangular components
In all vector problems a natural system of axes presents itself. In many cases the axes are at right angles to one another. Components parallel to the axes of a rectangular system of axes are called
In general it is convenient to call the horizontal axis X and the vertical axis Y. The direction of a vector is given as an angle counter-clockwise from the X-axis.
The magnitude of A, |A|, can be calculated from the components, using the Theorem of Pythagoras:
and the direction can be calculated using
Note that if a vector is directed along one of the axes, then the component along the other axis is zero.
6. Additional questions
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Christmas Chocolates
Copyright © University of Cambridge. All rights reserved.
'Christmas Chocolates' printed from http://nrich.maths.org/
Penny, Tom and Matthew were each given mint chocolates in a hexagonal box:
Penny ate $10$ chocolates and then quickly worked out that there must have been $61$ chocolates at the start.
Tom ate $20$ chocolates and then also managed to work out very quickly that there were originally $61$ chocolates:
Matthew ate $24$ chocolates and could also see very easily that he must have started with $61$ chocolates:
Can you see how each child managed to work out that there were $61$ chocolates in the full box?
Penny, Tom and Matthew have been promised a larger box of chocolates as a Christmas present from their grandmother. The box will have $10$ chocolates along each edge, instead of just $5$.
How would each child work out how many chocolates the larger box will contain?
Can you describe any other ways to work it out?
Here are some more questions you might like to consider
• For which sizes of chocolate box will the three children be able to share the chocolates equally?
• For which sizes of chocolate box will the boys be able to share the chocolates equally?
• Can you describe how each child would work out the number of chocolates in a box with $n$ chocolates along each edge?
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Solid Geometry
of a pyramid (cone) is a portion of pyramid (cone) included between the base and the section parallel to the base not passing through the vertex.
Formula for Volume of a Frustum
The volume of a frustum is equal to one-third the product of the altitude and the sum of the upper base, the lower base, and the mean proportional between the bases. In symbols
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Basic idea of an implementation
In conceptual terms, to say that a system implements a given computation is to say that something about that system - some aspect of its behavior - is described by the computation. It is thus a way
of characterizing the system.
For example, if a system can take two numbers as inputs and produces their sum as an output, we could characterize it as an "adder". If instead it can take two numbers as inputs and produces their
product as an output, we could characterize it as a "multiplier". "Adding" and "multiplying" are two different _types_ of computations. These behaviors are fairly general, so they can be can be
further divided into specific computations if more details are specified, such as what format the inputs are in, internal functioning during intermediate steps, and so on.
Two "multipliers" with different intermediate steps would implement different computations. Thus, while the word "computation" suggests something that is done to get a result, that is misleading in
the context of the computationalist view of consciousness, which instead focuses on internal functioning of the system in question.
If we find a third system and discover that it behaves like an "adder", then it has a lot in common with our other "adder": we now know a lot about how it can behave. But there's also a lot that this
characterization does NOT tell us. It doesn't tell us what the system is made out of; or what its other behaviors might be; or what internal processes it uses to perform the additions. It also does
not tell us whether it performs the addition multiple times, redundantly, perhaps performing the same addition again but sending that (same) result to some other person.
In addition to its capabilities, a system's behavior is further characterized by the exact sequence of computer states that are actually involved in its behavior. For example, an adder which added
the numbers 45 and 66 and got 111 is characterized by those numbers, in addition to just being an adder. Such a sequence of numbers (or more precisely the sequence that includes all intermediate
states of the computation) can be called the "activity" of the computation, and together with the computation it specifies what is known as a
of the computation.
It is important to note that neither the computation alone nor the activity alone are enough to adequately describe the behavior of the system; both are needed.
Often, I will speak only of computations, but it should be understood by the context that "runs of computations" are often meant as well. For example, if I say that a particular computation gives
rise to a particular conscious observation, I mean that a particular run of that computation - corresponding to a particular starting state - gives rise to that observation. A different run of the
same computation (such as the same type of brain except in a different starting state) might then give rise to a different kind of observation or to none at all.
In the case of programmable systems, the distinction between computations and runs becomes somewhat arbitrary, but this is not a problem as we can always specify what computations we are interested
in and use that to make the choice.
A system could be both an adder and a multiplier; for example, the universe is both if at least one of each exists as a sub-system of it. Yet a hypothetical universe with 100 adders and 1 multiplier
would be significantly different from one with 1 adder and 100 multipliers. A more detailed way of characterizing such systems would be to state how many instances of each kind of computation is
performed by it: in other words, to give a measure distribution on computations. Such a measure distribution (or more exactly, a measure distribution on runs of computations) is the main tool needed
to evaluate the computationalist version of the Many Worlds Interpretation of QM, and will be addressed in later posts.
While artificial computers tend to operate on command, there is nothing about the concept of implementing computations that requires that. For example, a robot that walks around and listens for
numbers, then once it has heard two of them, adds them and says the result, then starts again, would still be an "adder". Such a robot would behave according to its internal agenda, ignoring the
desires of the people around it even if they beg it to stop.
It's also important to note "outputs" need not be distinguished from internal parts of the system. Any parts of the system that produce the behavior in question are the relevant parts, regardless of
how they interact with things outside the system.
For a computation with more than one step, it is useful to define "inputs" as substates that are affected by influences outside the scope of the computation in such a way that their values are not
determined by the previous state of the computation. Influences outside the scope of the computation are not necessarily outside of the system (such as the universe) which performs that computation.
Those familiar with computer science might be surprised that the concept of the Turing Machine will play no role in using computations to characterize systems. A Turing Machine is a type of computer
that is useful for specifying which functions could be calculated by in principle by digital computers (if unlimited memory - an infinite number of substates - is available, but the transition rule
for each substate depends on a finite number of substates) and how easily (setting bounds for how memory needs and processing time scale with the size of a problem).
Because this class of computers is so ubiquitous in computer science, those computer scientists who dare venture into the swamps of philosophy far enough to read a paper or attend a lecture on the
implementation problem sometimes completely lose interest in the problem when they realize that no one is mentioning Turing Machines. In fact, it would be quite easy to describe Turing Machines as a
special case of the computers that I will describe.
But here we are concerned with characterizing the behavior of actual systems as they exist, not with finding what size of problems they could be programmed to handle. It is also important to note
that digital computing is just a special case of the behaviors that could be characterized as computation; analog computing can certainly be considered as well, and most of the definitions I will
make are general enough to cover all cases. However, I will focus on digital computation in most of the examples I look at.
The next step is to formalize the idea of implementation by giving a mathematical criterion for whether a computation is implemented by a given mathematically described system. This will be done by
requiring a mapping from states of the underlying system to formal states of the computation, and requiring the correct behavior of the states as they change over time.
However, as will be seen in
the next post
, this approach quickly runs into a problem: without restrictions on allowed mappings, 'almost any' physical system would seem to implement 'almost any' computation. This absurd result would imply
that a rock implements the same computations as a brain. The likely solution is to impose restrictions on the allowed mappings, but finding a fully satisfactory set of restrictions has proven to be a
difficult task. My proposals for this will be presented in subsequent posts; I think that I have been successful in finding (at least close to) the right set of restrictions.
1 comment:
1. This is really interesting. Keep up the good work.
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Past Events
Fall 1999 Events
The activites below took place during the Fall 1999 semester as part of a joint program between the Math Club @ WMU and the mathematics honors society Pi Mu Epsilon. Unless otherwise stated, events
were held in the Alavi Commons Room on the 6th floor of Everett Tower.
Friday, September 17, 4pm
Pi Mu Epsilon Pizza Party
Friday, October 8, 4pm
"Mathematical Mindsets and Toolsets"
Ed Moylan, Ford Motor Company
Friday, October 15, 4pm
in 1104 Rood Hall
"Cannonballs and Honneycombs"
Thomas Hales, University of Michigan
Tuesday, October 19 - Thursday, October 21
in 3368 Rood Hall
Annual Pi Mu Epsilon Book Sale
from 9:00 a.m. until 3:00 p.m.
Friday, November 5, 4pm
in 1104 Rood Hall
"The Soap Bubble Geometry Contest:
Questions, Answers, Demonstrations, and Prizes"
Frank Morgan, Williams College
Friday, November 5
in the Presidents' Dining Room
Pi Mu Epsilon Initiation Banquet
After-dinner talk by Frank Morgan, Williams College:
"Students and Mathematics in the News and on TV"
Friday, November 19, 4pm
in 1104 Rood Hall
"Using the Right Directions to Find Trees,
Stars, and Other Interesting Objects"
Mark Crawford and Oscar Neal, WMU
Thursday, December 2, 7pm
in 1104 Rood Hall
Movie and Popcorn
with the Physics Club.
To download a copy of the Fall 1999 schedule, click here.
< Previous Calendar Archives Next >
Fall 1999 Officers
President Vice President
Treasurer Secretary
Graduate Representative
Faculty Advisor
Chapter History
Our chapter was established in 1988. Before a chapter can be established at an institution, the PME National Council requires clear evidence of an ongoing history of student mathematical activity
outside the classroom, and adequate support for these activities.
We were the fifth chapter to be established in Michigan, and hence were assigned the fifth letter of Greek alphabet: epsilon. Thus, our official name is:
The Michigan Epsilon Chapter
of Pi Mu Epsilon
Do You Know More?
Some information about previous semesters is incomplete. We ask anyone with photos, documents, or details about missing events to inform us using our Contact Information.
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If the Earth Was Shrunk to the Size of a Squash Ball Would It Be Smoother Than a Squash Ball and Why?
We first need to establish the scale factor we would have to shrink the Earth by in order to reduce it to the size of a squash ball.
The Earth is 7,926 miles in diameter at the equator, and a regulation squash ball has a diameter of 1.7 inches. This means that to shrink the Earth down to the size of a squash ball, its size would
have to be multiplied by a scale factor of 3.45 x 10-9.
To compare the smoothness of the two surfaces, we need to know the variation of the surface, that is, the difference between the highest and lowest points.
For a squash ball, this is a simple process, because there are very few areas where the surface is higher than the average, but there are many small indentations or depressions.
Since these depressions are roughly 0.004 inches in depth, the variation in surface height can be taken to be roughly 0.004 inches.
For the Earth, the lowest point below the surface is in the Mariana Trench, which is 36,200 feet below sea level at its deepest point, known as the Challenger Deep. The highest point is, of course,
the summit of Mount Everest, which is estimated at 29,029 feet above sea level.
Therefore, the variation in the height of the Earth’s surface is 65,229 feet.
If we scaled the Earth down to the size of a squash ball, using the scale factor calculated above, the variation of its surface would be 2.25 x 104 feet, or 0.0027 inches.
This figure is in fact about two-thirds of the figure for the squash ball, so what your correspondent heard is actually true: if Earth were scaled down to this size it would indeed be smoother than
the average regulation squash ball.
Now for the second part of the question. The lack of any raised areas on the squash ball’s surface means that there are in fact mostly depressions or indentations in the surface.
So if a squash ball were scaled up to the size of the Earth, there would be no mountains as such.
There would, however, be a lot of large craters. In fact, if we scaled up the indentations in the ball’s surface we would end up with some immense depressions that were almost 18 miles deep.
If these depressions were ocean trenches like those that are found on the Earth’s surface, they would penetrate the 3.7-mile-thick oceanic crust, and extend right through the Mohorovicic
discontinuity where the crust meets the mantle and well into the mantle beneath.
These craters would not only be deep but could be anything up to about 37 miles wide.
If the mass of the Earth were crushed down to the size of a squash or racketball, then it would be dense enough to be either neutronium or a black hole.
In the case of neutronium, the gravity at the surface would be in the order of at least a million times the gravity you are feeling now, more than enough to smooth out any irregularities in the
surface. In the case of a black hole there wouldn’t be a surface, just an event horizon which would be smooth.
Expanding a ball to the size of the Earth is somewhat different. If we assume that the ball is made mainly of carbon atoms and weighs 2 pounds, then there would only be around 352 atoms in each cubic
This is actually less dense than the Earth’s upper atmosphere on the edge of outer space.
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Contraction Mapping Principle
January 29th 2009, 04:44 PM
Contraction Mapping Principle
Let M be a compact metric space. Let $\Phi : M \rightarrow M$ be such that $d(\Phi (x), \Phi (y)) < d(x, y), \forall x, y \in M, x eq y$. Show that $\Phi$ has a unique fixed point.
I'd like to use the Contraction Mapping Principle. I can see that M is complete (as it is a compact metric space), but am not sure where to find a constant $c \in [0,1)$ such that $d(\Phi (x), \
Phi (y)) \leq c \cdot d(x, y)$. Any advice?
January 29th 2009, 06:22 PM
Let M be a compact metric space. Let $\Phi : M \rightarrow M$ be such that $d(\Phi (x), \Phi (y)) < d(x, y), \forall x, y \in M, x eq y$. Show that $\Phi$ has a unique fixed point.
I'd like to use the Contraction Mapping Principle. I can see that M is complete (as it is a compact metric space), but am not sure where to find a constant $c \in [0,1)$ such that $d(\Phi (x), \
Phi (y)) \leq c \cdot d(x, y)$. Any advice?
It seems like you don't need to find a "c" to show the existence of a fixed point of x such that $\Phi (x) = x$.
First, choose a point $x_{1}$ in M and define
$x_{2} = \Phi (x_{1}), x_{3} = \Phi (x_{2}),...,x_{n}= \Phi (x_{n-1})$ for $n \ge 2$.
You might need to mark some points in M (to figure out the points indeed converge) and make sure each $d(x_{k-1}, x_{k}) > d(\Phi (x_{k-1}), \Phi (x_{k}))$ where $k=2,3,...,n$.
Since $\Phi$ is a contractive function, $\{x_{n}\}_{n=1}^{\infty}$is a Cauchy sequence. Since M is a complete metric space, the sequence has a limit in M and we call it x. A contractive function
in a metric space is a continuous, so $\{\Phi (x_{n})\}_{n=1}^{\infty}$ converges. We know that $\{\Phi (x_{n})\}_{n=1}^{\infty}$ is simply $\{x_{n}\}_{n=2}^{\infty}$ whose limit is x. Thus $\Phi
(x) = x$.
To show the uniqueness, suppose on the contrary that you have another point $\Phi (y) = y$. Now you can draw a contradiction if you check your distance function formula $d(\Phi (x), \Phi (y)) < d
(x, y)$
January 30th 2009, 10:47 AM
It seems like you don't need to find a "c" to show the existence of a fixed point of x such that $\Phi (x) = x$.
First, choose a point $x_{1}$ in M and define
$x_{2} = \Phi (x_{1}), x_{3} = \Phi (x_{2}),...,x_{n}= \Phi (x_{n-1})$ for $n \ge 2$.
You might need to mark some points in M (to figure out the points indeed converge) and make sure each $d(x_{k-1}, x_{k}) > d(\Phi (x_{k-1}), \Phi (x_{k}))$ where $k=2,3,...,n$.
Since $\color{blue}\Phi$ is a contractive function, $\color{blue}\{x_{n}\}_{n=1}^{\infty}$ is a Cauchy sequence. Since M is a complete metric space, the sequence has a limit in M and we call it
x. A contractive function in a metric space is a continuous, so $\{\Phi (x_{n})\}_{n=1}^{\infty}$ converges. We know that $\{\Phi (x_{n})\}_{n=1}^{\infty}$ is simply $\{x_{n}\}_{n=2}^{\infty}$
whose limit is x. Thus $\Phi (x) = x$.
To show the uniqueness, suppose on the contrary that you have another point $\Phi (y) = y$. Now you can draw a contradiction if you check your distance function formula $d(\Phi (x), \Phi (y)) < d
(x, y)$
Unless I'm missing something, the sentence in blue needs some justification. The usual proof that the sequence is Cauchy relies on estimating $d(x_m,x_n)$ (where n>m) by using the triangle
inequality to get $d(x_m,x_n) \leqslant d(x_m,x_{m+1}) + d(x_{m+1},x_{m+2}) + \ldots + d(x_{n-1},x_n)$. If the stronger condition $d(\Phi (x), \Phi (y)) \leqslant c\cdot d(x, y)$ holds, for some
c<1, then you can estimate the sum of those distances by using a geometric series. But with the weaker condition $d(\Phi (x), \Phi (y)) < d(x, y)\ (\forall x eq y)$, that method will not work.
A mapping satisfying the stronger condition is called a contraction map. A mapping satisfying the weaker condition is sometimes called a strictly metric map. A strictly metric map on a compact
space need not be a contraction map. For example, the map $\Phi(x) = \tfrac12x^2$ is strictly metric by not contractive on the closed unit interval [0,1]. It is not clear to me that a strictly
metric map on a compact space needs to have a fixed point. (If it has, then the fixed point is certainly unique.)
January 31st 2009, 12:30 AM
I can now see how to do this problem. Let $\delta = \inf\{d(x,\Phi(x)):x\in M\}$. It follows from the compactness of M that the infimum is attained, so there exists $x_0\in M$ with $d(x_0,\Phi
(x_0)) = \delta$. If $\deltae0$ then $\delta \leqslant d(\Phi(x_0),\Phi(\Phi(x_0)))<d(x_0,\Phi(x_0)) = \delta$. That contradiction shows that $\delta=0$ and $x_0$ is a fixed point of $\Phi$.
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Time Series Regression of Temperature Anomaly Data: 1 – Don’t Use OLS
Phil Jones Statement (February , 2010)
Phil Jones of the Climate Research Unit (CRU) responded to a series of questions from the BBC in early February, 2010 (link). Question B dealt with global warming trends in the 1995 – 2009 period.
Here’s the BBC question and Phil Jones answer:
BBC Question B: B – “Do you agree that from 1995 to the present there has been no statistically-significant global warming?”
Phil Jones Answer: “Yes, but only just. I also calculated the trend for the period 1995 to 2009. This trend (0.12C per decade) is positive, but not significant at the 95% significance level. The
positive trend is quite close to the significance level. Achieving statistical significance in scientific terms is much more likely for longer periods, and much less likely for shorter periods.”
Phil Jones’ statement provided a time series regression learning moment for many of us citizen climate observers who quickly checked his statement with our Excel, R or other handy regression
analysis tools. I sure did. Two readers, J and S, contacted me with questions – comments:
J: “Hi Kelly,I’ve got some really basic statistics questions – do you mind if I ask you? Forgive me if they seem rudimentary – it’s been 20 years since I did statistics at …
I’m looking at the recent Phil Jones statement of no statistically significant warming since 1995. So I was trying to reproduce his results and used the LINEST function in Excel on the HadCRUT
temperature data. The LINEST function gave me a slope of 0.11 with an uncertainty of 0.058. Is the uncertainty value from LINEST equal to one standard deviation? If so, would that mean the 95%
confidence interval = ± 1.96 * standard deviation = ± 0.114
So the 95% confidence interval of the slope is 0.11 ± 0.114 hence it’s not a statistically significant trend.
Is this all correct? Just want to make sure I’ve got my figures correct. Thanks, J”
S: “Kelly, Thanks for your site. I have been learning R using your site (and other sources).
I am somewhat confused about the statistical significance issue. It is claimed that Hadley (and RSS and UAH) temperature data are not (quite) statistically significant from 1995.
My understanding is that statistical significance largely involves the p value. My R script of Hadley data from 1995 gives a p value of 5.538e-07. This is statistically significant at the 99% level!
Am I misunderstanding statistical significance? Or Is there something wrong with my script … Cheers, S”
I’ve written this tutorial to make sure that I fully understand how to properly conduct time series regression of temperature anomaly data. I also hope that it helps J, S and any other interested
readers who need a little information and/or refresher on time series regression.
Working With Hadley Anomaly Time Series Data
Let’s retrieve and look at the Hadley temperature anomaly data to see if we can reproduce Phil Jones results. I regularly update a consolidated monthly temperature anomaly data file for the 5 major
series on my ProcessTrends website. You can download the data file and follow along in Excel or R if you’d like.
Here’s my R script that reads in the monthly temperature anomaly data.
# Retrieve temperature anomaly data and print head() to console
options(digits = 6)
l<- "http://processtrends.com/Files/RClimate_consol_temp_anom_latest.csv"
df<- read.csv(link)
Here’s the first 6 lines of the df data.frame. Notice the 5 temperature anomaly data series start in 1880. The satellite based RSS and UAH series are NA until 1979.
Let’s subset the data to the 1995-2009 period for just the HAD data series.
## subset HAD series for 1995 - 2009 period
df <- subset(df, df$yr_frac > 1995 & df$yr_frac < 2010 )
HAD_df <- df[,c(1,4)]
Here’s the first 6 lines of the HAD_df data.frame.
Now we are ready to generate an ordinary least squares (OLS) linear model of the HAD anomaly data and look at the regression summary(ols_mod).
## Create OLS Linear model
ols_mod <- lm(HAD ~ yr_frac, data = HAD_df)
Here’s our summary(ols_mod).
A few key points on summary(ols_mod):
• HAD = -21.01643 + 0.01069 * yr_frac
• Adjusted R-squared reported to be 0.116
• Intercept and slope t values indicate that they are statistically significant @ 0.01 level
• F-statistic indicates that results are statistically significant @ 0.01 level
Let’s plot the HAD anomalies and the OLS trend line to see how they look.
Title <- "Hadley Temperature Anomaly Trend Chart and\n OLS Linear Regression: 1995-2009"
plot(HAD ~ yr_frac, data = HAD_df, type="l", col = "grey", main = Title, cex.main=0.85,
ylab ="HAD Anomnaly - C", xlab="", las=1)
abline(ols_mod, col = "red")
Here’s our 1995 – 2009 trend chart with OLS regression line.
Everything looks fine, right? The OLS model says that the 1995-2009 Hadley temperature anomaly trend of 0.01069 is statistically significant. Why does Phil Jones say “.. trend (0.12C per decade) is
positive, but not significant at the 95% significance level“? Readers please note that I am not sure exactly which data set Phil Jones used, I have used the HADCRUT3 nh+sh data set. Our results are
close to his, but not exactly the same.
Before we do a diagnosis of the OLS linear regression model, let’s briefly review what is well known about using OLS for time series data.
Brief Review of OLS and Time Series Data Literature
George Udny Yule introduced the concept of nonsense correlation in his 1926 paper, “Why Do We Sometimes Get Nonsense Correlations between Time-series?” published in the Journal of the Royal
Statistical Society 89, 1–64. PDF available at this link.
Granger & Newbold, in their1974 Spurious Regressions in Econometrics article (link) , said:
“It is very common to see reported in applied econometric literature time series regression equations with an apparently high degree of fit, as measured by the coefficient of multiple correlation R2
or the corrected coefficient R2, but with an extremely low value for the Durbin-Watson statistic. We find it very curious that whereas virtually every textbook on econometric methodology contains
explicit warnings of the dangers of autocorrelated errors..”
Granger and Newbold go on to discuss the consequences of autocorrelated errors in regression analysis:
Three major consequences of autocorrelated errors in regression analysis :
(i) Estimates of the regression coefficients are inefficient.
(ii) Forecasts based on the regression equations are sub-optimal.
(iii) The usual significance tests on the coefficients are invalid.
Is OLS suitable for times series data? Here’s what SAS (major statatisitcal software vendor) has to say:
“Ordinary regression analysis is based on several statistical assumptions. One key assumption is that the errors are independent of each other. However, with time series data, the ordinary regression
residuals usually are correlated over time. It is not desirable to use ordinary regression analysis for time series data since the assumptions on which the classical linear regression model is based
will usually be violated.” Source: SAS – link
This is not new information, statisticians and econometricians have known and written about this for years. While it may be new to some citizen climate observers, we need to learn or remember that R
and/or Excel will give OLS results for time series analysis that may look better than they actually are because the underlying OLS assumptions are violated.
Diagnosis of OLS_Mod Results
Now that we have been warned about autocorrelation, let’s see how our HAD ols_mod holds up to regression diagnosis.
As a first diagnosis step, let’s look at the ols_mod residuals.
plot(ols_mod$residuals, pch=16, col = "grey", cex=1, las=1,
main="Hadley Anomoly: 1995-2009 OLS Model Residuals",
ylab = "Residual")
Here’s our residuals plot.
These residuals do not look independent of each other, we see a number of runs of consecutive residuals on the same side of the 0 line. This tells me the temperature anomalies tend to persist. The
t+1 anomaly tends to stay on the same side of the 0 line as the t anomaly. This raises a red flag for me.
Let’s use R’s acf() function to look at the autocorrelation function of the HAD anomaly data series.
acf(HAD_df$HAD, las=1)
Here’s our acf() report.
The acf() shows the correlation coefficient (r) between values t & t-lag. The dashed blue horizontal lines correspond to the 95% confidence limits. This plot shows that the Hadley anomaly
observations are autocorrelated for lags of 1-10 and 20-23.
We can also run a Durbin Watson test which gives us these results:
# Install the car library if you don't already have it installed
durbin.watson(ols_mod, max.lag=12)
Here’s our durbin.watson() report.
Clearly our ols_mod contains autocorrelation errors, meaning that the “.. usual significance tests on the coefficients are invalid”, as Granger & Newbold said in 1974.
There is an R package that tests underlying OLS Linear Model Assumptions (gvlma). Let’s use this package to assess how our ols_mod meets the underlying OLS assumptions.
# You'll need to install the gvlma package if you don't have it installed
gvmodel <- gvlma(ols_mod)
Here’s our gvlma() report.
The gvlma() results are very clear, the ols_mod regression does not meet the underlying OLS assumptions! Even though our OLS regression looks good, we have to conclude that the Hadley anomaly series
for 1995-2009 is serially correlated, meaning that we can not rely on the OLS regression significance tests.
My take home message from this example, don’t use OLS for climate time series data!
In my next post in this series, I’ll show how to use a generalized least square (gls) regression for this autocorrelated time series data.
6 responses to “Time Series Regression of Temperature Anomaly Data: 1 – Don’t Use OLS”
1. Should the call to acf use ols_mod$residuals as it’s argument instead of HAD_df$HAD?
Otherwise the long lag autocorrelations go through the roof when you use a period long enough to show a significant trend. If you do acf on the residuals, then the results are somewhat similar
for any reasonably linear period.
2. Thanks for the great analysis of OLS with time-series data; I look forward to the next installment.
I would like to suggest that the second plot should be replaced with a process behavior chart of the residuals (individuals and moving range), which would provide tests for trends and
non-homogeneity in the data. You are already using the residuals chart to eyeball the randomness; the ImR chart would simply provide a clearer gauge of (non)randomess. Looking at the data, it’s
clear that the residuals would at least fail the Western Electric rule 4 and probably rule 3 (nine consecutive points on the same side of the mean and probably four out of five points > 1 sigma
from the mean, respectively).
3. Following along using the R code provided, my plot of the trend line did not look the same as the one provided in the post. Then I noticed that, despite the figure heading indicating the range is
1995-2009, the posted graph has the full range of data from 1880.
I assume you have linked to the wrong .png file?
□ Terry
Thank you for catching that. I’ve corrected the link, it now show the 1995-2009 data, not the entire data series.
4. Kelly,
Thanks for your clear analysis on this topic, which has purplexed me for some time.
I am looking forward to your next post.
5. Remember the autocorrelation is in the errors, not the original data points.
You see the problem with the model in the first plot – it’s clear that a linear trend is *not* a good fit to this data. Why worry about the significance of a clearly incorrect model?
This entry was posted in Global Warming, R Climate Data Analysis Tool, RClimate Script, Time Series Charts and tagged Climate Trends, R scripts, Time Series Regression, Trend Chart. Bookmark the
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Why Study Statistics?
Statistics is a very practical discipline, concerned about real problems in the real world. It has applications in:
• bioinformatics;
• biology (biostatistics or biometrics);
• climatology;
• computing or computer science (statistical computing is a highly sought-after skill);
• economics (econometrics);
• finance (financial statistics);
• psychology (psychometrics);
• physics (statistical physics is a modern discipline in physics);
• the health industry (medical statistics).
Studying statistics provides a wealth of
job opportunities
In most disciplines, it is almost never possible to examine or study everything of interest (such as an item, person, business or animal). For example, suppose we are interested in the effect of a
certain drug claimed to improve the circulation of blood. We cannot test the drug on everyone!
However, we can test the drug on a selection of people. This rasies a number of questions, however:
• What group of people do we choose? Who should be in the group?
• What can information gathered from a small group of people tell us about the effect of the drug in the population in general?
• Won't the measured effect depend on which people are in the group? Won't it change from one group to the next? So how can any useful information be found?
• How many people should be in such a group to obtain useful information?
To answer these questions, we need statistics.
Statistics appears in almost all areas of science, technology, research, and wherever data is obtained for the purpose of finding information. Statistics has been described as the science of making
conclusions in the presence of uncertainity.
Statistics can provide answers to all the questions listed above.
Despite the affinity of statistics with real situations, it has a strong mathematical foundation.
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University Libraries > URI Faculty Publications > 1991 > MathematicsUniversity of Rhode Island LibrariesUniversity of Rhode Island Libraries Web Site HeaderUniversity of Rhode Island Libraries
Faculty Publications
Compiled by Margaret J. Keefe
Faculty Publications 1991
ARTS & SCIENCES
Anderson, Ian and Norman J. Finizio. "An Infinite Class of Cyclic Triplewhist Tournaments." Congressus Numerantium. 91:7-18. November 1991.
Chuanxi, Q. and G. Ladas. "Oscillations of Higher Order Neutral Differential Equations With Variable Coefficients." Mathematische Nachrichten. 150:15-24. 1991.
Chuanxi, Q., S. A. Kurkulis and G. Ladas. "Oscillations of Systems of Difference Equations With Variable Coefficients." Journal of Mathematical and Physical Sciences. 25(l):1112. 1991.
Clark, D. S. and E. R. Suryanarayan. "Quasiperiodic Tilings With Low Order Rotational Symmetry." Acta Crystallographica. Section A. Foundations of Crystallography. 46(5):498-502. 1 September 1991.
"Erratum." Acta Crystallographica. Section A. Foundations of Crystallography. 46(6):852. 1991.
Driver, Bruce K. and Rodney D. Driver. "Simplicity of Solutions of x1(t)=bx(t 1)." Journal of Mathematical Analysis and Applications. 157(2):591-608. May 15, 1991.
Driver, Rodney D. SEE also: Driver, Bruce K.
Driver, Rodney D. "Bombing Brutal to Iraqis (Letter)." Narragansett Times. February 16, 1991. p.17A.
Driver, Rodney. "Chariho Exeter Depositors Get Explanation (Letter)." Narragansett Times. December 4, 1991. p.18A.
Driver, Rod. "A Few Reasons Why Bush Shouldn't Receive an Honorary Degree (Letter)." Good 5c Cigar. December 6, 1991. p.6.
Driver, Rod. "The Rules of the Game: Play Dumb." Providence Sunday Journal. December 29, 1991.
Driver, Rodney D. "State Lawmakers Live by Unusual Ethics (Letter)." Narragansett Times. June 14, 1991. p.19 A.
Driver, Rod. "Teachers Not Afraid to Tell Students Bad Work is Unacceptable (Letter)." Westerly Sun. October 20, 1991.
Driver, Rod. "Testing of Teachers Is a Necessity (Letter)." Westerly Sun. May 16, 1991.
Driver, Rodney. "Viewpoint: Driver: Last Minute Bills Made No Sense." Narragansett Times. December 27, 1991. p.24 A.
Driver, Rodney. "Viewpoint: Unwritten Rules at the Assembly." Narragansett Times. July 31, 1991. p.15A.
Finizio, Norman J. SEE also: Anderson, Ian.
Finizio, Norman J. "Orbits of Cyclic Wh[v] of Zn Type." Congressus Numerantium. 82:15-28. December 1991.
Georgiou, D. A., E. A. Grove and G. Ladas. "Oscillation of Neutral Difference Equations With Variable Coefficients." IN: Differential Equations: Stability and Control, Edited by Saber Elaydi. M.
Dekkar. (Lecture Notes in Pure and Applied Mathematics v.127) 1991. p.165-173.
Gopalsamy, K., M. R. S. Kulenovic and G. Ladas. "On a Logistic Equation With Piecewise Constant Arguments." Differential and Integral Equations. 4(l):215-223. January 1991.
Gopalsamy, K. and G. Ladas. "Oscillations of Delay Differential Equations." Journal of the Australian Mathematical Society. Series B - Applied Mathematics. 32(4):377-381. April 1991.
Grove, Edward A. SEE: Georgiou, D. A.
Gyori, I., G. Ladas and L. Pakula. "Conditions for Oscillation of Difference Equations With Applications to Equations With Piecewise Constant Arguments." SIAM Journal of Mathematical Analysis. 22
(3):769-773. May 1991.
Gyori, I., G. Ladas and P. N. Vlahos. "Global Attractivity in a Delay Difference Equation." Nonlinear Analysis, Theory, Methods & Applications. 17(5):473-479. 1991
Gyori, I and G. Ladas. Oscillation Theory of Delay Differential Equations: With Applications. Oxford. (Oxford Mathematical Monographs) 1991. 386p.
Gyori, I. and G. Ladas. "Positive Solutions of Integro-Differential Equations With Unbounded Delay." Journal of Integral Equations and Applications. 4:1113 1120. 1991.
Jaroma, J. H., S. A. Kurkulis and G. Ladas. "Oscillations and Stability In a Discrete Delay Logistic Model." Ukrainskii Matematicheskii Zhurnal. 43(6):734-744. 1991.
Johnson, Diane. SEE: Lamagna, Edmund.
Kaskosz, B. and S. Lojasiewicz, Jr. "Boundary Trajectories of Systems With Unbounded Controls." Journal of Optimization Theory and Applications. 70(3):539-559. September 1991.
Ladas, Gerasimos. SEE also: Chuanxi, G,; Georgiou, D. A.; Gopalsamy, K,; Gyori, I.; Jaroma, J. H.; Qian, C.
Ladas, G. and C. Qian. "A Comparison Theorem for Odd Order Neutral Differential Equations." Journal of the Nigerian Mathematical Society." 10:99+. 1991.
Ladas, G. and C. Qian. "Linearized Oscillations for Even Order Neutral Differential Equations." Journal of Mathematical Analysis and Applications. 159(l):237-258. July 15, 1991.
Ladas, G. "Necessary and Sufficient Conditions for oscillation; Open Problems and Conjectures." IN: International Symposium on Functional Differential Equations (1990. Kyoto. Japan, 30 August 2 -
September 1990. Edited by T. Yoshizawa and J. Kato. World Scientific. 1991. p.196-203.
Ladas, G., Ch. G. Philos and Y. G. Sficas. "Oscillations in Neutral Equations With Periodical Coefficients." Proceedings of the American Mathematical Society. 113(l):123-134. September 1991.
Ladas, G., C. Qian and J. Yan. "Oscillations of Higher Order Neutral Differential Equations." Portugaliae Mathematica. 48(3):291-307. 1991.
Ladas, G., Ch. G. Philos and Y. G. Sficas. "Oscillations of Integro-Differential Equations." Differential and Integral Equations. 4(5):113-1120. September 1991.
Ladas, G. "Recent Developments in the Oscillation of Delay Difference Equations." IN: Differential Equations: Stability and Control, Edited by Saber Elaydi, M.. Dekker. (Lecture Notes in Pure and
Applied Mathematics, v.127) 1991. p.321-332.
Ladas, G., C. Qian, P. N. Vlahos and J. Yan. "Stability of Solutions of Linear Nonautonomous Difference Equations." Applicable Analysis. 41(1 4):183-191. 1991.
Lamagna, E. and D. Johnson. "The Calculus Companion: A Computational Environment for Exploring Mathematics." Computer Algebra Systems in Education Newsletter. 9:1-4. February 1991.
Lapidot, Eitan and James T. Lewis. "Best Approximation using a Peak Norm." Journal of Approximation Theory. 67(2):174-186. November 1991.
Lewis, James T. SEE: Lapidot, Eitan.
Lin, Michael and Robert Sine. "Contractive Projections on the Fixed Point Set of Loo Contractions." Colloquia Mathematica. 62(1):91-96. 1991.
Newman, D. J. and O. Shisha. "Magnitude of Fourier Coefficients and Degree of Approximation by Riemann Sums." Numerical Functions Analysis and Optimization. 12(5-6):545-550. 1991.
Pakula, Lewis I. SEE also: Gyori, I.
Pakula, Lewis and Sol Schwartzman. "Independence for Sets of Topological Spheres." Canadian Mathematical Bulletin. 34(4):520-524. 1991.
Qian, C., S. A. Kurkulis and G. Ladas. "Oscillations of Systems of Difference Equations With Variable Coefficients." Journal of Mathematical and Physical Sciences. 25(l):1-12. 1991.
Roxin, E. "Some Applications of the Attainable Set of X = A(u(t)x." IN: Differential Equations: Stability and Control, Edited by Saber Elaydi. M. Dekker. (Lecture Notes in Pure and Applied
Mathematics, v.127) 1991. p431-434.
Schwartzman, Sol. SEE: Pakula, Lewis I.
Shisha, Oved. SEE: Newman, D. J.
Sine, Robert. SEE also: Lin, Michael.
Sine, Robert. "Constricted Systems." Rocky Mountain Journal of Mathematics. 21(4):1373-1383. Fall 1991.
Suryanarayan, E. Ramnath. SEE: Clark, Dean S.
Return to Faculty Publications 1991 Table of Contents.
Return to Faculty Publications homepage.
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ocally regular categor
locally regular category
Locally regular categories
A locally regular category is a relative of a regular category in which the condition of finite limits is weakened to finite connected limits. It is so named because every slice category of a locally
regular category is a regular category, although the converse is not quite true.
A category $C$ is locally regular if
1. It has finite connected limits — equivalently, it has pullbacks and equalizers;
2. It has (extremal epi, mono) factorizations which are stable under pullback; and
3. Every span factors as an extremal epi followed by a jointly-monic span.
• Every regular category is locally regular. Factorizations of spans may be obtained by factorizations of single morphisms into a binary product.
• The category $LH$ of topological spaces (or locales) and local homeomorphisms is locally regular, but not regular. Its slice categories are precisely the sheaf toposes of spaces (or locales).
• A locally regular category is regular if and only if it has a terminal object.
• The factorization axiom for spans implies, by induction, a similar factorization property for nonempty finite cosinks. However, similar factorizations for empty cosinks (i.e. “supports”) do not
necessarily exist.
• Every locally regular category gives rise to a tabular allegory of binary relations (where we define a “binary relation” to mean a jointly-monic span). For composition of relations, we require
pullbacks and stable factorizations of spans. For intersection of binary relations, we require equalizers.
• Conversely, every tabular allegory has a locally regular category of maps (left adjoints). So locally regular categories are essentially the same as tabular allegories.
Revised on March 4, 2012 13:37:16 by
Mike Shulman
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Algebras over $\cobar(\cofrob)$
Gabriel C. Drummond-Cole, John Terilla and Thomas Tradler
We show that a square zero, degree one element in $W(V)$, the Weyl algebra on a vector space $V$, is equivalent to providing $V$ with the structure of an algebra over the properad $\cobar(\cofrob)$,
the properad arising from the cobar construction applied to the cofrobenius coproperad.
Journal of Homotopy and Related Structures, Vol. 5(2010), No. 1, pp. 15-36
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Re: st: Calculate weighted average across variables with externally give
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Re: st: Calculate weighted average across variables with externally given weights - controlling for missing values
From Andreas Karpf <andreas.karpf@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: Calculate weighted average across variables with externally given weights - controlling for missing values
Date Mon, 3 Oct 2011 11:16:58 +0200
Thank you very much for your reply Nick, and please excuse my claims
of urgence. I thought a while about the solution you gave me and I
don't really see how I can handle my problem with that. Setting the
missing values to zero is quite obvious and was also my first idea.
This solves the problem in the enumerator but unfortunately not in the
denominator. If one value is missing I don't want to divide by the
whole sume of weights (see problem 2)). A solution via dummy variables
seems quite cumbersome and I don't really know how I would have to do
it because I am having time series data and the weights apply for
different variables and not for different values of one variable.
isn't there any stata command for exactly this purpose (cross-variable
weighted averages with user supplied weights) which I could combine
with cond(missing(x), 0, x).
Thank you very much for your help,
2011/10/3 Nick Cox <njcoxstata@gmail.com>:
> To get Stata to treat values of -x- as 0 if they are missing you can
> supply instead
> cond(missing(x), 0, x)
> or use -mvdecode-.
> Nick
> On Mon, Oct 3, 2011 at 2:16 AM, Andreas Karpf <andreas.karpf@gmail.com> wrote:
>> I have a couple of time series variables for different industrial
>> sectors like manufacturing, services industry, communication industry
>> etc.
>> t ; var_sect_1 ; var_sect_2 ; var_sect_3 ; var_sect_4;
>> jan ; ; ; ; ;
>> feb ; ; ; ; ;
>> mar ; ; ; ; ;
>> apr ; ; ; ; ;
>> What I want to do (example january):
>> weight_avg_january= var_sect_1[jan] *weight_sect_1 +
>> var_sect_2[jan]*weight_sect_2 + var_sect_3[jan]*weight_sect_3 +
>> var_sect_4[jan]*weight_sect_4/(weight_sect_1+weight_sect_2+weight_sect_3+weight_sect_4)
>> if there is however a missing value for january sector 1 it should look like:
>> weight_avg_january= var_sect_2[jan]*weight_sect_2 +
>> var_sect_3[jan]*weight_sect_3 +
>> var_sect_4[jan]*weight_sect_4/(weight_sect_2+weight_sect_3+weight_sect_4)
>> these data relates to a kind of business monitor survey and i would
>> like to calculate the aggregate indicator by using sectorial weights,
>> this means weights
>> which correspond to the contribution of each sector (services,
>> manufacturing) to the gdp. i at first though i could do that by hand
>> but than i realized that 1) if there is
>> one missing value in e.g. sector 1 in january stata outputs a missing
>> value for the weighted average for january. so it doesn't just ignore
>> the mv but it refuses to calculate the datapoints which are there. 2)
>> even if problem number one would be solved of course the denominator
>> would not be correct because if the sector 1 data in january is
>> missing also the weight in the denominator for sector 1 should be
>> omitted.
>> The weights i am referring to are from an external statistic.
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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Decoding a message
November 9th 2009, 07:37 AM #1
Nov 2009
I could use all of the help I can get with this...it did not make any sense to me when my teacher explained it. Ok so here's the problem
Let p=13 and q=23. Let e=5. Decode the message: When I think about my professor, I think about 69,214,69.
1. First find d with de congruent to 1 mod((p-1)(q-1)).
2. Now, find 69^d mod((13)(23)) and 214^d mod((13)(23)). I'l give you 214^d is congruent to 15 mod((13)(23)).
3. Write decoded message in terms of letters.
Thank you SO MUCH for any help with this.
***UPDATE: Ok, so I think that d=53. Verify this if you'd like. So now, what I need to do is figure out 69^(53)=____ mod((13)(23)). I am not sure how to do this...the large exponent is throwing
me off, so it'd be nice if someone could explain that.
Last edited by steph3824; November 9th 2009 at 11:06 AM. Reason: Update
I could use all of the help I can get with this...it did not make any sense to me when my teacher explained it. Ok so here's the problem
Let p=13 and q=23. Let e=5. Decode the message: When I think about my professor, I think about 69,214,69.
1. First find d with de congruent to 1 mod((p-1)(q-1)).
2. Now, find 69^d mod((13)(23)) and 214^d mod((13)(23)). I'l give you 214^d is congruent to 15 mod((13)(23)).
3. Write decoded message in terms of letters.
Thank you SO MUCH for any help with this.
***UPDATE: Ok, so I think that d=53. Verify this if you'd like. So now, what I need to do is figure out 69^(53)=____ mod((13)(23)). I am not sure how to do this...the large exponent is throwing
me off, so it'd be nice if someone could explain that.
What you need is modular exponentiation
Essentially, you use $69^{a+b}=69^{a}69^b$ and substitute each term with its residue modulo 13*23, so you have smaller numbers. Optimally, using decomposition in base 2, you reduce to power 2
(and powers of 2, by induction). Read the wikipedia (or google for modular exponentiation if it is not clear enough).
November 9th 2009, 02:01 PM #2
MHF Contributor
Aug 2008
Paris, France
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Math Help
August 6th 2009, 12:04 AM #1
Junior Member
Jul 2009
11. If -1 < y < 0, then which of the following is true
A. y <
B. y
& so on……
for pentagon the diagonals are 4 what is it for heptagon
Last edited by mr fantastic; August 6th 2009 at 05:17 AM. Reason: Merged posts
For your first post.Note;
if 0<|y|<1
then |y^n|<|y^m| and n>m
if -1<y<0
then |y^3|<|y^2| and y^3<y^2 regardless of sign
|y^4|<|y^2| and y^4<y^2 regardless of sign
y,y^3 are -ve, y^2,y^4 are positive
so y<y^3<y^4<y^2
You could just plug in y=-0.5 and it will tell you which is bigger
August 6th 2009, 05:13 AM #2
Jul 2009
August 6th 2009, 05:18 AM #3
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Return from Digression
September 22, 1999 Perl and the Lambda Calculus Slide #32
• Our addition function is (Y R).
• We can work through this by hand and compute
((Y R) (SUCC ZERO) (SUCC ZERO))
• The result is indeed (SUCC (SUCC ZERO))
• It's totally straightforward
• I'd do it now, but it won't fit on the slide.
• It's tedious but instructive to work it out for yourself
• If you do it, you'll see how the miraculous Y works
Next Copyright © 1999 M-J. Dominus
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North Attleboro Geometry Tutor
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Here's the question you clicked on:
Salinity of the ocean: is generally very low changes with climate decreases with increase in temperature increases where a river meets the ocean
• one year ago
• one year ago
Best Response
You've already chosen the best response.
You should use process of elimination for this: identify any which strike you as being impossible right off the bat. Which ones do you think it probably isn't?
Best Response
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i would eliminate the first and last options
Best Response
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Me too. Of the remaining ones, salinity is not affected by temperature. You can make a glass of salt water, heat it, and it is still as salty as it was when you started. So you can scratch the
third option off the list as well. Which leaves...
Best Response
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so it changes with climate?
Best Response
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That is what you have got left. :D So you see the general process for approaching questions like this - scratch out the impossible ones, then consider the possibilities that leaves for scientific
Best Response
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yeah, thanks. :) i figured it would be that one but wasnt sure. thanks for your help.
Best Response
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Not a prob. :)
Your question is ready. Sign up for free to start getting answers.
is replying to Can someone tell me what button the professor is hitting...
• Teamwork 19 Teammate
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• ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.
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03Exx Set theory
• 03E02 Partition relations
• 03E04 Ordered sets and their cofinalities; pcf theory
• 03E05 Other combinatorial set theory
• 03E10 Ordinal and cardinal numbers (1)
• 03E15 Descriptive set theory [See also 28A05, 54H05]
• 03E17 Cardinal characteristics of the continuum
• 03E20 Other classical set theory (including functions, relations, and set algebra)
• 03E25 Axiom of choice and related propositions
• 03E30 Axiomatics of classical set theory and its fragments
• 03E35 Consistency and independence results
• 03E40 Other aspects of forcing and Boolean-valued models
• 03E45 Inner models, including constructibility, ordinal definability, and core models
• 03E47 Other notions of set-theoretic definability
• 03E50 Continuum hypothesis and Martin`s axiom [See also 03E57]
• 03E55 Large cardinals
• 03E57 Generic absoluteness and forcing axioms [See also 03E50]
• 03E60 Determinacy principles
• 03E65 Other hypotheses and axioms
• 03E70 Nonclassical and second-order set theories
• 03E72 Fuzzy set theory
• 03E75 Applications of set theory
• 03E99 None of the above, but in this section
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Exponential distribution qn
September 22nd 2009, 11:18 PM #1
Aug 2009
Exponential distribution qn
Hi, could someone help me out with this question?
Calls to an emergency ambulance service in a large city are modeled by a Poisson process with an average 3.2 calls per hour.
(a) what is the probability distribution of X where X is the length of time between successive calls?
I'm saying it's an exponential distribution? But I'm not entirely sure...
(b) find the E(X) = miu and Var(X) = sigma^2
(c) find the probability that X is in the interval miu +/- 0.5sigma.
I'm thinking if it were an exponential distribution, then I integrate the density function from the lower to the upper limits?
(d) what is the probability that the time between successive calls to the ambulance service is longer than 30 minutes?
(e) What is the probability that there will be at least one emergency ambulance service call in the next 45 minutes, given that there was no call in the last 15 minutes?
Thanks in advance!
Hi, could someone help me out with this question?
Calls to an emergency ambulance service in a large city are modeled by a Poisson process with an average 3.2 calls per hour.
(a) what is the probability distribution of X where X is the length of time between successive calls?
I'm saying it's an exponential distribution? But I'm not entirely sure... Mr F says: Correct. In a poisson process the waiting times are exponentially distributed.
If the number of arrivals in a given time interval [0,t] follows the Poisson distribution, with mean = λt, then the lengths of the inter-arrival times follow the Exponential distribution, with
mean 1 / λ.
(Quoted from Poisson distribution - Wikipedia, the free encyclopedia)
(b) find the E(X) = miu and Var(X) = sigma^2
(c) find the probability that X is in the interval miu +/- 0.5sigma.
I'm thinking if it were an exponential distribution, then I integrate the density function from the lower to the upper limits?
(d) what is the probability that the time between successive calls to the ambulance service is longer than 30 minutes?
(e) What is the probability that there will be at least one emergency ambulance service call in the next 45 minutes, given that there was no call in the last 15 minutes?
Thanks in advance!
With the pdf found in (a), the rest should follow.
For (e), calculate $\Pr\left( \frac{1}{4} \leq X \leq 1 | X > \frac{1}{4} \right)$.
September 23rd 2009, 01:59 AM #2
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Math Forum Discussions - Re: Matheology � 203
Date: Jan 30, 2013 5:15 PM
Author: Virgil
Subject: Re: Matheology � 203
In article
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 30 Jan., 10:31, William Hughes <wpihug...@gmail.com> wrote:
> > For a potentially infinite list L, the
> > antidiagonal of L is not a line of L.
> Of course. Every subset L_1 to L_n can be proved to not contain the
> anti-diagonal
> >
> > Does this imply
> >
> > There is no potentially infinite list
> > of 0/1 sequences, L, with the property that
> > any 0/1 sequence, s, is one of the lines
> > of L.
> Do you mean potentially infinite sequences?
> Look, everything Cantor does, concerns only finite initial segments.
> You could cut off the sequences behind the digonal digit.
But the anti-diagonal does not nave any merely "diagonal" digit, so must
be endless.
> The only thing not terminating, then could be the diagonal itself. But
> then you would claim that the diagonal differs from every entry,
> because it has more digits. In the original argument, the diagonal
> differs at the same places that also exist in the entries. Therefore
> the argument with the diagonal "being longer" is wrong.
Any anti-diagonal differs from each listed entry of the list from which
it is derived in AT LEAST one digit position, so is not a listed entry
of that list. Thus those. like WM, who claim existence of a complete
lists of functions from |N to any set of more than one member are wrong.
> So in fact, Cantor shows that the countable set of all terminating
> decimals is uncountable.
WM claims to be able to prove it but his "proof" is only invalid in his
wild weird world of in Wolkenmuekenheim.
What Cantor actually showed was that given a list of functions each
having domain |N and a codomain of at least two members there is no
surjection from |N to that set of functions. Thus any such set satisfies
the standard definition of uncountability.
That WM does not like it is his problem, not ours.
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Lvov-Warsaw School
First published Thu May 29, 2003; substantive revision Wed Aug 28, 2013
The Lvov-Warsaw School (LWS) was the most important movement in the history of Polish philosophy. It was established by Kazimierz Twardowski at the end of the 19^th century in Lvov, a city at that
time belonged to the Austro-Hungarian Empire. The LWS flourished in the years 1918–1939. Kazimierz Ajdukiewicz, Tadeusz Kotarbiński, Stanisław Leśniewski, Jan Łukasiewicz and Alfred Tarski are its
most famous members. It was an analytical school similar to the Vienna Circle in many respects. On the other hand, the attitude of the LWS toward traditional philosophy was much more positive than
that of logical empiricism. Although logic became the most important field in the activities of the LWS, its members were active in all spheres of philosophy. World War II and political changes in
Poland after 1945 caused the end of the LWS as an organized philosophical enterprise. One can consider it to have later been continued individually by its representatives.
Kazimierz Twardowski (1866–1938) began his post as professor of philosophy at Lvov University in 1895. He came to Lvov from Vienna, where he had studied philosophy under Franz Brentano and Robert
Zimmermann. Twardowski belonged to the last group of Brentano's students. His Habilitationschrift (1894) concerned the concepts of the content and the object of presentations; it clarified and
sharpened this important distinction. This work strongly influenced Meinong and Husserl.
Twardowski appeared in Lvov with the ambitious plan of creating a scientific philosophy (in Brentano's spirit) in Poland (at that time, Poland was partitioned between Austro-Hungary, Germany and
Russia; Lvov belonged to the Austro-Hungarian Empire.) In fact, he subordinated all his activities to achieving this task and considerably limited his own scientific work. Twardowski was an
extraordinary and charismatic teacher. He very soon attracted many young people to philosophy. After ten years of teaching he sometimes had about 200 candidates for seminars and 2000 attendants of
his lectures. He propagated a clear style of writing and speaking about philosophical matters, insisted upon justification of philosophical theses and sharply distinguished philosophy as a science
from world-views. Following Brentano, he favoured problems on the borderline of descriptive psychology, grammar and logic (he supplemented his object/content distinction by that of actions/products).
A photo of Twardowski's last seminar participants taken during the 1936–1937 academic year is available (see the supplement), with most of the participants identified.
Although Twardowski was not a logician and did not consider himself as such, his program formed a friendly environment for logic in all its subdomains: formal logic, semantics and methodology of
science. Jan Łukasiewicz (1878–1956) was the first of Twardowski's students to be interested in logic. He began his lectures in logic in Lvov in 1906. Kazimierz Ajdukiewicz (1890–1963), Tadeusz
Czeżowski (1889–1981), Tadeusz Kotarbiński (1886–1981) and Zygmunt Zawirski (1882–1948) studied mainly under Twardowski, but they also attended courses conducted by Łukasiewicz. Stanisław Leśniewski
(1886–1939) joined this circle in 1910. Warsaw appeared on the stage exactly in 1915, when the University of Warsaw was reopened. The academic staff was mainly imported from Lvov; Łukasiewicz was
appointed professor of philosophy.
Poland recovered its independence in 1918 and Polish scholars began to build national academic life. The program for the development of mathematics elaborated by a mathematician Zygmunt Janiszewski
(the Janiszewski program) had a great importance for the subsequent development of the LWS. According the Janiszewski program, Polish mathematicians should concentrate on set theory, topology and
their applications to other branches of mathematics. In particular, the Janiszewski program attached great importance to mathematical logic and the foundations of mathematics. Two philosophers,
namely Leśniewski and Łukasiewicz, became professors of the University of Warsaw at the Faculty of Mathematical and Natural Sciences. Both began intensive teaching of mathematical logic, mostly among
mathematicians but also among philosophers. Thus, logic in the LWS had two parents: mathematics and philosophy.
Alfred Tarski (1901–1983) opened the list of young mathematicians and philosophers attracted by logic in Warsaw. The logical community in this city included (in alphabetical order and covering the
whole period 1918–1939: Stanisław Jaśkowski (1906–1965), Adolf Lindenbaum (1904–1941?), Andrzej Mostowski (1913–1975), Moses Presburger (1904?-1943), Jerzy Słupecki (1904-1987), Bolesław Sobociński
(1904–1980; a philosopher by training)) and Mordechaj Wajsberg (1902–1942?). The names of three other logicians who graduated shortly before 1939 or studied during War World II and began their
academic work after 1945 should be added, namely Jan Kalicki (1922–1953; a mathematician), Czesław Lejewski (1913–2001; a classicist and philosopher) and Henryk Hiż (1917; a philosopher).
The development of logic in Warsaw had two subperiods in 1918–1939, namely 1918–1929 and 1929–1939. The first decade consisted in intensive teaching and scientific work at the seminars of Leśniewski
and Łukasiewicz. Not many results were published at that time. The explosion of publications took place in 1929 and later. There are several factors which caused the development of mathematical logic
in Poland. The Warsaw school of logic appears to be model case, but the power of this circle influenced other places where the general environment was not so favourable to logic. The fruitful
co-operation of mathematicians and philosophers in Warszawa had the utmost significance. The founders of the Polish mathematical school made a brave experiment consisting in inviting two philosophers
with a modest mathematical background as professors at the Faculty of Mathematics and Natural Sciences; this did not happen in any other country. The gifts of Leśniewski and Łukasiewicz as teachers
and the abilities of the latter as an organizer attracted young mathematicians. In Poland mathematical logic was considered to be an autonomous science, not a part of mathematics or philosophy. From
the present-day point of view this might seem as an exaggeration, but this ideology contributed essentially to the strength of Polish logic. Their representatives were fairly conscious of the fact
that the propagation and defence of the autonomy of this field had to be confirmed by important scientific results and international recognition. Moreover, this view about logic motivated various
purely theoretical investigations on formal systems. On the other hand, Polish logicians strongly insisted that logic should not be restricted only to mathematics and required the co-operation of
representatives of all field in which logic might be used. Still another factor played an important role, namely the conviction about the social significance of logic as a weapon against all kinds of
irrationalism. Tarski once said “Religion [you can also say “ideology” — JW] divides people, logic brings them together.” According to Łukasiewicz, “Logic is morality of thought and speech”. Thus,
Polish logicians doing logic and teaching it were convinced that they were performing an important social service.
Kotarbiński was appointed professor of philosophy in Warsaw in 1919. His teaching activity resulted in a group of scholars working mainly in the philosophy of science, including Janina Hosiasson
(later Mrs. Lindenbaum; 1899–1942), Edward Poznański (1901–1976), Dina Sztejnbarg (later Mrs. Kotarbiński) (1901–1997) and Aleksander Wundheiler (1902–1957).
Twardowski and Ajdukiewicz (appointed professor in 1928) remained in Lvov. They trained a group that included Izydora Dąmbska (1904–1983), Maria Kokoszyńska (1905–1981), Henryk Mehlberg (1904–1978)
and Zygmunt Schmierer (?–1943). Although Twardowski's students also taught at other Polish universities (Czeżowski in Vilna, Zawirski in Poznań and Kraków), Lvov and Warsawa were the main centres of
the LWS. The school was also joined by a group of catholic philosophers, including Father Innocenty (Józef) M. Bocheński (1902–1995) and Father Jan Salamucha (1904–1944).
World War II had disastrous consequences for the LWS. Twardowski and Leśniewski died before September 1, 1939. Of the people mentioned above who lost their lives (mostly Jews murdered by the Nazis):
the Lindenbaums, Presburger, Salamucha, Schmierer and Wajsberg. Zawirski died in 1947. Many emigrated from Poland during World War II or shortly after it: Łukasiewicz (Dublin), Tarski (Berkeley), Hiż
(Philadelphia), Kalicki (Berkeley), Lejewski (Manchester), Mehlberg (Toronto, Chicago), Sobociński (Notre Dame) and Wundheiler (New York); Bocheński (Fribourge) and Poznański (Jerusalem, before
The situation in Poland in 1945–1948 was similar as before 1939. The Marxist ideological offensive against bourgeois philosophy started in 1949. The policy became more liberal after 1956. Although
many scholars of the LWS actively taught and worked in the new political reality, it would be difficult to say that the school continued its former manner of existence. The tradition of the LWS was
rather preserved in individual hands, but not as an organized enterprise.
Note: the present essay focuses on the logical wing of the LWS. In 1939, the entire school comprised about 80 scholars actively working in all branches of philosophy as well as in other academic
fields, like psychology, sociology, theoretical linguistics, history of art and literary studies.
Most philosophers of the LWS understood philosophy as a collection of disciplines, including logic, ethics, aesthetics, metaphysics, and epistemology. Philosophy is a science, like any other. All
members of the LWS inherited from Twardowski his main metaphilosophical claims concerning clarity, justification and the separation of philosophy from world-views. It also meant a radical rejection
of all kinds of irrationalism. A view, called anti-irrationalism by Ajdukiewicz, demanded that every rationally accepted proposition be intersubjectively communicable and testable. Although there was
no a priori list of meaningful questions subjected to philosophical work, one should be sceptical about so-called great metaphysical problems and their scientific status. Philosophical activity must
begin with a very careful linguistic analysis of investigated problems and their meaning.
Twardowski himself favoured descriptive psychology as basic, but many of his students found logic to be the most important source of methodological criteria for philosophy. Perhaps Łukasiewicz was
the most radical in this respect. According to him, a reform of philosophy was needed in order to avoid errors of the past. Philosophy should proceed as logic does, axiomatically starting from clear
concepts and evident principles. Other philosophers of the LWS were more modest and did not demand that philosophy should be axiomatized. However, logical analysis of philosophical discourse became
the standard method of analysis. Yet the task of philosophy is not limited to the analysis of language. Thus, philosophy, according to methodological claims of the LWS, was analytic but not purely
linguistic. Philosophy is concerned with the world but it mainly (though not “only”) performs its task also by the analysis of language, which is used in speaking about reality. This view about
philosophy is to be contrasted with that of the Vienna Circle. In particular, the LWS was not interested in a general metaphilosophical scheme that sharply divided philosophy into good and bad, but
rather with an analysis of concrete problems. Thus, the LWS was united more by a common methodological attitude and very general claims concerning rationality than by a commonly accepted
philosophical theory.
However, some general views were shared by most (“most” is very important here) members of the LWS. These include: anti-scepticism, anti-naturalism in humanities and axiology, realism in epistemology
and philosophy of science, absolutism in epistemology and axiology, and empiricism. These views were characteristic for Brentano and became implemented in Polish philosophy by Twardowski.
Łukasiewicz invented a parenthesis-free logical notation. The idea consisted in writing logical constants before their arguments. Łukasiewicz replaced the usual signs for logical operations by
letters: N (negation), K (conjunction), A (disjunction), C (implication) and E (equivalence). Any well-formed formula (the present explanations are restricted to propositional calculus) must begin
with a capital letter (propositional variables are symbolized by lower-case Latin letters) which is the main functor of the entire formula. The main connective has as its arguments variables or
formulas consisting of variables and constants. The following are examples: Cpp for (p → p), CCppNq for ((p → p) → ¬q). The structure of a formula (and hence also its meaning) in Polish notation is
uniquely determined by the position of the letters. The parenthesis-free notation is unambiguous in the sense that any finite sequence of symbols for connectives and variables is interpretable in a
unique way. This implies that any wff coded in Polish notation has only one translation into the standard symbolism. The main advantage of Polish notation is its economy, because it avoids special
punctuation devices such as brackets or dots. When Łukasiewicz met Turing in 1949, the latter remarked that Polish notation was much better for computers, because formulas with function-symbols in
front could be better elaborated by mechanical devices.
The parenthesis-free symbolism was closely associated with some ideas of Polish logicians concerning the good properties of formal systems. Of course, any correct logical system should be consistent
and, if possible, syntactically and semantically complete. It should also be based on independent sets of primitive terms and axioms. The Warsaw School of Logic strongly emphasized the last property,
often considered secondary. Thus, the dependence of primitive terms or axioms was regarded as an essential defect. Moreover, some additional structural properties of logical systems were recommended:
(a) a system with fewer primitive concepts is better; (b) a system with fewer axioms is better; (c) if we define the length of an axiom system as the number of symbols occurring in all of its axioms,
the shortest axiom system is the best; (d) a system with fewer different symbols is better; (e) if we define an organic theorem as one which has no other theorem inside it (for example, the formula
CpCqq is not an organic theorem), organic axioms are better than non-organic ones. Thus, the ideal axiom system consists of a sole organic axiom of the shortest possible length, provided that it is
consistent. Requirements (a)-(f) apply particularly well to propositional calculus. They became the guiding principles of many logical investigations in the Warsaw School of Logic. Logicians of this
school also made precise many important metalogical concepts, including those of the logical matrix, consequence operation, deductive system and model.
Łukasiewicz formulated several axiomatic bases for the functionally complete propositional calculus, that is PC in which all 16 binary connectives can be defined. The most popular is the N-C system,
having as axioms the formulas: CCpqCCqrCpr, CCNppp, CpCNpq, and the usual rules of inference (substitution, detachment). This system is consistent, independent, Post-complete (= semantically
complete): Łukasiewicz and his collaborators invented new methods for proving these properties. According to the criteria mentioned in the preceding section, one should look for the simplest axiom
The discovery of many-valued logic is commonly considered to be one of the major achievements of Łukasiewicz. He did it in 1918, a little earlier than Post. However, although Post's remarks were
parenthetical and extremely condensed, Łukasiewicz explained his intuitions and motivations carefully and at length. He was guided by considerations about future contingents and the concept of
Łukasiewicz observed that no functor of classical propositional calculus could be read as “it is possible that” and provided that the formula Mp (it is possible that p) is extensional (i.e., that its
value depends solely on the value of p). The difficulty can be solved if we admit a third value. Sentences about future contingent states of affairs are natural candidates for having the third value
(½). For example, the sentence “I will visit Warszawa next year”, is neither true nor false, it is merely possible and has the value ½. Its negation has the same value. This idea led to three-valued
logic. The usual equalities for N, A, Kand C are supplemented by (I list only some cases) p = ½ = Np, K ½½ = ½, A½½ = ½. Easy calculations shows that ApNp and NKpNp have the value ½ for p = ½. This
means that the laws of contradiction and excluded middle do not hold in three-valued logic. Later, Łukasiewicz generalized it to logics with an arbitrary finite number of values and finally to an
countably infinite number of values. The sense of implication is given by the equations:
Cpq = 1, for p≤q
Cpq = 1−(p + q), for p>q,
and the sense of negation by the equation:
Np = 1−p, where 0≤p≤1.
If we have only two values, these equations determine the usual truth-tables for C and N.
Three problems arose after discovering many-valued logic. The first concerned its axiomatization and metalogical properties, the second its philosophical foundations and intuitive interpretation, and
the third its applications. Due to the work of Łukasiewicz himself, Wajsberg and Słupecki, the first group of questions was largely solved. Wajsberg showed that the formulas: CpCqp, CCpqCCqrCpr,
CCNpNqCqp, CCCpNppp axiomatize Ł[3] (three-valued propositional calculus). The same author proved that a finite Ł[n] is axiomatizable if it includes the theorems: CCpqCCqrCpr, CCCqrCCpqCpr, CCqqCpp,
CCpqCNqNp, CNqCCpqNq. If n = ℵ[0], Ł[n] can be axiomatized by (Łukasiewicz's conjecture, proved by Wajsberg): CpCqp, CCpqCCqrCpr, CCCpqqqCCqpp, CCCpqCqpCqp, CCNpNqCqp. However, all of the above
axiom-sets are functionally incomplete. The problem was solved by Słupecki for Ł[3]. He introduced the new functor T defined by T1 = T½ = T0 = ½ and added the formulas CTpNTp, CNTpTp to Wajsberg's
axioms. All Łukasiewicz's many-valued logics are consistent. Słupecki proved that Ł[3] is Post-complete. Every Ł[n] (n>2) is contained in two-valued logic, although the converse does not hold; for
example, the formulas CCNpNp, CCNppp, CCpqCCpNqNp, CCpKNqNp, CcpEqNqNq are theorems in the two-valued system only. If n = ℵ[0], Ł[n] is contained in every finite Ł[n].
At first, Łukasiewicz called his three-valued logic “Non-Aristotelian”, but later he preferred the qualification “Non-Chrysippean”. According to Łukasiewicz, the Stagirite himself doubted the
validity of the principle of excluded middle in the domain of future contingents. On the other hand, the Stoics believed that every proposition is true or false, independently of its temporal
reference. Thus, the Stoics accepted the principle of bivalence in its unrestricted form. Now, the foundation of two or many-valued logic lies not in this or that logical theorem, but in metalogic;
in particular, it is determined by accepting or rejecting the principle of bivalence. Whoever, as Chrisippus did, accepts the validity of the principle of bivalence, opts for two-valued logic;
whoever even partially rejects this principle, as Aristotle did, thereby opens the door to many-valued logic. Łukasiewicz took the side of Aristotle. However, this did not close the problem of the
interpretation of other logical values. Łukasiewicz tried to go though indeterminism and causality. A typical difficulty is the following. Take p as valued by ½. Its negation also has the value ½.
The same holds for KpNp, contrary to the firm intuition that any pair of contradictory sentences is false. Difficulties with interpretation changed Łukasiewicz's primary view concerning the relation
of many-valued logic to reality. At first, guided by a realistic epistemology of logic, he maintained that one of the rival logics could be proved to be the correct description of the physical world.
Later, he was rather inclined to look at logical systems as formalisms having their own problems deserving research and as useful devices for solving various questions but not as something leading to
the only “true” ontological scheme. Yet he believed that many-valued logic would play a considerable role in the foundations of mathematics.
3.3.1 Leśniewski's systems
Leśniewski intended to a formulate the full logical system that would serve as the basis for the whole of science, and in particular for mathematics. This system consists of three parts (a)
protothetic (a generalized propositional calculus); (b) ontology (a logic of terms); (c) mereology (a theory of parts and wholes). Protothetic is a calculus in which quantifiers bind propositional
variables and variables referr to arbitrary functors constructible over the usual functors: that is, functors of propositional variables, functors of functors, etc. In general, if we start with the
category of sentences alone, protothetical quantifiers bind variables of all further definable categories. The shortest axiom of protothetic (written in the Russell-like symbolism) is the formula
[pq] :: p ↔ . q ↔ :.[f]:. f (pf (p [u]. u)). ↔ : [r]: f (qr). ↔ . q ↔ p
(Sobociński). Protothetic is an absolute propositional calculus in the sense that the principle of bivalence is its theorem. In fact, protothetic inspired Łukasiewicz's system with variable functors,
another absolute propositional logic.
If we add the functor ε (read as “is”) forming sentences from two names, we obtain Leśniewski's ontology (LO). The meaning of the constant ε is perhaps the most important matter for a proper
understanding of LO. The epsilon corresponds well to the sense of the copula “est” in Latin sentences of the type “Socrates est homo”. The epsilon has no spatio-temporal connotations and does not
indicate the membership relation or identity. The rendering of the epsilon by the English “is” may be misleading, because the latter is modified by articles. The axiomatic characterization of the
meaning of the epsilon is given by
(O) [Aa]:: (A ε a) ↔ :.[ΣB]. (B ε a):. [BC]: (B ε A). (C ε A). →
(B ε C):. [B]: (B ε A) →. B ε a.
Its simplified form (discovered by Sobociński) is:
(O′) [Aa] A εa: ↔.[ΣB]. (A εB). (B εa).
The right sides of (O) and (O′) are conjunctions. The intuitive content of (O) is simple in spite of its formal complexity. It establishes that the sentence “A is a” is equivalent to the following
conditions (a) A is not an empty term; (b) there is only one A; (c) whatever is A, is also a. Thus, “A is a” is a singular sentence which is true if and only if (a)-(c) hold. In particular, such a
sentence is false if A is a general or an empty term. On the other hand, (O) (or O′) is valid for all terms, even general or empty ones. Thus, LO is valid in all domains, including the empty one and
can be regarded as the first system of free logic. In LO we can define two important concepts, namely that of existence and that of being an object. This is done by (I use non-symbolic forms): (1)
for any A, A exists = for some x, x is A; (2) for any A, A is an object = for some x, A is x. LO performs functions usually provided by predicate logic. The meaning of the constant ε is sufficiently
general to define identity and the inclusion of classes. Since these concepts are definable in elementary ontology, it is stronger than first-order logic.
Mereology, assumes protothetic and ontology as logically prior theories, and has the term “part of” as its sole primitive concept. Being a part is a non-reflexive and transitive relation. There are
no empty classes. Moreover, the class consisting of a single element is identical with it. In general, mereology is a theory of sets in the collective (mereological) sense, contrary to ordinary set
theory, which describes sets in the distributive sense. The main difference between the two interpretations of the term “set” consists in the fact that the membership relation is transitive under the
mereological reading, but non-transitive under the distributive one. Leśniewski believed that his theory classes would perform all the tasks of ordinary set theory without generating paradoxes. In
fact, he invented mereology when he tried to solved the Russell paradox. Since there are no mereological classes which are not their own elements, the question which led to the Russell paradox simply
makes no sense in Leśniewski's systems. On the other hand, mereology is weaker than set theory.
Leśniewski's systems have some formal features, even very peculiar ones in some respects. All are axiomatic. According to his nominalistic preferences, they are concrete physical objects. Expressions
are always understood as sequences of concrete inscriptions. There are as many expressions as have been written; no expression exists merely potentially. This view is called constructive nominalism.
According to it, two intuitively equivalent systems, for example, protothetic based on equivalence and protothetic based on implication, are different systems. Every logical system, on Leśniewski's
view, is not finished at any time because there is always the possibility of adding new elements to it. Hence, the rules for constructing and developing formal systems are of crucial importance for
Leśniewski's logics. He understood this very well and devoted much attention to explaining the details of his formalization. Leśniewski formulated his procedural directives purely syntactically and
completely. Due to the role of equivalence, he was able to treat definitions as theorems. In general, Leśniewski's systems are commonly considered to be perfect from the point of view of the
requirements of correct formalization. The Leśniewski project is a version of logicism. Leśniewski's three systems form a grand logic and provide a universal language to capture the whole of
knowledge. It is certainly not an orthodox system and lies on the margin of contemporary research in logic. Yet it continues to attract many logicians and philosophers. In spite of their marginality,
Leśniewski's systems are investigated in all parts of the world.
Leśniewski suggested the theory of syntactic categories, later developed by Ajdukiewicz in the early 1930s. This theory takes the categories of sentences and names (for Ajdukiewicz, following
Leśniewski, there is no syntactic difference between proper names and common nouns) as fundamental and assigns the pointer s to sentences and n to names. Now functors have fractions as pointers. For
example, “is” has s/nn as its categorial index; it says that ”is“ is a two-placed functor of two nominal arguments which forms a sentence. The conjunction as a propositional connective forming
sentences from two other sentences has s/ss as its index. Consider now the expression “p and q”. Write the categorial indexes of its parts. We thus obtain the sequence: s s/ss s. Perform
simplifications by divisions similar to dividing algebraic fractions. The letter s is the result. A simple algorithm says that an expression is syntactically coherent if and only if s or n is its
index after performing all simplifications. Ajdukiewicz's quasi-arithmetical notation was the first system of categorial grammar.
Due to semantic paradoxes, Hilbert's formalistic metamathematics and the syntacticism of the Vienna Circle, the concept of truth was expelled from the domain of logic. It was Tarski who changed this
attitude. He was inspired by the Aristotelian tradition in philosophy, as well as the non-constructive style of working on the foundations of mathematics that was prevailed in Poland. In 1933, he
published a book on the concept of truth (in Polish), translated into German in 1936 and English in 1956.
Tarski's theory of truth (the semantic conception of truth) has two aspects: philosophical and formal. Philosophically, it is a version of Aristotle's idea that truth consists in saying that what is,
is and what is not, is not (it is related to the idea of correspondence). However, the main problem was formal. Tarski had to offer a construction free of semantic paradoxes, in particular the Liar.
He achieved this goal by postulating that the concept of truth must be defined for a definite, well-constructed formalized language L. However, the definition itself should be formulated in the
metalanguage ML. The definition is to be formally correct, that is, it cannot lead to contradictions and has to satisfy the usual conditions of correctness (non-circularity, etc.). It should also be
materially adequate. According to Tarski, the basic intuition is captured by the T-scheme: s is true if and only if P, where the letter s represents the name of a sentence and P is a translation of
this sentence into the metalanguage ML. Now the condition of material adequacy (the Convention T) says that a truth definition TD is materially adequate if and only if all equivalences (that is, for
all sentences in L) stemming from the T-scheme by appropriate replacements are provable from the definition. The conditions are satisfied by the following definition:
A sentence A of a language L is true if and only if it is satisfied by all infinite sequences of objects taken from the universe of discourse.
A more refined version is model-theoretical:
A sentence A is true in a model M if and only if A is satisfied by all infinite sequences of objects taken from the carrier of M.
This definition implies the metalogical principles of excluded middle and contradiction, which both are equivalent to the principle of bivalence.
Tarski's truth-definition is one of the most debated contemporary philosophical and logical ideas. It strongly influenced semantics, philosophy of language, philosophy of science and epistemology. In
particular, it became the first step toward model-theory, a central branch of mathematical logic. Two applications of this definition are worth mentioning. Firstly, Tarski succeeded in formulating an
exact definition of “logical consequence” (“following from” or “logical entailment”):
A sentence A logically follows from the set X of sentences if and only if every model of X is a model of A.
Secondly, Tarski proved the following limitative theorem:
If a formal system S captures Peano arithmetic, the truth-predicate (or: the set of S-truth) is not definable in it.
Łukasiewicz revolutionized the history of logic. He proposed to look at the history of logical ideas through the glasses of mathematical logic. The reason was that he was convinced about the
continuity of formal logic from Aristotle to modern mathematical logic, perhaps with a break from the 16^th century to Boole and Frege (of course, with the exception of Leibniz). Thus, old sound
logical theories should be considered as anticipations of ideas advanced in the 19^th and the 20^th centuries. Guided by this assumption, Łukasiewicz showed that the Stoics invented propositional
calculus, contrary to the prevailing view that Stoic logic was a part of Aristotle's logic. In particular, Łukasiewicz demonstrated that the Stoic logic of propositions was a system of rules, not
theorems. Another of Łukasiewicz's historical discoveries consisted in the rehabilitation of medieval logic, commonly neglected as fruitless scholasticism. He was joined in these investigations by
Bocheński and Salamucha.
Historical work inspired logicians of the LWS toward modern interpretations of traditional logical doctrines. The most famous is Łukasiewicz's formalization of the Aristotelian logic of categorical
sentences (the syllogistic plus conversion and other rules of so-called direct inference). Łukasiewicz interpreted this logic as a specific formal theory, not as a fragment of predicate logic, as it
was usually done (for example by Frege or Russell). Yet the logic of categorical sentences assumes propositional logic as prior. The logic of assertoric sentences (Łukasiewicz also considered its
modal extension) has the following form. Let the formulas (lower-case letters are term variables) Uab, Iab, Yab, Oab stand for the sentences “every a is b”, “some a are b”, “no a is b” and “some a
are not b”. We can define Yab as NIab and Oab as NUab. The axioms are as follows: (a) Uaa; (b) Iaa; (c) CKUmbUamUab (the Barbara modus); (d) CKUmbImaIab (the Datisi mode); the rules are: all the
rules of propositional calculus, substitution for term variables, the definitional replacement according to the definitions of Yab and Oab.
There was no official philosophy of logic and mathematics in the LWS. Most Polish logicians treated logical studies as independent of philosophical commitments. Only Leśniewski had explicit
philosophical views which influenced the form of his systems. This does not mean that concrete works were not influenced by philosophical ideas. Łukasiewicz's many-valued logic and Tarski's theory of
truth are perhaps model cases. The former had the problem of determinism as its background and the second was strongly inspired by the Aristotelian tradition in thinking about truth. It was also the
case that Polish logicians had inclinations to empiricism as a general epistemological attitude and this philosophy often resulted in sympathies to nominalism (Tarski), constructivism (Mostowski) and
scepticism concerning the sharp distinction between logical and extralogical truth (Tarski). However, the technical side of logical problems decided about investigations and sometimes forced changes
in philosophical standpoints. The example of Łukasiewicz is instructive once again. Although he at first thought of logic as a true or false description of reality, he later adopted a more
conventionalist and instrumentalistic standpoint. This attitude allowed him to accommodate various ideas coming from rival foundational directions, that is, logicism, formalism and intuitionism. In
fact, Leśniewski and Tarski contributed to the theory of logical types and combined it with the theory of syntactic categories; Tarski's version in his work on truth is particularly important. Tarski
also showed new perspectives for logicism by defining logical concepts as invariants under one-to-one transformations. He also contributed to general metamathematics (the theory of consequence
operation) and intuitionistic logic. However, a very special feature of logical investigations performed in the LWS consisted in the free admission of all fruitful mathematical methods, including
non-constructive ones. This was the main point of the set theoretical approach to the foundations of mathematics which replaced logicism.
The above survey does not do justice to many of the logical studies carried out in the LWS. Let me only mention some of them: particular historical studies of Bocheński and Salamucha, several
interpretations of traditional logic (Ajdukiewicz, Czeżowski), partial propositional calculi (all Warsaw logicians), propositional calculus with variable functors (Łukasiewicz), paraconsistency
(Jaśkowski), Ł-modal systems (Łukasiewicz), rejection rules, natural deduction (Jaśkowski), intuitionistic logic (Jaśkowski, Tarski, Wajsberg), free logic (Jaśkowski, Mostowski), the elimination of
quantifiers (Tarski, Presburger), undecidability (Tarski, Mostowski), the foundations of geometry (Tarski), the elementary theory of real numbers, the calculus of systems (Tarski), the
Kleene-Mostowski hierarchy, generalized quantifiers (Mostowski), as well as several particular results: the deduction theorem (Tarski), the upward Löwenheim-Skolem theorem (Tarski), the separation
theorem for intuitionistic logic (Wajsberg) or the Lindenbaum maximalization lemma.
The philosophy of science was a favourite field of the LWS. Since science is the most rational human activity, it was important to explain its rationality and unity. Since most philosophers of the
LWS rejected naturalism in the humanities and social sciences, the way through the unity of language (as in the case of the Vienna Circle) was excluded. The answer was simple: science qua science is
rational and is unified by its logical structure and by definite logical tools used in scientific justifications. Thus, the analysis of the inferential machinery of science is the most fundamental
task of philosophers science. Inductivism was a prevailing view about justification in empirical science. Hosiasson formulated an axiomatic system of inductive logic, anticipating Carnap's later
work. Other attempts to establish the foundations of inductive inference were undertaken by Ajdukiewicz (via statistics, decision theory and game theory (he mainly investigated the problem of the
rationality of modes of fallible inference), Czeżowski (via probability logic in the sense of Reichenbach) and Zawirski (via a combination of many-valued logic and probability theory).
Łukasiewicz worked on problems of the methodology of the empirical sciences in 1902–1910. At first, he tried to develop the inverse theory of induction (induction as inversed deduction) proposed by
Jevons and Sigwart. However, he abandoned this project very soon and adopted a radical deductionist standpoint. For him, induction plays no significant role in science. Deduction remains the only
credible mode of reasoning in all spheres of science. As applied in empirical science, it leads to negative results; that is, it can show that some hypotheses are false in the face of empirical data.
Łukasiewicz also offered a formal argument against induction derived from probability theory. Assume that H is a universal hypothesis. Its a priori probability is equal (or close) to zero and no
further empirical data can increase it. These ideas contain the main points of Popper's philosophy of empirical science.
Tarski's semantic ideas converted most members of the LWS to scientific realism. Formerly, under the influence of conventionalism, instrumentalism concerning scientific theories had adherents
(Ajdukiewicz, Łukasiewicz). A radical form of anti-realism was developed by Poznański and Wundheiler in the 1930s. They pointed out that verification in empirical science is cyclic and principally
anti-fundamentalistic. In particular, it is not possible to identify any data without a reference to theories. Hence, truth in science cannot consist in a correspondence with facts.
Of many investigations concerning special problems, let me only mention Mehlberg's version of the causal theory of time and some works on the causality problem in quantum mechanics. He admitted a
universal time as a synthesis of physical (intersubjective) and psychical (subjective) time. The causal theory does not lead to the anisotropy of time. It can be that universal time is symmetric, but
locally asymmetry is possible. Mehlberg and Zawirski defended a moderate causalism in quantum mechanics. In particular, Zawirski argued that the unpredictability of the future (Heisenberg) does not
entail that the principle of causality fails.
Kotarbiński developed a general doctrine, called reism. It has two aspects, ontological and semantic. Hence, we can speak about semantic reism and ontological reism, although this distinction was
clarified by Kotarbiński later. In general, reism goes against the acceptance of the existence of general (abstract) objects, that is, facts, properties, states of affairs, relations, etc. The main
ontological thesis of reism is as follows (it is subdivided into two subtheses): (R1) any object is a material, spatio-temporal, concrete thing; (R2) no object is a state of affairs, property of
relation (according to Kotarbiński, these three categories exhaust the domain of alleged abstract objects). Now (R1), that is, the positive thesis of reism has a rich content. Firstly, it marks a
formal feature of existing objects, namely their concrete character.
Secondly, it characterizes things as material and spatio-temporal entities, that is, as physical objects. Leibniz conceived monads as spiritual concreta. For the later Brentano, every object is
concrete, but there are souls and bodies. Thus, Leibniz's reism was monistic and spiritualistic, Brentano's was dualistic, and Kotarbiński's monistic and materialistic. Although the terminology
varies (one can equivalently speak about reism, concretism or nominalism), two claims of any theory going against general (abstract) objects should be very sharply distinguished. The first is
formal-ontological and points out the formal feature of existents, namely that they are individuals; but the second is material-ontological or metaphysical and focuses on their nature as physical or
psychical entities.
Semantic reism is parallel to the ontological aspect of this doctrine. The key idea consists in the distinction of genuine names and apparent names (onomatoids). A name is genuine if and only if
refers to things, that is, to concrete physical things. By contrast, onomatoids are words that allegedly refer to abstract entities, “allegedly” because their referents do not exist. At first glance,
apparent names are similar to empty terms. However, this resemblance is merely apparent, because empty nouns are genuine names and can always be decomposed into non-empty genuine names (e.g., "round
square"). This becomes evident when we try to formulate the conditions of meaningfulness for sentences. In general, a sentence is meaningful if and only if it consists (except for logical constants)
only of genuine names or is reducible to such sentences. For example, the sentence “all cats are animals” is reistically meaningful, but “properties are abstract objects” is not. Furthermore, “a
square triangle is rectangular” is good, but “sets exist outside of time and space” is not. The sentence “whiteness is a property of snow” can be reduced to “snow is white”. This example shows how to
translate some sentences with apparent names into purely reistic statements.
Matters become clearer if one remembers that Leśniewski's calculus of names is the underlying logic of reism. The copula “is” in “snow is white” functions with its meaning defined by the axiom of LO
(see above). Thus, this sentence is true if its subject refers to an individual object. The traditional interpretation of common nouns and adjectives, consistent with LO, as general terms referring
to many objects, saves their reistic character. Thus, one can say that the formal-ontological aspect of reism is adequately displayed by LO. Of course, reism as a metaphysical doctrine is an addition
to LO.
Kotarbiński recommended reism as a sound view. In particular, it defends philosophy and ordinary thinking before hypostases, that is, accepting the existence of abstract objects on the basis of using
apparent names. Thus, reism defends us against idola fori in Bacon's sense. Kotarbiński's reism is perhaps the most radical materialistic nominalism in the history of philosophy. Reism is exceptional
to the main tendency in the LWS in that it proposes a uniform language, proper everywhere, including the humanities, sociology and psychology (Kotarbiński supplemented reism by radical realism, that
is, the view that there are no mental contents). In this respect, reism resembles physicalism. The troubles of reism are typical those which are in the case of any reductive materialism and
nominalism, and concern the interpretation of mathematics, semantics, psychology, the humanities and social sciences.
Radical conventionalism is an epistemological theory developed by Ajdukiewicz in the early 1930s. It is based on a conception of language and meaning. The concept of meaning is taken as a primitive.
Now, the meaning of expressions in a language L induces the rules for accepting its sentences. Ajdukiewicz lists three kinds of meaning-rules (or sense-rules): (a) axiomatic (they demand the
unconditional acceptance of sentences, for example, “A is A”; (b) deductive (they demand the acceptance of a sentence relatively to the prior acceptance of other sentences, for example, ¬A follows
from A → B and ¬B), (c) empirical (they demand the acceptance of a sentence in a definite empirical situation, for example “it is raining” when it rains).
The special significance of meaning-rules and their relation to the meanings of expressions appears when special languages are taken into account, namely closed and connected. A language L is open if
it can be extended to a new language L′ without changes in the meanings of other expressions; otherwise L is closed. A language is disconnected if there is a non-empty subset X of L such that no
element of X is linked by the meaning-rules to other elements of L; otherwise, L is connected. It follows from the above definitions that if L is closed and connected, it cannot be enriched without
changing the meanings of the original expressions.
According to Ajdukiewicz, natural languages are open and disconnected. By contrast, scientific languages are closed and disconnected. Let L be closed and connected. The set of meanings of L is its
conceptual apparatus. If A and A′ are two conceptual apparatuses, they are either identical or mutually non-translatable. Since the acceptance and rejection of sentences is always related to a
language L, empirical data do no force us to accept or reject any sentences, because there always remains the possibility of changing a given conceptual apparatus. This is a considerable
radicalization of the conventionalism of Poincaré. The difference is the following. For Poincaré, since theoretical principles are conventions, we are free to modify them, but experiential reports
are perfectly stable. Ajdukiewicz extended conventionalism to all sentences, because any sentence, no matter whether experiential or theoretical, depends on a conceptual apparatus. It is why
Ajdukiewicz called this conventionalism radical.
In the middle of the 1930s Ajdukiewicz changed his view. He came to the conclusion that closed and connected languages are fictions. He was influenced by the semantic ideas of Tarski. Tarski also
argued that, contrary to Ajdukiewicz's hopes, the invariance of meaning-rules over permutations of expressions influences their meaning-relations. Gradually, Ajdukiewicz developed the program of
semantic epistemology, mainly directed toward the defence of realism against various forms of idealism. In particular, he criticized Rickert's transcendental idealism and Berkeley's subjectivism. For
Rickert, reality is only a correlate of the Transcendental Subject. Now the Transcendental Subject can be identified with the set T of true propositions obtainable on the basis of axiomatic and
deductive rules. However, due to incompleteness phenomena, T cannot be generated in this way. For Ajdukiewicz, that was a justification that transcendental idealism fails. Ajdukiewicz compared the
language used by Berkeley to the language of syntax, because the former reduces relations of the mind to its objects to relations between thoughts. On the other hand, the ordinary way of speaking
about objects employs semantic relations. Berkeley's claim esse = percipi is similar to an attempt to define semantics in a purely syntactic language. However, due to Tarski's results about the
relation between syntax and semantics, this is impossible. Finally, Ajdukiewicz argued that any idealistic language is understandable only if it is associated with a realistic language. Hence, any
attempt to consider idealistic language as self-sufficient cannot be successful.
The LWS acted in a country which never belonged to the philosophical superpowers. This circumstance is important for any assessment of the significance of the LWS. One can measure it on a national or
an international scale. The importance of the LWS for Polish philosophical culture was enormous. Twardowski fully realized his task. He introduced scientific philosophy in his sense into Poland and
created a powerful philosophical school. It did very much for the subsequent development of philosophy in the country. In particular, it popularized very high standards of doing philosophy. This was
important in the difficult times after 1945, when Marxism started an ideological and political offensive against bourgeois philosophy. In fact, due to the strong methodological tradition related to
the LWS, Polish philosophy did not lose its academic quality in 1945–1989.
As far as the matter concerns international importance, one thing is clear. The logical achievements of the LWS became the most famous. Doubtless, the Warsaw school of logic contributed very much to
the development of logic in the 20^th century. Other contributions are known but rather marginally. This is partially due to the fact that most philosophical writings of the LWS appeared in Polish.
However, this factor does not explain everything. Many writings of the LWS were originally published in English, French or German. However, their influence was very moderate, considerably lesser than
that of similar writings of philosophers from the leading countries. This is a pity, because radical conventionalism, reism or semantic epistemology are the real philosophical pearls. But perhaps
this is the fate of results achieved in cultural provinces.
The bibliography is divided into two sections. The first contains writings of the LWS in Western languages, the second writings on the LWS and its particular representatives.
Works of the LWS
A. Anthologies
• McCall, S. (ed.), 1967, Polish Logic 1920–1939, Oxford: Clarendon Press.
• Pearce, D. and Woleński, J. (eds.), 1988, Logischer Rationalismus. Philosophische Schriften der Lemberg-Warschauer Schule, Frankfurt am Main: Athenäum.
B. Books by Particular Philosophers
• Ajdukiewicz, K., 1958, Abriss der Logic, Berlin: Aufbau-Verlag.
• Ajdukiewicz, K., 1973, Problems and Theories of Philosophy, Cambridge: Cambridge University Press.
• Ajdukiewicz, K., 1974, Pragmatic Logic, Dordrecht: Reidel.
• Ajdukiewicz, K., 1978. The Scientific World-Perspective and Other essays, 1931–1963, Dordrecht: Reidel.
• Bocheński, I. M., 1961, A History of Formal Logic. Notre Dame: The University of Notre Dame Press.
• Czeżowski, T., 2000, Knowledge, Science and Values. A Program for Scientific Philosophy, Amsterdam: Rodopi.
• Kotarbiński, T., 1965, Leons sur l'histoire de la logique, Warszawa: Państwowe Wydawnictwo Naukowe.
• Kotarbiński, T., 1966, Gnosiology. The Scientific Approach to the Theory of Knowledge, Wrocław: Ossolineum.
• Leśniewski, S., 1988, Lecture Notes in Logic, Dordrecht: Kluwer.
• Leśniewski, S., 1992, Collected Works, Dodrecht: Kluwer.
• Łukasiewicz, J., 1957, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Oxford: Clarendon Press, 2^nd edition.
• Łukasiewicz, J., 1963, Elements of Mathematical Logic, Warszawa: Państwowe Wydawnictwo Naukowe.
• Łukasiewicz, J., 1970, Selected Works, Amsterdam: North-Holland.
• Łukasiewicz, J., 1993, Über den Satz des Widerspruchs bei Aristoteles, Hildesheim: Olms.
• Mehlberg, H., 1956, The Reach of Science, Toronto: University of Toronto Press.
• Mehlberg, H., 1980, Time, Causality, and the Quantum Theory, Dordrecht: Reidel.
• Mostowski, A., 1979, Foundational Studies, 2 vols., Amsterdam: North-Holland.
• Tarski, A., 1941, Introduction to Logic and to the Methodology of Deductive Sciences, Oxford: Oxford University Press, Oxford.
• Tarski, A., 1956, Logic, Semantics, Metamathematics Papers from 1923–1939, Oxford: Clarendon Press.
• Tarski, A., 1986, Collected Papers, 4 vols., Basel: Birkhäuser.
• Twardowski, K., 1999, On Actions, Products and Other Topics in Philosophy, Amsterdam: Rodopi.
• Wajsberg, M., 1977, Logical Works, Wrocław: Ossolineum.
• Zawirski, Z., 1994, Selected Writings on Time, Logic & Methodology of Science, Dordrecht: Kluwer.
Works on the LWS and its Particular Members
A. About the LWS
• Coniglione, F. Poli, R. and Woleński, J. (eds.), 1993, Polish Scientific Philosophy. The Lvov-Warsaw School, Amsterdam: Rodopi.
• Jadacki, J. J., 2009, Polish Analytical Philosophy, Semper: Warszawa.
• Jadacki, J. J., Paśniczek, J. (eds.), 2006, The Lvov-Warsaw School — the New Generation, Rodopi: Amsterdam.
• Jordan, Z., 1945, The Development of Mathematical Logic and of Logical Positivism in Poland between Two Wars, Oxford: Clarendon Press.
• Kijania-Placek, K. and Woleński, J. (eds.), 1996, The Lvov-Warsaw School and Contemporary Philosophy, Part II, Axiomathes, 7(3): 293–415.
• Kijania-Placek, K. and Woleński, J., 1998, The Lvov-Warsaw School and Contemporary Philosophy, Dordrecht: Kluwer.
• Krajewski, W. (ed.), 2001, Polish Philosophers of Science and Nature in the 20th Century, Rodopi: Amsterdam.
• Lapointe, S., Woleński, J., Mathieu, M., Miśkiewicz, W., 2009, The Golden Age of Polish Philosophy. Kazimierz Twardowski's Philosophical Legacy, Dordrecht: Springer.
• Skolimowski, H., 1967, Polish Analytical Philosophy, London: Routledge and Kegan Paul.
• Szaniawski, K. (ed.), 1989, The Vienna Circle and the Lvov-Warsaw School, Dordrecht: Kluwer.
• Woleński, J., 1989, Logic and Philosophy in the Lvov-Warsaw School, Dordrecht: Kluwer.
B. Works on Particular Members
• Sinisi, V. and Woleński, J. (eds.), 1995, The Heritage of Kazimierz Ajdukiewicz, Amsterdam: Rodopi.
• Gasparski, W., 1993, A Philosophy of Practicality: A Treatise on the Philosophy of Tadeusz Kotarbiński, Helsinki: Societas Philosophica Fennica.
• Woleński, J. (ed.), 1990, Kotarbiński: Logic, Semantics and Ontology, Dordrecht: Kluwer.
• Luschei, E., 1963, The Logical Systems of Leśniewski, Amsterdam: North-Holland.
• Miéville, D., 1984, Un développement des systmes logiques de Stanisław Leśniewski. Prototétique — Ontologie — Méreologie, Bern: Peter Lang.
• Miéville, D., 2001, Introduction à l’œvre de S. Leśniewski, F. I: La protothétique, Neuchâtel: Université de Neuchâtel.
• Srzednicki, J. (ed.), 1984, Leśniewski's Systems. Ontology and Mereology, The Hague: Nijhoff.
• Srzednicki, J. (ed.), 1998, Leśniewski's Systems. Prothotetic, Dordrecht: Kluwer.
• Vernant, D. and Miéville, D. (eds.), 1995, Stanisław Leśniewski aujourd'hui, Groupe de Recherches sur la philosophie et le langage/Centre de Recherches Sémiologiques, Grenoble/Neuchâtel.
• Ehrenfeucht, A., Marek, V. W., Srebrny, M. (eds.), 2008, Andrzej Mostowski and Foundational Studies, Amsterdam: IOS Press.
• Feferman, A., and Feferman, S., 2004, Alfred Tarski. Life and Logic, Cambridge: Cambridge University Press.
• Moreno, L. F., 1992, Wahrheit und Korrepondenz bei Tarski. Eine Untersuchung der Wahrheitstheorie Tarskis als Korrespondenztheorie der Wahrheit, Würzburg: Königshausen&Neumann.
• Patterson, D. (ed.), 2008, New Essays on Tarski and Philosophy., Cambridge: Cambridge University Press.
• Stegmüller, W., 1957, Das Wahrheitsproblem und die Idee der Semantik. Eine Einführung in die Theorien von A. Tarski und R. Carnap, Wien: Springer.
• Woleński, J. and Köhler, E. (eds.), 1999, Alfred Tarski and the Vienna Circle, Dordrecht: Kluwer.
• Cavallin, J., 1997, Content and Object. Husserl, Twardowski and Psychologism, Dordrecht: Kluwer.
How to cite this entry.
Preview the PDF version of this entry at the Friends of the SEP Society.
Look up this entry topic at the Indiana Philosophy Ontology Project (InPhO).
Enhanced bibliography for this entry at PhilPapers, with links to its database.
• Archives of the Lvov-Varsovie School, maintained by L'Institut d'Histoire et de Philosophie des Sciences et des Techniques (IHPST), Paris and University of Warsaw, Institute of Philosophy.
logic: many-valued | mereology | Tarski, Alfred | Tarski, Alfred: truth definitions
The author and editors would like to thank Branden Fitelson for supplying a correction to this entry after publication.
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Finding Lift Force - Glider Wing
I found the equation for Horizontal Velocity. Here it is for anyone else who might be stuck in the same place.
E is energy
m is mass
In my case Energy was elasticity since my glider was flung from a rubber band. So I had to find ft-lbf, then use different conversion factors to get to a suitable velocity unit.
My work:
V[H]=[tex]\sqrt{\left[\frac{2(4.1625)ft-lbf}{27.999g}\right]\left[\frac{453.6g}{1 lbm}\right]\left[32.2\frac{lbm-ft}{lbf-s}\right]}[/tex]
Leaving me with Velocity in Ft/s
V[H]= 65.899 ft/s
Additional Information:
I was given a chart, from whence I figured that 15 in displacement gave 3.33 lbf, and converted to ft-lbf, giving me my E.
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Convolution \[sin\omega t * cos \omega t\]
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\[sin\omega t * cos \omega t\]\[=\int_0^tsin\omega u \ cos \omega (t-u)du\]\[=-\frac{1}{\omega}\int_0^t\ cos \omega (t-u)d(cos\omega u)\]\[=\frac{1}{2\omega}[ cos ^2\omega (t-u)]_0^t\] Doesn't
seem right
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\[ =\int_0^t \sin\omega u \; \cos \omega (t-u)du \\ =\int_0^t \sin\omega u \; (\cos \omega t \cos \omega u - \sin \omega t \sin \omega u)du \\ = \int_0^t \sin\omega u \; \cos \omega t \;\cos \
omega u \;du - \int_0^t \sin \omega u \;\sin \omega t \; \sin \omega u \;du \]
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\[ =\int_0^t \sin\omega u \; \cos \omega (t-u)du \\ =\int_0^t \sin\omega u \; (\cos \omega t \cos \omega u - \sin \omega t \sin \omega u)du \\ = \cos \omega t \; \int_0^t \sin\omega u \;\cos \
omega u \;du - \sin \omega t \;\int_0^t \sin \omega u \; \sin \omega u \;du \]
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How to calculate a Fredholm index numerically
up vote 3 down vote favorite
How can one calculate the index of a Fredholm operator numerically ?
In numerically calculations one uses always finte dimensional spaces. But linear operators on finite dimensional spaces have always index zero.
fa.functional-analysis na.numerical-analysis
3 If you happen to be lucky enough for the operator to be an (ordinary) differential operator, e.g. $Ly = y' - A(x)y$, and you are even more lucky so that the limits $A(\pm \infty)$ exist and have
spectrum away from the imaginary axis, then the Fredholm index of L is the difference between the Morse index of A(\infty) and the Morse index of A(-\infty). – Aaron Hoffman Aug 24 '11 at 20:03
Are you thinking of operators on particular (function) spaces, or a Fredholm operator in full generality? – Yemon Choi Aug 24 '11 at 20:50
1 A linear operator between two DISTINCT finite-dimensional vector spaces does not have index zero, and that may help. – Alain Valette Aug 25 '11 at 7:27
@Aaron: of course, assuming $A(x)$ is a path of operators on a finite dimensional space. – Pietro Majer Aug 25 '11 at 7:29
Alain ... but it has non-zero index for a TRIVIAL reason! – Helge Aug 25 '11 at 11:06
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1 Answer
active oldest votes
The two key properties of the Fredholm index are
• It is a (norm)-continuous function from the bounded linear operators to the integers. In particular, if $A$ is a Fredholm operator, then there exists $\delta > 0$ such that for $\
|A - B\| < \delta$, we have $index(A) = index(B)$. This tells you that you can approximate your problem.
• The Fredholm index doesn't see compact perturbations. So if $A$ is Fredholm and $K$ is compact, then $index(A +K ) = index(A)$. This tells you that you cannot do naive
computations like picking some finite orthonormal set $\psi_{j}$ with $j=1,\dots,N$ and hope that the $N \times N$ matrix $$ A_{j,k} = \langle \psi_j, A \psi_k\rangle $$ tells you
up vote 2 down anything about the Fredholm index of $A$.
vote accepted
So you will now need to do something smarter. This is possible in many particular cases, for example for Toeplitz operators. The first property allows one to reduce the computation of
the index to the computation of the winding number of a polynomial. Or the Atiyah--Singer index theorems reduces computing the index to some topological information ...
So to get a more meaningful answer, you will need to be more specific about the problem.
For more refined invariance theorems one the Fredholm index check Hoermander's The Analysis of Linear Partial Differential Operators. – Pietro Majer Aug 25 '11 at 7:32
Which volume? There are 4 if I remember correctly... – Helge Aug 25 '11 at 11:06
It's Vol III, Chapter 19 (Elliptic Operators on a Compact Manifold Without Boundary), Sec.19.1 Abstract Fredholm Theory. – Pietro Majer Aug 25 '11 at 12:07
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Figure 1: Numerical simulation of how superfluid turbulence forms during the strongly nonequilibrium kinetics of Bose-Einstein condensation governed by the Gross-Pitaevskii equation. The panels show
the formation of well-structured topological defects from a structureless weakly turbulent state, and subsequent decay of the vortex tangle (i.e., superfluid turbulence) with increasing time from
left to right. The state in the central panel is close to what Henn et al. observe in their experiment (adapted from Ref. [5]).
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My neighbour Paul Dirac
The first Dirac House
By Hamish Johnston
Did he live here for 21 years?
The salmon-coloured house in the centre of the photo is 15 Monk Road, the birthplace of Paul Dirac.
This view of suburban Bristol is from our back window and would have been somewhat different in 1902 when Dirac was born. The houses had been built a year earlier and I’m guessing the gardens would
have been devoid of trees and large shrubs.
Dirac lived in this house until he was about ten (according to his latest biographer Graham Farmelo or 21 — according to the historical plaque on the house (right).
I sometimes wonder what the current owners make of this plaque — are they worried that Dirac left some antimatter lurking under the floorboards?
In the April issue of Physics World the Bristol physicist Sir John Enderby reviews Graham Farmelo’s book The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius.
I asked Sir John about the discrepancy between Farmelo and the blue plaque and he isn’t sure which account is correct. He’s doing a little digging now, so stay tuned for an update.
Farmelo says that the Diracs moved to nearby Julius Rd (photo below) in 1912, so it is fairly certain that Paul spent his formative years in the same Bristol neighbourhood.
Indeed, Dirac went to nearby Bishop Road School , which also counts the actor Cary Grant as one of its former pupils. And if you continue along Bishop Rd, turn left at Gloucester Rd and then right on
Ashley Down Rd you will reach the new Dirac Rd
One thing I can tell you about 15 Monk Road is that it has a tiny garden (much smaller than the garden in the foreground of the photo) so there wouldn’t have been much room for young Paul to play.
However, the road would have been devoid of cars so he could have been tearing around out there — although after reading Sir John’s review I can’t imagine young Paul tearing around anywhere.
I have to admit a strange kinship to the Diracs. Like Paul’s father I am a foreigner (he was Swiss, I am Canadian); we are also bringing up three children in an identical house; and we are thinking
of moving in the general direction of Julius Rd — the gardens are bigger over there!
Julius Rd in Bristol: did Paul Dirac move here in 1912?
9 comments
1. I’m afraid the information on the plaque is incorrect. The month in which the Dirac family moved to Julius Road (from memory, April 1913) is recorded in Charles Dirac’s nationalisation papers,
stored in the National Archives. It took me years to pin down this date! By the way, the Julius Road address is used in family correspondence long before 1923.
Graham Farmelo
2. I wonder how aware are the general public of PAM Dirac. A few years back I was visiting Westminster Abbey with my son, and since I hadn’t been there after the plaque of Dirac had been placed, I
asked the guide where it was. At first he didn’t know what I was talking about. After some thought he pointed the small plaque on the floor.
3. As I say in the book, Dirac was – according the convention – Swiss until he was about 17. So there is a case for saying that he was partly Swiss but, in my opinion, the case is weak. He was born
and bred an English person, for a start. Perhaps more important, he certainly did not consider himself Swiss as he disliked his Swiss father so much. After a childhood visit, Paul Dirac did not
consent to visit Switzerland until he visited CERN when he was in his 70s, as I recall.
4. Dirac was a unique Black Swan!
Back in 1974 I remember writing from Burma to Dirac then at Florida State University, Tallahasee requesting for his papers on Large Numbers Hypothesis which he sent me through his secretary in a
very short period of a few weeks. I also received his 1942/43 publications on Quantum Electrodynamics from the Dublin Institute of Advanced Studies .
Many physicists of his stature would simply not have time for a relatively unknown university physicist from Burma! I think Dirac did care about other people but he cherished his freedom so very
much that he would not like anything to interfere with it!
During the war years he did a good deal of consulting work for both Peierl’s group at Birmingham and Simon’s group at Oxford(specifically on atomic bomb design and development and also on the
general problem of isotope separation).
According to Eugene Wigner a fellow physicist and Dirac’s brother-in-law “….During the (second world) war Paul, as were many others, was greatly concerned not only about the future of England,
but also of the whole world, of freedom,democracy and a diversity of cultures. When Hitler attacked the western part of Europe, and in particular also England, he volunteered to do scientific and
technical work for the defence of his country and the freedom of the world. What is very memorable in this connection is that he considered this to be his duty and refused all compensation for
his work. ..the special subject he worked on..seems to have been a secret-but the fact that he did it so unselfishly, as a gift to freedom, is worth remembering. And the Free World as we now have
some of it has a proud recollection of Paul Adrien Maurice Dirac. His wife is also justly proud of it. So are many of us!”
I was simply delighted to hear and read the reviews and comments –many good ones – about a new book on Dirac’s life and work! There exists a previous biography of Dirac which I have read but was
not too pleased with it- a feeling shared by the members of the Dirac family particularly Dirac’s wife, Wigner’s sister. I am particularly pleased that a theoretical physicist has written it!
I still need to read Farmelo’s book! It is still being ordered!
I realize that there also exists “Collected Works of Dirac” . All the same I tend to agree with Chandrashekhar, that a book on the work of such a creative scientist should cover all his life’s
work! Silence should not shroud his post 1948 work.
Many would like to say that Dirac’s last important paper was that on magnetic monopole that he wrote he wrote in 1948 Physical Review in October. But according to Freeman Dyson, the memorable
1948 issue of Physical Review (vol 73) contains a number of “wonderful papers on physics ” ; Bloembergen,Purcell and Pound on relaxation effects in nuclear magnetic absorption;Lewis,Oppenheimer
and Wouthuysen on the multiple production of mesons; Foley and Kusch on the experimental discovery of the magnetic moment of the electron;J Schwinger on the theoretical explanation of the
anomalous magnetic moment and Dirac’s paper on quantum theory of localizable dynamical systems. Dyson said that “Dirac’s is the only one concerned with quantum field theory”
Many believe Dyson did for quantum field theory in 1948/1950 what Dirac did for quantum mechanics in the late 1925/1926 so he should be in a better position to make judgement on the importance of
Dirac’s paper!
In 1950’s in his search for a better QED he developed the Hamiltonian theory of constraints (Cand J Math 1950 1 129; 1951 2 1) based on lectures that he delivered at the 1949 International
Mathematical Congress in Canada. In his second paper using Hamiltonian methods he derived the Tomonaga-Schwinger equation for mesons in Schroedinger representation.
In the late 50’s he applied the Hamiltonian methods he had developed to cast Einstein’s general relativity in Hamiltonian form (Proc Roy Soc 1958,A vol 246, 333,Phys Rev 1959,vol 114, 924) and to
bring to a technical completion the quantization problem of gravitation according to Salam and DeWitt. In 1959 also he gave an invited talk on “Energy of the Gravitational Field” at the New York
Meeting of the American Physical Society later published in 1959 Phys Rev Lett vol 2, 368.
In 1964 he published his “Lectures on Quantum Mechanics” which deals with constrained dynamics of nonlinear dynamical systems including quantization of curved spacetime. He also published a paper
entitled “Quantization of the Gravitational Field” in 1967 ICTP/IAEA Symposium on Contemporary Physics.
In 1961 he apparently found from his old notes a rather a novel method of deriving the Schwinger term and the Lamb shift (1056.17MHz)without using the usual “joining technique” of utilizing
Bethe’s non-relativistic result adopted by a number of workers-including Weisskopf and French, Feynman and Schwinger-in quantum electrodynamics.
This work is also based on his theory of constrained dynamics. He also gave a series of lectures on quantum electrodynamics in 1962/1963 at the Belfer Graduate School of Science, Yeshiva
University and at Ban-Ilan University Israel, in 1965. He again gave an invited talk on quantum electrodynamics at the New York Meeting of the American Physical Society in January 1965 which was
later published in 1965 Phys Rev vol 139, 684.
In 1962 Dirac put forward the idea that the elementary particles might correspond to modes of vibrating membrane (Proc Roy Soc A 268, 57; Proc Roy Soc A 270, 354 ).Within the context of the
string theory, the membrane idea could not be revived. In 1986, however, Hughes,Liu and Polchinski showed that a super-membrane could be introduced by combining membrane and supersymmetry.
In 1963 he published a paper entitled “A Remarkable Representation of 3+2 deSitter Group” in J Math Phys vol 4, 901 which was followed by two related papers on “Positive Energy Wave Equation I &
II ” in the 1971 Proc Roy Soc A 322, 435 and 1972 Proc Roy Soc A 328, 1 respectively. In the meantime he had also published in 1971 “Spinors in Hilbert Space” in which he showed amongst other
things that starting with fermion variables one can end up with boson variables particularly when dealing with creation and annihilayion operators in Hilbert space.
He put down his ideas of formulating Einstein’s general relativity in a form of a slim book entitled “General Theory of Relativity” published in 1974 soon after his formulation of a scalar
–tensor theory of gravity using Weyl’s geometry as opposed to pseudo-Riemann geometry(“ Long Range Forces and Broken Symmetry” 1973 Proc Roy Soc A 403, 333). In his book on general relativity he
deals with polarization of gravitational waves and localization of gravitational energy making use of harmonic coordinates. He also introduced a metric now known as Dirac-Lemaitre metric to
discuss black holes and singularities. The slim book carries as usual the Dirac stamp of originality.
I believe typical “ Black Swans” like Schwinger and Einstein had the “not unusual” difficulties with the publication of their later research works . Schwinger apparently resigned from the
Fellowship of the American Physical Society when Phys Rev refused to publish his later research relating to “source theory”. Although Dirac would certainly qualify as a “Black Swan” and despite
his rather overdue and late election to the Honorary Fellowship of the Institute of Physics in 1971 in contrast to his very early election in 1948 to the Honorary Fellowship of the American
Physical , the apparent ease with which he managed to publish his post 1948 research work in research journals, books and proceedings is quite unique – and not at all strange-amongst “Black
5. When I was an undergrad at the U.of Florida in 1975, I had the pleasure of meeting Dirac after his colloquium, and coercing him into autographing my QM text.
He was of course quite old by then, and until now, the youngest picture I had ever seen of him was in his 30′s. Farmelo’s book has unveiled a stunning, but grainy photo of a very young Dirac, ~
age 7 I’m guessing. I am curious why there seems to be this 25+ year gap in photographs of him ?
One would think the U.of Bristol would have photos of him in his early 20′s…
6. Hi, just to say does anyone know which number Julius Road he lived in? I lived at Julius Road for 20 + years, the houses are big but the gardens are not especially huge.
□ Annie,
According to Farmelo’s book, Dirac lived at Six Julius Road.
7. Normally I compile guides to contemporary art, but I was inspired by Graham Farmelo’s gripping book to make a guide and Google map to the rather few Dirac-related spots in Bristol: places he
lived and studied, plus memorials and places named after him. In researching it I came across Hamish Johnston’s fascinating blog post (ie this) and the associated learned comments, which is
recommended on the blog. It’s only a small guide, but I hope some people may find it useful if planning a Dirac pilgrimage: it’s at http://artanorak.tumblr.com/post/1167260633/dirac
8. Pretty! This was a really wonderful article.
Thanks for supplying this info.
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CRC polynomal calculation
On Wed, 28 Apr 2004 18:29:29 GMT, Kai Harrekilde-Petersen
<(E-Mail Removed)> wrote:
>"Runar Gjelsvik" <(E-Mail Removed)> writes:
>> I was looking at this site posted somewhere earlier:
>> The thing I'm wondering is how do you calculate what polynom to use? I'm
>> planning on serially send a data frame of about 72 bits. I was thinking of
>> using manchester encoding and CRC.
>The polynomials are selected from a pool of mathematically test
>polynomials. In your case, I'd pick one of the wellknown ones,
>e.g. the CRC32 used in Ethernet (and a lot of other places).
>The longer CRC that you use, the better coverage (error detection) you
Good advice so far.
>In general, the probability for an undetected single-bit error is
>(2^N)-1:1 for an N bit CRC. This requires that the length of the frame
>you send is less than 2^(N-1)*N bit, IIRC.
Any CRC will detect a single bit error in a frame of any length.
Even parity (equiv to 1 bit CRC) will detect a single bit error in
arbitrarily long frames.
I listed some CRC error detection properties here:
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Number of results: 27,413
What are some ethical issues that could surface in the business world when using linear optimization techniques
Tuesday, June 15, 2010 at 11:45pm by Hickey
Optimization - Calculus
Thanks both of you.
Tuesday, December 11, 2007 at 7:41pm by Anonymous
Calculus optimization problem
Sunday, October 30, 2011 at 4:40pm by sean
Calculus 12 Optimization
Thursday, May 19, 2011 at 9:10pm by sierra
Calculus 12 Optimization
Thursday, May 19, 2011 at 9:10pm by Anonymous
Calculus Optimization
did you mean your equation to be y = -x^2/2e + e^x ??
Tuesday, January 18, 2011 at 5:05pm by Reiny
Calculus (Optimization)
I mean length + 4x
Friday, December 16, 2011 at 6:23pm by Damon
Calculus Optimization Problem
I agree. You are welcome.
Wednesday, March 6, 2013 at 1:10pm by Dr. Jane
Calculus - Optimization
Good catch, Steve. Thank you.
Saturday, November 16, 2013 at 4:24pm by MathMate
Calculus (Optimization)
Now I'm lost, I don't get why you changed the signs.
Friday, December 16, 2011 at 8:20pm by Mishaka
Optimization - Calculus
Yes, and take the derivative and set to zero for the min distance.
Tuesday, December 11, 2007 at 7:41pm by bobpursley
Optimization - Calculus
Yes, and take the derivative and set to zero for the min distance.
Tuesday, December 11, 2007 at 7:41pm by bobpursley
i am struggling with the concept of optimization. does anyone have any hints on how to solve these problems???
Wednesday, December 10, 2008 at 3:54pm by Hannah
Calculus (Optimization)
I dont see how you got the signs as you did. Please recheck
Friday, December 16, 2011 at 8:20pm by bobpursley
Calculus (Optimization)
Ok, your equation is right. Recheck your final signs as I stated.
Friday, December 16, 2011 at 8:20pm by bobpursley
Applied Calculus
I had to miss my last week of Calculus due to some personal things, so I had not been through any lectures of optimization. It hurt me greatly, now I have a few of these that I don't know how to set
up at all.
Wednesday, March 13, 2013 at 5:04pm by Jacob
Calculus Optimization Problem
Thank you! I solved it out, and I got x=5 and y= 10 with a product of 500. Is this correct
Wednesday, March 6, 2013 at 1:10pm by Mary
Optimization - Calculus
You forgot the squares: d = [(x - 3)^2 + y^2]1/2 It is easier to minimize d^2: d^2 = (x-3)^2 + y^2 Insert in here y^2 = x+1 and set the derivative of d^2 w.r.t. x equal to zero.
Tuesday, December 11, 2007 at 7:41pm by Count Iblis
calculus optimization
i mean find the dmensions of a drum that has a volume of 10 cubic feet and minizes the total cost
Tuesday, April 6, 2010 at 7:20pm by MILEY
Find the point on the graph of y=2x-4 that is closest to the point (1,3). (Optimization equation)
Monday, December 6, 2010 at 7:28pm by Michelle
Calculus I
section is on Optimization: Find the point on the curve y = x^2 closest to the point (3, 4)
Saturday, April 14, 2012 at 11:11am by Sandra Gibson
i get a derivcative of 24w^2+24w^-1 is that correct?
Wednesday, November 23, 2011 at 5:34pm by Kay
Calculus (Optimization)
hang on, I reread the problem statement. In the first response I gave, I took your equation. I dont think it is right. give me a minute.
Friday, December 16, 2011 at 8:20pm by bobpursley
find the derivatives, then plug it in to the orginal questioin.then use the length to find the area as it is an optimization question
Monday, January 16, 2012 at 12:27am by saphire
Calculus (Optimization)
Reading the question again, I think that I took the wrong interpretation and Damon took the right one.
Friday, December 16, 2011 at 6:23pm by Reiny
Calculus (Optimization)
Nevermind, that 4.42 was a mistake and my very original answer of 1.105940354 was absolutely correct!!! This is the right answer, I know it!
Friday, December 16, 2011 at 8:20pm by Mishaka
I don't understand how to solve optimization problems, (like here's the volume of a box, find the least amount of material it would take to make such a box). Is there a tutorial or some general step
by step instruction on how to do these? Thanks in advance, Amy :) http://...
Sunday, April 1, 2007 at 8:10am by Amy
Calculus optimization problem
Check the related question. I think you'll find that this problem has been answered many times, using different numbers.
Sunday, October 30, 2011 at 4:40pm by Steve
Calculus (Optimization)
Okay, so does this change my original answer of approximately 1.64 to 4.42??? The 4.42 came from putting your new values in the quadratic equation.
Friday, December 16, 2011 at 8:20pm by Mishaka
optimization find the point on the graph of the function that is closest to the given point f(X)= square root of x point:(8,0)
Sunday, December 9, 2012 at 1:12am by Anonymous
AP Calculus
A cardboard box of 108in cubed volume with a square base and no top constructed. Find the minimum area of the cardboard needed. (Optimization)
Sunday, October 31, 2010 at 5:22pm by Anonymous
You might try some of the following links: http://search.yahoo.com/search?fr=mcafee&p=ethical+issues+in+the+business+world+using+linear+optimization+techniques Sra
Tuesday, June 15, 2010 at 11:45pm by SraJMcGin
calculus optimization max min
That is a problem you do when refreshed, and have some time. Here it is worked given the triangle vertexes. You are given two point, and the altidude. For your upper vertex, x,y, choose it such that
the altitude (y) is 4. http://www.analyzemath.com/calculus/Problems/...
Tuesday, October 19, 2010 at 2:48pm by bobpursley
Optimization - Calculus
Find the point closest to the line sqroot(X+1) from the point (3,0). d = [(x - 3) + (y - 0)]^1/2 d = [(x - 3) + (y)]1/2 Do I now substitute in the equation y = sqroot(X+1) and solve?
Tuesday, December 11, 2007 at 7:41pm by Anonymous
Calculus - Optimization
Let L=length, then girth=84-L=2*radius=2R, or R=(84-L)/2 Volume, V = πR²L Express V in terms of L using R=(84-L)/2 Equate dV/dL=0 and solve for L.
Thursday, November 24, 2011 at 1:53am by MathMate
Calculus Optimization Problem
You need to substitute y = 15-x x(15-x)^2 x(225 -30x+x^2) 225x -30x^2 + x^3 Now you can take the derivative and set it equal to zero.
Wednesday, March 6, 2013 at 1:10pm by Dr. Jane
Math - Calculus I
Optimization Problem: Find the dimensions of the right circular cylinder of greatest volume inscribed in a right circular cone of radius 10" and height 24"
Thursday, December 5, 2013 at 9:36pm by Alex
Calculus 12 Optimization
A farmer wishes to make two rectangular enclosures with no fence along the river and a 10m opening for a tractor to enter. If 1034 m of fence is available, what will the dimension of each enclosure
be for their areas to be a maximum?
Thursday, May 19, 2011 at 9:10pm by K.lee
Calculus (Optimization)
The U.S. Post Office will accept a box for shipment only if the sum of the length and girth (distance around) is at most 108 inches. Find the dimensions of the largest acceptable box with square
Friday, December 16, 2011 at 6:23pm by Mishaka
Calculus Optimization
A model space shuttle is propelled into the air and is described by the equation y=-x^2/2e + ex (in 1000 ft), where y is its height above the ground. What is the maximum height that the shuttle
Tuesday, January 18, 2011 at 6:18pm by jennifer
calculus test corrections. 1 question
Review your material on linear optimization. The max or min value of f(x,y) is found to be at the vertices of the region defined by the constraints. Here, the region is a triangle, with vertices at
(0,0) (2,0) (0,4) f(0,0) = 1 f(2,0) = -1 f(0,4) = 5
Tuesday, April 10, 2012 at 7:48pm by Steve
Calculus Optimization
A model space shuttle is propelled into the air and is described by the equation y=(-x2/2e)+ex in 1000 ft, where y is its height in feet above the ground. What is the maximum height that the shuttle
Tuesday, January 18, 2011 at 5:05pm by jennifer
Calculus-Applied Optimization Problem
If a total of 1900 square centimeters of material is to be used to make a box with a square base and an open top, find the largest possible volume of such a box.
Thursday, October 31, 2013 at 5:09pm by Ashley
Calculus - Optimization
The cost of fuel for a boat is one half the cube of the speed on knots plus 216/hour. Find the most economical speed for the boat if it goes on a 500 nautical mile trip.
Wednesday, March 20, 2013 at 6:17pm by Sam
Calculus-Applied Optimization Problem:
Find the point on the line 6x + 3y-3 =0 which is closest to the point (3,1). Note: Your answer should be a point in the xy-plane, and as such will be of the form (x-coordinate,y-coordinate)
Wednesday, October 30, 2013 at 12:42pm by Sara
Calculus-Applied Optimization Quiz Problem
A rancher wants to fence in a rectangular area of 23000 square feet in a field and then divide the region in half with a fence down the middle parallel to one side. What is the smallest length of
fencing that will be required to do this?
Thursday, October 31, 2013 at 6:38pm by Riley
Calculus (Optimization)
Let me do some thinking... if 0=V' = 112 - 88x - 12x^2 multipy both sides by -1, and 12x^2+88x-112=0 I dont see those as your signs....
Friday, December 16, 2011 at 8:20pm by bobpursley
Optimization An offshore oil well is 2km off the coast. The refinery is 4 km down the coast. Laying a pipe in the ocean is twice as expensive as on land. What path should the pipe follow in order to
minimize the cost?
Sunday, November 6, 2011 at 1:07am by lele
Calculus (Optimization)
Both of you, thank you very much!!! I arrived at the correct answer width = 18 and length = 36, but I just got that answer by chance and wasn't sure how I could prove (mathematically) that it was
indeed correct, your explanations helped tremendously!
Friday, December 16, 2011 at 6:23pm by Mishaka
A 100 inch piece of wire is divided into 2 pieces and each piece is bent into a square. How should this be done in order of minimize the sum of the areas of the 2 squares? a) express the sum of the
areas of the squares in terms of the lengths of x and y of the 2 pieces b) what...
Tuesday, November 16, 2010 at 9:17pm by katie
AP Calculus
The sum of the two bases and the altitude of a trapezoid is 16ft. a) Define the area A of the trapezoid as function of its altitude. b) Find the altitude for which the trapezoid has the largest
possible are. (Optimization)
Sunday, October 31, 2010 at 5:26pm by Anonymous
optimization calculus
a net enclosurefor practisinggolf shots is open at one end, as shown, find the dimensions that will minimize the amount of netting needed and give a volume of 144 m^3(netting is required only the
sides, the top, the far end.)
Tuesday, March 8, 2011 at 1:01pm by Anonymous
Calculus - Optimization
but an 8x8x8 box has length+girth = 8+32 = 40 inches, so it will not work. We need to optimize s^2(24-4s) since a square has 4 sides. v = 24s^2 - 4s^3 v' = 48s - 12s^2 v'=0 when s=4 So, a 4x4x8 box
has max volume. Do (B) similarly
Saturday, November 16, 2013 at 4:24pm by Steve
calculus optimization max min
find the dimensions of the rectangular area of maximum area which can be laid out within a triangle of base 12 and altitude 4 if one side of the rectangle lies on the base of the triangle thanks
Tuesday, October 19, 2010 at 2:48pm by Oswaldo
Calculus I Quick Optimization Problem
Could you please explain this problem step by step, thank you! You are planning to make an open rectangular box that will hold a volume of 50 cubed feet. What are the dimensions of the box with
minimum surface area?
Tuesday, January 3, 2012 at 10:17pm by Lisa
Calculus (Optimization)
You are lost. This is algebra. if 0=112 - 88x - 12x^2 do whatever you know to put it in standard form, ax^2+bx+c=0 when you do that a and b will have the SAME signs. Surely you can do that. if a=-12,
then b=-88, and c=112 if a=12, then b=88, and c=-112
Friday, December 16, 2011 at 8:20pm by bobpursley
calculus optimization problem
L + 2 W = 460 so L = (460 - 2 W) L * W = A A = (460-2W)W = 460 W - 2 W^2 DA/DW = 460 - 4 W ZERO FOR MAX OR MIN 4 W = 460 W = 115 L = 460 - 2*115 = 230
Saturday, March 30, 2013 at 1:22am by Damon
optimization calculus
sandy is making a closed rectangular jewwellery box with a square base from two different woods . the wood for top and bottom costs $20/m^2. the wood for the side costs $30/m^2 . find dimensions that
minimize cost of wood for a volume 4000cm^3?
Thursday, March 10, 2011 at 12:07pm by Anonymous
Calculus (Optimization, Still Need Help)
I just wanted to correct something for my equation, it should be: V = (14 - 2x)(8 - 3x)(x), which simplifies to V = 112x - 44x^2 - 4x^3. Take the derivative: V' = 112 - 88x - 12x^2 Now all I need are
the roots, any help? I think I found one around 1.10594, possibly?
Friday, December 16, 2011 at 8:20pm by Mishaka
Calculus (Optimization)
I rechecked and found that 3x^2-22x+28 has the correct signs. Knowing this equation and the values I found from the quadratic equation, would you say that the 1.639079157 term is correct? (The 2.69
square inches came from squaring the 1.639079157).
Friday, December 16, 2011 at 8:20pm by Mishaka
Calculus Optimization Problem
Find two positive numbers whose sum is 15 such that the product of the first and the square of the second is maximal. I came up with this so far: x + y = 15 xy^2 is the maximum derivative of xy^2=
2xyy' + y^2 Now how do I solve this ^ after I set it to zero? I am stuck on that...
Wednesday, March 6, 2013 at 1:10pm by Mary
Calculus I Quick Optimization Problem
For symmetry reasons, the optimum must have a square base. Let the height of the walls be x and the side length be y Volume = x^2*y = 50 Area = x^2 +4xy = x^2 + 4x*50/x^2 = x^2 + 200/x dA/dx = 0 = 2x
-200/x^2 x^3 = 100 x = 4.64 ft y = 2.32 ft
Tuesday, January 3, 2012 at 10:17pm by drwls
calculus optimization problem
by cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. if the cardboard is 30 inches long and 14 inches
wide find the dimensions of the box that will yield the maximum volume.
Wednesday, April 3, 2013 at 3:22am by sasha
Calculus - Optimization
Cost=.10*(pi*r^2+2Pi*r*h)+.20PIr^2 volume= PIr^2h or h= volume/PIr^2 h=3000/(PIr^2) Put that into the cost function for h. Then take the derivative of cost with respect to r (dCost/dr), set equal to
zero, solve for r.
Sunday, June 3, 2012 at 7:55pm by bobpursley
calculus optimization max min
The height of the rectangle can be anything from 0 to 4. Call it h. The width (4) of the rectangle varies linearly from 12 to 0, with w = 12 (1 - h/4)= 12 - 3h Area = f(h) = h*w = 12h - 3h^2 dA/dx =
0 when 6h = 12 h = 2; w = 12 - 6 = 6 Amax = 12 The triangle does not have to ...
Tuesday, October 19, 2010 at 2:48pm by drwls
Calculus-Applied Optimization Problem
N(x) = 100-(x-425)/9 R(x) = x*N(x) = x(100-(x-425)/9) = 1/9 (1325x-x^2) dR/dx = 1/9 (1325-2x) dR/dx=0 at x=1325/2 = $662.50
Thursday, October 31, 2013 at 9:27pm by Steve
Calculus (Optimization)
set V'=0, and you have a quadratic. Why not use the quadratic equation.. 12x^2+88x-112=0 3x^2+22x-28=0 x= (-22+-sqrt (22^2+12*28))/6 doing it in my head, I get about.. (-22+-28)/6= 4/6, 7.5 in my
head. check my work and estimates.
Friday, December 16, 2011 at 8:20pm by bobpursley
Explain the global optimization process for a continuous function over a closed interval. Be sure to identify all steps involved and clearly explain how the derivative is utilized in this process.
Does this have to do with the first derivative rule or second derivative rule ...
Saturday, October 25, 2008 at 2:40pm by George
Explain the global optimization process for a continuous function over a closed interval. Be sure to identify all steps involved and clearly explain how the derivative is utilized in this process.
Does this have to do with the first derivative rule or second derivative rule ...
Monday, October 27, 2008 at 9:58am by George
Calculus-Applied Optimization Problem
maximize xyz subject to xy + 2xz + 2yz = 1900 If it were a complete cube, max volume is where all faces are square. In this case, missing the top, the area must be divided 1/3 to base and 2/3 to
sides. So, x=y=√(1900/3) and z=x/2 The box is 25.1661 x 25.1661 x 12.5831
Thursday, October 31, 2013 at 5:09pm by Steve
calculus optimization
a company manufactures large cylindrical drums.the bottom and sides are made from a metal that costs $4.00 a square foot, while the reinforced lid costs $6.00 a square foot. ind thedmensions ofa drm
that hasa volume of 10cubic feet and minizes the total cost
Tuesday, April 6, 2010 at 7:20pm by MILEY
Calculus (Optimization)
I think that you might have gotten the equation wrong, I think that it should be: 3x^2 - 22x + 28. When I put this equation into the quadratic equation, I got 5.694254177 and 1.639079157. So the
squares that need to be cut out should have an area of approximately 2.69 square ...
Friday, December 16, 2011 at 8:20pm by Mishaka
Saturday, October 3, 2009 at 10:55pm by Anonymous
Calculus (optimization problem)
A cyclinderical tank with no top is to be built from stainless steel with a copper bottom. The tank is to have a volume of 5ð m^3. if the price of copper is five times the price of stainless steel,
what should be the dimensions of the tank so that the cost is a minimum?
Wednesday, March 24, 2010 at 5:06pm by Joey
calculus optimization problem
A farmer has 460 feet of fencing with which to enclose a rectangular grazing pen next to a barn. The farmer will use the barn as one side of the pen, and will use the fencing for the other three
sides. find the dimension of the pen with the maximum area?
Saturday, March 30, 2013 at 1:22am by lori
calculus (optimization)
xy=384 so, y = 384/x f = 2x+3y = 2x + 1152/x df/dx = 2 - 1152/x^2 min f is when df/dx=0, at x=24 So, the field is 24x16 f = 48+48=96 as usual in these problems, the fencing is divided equally among
lengths and widths.
Saturday, November 30, 2013 at 2:54am by Steve
Calculus (Global Max)
Explain the global optimization process for a continuous function over a closed interval. Be sure to identify all steps involved and clearly explain how the derivative is utilized in this process.
Does this have to do with the first derivative rule or second derivative rule ...
Saturday, October 25, 2008 at 12:37pm by George
Calculus - Optimization
A fence is to be built to enclose a rectangular area of 800 square feet. The fence along 3 sides is to be made of material $4 per foot. The material for the fourth side costs $12 per foot. Find the
dimensions of the rectangle that will allow for the most economical fence to be...
Sunday, November 17, 2013 at 3:38pm by Jess
Calculus - Optimization
Let the size of the square (cross-section) be s. Then we need to maximize V=s²(24-2s) with respect to s. First find the derivative and equate to zero: dV/ds = 48s-6s²=0 means s=0 or s=8 s=0
corresponds to a minimum volume and s=8 corresponds to a maximum volume. So ...
Saturday, November 16, 2013 at 4:24pm by MathMate
calculus (optimization)
a rectangular study area is to be enclosed by a fence and divided into two equal parts, with the fence running along the division parallel to one of the sides. if the total area is 384 square feet,
find the dimensions of the study area that will minimize the total length of ...
Saturday, November 30, 2013 at 2:54am by yareli
Math OPTIMIZATION
hey thanks! but where did you get 16 in b?
Saturday, September 3, 2011 at 1:47am by Willoby
Mathematics optimization
Saturday, September 3, 2011 at 7:33pm by Mgraph
Thank you so much! I appreciate it!
Monday, November 21, 2011 at 8:47pm by Maria
This is a optimization problem !!
Friday, October 19, 2012 at 5:42pm by some one
Basic Calculus-Optimization Problems
Since both x and y have to be positive numbers, you simply have to find for what values 200 = 4x+3y lies in the first quadrant. Find the x and y intercepts let x = 0 , y = 200/3 let y = 0 , x = 50 so
0 < x < 50 0 < y < 200/3
Friday, March 16, 2012 at 1:55am by Reiny
Calculus-Applied Optimization Quiz Problem
If the length and width are x,y, then we want to minimize f = 3x+2y subject to xy = 23000 so, y=23000/x, and we want the minimum of f = 3x+2(23000/x) df/dx = 3 - 46000/x^2 df/dx =0 at x = 20/3 √345)
f = 40√345
Thursday, October 31, 2013 at 6:38pm by Steve
i almost thought this was an optimization problem
Monday, February 15, 2010 at 7:04pm by mike
Related rates
i'm sorry, this is about optimization problems.
Sunday, May 20, 2012 at 9:02pm by :)
what is sub optimization need an example of it to
Sunday, November 4, 2012 at 12:40pm by at
Calculus - Optimization
if the expensive side is x and the other dimension is y, then the cost c is c = 4(x+2y) + 12x But, we know the area is xy=800, so y = 800/x and the cost is now c = 4(x+1600/x) + 12x minimum cost when
dc/dx=0, so we need dc/dx = -16(400-x^2)/x^2 dc/dx=0 when x=20, so the fence ...
Sunday, November 17, 2013 at 3:38pm by Steve
Optimization At 1:00 PM ship A is 30 miles due south of ship B and is sailing north at a rate of 15mph. If ship B is sailing due west at a rate of 10mph, at what time will the distance between the
two ships be minimal? will the come within 18 miles of each other? The answer is...
Tuesday, December 22, 2009 at 9:20pm by Jake
This is pretty easy. area sides=2L*m+2Wm area bottom= LW area lid= LW Volume=lwm But l=2w volume=2w^2 m or m= 4/w^2 costfunction= 4*basearea+8(toparea+sides) Now, write that cost function in terms of
w (substitute) take the derivatative. with respect to w, set to zero,and ...
Wednesday, November 23, 2011 at 5:34pm by bobpursley
Optimization (Math)
I understand now, thanks a lot steve!
Tuesday, November 1, 2011 at 12:08pm by Tommy
Given y=(x)^1/2, find the closest point to (3/2,0)
Saturday, December 15, 2012 at 9:28pm by Daryl
Calculus - Optimization
A cylindrical container with a volume of 3000 cm^3 is constructed from two types of material. The side and bottom of the container cost $0.10/cm^2 and the top of the container costs $0.20/cm^2. a)
Determine the radius and height that will minimize the cost. b) Determine the ...
Sunday, June 3, 2012 at 7:55pm by Nevin
Calculus (Optimization)
v = vol = x^2 y girth = 4 x length = y so y + 4x </=108 since maximizing y + 4x = 108 or y = 108 - 4x v = x^2 (108-4x) v = 108 x^2 - 4 x^3 dv/dx = 216 x - 12 x^2 = 0 for max or min so x(216 - 12x) =
0 x = 18 for max then y = 108 -4(18) = 36
Friday, December 16, 2011 at 6:23pm by Damon
Calculus (optimization problem)
Volume = V = pi r^2 h = constant so pi r^2 = V/h and r =(V/[pi h])^.5 and cost = 5 pi r^2 + 2 pi r h cost = 5 V/h + 2 pi r h cost = 5 V/h + 2 pi (V/pi)^.5 h^.5 d cost/dh = -5 V/h^2 + 2 pi (V/pi)^.5
(.5)(h^-.5) 0 when 5 V/h^2 = pi^.5 V^.5 h^-.5 h^1.5 = 5 V^.5/pi^.5 h^3 = 25 V/pi
Wednesday, March 24, 2010 at 5:06pm by Damon
Calculus (Optimization)
A rectangular piece of cardboard, 8 inches by 14 inches, is used to make an open top box by cutting out a small square from each corner and bending up the sides. What size square should be cut from
each corner for the box to have the maximum volume? So far I have: V = (14 - 2x...
Friday, December 16, 2011 at 8:20pm by Mishaka
What is YOUR answer? Have you researched the definition for each of the terms here? You might try some of the following links: http://search.yahoo.com/search?fr=mcafee&p=
ethical+issues+in+the+business+world+using+linear+optimization+techniques (I would have said a Civil Rights...
Wednesday, June 16, 2010 at 9:16am by SraJMcGin
Calculus - Optimization
UBC parcel post regulations states that packages must have length plus girth of no more than 84 inches. Find the dimension of the cylindrical package of greatest volume that is mailable by parcel
post. What is the greatest volume? Make a sketch to indicate your variables. I ...
Thursday, November 24, 2011 at 1:53am by Ass11
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Performance Analysis of MIMO-STBC Systems with Higher Coding Rate Using Adaptive Semiblind Channel Estimation Scheme
The Scientific World Journal
Volume 2014 (2014), Article ID 304901, 17 pages
Research Article
Performance Analysis of MIMO-STBC Systems with Higher Coding Rate Using Adaptive Semiblind Channel Estimation Scheme
Department of Electronics & Communication Engineering, Jaypee University of Engineering & Technology, A.B. Road, Raghogarh, Guna, Madhya Pradesh 473226, India
Received 23 August 2013; Accepted 11 October 2013; Published 12 February 2014
Academic Editors: T. Guo and S. Savazzi
Copyright © 2014 Ravi Kumar and Rajiv Saxena. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Semiblind channel estimation method provides the best trade-off in terms of bandwidth overhead, computational complexity and latency. The result after using multiple input multiple output (MIMO)
systems shows higher data rate and longer transmit range without any requirement for additional bandwidth or transmit power. This paper presents the detailed analysis of diversity coding techniques
using MIMO antenna systems. Different space time block codes (STBCs) schemes have been explored and analyzed with the proposed higher code rate. STBCs with higher code rates have been simulated for
different modulation schemes using MATLAB environment and the simulated results have been compared in the semiblind environment which shows the improvement even in highly correlated antenna arrays
and is found very close to the condition when channel state information (CSI) is known to the channel.
1. Introduction
The need for speed, reliability, and quality in the wireless digital communication has lured the interest of vast community of technologist and communication forums to have a look at the
multiple-input multiple-output (MIMO) technology that has proved a vital role in satisfying their needs [1]. Nowadays wireless networks are utilizing many techniques such as wireless local area
network (WLAN), wireless sensor networks (WSN), and personal area network (PAN) which demands more spectrum and makes it costlier and precious resource. This technology has comprehensive uses like in
worldwide interoperability for Microwave Access, providing a wireless alternative to cable and the digital subscriber line (DSL) for the leased line last mile access without compromising the speed as
the MIMO provides high data rate. This has attracted the attention of the researchers towards the use of multiple antennas at both transmitter and receiver which has been known as MIMO antenna array
systems. MIMO nowadays has been widely used for enhancing the channel capacity using multiple antenna configurations in both 3G and 4G technologies adopting the influence over the wireless local area
network (WLAN) as the IEEE 802.11 standard which has reached the acquired level as high as 600Mbps [2]. The advantages are not only bound to the communication field but they are also injected in
the medical field in sensing of the cardiopulmonary activity [3]. In MIMO systems, the data rate may be enhanced by using spatial multiplexing whereas the reliability may be enhanced by using space
time coding (STC). Space time block codes (STBCs) have been implemented in MIMO systems for increasing the diversity gain or coding gain by coding across multiple antennas over multiple symbol
durations. Initially, STBC was analyzed by [4], which further was modified by [5] using maximum likelihood (ML) technique. Each antenna element on a MIMO system operates on the same frequency and
therefore requires no extra bandwidth and it also requires less power than that required by a single antenna. Advantage of MIMO systems has been taken as Beamforming, spatial diversity and spatial
multiplexing [6, 7].
Transmit diversity has been studied extensively for combating the fading channels because of its simple implementation using multiple antenna at the transmitter end. The first bandwidth efficient
transmit diversity scheme was proposed by [8] with the delay diversity scheme proposed by [9] which further was modified by [10] giving multilayered space time architecture. Alamouti introduced the
well-known STBC in [11] for two transmit and one receive antennas. STBC consists of data coded with the space and time to improve the reliability of the transmission. Later, [12, 13] introduced
orthogonal space time block coding which generalizes Alamouti’s transmission scheme to an arbitrary number of transmit antennas and is able to achieve the full diversity promised by the multiple
transmit and receive antennas. Like Alamouti scheme, these generalized codes have a very simple maximum likelihood decoding algorithm based only on linear processing at the receiver. Space time block
coding generalizes the transmission and arbitrary number of transmit antennas and is able to achieve full diversity with respect transmit and receive antenna. It is a set of practical signal design
techniques to which approaches the information theoretic capacity limits of MIMO systems. In last few years, some of the techniques have been evolved like orthogonal space time block coding [5, 14],
quasiorthogonal space time block coding (QOSTBC) [15, 16], and nonorthogonal STBC (NOSTBC) [17]. Generally all the MIMO-STBC systems require space time equalizer at the receiver end for decoding
purpose which also requires the channel state information (CSI) which is basically obtained through training based techniques at the expense of bandwidth efficiency. These training based techniques
are also in research for utilizing minimum bandwidth and increasing robustness to detect the signal with minimum number of transmitted training symbols for which the semiblind techniques are being
utilized. Pilot assisted semiblind technique has also evolved in [18], which has been successfully applied to MIMO systems for obtaining remarkable enhancement. The performance comparison among
various designs of pilot assignments using different kinds of modulation schemes and interpolation techniques for frequency offset estimation was proposed by [19, 20]. Some of the techniques do not
require CSI [21–24] and pay the penalty in performance of at least 3dB as compared to coherent maximum-likelihood (ML) receivers. The drawback of completely blind channel estimation is the inability
to detect the signals appropriately from the channel to the receiver. The desired signal can be easily identified and extracted with the help of some training sequences. The aim of identifying the
signal is to recognize the signal with strength with as much less training symbols as possible, for which the semiblind channel estimation technique has become the prominent technique these days.
Blind channel estimation algorithms based on ML technique have been proposed by [25, 26]. Iterative methods have been utilized to avoid the computational complexity of the ML detection technique, the
cyclic ML [26, 27], and the expectation maximization (EM) [27, 28] algorithms. These iterative methods require a careful initialization of the channel or symbol in case otherwise poor initializing
can strongly affect the symbol-error rate (SER). However, excluding some specific low rate codes, different approaches fail to extract the channels in a full blind manner whereas these can be
implemented using the semiblind channel estimation technique. Several approaches have been suggested in the literature to solve this problem, including the transmission of the short training sequence
[29, 30] or the use of precoders [31–33]. However these semiblind techniques cannot be employed in a noncooperative scenario since they need to modify the transmitter, which has been implemented
utilizing the adaptive pilot assisted channel estimation scheme (APACE) proposed by [18] and by some modification at the precoder and decoder in the proposed scheme, based on which the analysis has
been shown in [34] for the STBC with code rate up to 1. Despite the literature available universally, none of the algorithms is able to estimate the channel matrix for general STBCs without
modification of the pilot training symbols and precoding. In this paper, a new approach has been proposed to implement the MIMO-STBC system with the implementation of the proposed adaptive pilot
assisted semiblind channel estimation (APASBCE) scheme [18]and modifying the code rate to some higher levels. This paper describes the brief extract on MIMO systems, with respect to channel capacity,
system model, and channel models including the focus on spatial diversity, that is, STBC.
2. Space Time Block Coding (STBC)
2.1. STBC for Real Constellations
Considering transmission matrix with variables satisfying [35, 36] where is a constant and is a identity matrix, STBC can achieve a full diversity of order of 1. Square STBC matrix with real
constellation can be found if and only if the number of transmit antennas is , or . These codes offer full transmit diversity of due to their full rate . The real transmission matrices for 2, 4, and
8 transmit antennas are given by At the receiver end, the received expressions are based on Alamouti’s model with the simplicity of having only real symbols and therefore no conjugate symbol in the
equations. Thus the received expressions for any number of received antennas become where , are independent noise samples, is the channel transfer function from the th transmit antenna and denotes
the receive antenna. Received signals are then combined for two transmit antennas as Similarly, the received signal for four and eight transmit antennas can also be derived. Alamouti STBC do not
require CSI at the transmitter and can be used with two transmit antennas and 1 receive antenna with accomplishment of full diversity of 2. It reduces the effect of fading at receiver station at the
cost of some additional antenna elements at the transmitter end. If having more antennas is not a problem, then this scheme is appropriate for getting full diversity of with two transmit antennas.
2.2. STBC for Complex Constellation
For STBC with complex constellation, if transmission matrix with variables satisfies [35, 36] where is a constant and is a identity matrix, STBC can achieve full diversity of the order of 1. The full
rank diversity that was introduced by Alamouti is considered the simplest STBC with complex constellation and it is also the only STBC code with complex constellation, which is the only STBC
achieving full rate of 1 for a full diversity of 2. For the case of 3 transmit antennas, [4] made block codes with 1/2 and 3/4 code rates and full diversity 3 . The aim of using higher number of
transmit antennas on generalized STBC is to achieve high rate with full diversity, minimum coding delay , and minimum decoding complexity. Examples of half rate complex transmission matrices
achieving full diversity for three and four transmit antennas are given as where denotes the complex conjugate of the element. The matrix code transmits 4 symbols every 8 time intervals and therefore
has rate 1/2. For both schemes, flat fading channel are assumed to be constant over 8 symbol periods. Thus the received constellation derivations and the received signals can be formed as in (4) and
then combined to retrieve the original transmitted symbols using maximum likelihood detection to minimize the decision metric which can also be formed for four transmit antenna. If three transmit
antennas are considered and, three symbols are transmitted every four time intervals, and therefore has code rate 3/4. Example of 3/4 code rate complex transmission matrix for three transmit antennas
is given as It is known that the complexity at the receiver end increases linearly with the number of transmit antennas and the receive antennas. Indeed, for receiving antennas, the expression of
matrix will have times more terms than that it has now. Performance of STBC for complex constellation matrices of 1bit/s/Hz, 2bits/s/Hz, and 4bits/s/Hz for two transmit antennas and 1/2bit/s/Hz,
1bit/s/Hz, and 2bit/s/Hz for three and four transmit antennas has already been analyzed.
2.3. Orthogonal Space Time Block Codes
As shown earlier, examples of 1/2 and 3/4 code rate complex transmission matrices for four transmit antennas have been proposed by [36] which gave full diversity of . With four transmit antennas and
code rate of 1/2 and 3/4, complex transmission matrices have been given as
2.4. Quasi-Orthogonal Space Time Block Codes
Full rate STBCs, using complex symbols in their transmission matrix, are not possible to achieve as we have seen in previous section. Indeed, the particular case of Alamouti code presented can only
achieve full rate with full diversity which follows the rules of orthogonal design for simple decoding. The new STBC technique called quasiorthogonal STBC (QOSTBC) is proposed by [15], which achieved
full rate at the cost of higher complexity decoding. Quasiorthogonal designs are attractive because of their achievement of higher code-rate than orthogonal designs and lower decoding complexity than
nonorthogonal designs. As suggested in [15], Now, , , is defined as the th column of ; it is easy to see that , where is the inner product of vectors and . Therefore, the subspace created by and is
orthogonal to the subspace created by and . This orthogonality allows the calculation of the maximum likelihood decision metric. Indeed the maximum likelihood detection is to find the pair () and ()
that minimizes over all possible values of () and minimizes over all the possible values of () pairs. It seems that the complexity of the decoder increases as compared to the STBC decoder presented
earlier. However, complexity of QOSTBC does not grow linearly as for STBC but exponentially with the number of transmit and receive antennas. Similarly, the QOSTBC code with different rate and higher
number of transmit antennas has also been proposed.
3. System Model
Consider an quasistatic Rayleigh flat fading MIMO channel, where and denote the number of transmit and receive antennas. The system is described by , where is which is the transmitted symbol vector
of transmitter, denotes the received vector , and is the complex valued Gaussian white noise vector at the receiving end for MIMO channels with energy distributed according to assumed to be zero
mean, white, and independent of both channel and data fades. The channel model considered here is denoted by [37] with and representing the normalized transmit and receive correlation matrices with
identity matrix. The entries of are independent and identically distributed (...) (0, 1). A system block diagram using Alamouti’s method is shown in Figure 1. The transmitting symbols are encoded
according to orthogonal STBC scheme. A pilot sequence is inserted in the transmission of every symbol which will be reduced with the implementation of the proposed adaptive semiblind estimation
scheme in [18]. A different pilot scheme has been used for each channel and these orthogonal pilot sequences enable the receiver to decouple pilot sequences from the combined signals for each channel
at a receive antenna. The transmitted symbols have been considered having empty slots left in their codeword matrix for maintaining the orthogonality between the symbols of the vector.
Assuming the block transmission scheme with block length , the th received data block can be expressed as where Here, is defined as where the denotes the transpose in the second exponential term. If
a slow fading environment is considered, the time becomes much longer than the data block length . The matrix can be treated as a mapping transforming the th block to complex matrix of transmit
signals, where is the th symbol vector alphabet set of length , that is, set of all possible symbol vectors. The matrix is called an OSTBC if all elements of this matrix are linear functions of the
complex variables and their complex conjugates. The calculation of the basis function of OSTBC can be denoted by where where and denote the real and imaginary parts. It is known that OSTBC is
completely defined by its basis matrices . If the channel frequency offset is not available, then (10) can be rewritten in vectorized form and the real valued matrix can be denoted by, . The matrix
follows the decoupling property; that is, its columns have identical norms and are orthogonal to each other. where denotes the Frobenius norm of a matrix. follows the basis matrices, and is referred
to as a time varying OSTBC [34].
4. Design Condition and Decoding Method
It has been found that denotes an OSTBC for transmit antennas which transmit information symbols with having empty slots left in its codeword matrix for orthogonality; we then found information
symbols transmitting high code rate with full diversity from as where is the codeword matrix with additional information symbols to be transmitted from empty slots of and is the optimization
matrix-wise entries that are with both and being nonoverlapping entries. Owing to the nonorthogonal structure of the information symbols, as unknown deterministic parameters, it is required to apply
ML estimation approach to jointly estimate both the symbols and pilots. To obtain the ML estimates of all these parameters, the log-likelihood function needs to be maximized. Hence the parameter
estimates can be found by solving the following optimization problem: where is the likelihood function computed for snapshots and is the set of all possible values of the transmitted symbols
received. It is not easy to solve (17) because its computational cost grows exponentially in . To simplify the optimization problem in (17), we have to maximize the expectations and minimize the
error in the estimates for which Now, the elimination of terms coming from additional transmitted symbols from empty slots of will be tried by computing intermediate signals from the received signals
for all possible values of the additional symbols in as and the optimization problem in (17) can be rewritten as The likelihood function for any can be expressed as where denotes the statistical
expectation. Taking into account that all are independent random vectors, the obtained value is given as Using (21) and (22), the problem in (20), can be formulated as shown in (18) and can also be
written as It is known in (23) that the th term of the sum is minimized with where (24) follows from the fact that satisfies the decoupling property that has been discussed earlier in (15). Using
this equation, the objective function in (23) can be concentrated with respect to and, after such concentration, the latter optimization problem can be shown as This function can further be solved in
a simple manner and found with the existence of traces of matrix as Now, with the little replacements in the expression for convenience, equation can be denoted by where is matrix whose th column can
be defined as , where is the th coloumn of the identity matrix and is the Kronecker matrix product. Now, putting (27) in (26), the concentrated optimization problem can be denoted by where The above
expression (29) is real matrix which depends on the received data vectors and the carrier frequency offset . Further, this can be solved and the carrier frequency offset can be derived using (28):
where denotes the largest eigenvalues of matrix. And further for the estimates of channel one has where denotes the normalized principal eigenvector of a matrix with the assumption of no multiplicity
in the largest eigenvalues of . Now for those specific OSTBCs that result in with multiple largest eigenvalues, h belongs to the subspace spanned by the corresponding multiple principal eigenvectors
of , and, as a result, the blind technique is not applicable using this method of detection. Hence, the semiblind technique proposed in [18] will be utilized which uses the small number of training
symbols both in time and frequency axis adaptively and decoded at the receiver end according to the requirement of the channel as shown in [34]. Using this method, it searches for all the possible
combinations of and we use the decoding procedure of that is used to obtain conditional estimates to get the weight vectors in (18) of [18]. An adaptive method of increasing pilot symbols in the
empty slots has been proposed and implemented in the same and then the robust estimation method has been found for getting the correct combination of . Finally, we minimize the decision metric in (18
) for is minimized over all possible values of . This method of estimating and detecting is somewhat similar to ML detection technique and therefore the total decoding complexity of is obtained. Now,
it is known that belongs to the subspace spanned by , where and . The proposed semiblind channel estimation scheme has been utilized to obtain the estimate of in a blind way and meanwhile estimating
the vector using the training symbols as low as possible. It is known that the number of entries in is much less than that in , and this semiblind estimator will require very less training data than
the direct training based channel estimator obtaining all entries of in a nonblind way. For this purpose, it is required to estimate the value of and take short time average of the detected estimates
and then further process it to give the branch metric which then further will proceed for giving the selected estimates with minimum branch metric which gives the minimum surviving states with
minimum value from the and eventually the possible block of transmitted sequence. The ML estimate for the STBC system of the vector can be written as This further will give where . This estimate can
be used to obtain the coefficients from these few training symbols to resolve the ambiguity in the channel vector estimate. To ensure that the ML estimate in (34) is unique, it is required that and,
for known nonidentifiable OSTBCs, holds true and therefore, as , the condition is satisfied for any number of receive antennas for which it is required to have code rate of STBC that should be higher
than 1 which will be further designed in the next section [34].
5. High Code Rate Design Method
In order to achieve energy efficient STBC codes with high code rates, it is required to construct the , a rotated version of the complex lattice with source information , where is a complex unitary
matrix, so that there is no shaping loss in the signal constellation emitted by the transmitting antenna as shown in [38].
For any given and column groups of the matrix and being the block length, then . Assuming that is even, the higher code rate STBC will be designed as where the real and imaginary matrices and of size
are given as where and are real and imaginary parts of , where , that is given as where and the th diagonal layer from left to right written as vector is given as The symbol rate of the STBC code is
given as which is the same as that of STBC decoding proposed in [39, 40]. For a large value of , the code rate can be up to and similarly for large elements on transmitting side, that is, , the code
rate can be up to . For the design of STBC with odd antenna elements, it is supposed to design an STBC for transmit antennas with the last antenna to be shut down; that is, when the is odd, the STBC
is obtained by selection of first columns of the STBC designed for antennas.
6. Proposed Code Designs
6.1. For Three Antenna Elements
In this section, new STBC code with code rates of 1.5 and 2 has been achieved for three transmit antennas with the use of the design procedure shown in the previous section. According to (16), using
the design method as shown in previous section, for the transmitting six symbols using three antennas, that is, code rate 2 can be found using the value of as with the optimization matrix, where and
denote the real and imaginary matrices for the 6 symbols per 3 transmit antennae with code rate 2. This matrix has been derived from the optimization matrix of , where is the identity matrix. After
continuous simulation search, maximum coding rate of has been found by sacrifice of some constellation angle for the optimum value of 65.49°, which gave minimum determinant value of 0.15 for the QPSK
modulation technique.
Now, secondly, the real and imaginary matrices for code rate 1.5 which transmits six information symbols per three time intervals are obtained as with the optimization matrix Maximum coding rate of
has been found by making constellation angle for the optimum value of 44.96°, which gave minimum determinant value of 0.32 for the QPSK modulation technique. Hence in this, it can be easily seen that
the complexity has been reduced upto for and for .
6.2. For Four Antenna Elements
In this section, another STBC code with higher code rates of 1.3, 1.5, and 2 has been achieved for four transmit antennas with the use of the design procedure shown in the earlier section. According
to (16), using the design method as shown in previous section, for the transmitting six symbols using four antennas, that is, code rate 1.3 can be found using the value of as Maximum coding rate of
has been found by making constellation angle for the optimum value of 1.04°, which gave minimum determinant value with the QAM modulation technique.
With the QPSK modulation technique, for code rate 2, we found that the following matrix is formed as with optimization matrix given as Maximum coding rate was found for the code rate of 2, with the
angle of QPSK signal constellation with symbols on the two-axis rotation to achieve full diversity.
Similarly, for the code rate 1.5 with the four antennas, using QPSK modulation rotation at can be obtained by making which resulted as with the optimization matrix
6.3. Decoding and Estimation Method
A decoding method for four antenna elements is being described here, in which the receiver calculates the intercepted received signals from the channel using (19) for all the combinations of to
obtain the ML estimates of . Therefore for the given values of , and which can only be obtained with the help of reduced form of (19), It is required to minimize the decision metric obtained with the
help of (18) for all possible values and obtained conditional ML estimates of which need additional decoding complexity of per each step of calculations. Therefore we get a total decoding complexity
of . The receiver follows the decoding procedure of , and it is observed that which is component of is calculated from which is component of ; here and are the column and row of the corresponding
matrix. The receiver combines the received intercepted signals to obtain These received estimates can now be utilized for the estimation of ML estimates for , where manipulated for estimating higher
code rate estimates: where and denote the real and imaginary parts of STBC codes , , and .
It is observed that a total decoding complexity of rather than by minimizing (18) for all the possible values as discussed earlier has been achieved.
Now, for the implementation of the estimated symbols with the semiblind channel estimation, it is required to deploy the scheme proposed in [41], although it can also be implemented with [18], but
the procedure to estimate has been refined in the second method by modifying precoder and decoder at the transmitter and receiver side implementing the same estimation technique used in [18]. It can
be seen in (29) of [41] that the weight vectors are not sufficient to estimate the symbols correctly in the semiblind environment with partial CSI conditions and an adaptive estimation method was
tried as shown in [18] for getting optimal linear minimum mean square error (MMSE) estimate for the channel path gain at the th symbols period where the weighting coefficients explicitly depend on
the symbol position. For each , can be obtained by solving the adaptive method as discussed in [18] which provides the unknown estimate sequence to avoid sacrifice of tracking ability of channel.
These estimated symbols can be used to obtain the coefficients from these few training symbols to resolve the ambiguity in the channel vector estimate. The minimum path metric with its short time
average of long detected sequence was detected which has then been utilized to calculate the minimum branch metric for all possible estimated vectors for tracking surviving state with minimum value
of channel coefficients . Then these metrics are utilized to update the weight vectors in (29) of [41] of th spatial equalizer at each step with increase in processing steps . Reference [41] has
discussed the capacity analysis of the proposed semiblind channel estimation scheme with modified precoder and decoder. In this paper, Bit Error Rate performance analysis has also been taken care of.
A comparative chart has been shown in Table 1 for showing the used number of antennas for different schemes and their symbol transmission rate with different coding rates.
7. Results Analysis and Conclusion
Performance analysis and improvement observed in the MIMO systems using different antenna configurations utilizing STBC using different modulation schemes with the implementation of Adaptive pilot
assisted semiblind channel estimation scheme for the partial CSI condition proposed earlier in [18] have been shown in this section. It is known that the performance for the different STBC coding
schemes degrades when more bits per symbol are transmitted, but we have simulated up to 9bps/Hz with higher code rate STBCs which has shown relatively good results. For the general simulation case
for known channel models, it is obtained that the best performance is obtained by using higher number of transmitting and receiving antenna elements. However, for any modulation case with low SNR
values, three-ransmitting-antenna STBC system with code rate 1/2 gives better results than the four-antenna-element system STBC with code rate 3/4 even though the gain for the said is higher. When
the simulation was tried with more numbers of antenna elements at the transmitting side with code rate 1/2, they gave better results than the 3/4 code rate type STBC systems. The possible reason for
this is that the higher rate of four-transmitting-antenna element system causes lower channel gain per symbol and therefore BER for particular SNR. If we consider equal data rates, and simulate the
16-QAM scheme and 64-QAM modulation scheme for the code rate that is, 1/2 and 3/4, for three and four antenna element systems, it is easily observable that the and with code rate 3/4 using 16-QAM
(4bits/symbol) gave the same data rate, as given by and with code rate 1/2 using 64-QAM (6bits/symbols). Hence we decided to show the comparative analysis of QPSK and 16-QAM modulation schemes for
different antenna configurations with maintaining the correlation coefficient of 0.5.
In this section, we have evaluated the BER performance and the received constellation comparisons for different modulation schemes using STBC with different code rates, for their constellation angle
maintaining the appropriate modulation for achieving the exact code rate and diversity. Also the capacity comparison is shown for the improvement seen with different antenna configurations with
different channels. Throughout the simulations, the noise variance has been considered between the 3dB and 20dB level for different scenarios.
The comparative result analysis has been shown in Table 2 for different STBCs with different transmitting antenna configurations with their respective code rates. The APASBCE scheme has been
implemented with the Alamouti’s model using QPSK and 16-QAM modulation technique for antenna configuration using STBC found in (6) with code rate 1 and diversity order 1. In Figure 2, the improvement
for antenna configuration with QPSK has been observed after the 17.4dB SNR level and 12dB SNR level for 16-QAM. It is seen that the semiblind result gave better result than the existing results
available in the literature as the number of iterations reaches up to the level when the symbols are easily identifiable at the receiver end. Figures 3 and 4 show that code rate 1/2 is performing
better than code rate 3/4 for both and antenna configurations, as discussed earlier in this section, but Figure 4 shows significant improvement in the BER between the simulated semiblind results as
compared with Figure 3. This happens because of the increase in number of antenna elements; as the number of elements increases, the symbol rate increases, and further by utilizing the proper code
rate STBC, the BER may be enhanced upto some extent which has been shown in Figure 4. Similarly, Figure 5 is showing the comparison of OSTBC for QPSK modulation for antenna configuration with
proposed estimation scheme with the block codes found in (44) with code rate 1 and diversity 1 in which the improvement has been observed after 18.8dB SNR level.
In Figure 6, comparisons for QOSTBC and OSTBC using antenna configuration with proposed scheme has been done using in (9) and and in (44) using QPSK and 16-QAM modulation scheme for code rates 1 and
1.3, respectively, with same bit rate of 8Bps/Hz. The improvements in this figure has been observed after 14.2dB SNR level for 16-QAM and after 11.4dB for QPSK modulation scheme but before
reaching the BER level of which is the advantage in this category. Similarly, Figure 7 is showing the improvement of OSTBC comparison for antenna configuration using QPSK modulation with APASBCE
scheme After 11.8dB SNR level for 8Bps/Hz bit rate and after 16.7dB for 9Bps/Hz bit rate before reaching the BER level of with the existing results for code rate 1.3 using block codes found in
and of (44). As discussed earlier, the higher rate of transmitting antenna element system causes lower channel gain per symbol and therefore BER for particular SNR value it is seen in this figure
that low rate, that is, 8Bps/Hz, for QPSK modulation is performing better than the higher rate, that is, 9Bps/Hz. It is observed in Figure 8, for QPSK modulation, that the system is able to
maintain 1.2dB gain at the level of 25dB of SNR, for the STBC with code rate 2, whereas in case of 1.5 code rate, STBC simulations, as depicted in Figure 9, were able to maintain less amount of
gain nearly of 0.8dB but at the SNR level of 20dB. Secondly, for 16-QAM modulation, as evident in Figures 8 and 9, the effect of maintaining gain is not of the same quantum, and maintains the gains
of 2.8dB and 2dB for APASBCE based STBC with code rates 2 and 1.5, respectively, at the higher level of SNR that is, 25dB, and 30dB. The decrease in maintaining less SNR gain has occurred because
of the loss of training symbols at the receiver end for which the algorithm again started to track the symbols, and after finding the sufficient amount of training symbols, it kept maintaining the
gain in SNR levels again.
The received constellation for different antenna configurations, with different modulation schemes utilizing the APASBCE technique, has been shown in further figures and their result analysis has
been shown in Table 3. These received constellations show how much rotation is required for achieving the particular value of code rate, required for these STBC techniques to modulate through the
channel using APASBCE scheme and to receive the ISI free symbols perfectively at the receiver. Figures 10(a) and 10(b) have been simulated for both the 16-QAM and QPSK whereas remaining figures from
Figures 11–13 has been simulated for QPSK modulation only with their required rotation angle with different antenna configurations.
It is also observable that the received signal improves with the increase in the number of antenna elements at the receiving end. We have used QPSK and 16-QAM modulation schemes using the gray
constellation mapping for the comparative study of the existing results available in literature with the proposed APASBCE [18] scheme results for these mentioned modulation schemes. The capacity
analysis and the improvement have already been discussed in [41] for the APASBCE based scheme using the existing MIMO systems available in the literature. Comparative study of the capacity
improvement has been shown in Figure 14 using (48), (49), and (52) in [41], where the analysis has been done using , , , and antenna systems for different channel numbers and obtained the improvement
with the increase in the number of partial CSI channels.
Figures 14(a) and 14(b) show the improved results of APASBCE based capacity which shows that, for antenna systems with 8 channels, the proposed system started to enhance the capacity at SNR level of
19.3dB. Similarly for antenna systems with 4 channels and 8 channels, it gave the improved enhancement of capacity at the SNR level of 17.9dB, and 17.2dB respectively. It also shows that the
capacity improvement is related to the increase in the number of the channels. Again, for antenna systems with 8 channels, the improvement started at the level of 16.4dB, whereas for antenna systems
with 4 channels, it shows the improvement after 19.2dB. For antenna systems, the improvement in capacity has been found initially at the level of SNR 12.8dB but then decreases and again improved
after the level of 15.7dB which then maintained its improvement after 20.6dB. This variation was caused because of the fading effect and channel estimation adaptively using APASBCE method for
stabilizing the channel state information.
Conclusively, this paper shows the comparison of the existing 16-QAM and QPSK modulation schemes for different code rates with the result of STBC with code rate higher than 1 using different STBC
techniques with the new improved results found using the proposed APASBCE scheme, as discussed earlier, which shows the better BER result and improved capacity with less number of used training
symbols and increasing the stability of the system by utilizing the minimum required number of training symbols. Also the constellation rotation for required angle has also been discussed for
different higher code rate values for both 16-QAM and QPSK modulation schemes. These high code rate STBCs have been obtained and analyzed with improved results and the quantitative improvement has
been discussed in this section.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The author thankfully acknowledges the support provided by the authorities and management of Jaypee University of Engineering & Technology, Guna, India.
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How (and when) to factor a function defined on a product of metric spaces?
up vote 1 down vote favorite
Suppose we have a set of regular functions defined on a product of metric spaces, for instance the Banach space of the smooth functions from $\mathbb R^n$ to $\mathbb C$. We know, thanks to the
Taylor series, that such a function can 'almost' be written as a linear combination of products $f_1 \cdot \ldots \cdot f_n$, each of the $f_i$ a continuous function from $\mathbb R$ to $\mathbb C$,
where the almost means we are using a density argument in the sup norm.
Is there a way to generalize such a situation to a generic product of metric spaces? In particular I am interested in the space of locally constant, compactly supported functions on powers of the
$p$-adic field $\mathbb Q_p$ but also the general question sounds interesting. Any reference is greatly appreciated, thanks.
gn.general-topology banach-spaces reference-request
I can not think of a proof for $\mathbb{R}\times \mathbb{R}$ which can not be generalized directly to product of two metric spaces. – Anton Petrunin Jul 2 '12 at 14:28
The proof of Taylor in $\mathbb R \times \mathbb R$ that I know goes like this: Let $f : \mathbb R \times \mathbb R \to \mathbb R$ be given. Let $a\in \mathbb R \times \mathbb R$, and suppose $f$
is sufficiently regular at $a$. Let $h \in \mathbb R \times \mathbb R$ be small enough. Consider the function of one variable $g(t) = f(a+th)$. Apply the one-dimensional Taylor theorem to $g$. –
Gerald Edgar Jul 2 '12 at 15:12
The Taylor series of a function doesn't always converge to it in the sup norm even in one dimension (take $e^x$). What exactly is the precise statement you want? – Qiaochu Yuan Jul 2 '12 at 15:28
A reasonable construction (for Banach space-valued functions defined on a product of metric spaces) is via partitions of unity. Given a continuous function $f:X\times Y\rightarrow (E, \|\cdot\|)$,
and $\epsilon >0$, there is a uniform approximation $\epsilon$-close to $f$ of the form $f_\epsilon(x,y):=\sum_{i,j} f(x_i,y_j)\phi_i(x)\psi_j(y)$. – Pietro Majer Jul 2 '12 at 17:59
You may look at topological tensor products which yield representations of function spaces on products like $C(K \times L) = C(K) \tilde{\otimes}_\epsilon C(L)$ or $C^\infty (\Omega_1 \times \
Omega_2)= C^\infty (\Omega_1)\tilde{\otimes} C^\infty (\Omega_2)$ (in the latter case you can choose any tensor topology due to nuclearity). – Jochen Wengenroth Jul 3 '12 at 7:35
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1 Answer
active oldest votes
Is this the statement you want: any locally constant, compactly supported function from ${\bf Q}_p^n$ to ${\bf C}$ is uniformly approximated by linear combinations of products $f_1\cdots
f_n$ where each $f_i$ is a locally constant, compactly supported function from ${\bf Q}_p$ to ${\bf C}$? Yes, this follows from the Stone-Weierstrass theorem for locally compact
up vote 2 Hausdorff spaces. The linear combinations in question constitute a self-adjoint algebra of functions which separate points and separate each point from infinity. Therefore they are
down vote uniformly dense in the space of continuous functions vanishing at infinity on ${\bf Q}_p^n$.
This is exactly what I was looking for, thanks a lot. – Niccolo' Jul 3 '12 at 10:34
You're welcome! – Nik Weaver Jul 3 '12 at 13:49
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A first order difference equation
March 5th 2010, 08:58 PM #1
Mar 2010
A first order difference equation
I'm reading a chapter on first order difference equations in which they use the example of population growth given by the model:
$y_{n}=\rho^{n}y_{0}+(1+\rho+\rho^{2}+\cdot\cdot\cd ot+\rho^{n-1})b$
Where $\rho$ is the reproduction rate and $b$ is the rate of immigration.
They then go on to assume that if $\rhoeq1$ they can simplify the equation to the compact form:
but I'm completely lost on the jump between the two. If anyone could shed some light on this it would be much appreciated.
I'm reading a chapter on first order difference equations in which they use the example of population growth given by the model:
$y_{n}=\rho^{n}y_{0}+(1+\rho+\rho^{2}+\cdot\cdot\cd ot+\rho^{n-1})b$
Where $\rho$ is the reproduction rate and $b$ is the rate of immigration.
They then go on to assume that if $\rhoeq1$ they can simplify the equation to the compact form:
but I'm completely lost on the jump between the two. If anyone could shed some light on this it would be much appreciated.
It is the sum of a geometric series
$\sum_{n=0}^{k}r^{n}=\frac{1-r^{k+1}}{1-r}, |r|<1$
Geometric series - Wikipedia, the free encyclopedia
Thanks...I actually figured it out right after I posted it and felt really dumb.
I guess that's what you get for not doing any math for 5 years and then cracking open a diff eq book.
March 5th 2010, 09:05 PM #2
March 5th 2010, 09:09 PM #3
Mar 2010
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[Numpy-discussion] array vs matrix
Anne Archibald peridot.faceted@gmail....
Sat Jun 7 20:54:44 CDT 2008
2008/6/7 Robert Kern <robert.kern@gmail.com>:
> On Sat, Jun 7, 2008 at 14:37, Ondrej Certik <ondrej@certik.cz> wrote:
>> Hi,
>> what is the current plan with array and matrix with regard of calculating
>> sin(A)
>> ? I.e. elementwise vs sin(A) = Q*sin(D)*Q^T? Is the current approach
>> (elementwise for array and Q*sin(D)*Q^T for matrix) the way to go?
> I don't believe we intend to make numpy.matrix any more featureful. I
> don't think it's a good idea for you to base sympy.Matrix's
> capabilities in lock-step with numpy.matrix, though. There are very
> different constraints at work. Please, do what you think is best for
> sympy.Matrix.
Let me elaborate somewhat:
We recently ran across some painful quirks in numpy's handling of
matrices, and they spawned massive discussion. As it stands now, there
is significant interest in reimplementing the matrix object from
scratch, with different behaviour. So emulating its current behaviour
is not a win.
For consistency, it makes a certain amount of sense to have sin(A)
compute a matrix sine, since A**n computes a matrix power. But looking
at the matrix exponential, I see that we have several implementations,
none of which is satisfactory for all matrices. I would expect the
matrix sine to be similar - particularly when faced with complex
matrices - so perhaps needing an explicit matrix sine is a good thing?
Also worth noting is that this problem can be evaded with namespaces;
matrix sin could be scipy.matrixmath.sin, abbreviated perhaps to
mm.sin, as opposed to np.sin.
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division algorithm for integers
division algorithm for integers
Given any two integers $a,b$ where $b>0$, there exists a unique pair of integers $q,r$ such that $a=qb+r$ and $0\leq r<b$. $q$ is called the quotient of $a$ and $b$, and $r$ is the remainder.
The division algorithm is not an algorithm at all but rather a theorem. Its name probably derives from the fact that it was first proved by showing that an algorithm to calculate the quotient of two
integers yields this result.
Mathematics Subject Classification
no label found
Added: 2001-11-17 - 01:36
Attached Articles
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Summary: Homogeneous Semantics Preserving
Deployments of Heterogeneous Networks of
Embedded Systems
Aaron D. Ames, Alberto Sangiovanni-Vincentelli and Shankar Sastry
Center for Hybrid and Embedded Software Systems
Department of Electrical Engineering and Computer Sciences
University of California at Berkeley
Summary. Tagged systems provide a denotational semantics for embedded sys-
tems. A heterogeneous network of embedded systems can be modeled mathemati-
cally by a network of tagged systems. Taking the heterogeneous composition of this
network results in a single, homogeneous, tagged system. The question this paper
addresses is: when is semantics (behavior) preserved by composition? To answer this
question, we use the framework of category theory to reason about heterogeneous
system composition and derive results that are as general as possible. In particular,
we define the category of tagged systems, demonstrate that a network of tagged
systems corresponds to a diagram in this category and prove that taking the com-
position of a network of tagged systems is equivalent to taking the limit of this
diagram--thus composition is endowed with a universal property. Using this univer-
sality, we are able to derive verifiable necessary and sufficient conditions on when
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Finding connected components of adjacency matrix graph
up vote 9 down vote favorite
I have a random graph represented by an adjacency matrix in Java, how can I find the connected components (sub-graphs) within this graph?
I have found BFS and DFS but not sure they are suitable, nor could I work out how to implement them for an adjacency matrix.
Any ideas?
algorithm graph matrix graph-theory
How is the adjacency matrix stored? Additional trickery can be used for some data formats. – harold Nov 14 '11 at 16:28
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2 Answers
active oldest votes
You need to allocate marks - int array of length n, where n is the number of vertex in graph and fill it with zeros. Then:
1) For BFS do the following:
Components = 0;
Enumerate all vertices, if for vertex number i, marks[i] == 0 then
Put this vertex into queue, and
while queue is not empty,
pop vertex v from q
marks[v] = Components;
Put all adjacent vertices with marks equal to zero into queue.
2) For DFS do the following.
Components = 0;
Enumerate all vertices, if for vertex number i, marks[i] == 0 then
Call DFS(i, Components), where DFS is
DFS(vertex, Components)
marks[vertex] = Components;
Enumerate all vertices adjacent to vertex and
for all vertex j for which marks[j] == 0
call DFS(j, Components);
After performing any of this procedures, Components will have number of connected components, and for each vertex i, marks[i] will represent index of connected component i belongs.
Both complete on O(n) time, using O(n) memory, where n is matrix size. But I suggest you BFS as far as it doesn't suffer from stack overflow problem, and it doesn't spend time on
recursive calls.
up vote 12 BFS code in Java:
down vote
public static boolean[] BFS(boolean[][] adjacencyMatrix, int vertexCount, int givenVertex){
// Result array.
boolean[] mark = new boolean[vertexCount];
Queue<Integer> queue = new LinkedList<Integer>();
mark[givenVertex] = true;
while (!queue.isEmpty())
Integer current = queue.remove();
for (int i = 0; i < vertexCount; ++i)
if (adjacencyMatrix[current][i] && !mark[i])
mark[i] = true;
return mark;
public static void main(String[] args) {
// Given adjacencyMatrix[x][y] if and only if there is a path between x and y.
boolean[][] adjacencyMatrix = new boolean[][]
// Mark[i] is true if and only if i belongs to the same connected component as givenVertex vertex does.
boolean[] mark = BFS(adjacencyMatrix, 5, 0);
for (int i = 0; i < 5; ++i)
If you need the exact code, I can add it for you. – Wisdom's Wind Nov 14 '11 at 16:42
Thankyou, I've realised my original question wasn't too clear. I need to find the connected component (so other reachable vertices) for a given vertex. I realise this is probably
similar but I can't visualise it, could you give me a similar pseudo code? – Denti Nov 14 '11 at 16:47
1 All you need is to drop top enumeration circle, and start from Components = 1: 1) For BFS you need to put your given vertex into queue and follow the algorithm. 2) For DFS just call
DFS(your vertex, 1). After that for all vertices i belongs to the same connected component as your given vertex you will have marks[i] == 1, and marks[i] == 0 for others. – Wisdom's
Wind Nov 14 '11 at 16:52
Sorry I don't quite understand what you mean by dropping the top enumeration circle, do you mind writing it again? Thanks. Also which is best to use for this problem BFS or DFS? –
Denti Nov 14 '11 at 16:56
When I said "drop", I meant that is redundant in your case, you don't have to write it. Read the last two lines I've add to the original answer, I suggest you to use BFS. – Wisdom's
Wind Nov 14 '11 at 16:59
show 6 more comments
You can implement DFS iteratively with a stack, to eliminate the problems of recursive calls and call stack overflow. The implementation is very similar to BFS with queue - you just
up vote 1 down have to mark vertices when you pop them, not when you push them in the stack.
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Reference for elliptic 3-folds
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
I was looking for a reference which studies elliptic 3-folds (Their canonical bundle, second Chern class, singular fibers,...), similar to one for surfaces (Which is available
in many books including Griffiths-Harris).
up vote 2 down vote
favorite ag.algebraic-geometry dg.differential-geometry complex-geometry
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I was looking for a reference which studies elliptic 3-folds (Their canonical bundle, second Chern class, singular fibers,...), similar to one for surfaces (Which is available in many books including
Rick Miranda's text (mentioned by Artie) describes how one can obtain an elliptic threefold from a Weierstrass equation and how the singular fibers behave in the particular model he
constructed. (In the surface case there one has a unique smooth projective surface associated with a Weierstrass equation, in the threefold case this is false.)
There are several long texts by Noboru Nakayama on elliptic fibrations which contains several standard facts. (Mathscinet list 4 papers in total with the word "elliptic" in the title.)
Rania Wazir's text contains (besides the arithmetical part) a proof for the Shioda-Tate formula for elliptic threefolds.
There are texts by Grassi and Morrison which discuss the role of elliptic threefolds in string theory.
up vote 4
down vote However, in the elliptic surface case you can obtain all the hodge numbers, chern classes etc. once you know a) the genus of the base curve and b) the degree of the discriminant (as a
divisor on the base curve). Such a clean statement seems hard/impossible to obtain in the elliptic threefold case, unless you restrict yourself to certain classes of elliptic threefolds
(e.g. one takes a Weierstrass model in some P^2-bundle and assumes that this is smooth).
Concerning the singular fibers: the discriminant is a (possible reducible) curve on the base surface. The fiber type over the generic point of an irreducible component is one of the fiber
types from Kodaira's list. On each irreducible component there are finitely many points with a different type. Miranda calculated which possibilities there are for there special points,
under the condition that you allow to replace your base surface with a birational one.
add comment
Rick Miranda's text (mentioned by Artie) describes how one can obtain an elliptic threefold from a Weierstrass equation and how the singular fibers behave in the particular model he constructed. (In
the surface case there one has a unique smooth projective surface associated with a Weierstrass equation, in the threefold case this is false.)
There are several long texts by Noboru Nakayama on elliptic fibrations which contains several standard facts. (Mathscinet list 4 papers in total with the word "elliptic" in the title.)
Rania Wazir's text contains (besides the arithmetical part) a proof for the Shioda-Tate formula for elliptic threefolds.
There are texts by Grassi and Morrison which discuss the role of elliptic threefolds in string theory.
However, in the elliptic surface case you can obtain all the hodge numbers, chern classes etc. once you know a) the genus of the base curve and b) the degree of the discriminant (as a divisor on the
base curve). Such a clean statement seems hard/impossible to obtain in the elliptic threefold case, unless you restrict yourself to certain classes of elliptic threefolds (e.g. one takes a
Weierstrass model in some P^2-bundle and assumes that this is smooth).
Concerning the singular fibers: the discriminant is a (possible reducible) curve on the base surface. The fiber type over the generic point of an irreducible component is one of the fiber types from
Kodaira's list. On each irreducible component there are finitely many points with a different type. Miranda calculated which possibilities there are for there special points, under the condition that
you allow to replace your base surface with a birational one.
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I've been solving this one.. Formula for vC(t): (1/C)*(1/(1000*2*PI))*(-cos(1000*2*PI*t) + 1) Computed value for v(0.0005) in volts:: 0.318310 Measured value for v(0.0005) in volts: 0.317
Estimated value for electrical energy stored at time 0.0005, in joules: 0.0000502 Power delivered to the load in watts: 2 Average current supplied by bridge rectifier, in amps: 0.205
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What is the length of the portion BC of the cloth?
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8cos35 degrees 8sin 35 degrees sin35 degrees/8 cos35 degrees/8
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use sin ratio.
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idk what that is..
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\[\sin \theta = \frac{opposite}{hypotenuse}\]
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idk what all that means..
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|dw:1333128731997:dw|Dont sleep in class. ;)
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I'm home schooled.. I can't sleep in class , haha . I just don't understand this stuff & I have HORRIBLE memory :) Ha !
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lol. Ok. this is easy i can teach you. :)
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So what is it exactly that I'm supposed to do here ??
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There are three basic ratios. Sin ratio, cos ratio, tan ratio. you have to remember these.. Not them down somewhere.
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Now these are the three sides. now it depends upon the question which two ratio you have to use.. you have to judge it by yourself.
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Now in your question.. you are given hypotenuse and you have to find the opposite side.. so u will be using sin ratio.
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i still don't get it though . like , what #'s do i plug in & where ?? it's so confusing to me!!
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Ok, so the basic idea is that if you have a right angle, and You know the second angle, you know the ratio of the two sides. Now all of the trig functions essentially take in an angle, and out
put a ratio.
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i'm still confused :(
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Do you want to talk it over in chat?
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No.. too overwhelming . Just mssg me .
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Can we do something like tiny chat? Messaging is too slow. If you have a skype, you could chat with me there (or create an anonymous one)
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i cant , im also takin a test & i can only be on here & my test (im bein supervised)
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Wait, you are allowed to go here and on your test?
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Alright, Trigonometry is like an extension of triangle congruence.
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*triangle similarity
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For triangle similarity, we can prove that two triangles are similar given certain info (SAS, AA, SSS, etc). But if they are similar, then how can we figure out the side lengths they have?
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The answer is that we use trigonometry. The basic idea of trig is that in a right triangle where we know all the degree measures, we can determine the ratio between the sides.
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Tell me if you are following what I'm saying, ok
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I'm not.. haha sorry . I am so confused !! I'm a visual/audio learner..
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ok, who is supervising you? I have file that will help you a lot, but it's for a geometry software called "GeoGebra"
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i have geogebra
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Alright, take a look at this file. Tell me if it helps.
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I'm just gonna guess cos theres like 4 or 5 probs like this & I have 15 mins to finish 20 more problems !!!!!
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sigh. if they are typable, put them into wolfram alpha dot com or something,
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Total # Posts: 7
managerial economics
A firm offers two differentiated products, X and Y and faces two types of consumers, types A and B. There are equal numbers of each type of consumers ¡V so, for simplicity, assume there is just one
of each type. The valuations of the two types of customers of the two pro...
managerial economics
A firm offers two differentiated products, X and Y and faces two types of consumers, types A and B. There are equal numbers of each type of consumers ¡V so, for simplicity, assume there is just one
of each type. The valuations of the two types of customers of the two pro...
managerial economics
A firm offers two differentiated products, X and Y and faces two types of consumers, types A and B. There are equal numbers of each type of consumers ¡V so, for simplicity, assume there is just one
of each type. The valuations of the two types of customers of the two pro...
managerial economics
BigBook is a monopolist book publishing company, which sells books in Australia and New Zealand. Assume there is a 1:1 exchange rate between Australia and New Zealand. The inverse demand equations
for Australia and New Zealand are as follows: Australia: PA = 100 - 2.5QA New Ze...
managerial economics
Players A and B are playing a simultaneous moves game and both can choose either strategy S1 or strategy S2. If both choose S1 both receive 0. If both choose S2 both receive -2. If their chosen
strategies differ they both receive -4. (a)Write out a table representing each play...
managerial economics
Consider the one-shot, simultaneous move game below, and answer the accompanying questions: Player & Strategy Firm B Left Right Firm A Up 4,4 0,0 Down 0,0 2,2 (a)List the strategies for Firm A and
Firm B (b)State the set of strategy profiles. (c)Suppose Firm A plays Up. What i...
managerial economics
Problem One5 Consider the following simultaneous moves game in normal form: Player Two t1 t2 Player One r1 -2,4 0,-2 r2 -4,5 0,1 (a)State the set of Nash equilibrium strategies. (b)State the payoffs
to each player in the Nash equilibrium.
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the first resource for mathematics
Compact failure of multiplicativity for linear maps between Banach algebras.
(English) Zbl 1201.47038
Let $𝒜$ and $ℬ$ be Banach algebras and let $T:𝒜\to ℬ$ be a linear map. Set
${S}_{T}\left(a,b\right)=T\left(ab\right)-T\left(a\right)T\left(b\right),\phantom{\rule{1.em}{0ex}}a,b\in 𝒜·$
For $\delta >0$, the map $T$ is called $\delta$-multiplicative if $\parallel {S}_{T}\parallel <\delta$. In B. E. Johnson [J. Lond. Math. Soc., II. Ser. 34, 489–510 (1986; Zbl 0625.46059)] and
[J. Lond. Math. Soc., II. Ser. 37, No. 2, 294–316 (1988; Zbl 0652.46031)], pairs $\left(𝒜,ℬ\right)$ which are AMNM (almost multiplicative bounded linear maps are near multiplicative bounded linear
maps) were investigated. Since then, many authors have contributed to the study of these algebras.
In the paper under review, the author considers other concepts of smallness of ${S}_{T}$. First of all, he introduces notions of compactness and weak compactness for multilinear maps from a product
of normed spaces to a topological space. As pointed out by the author, compact multilinear maps were also considered in the normed case by N. Krikorian [Proc. Am. Math. Soc. 33, 373–376 (1972; Zbl
0235.46068)]. Now a map $T$ is said to be a cf-homomorphism (compact from homomorphism) if ${S}_{T}$ is compact.
On the other hand, $T$ is called semi-cf-homomorphism if, for each $a\in 𝒜$, ${S}_{T}\left(a,·\right)$ and ${S}_{T}\left(·,a\right)$ are compact linear maps. In a similar way, the author defines
weakly compact, $n$-dimensional (resp., semi weakly compact, semi $n$-dimensional) from homomorphism.
The author studies general properties of such maps. Moreover, he gives a characterization of some Banach function algebras where such maps are automatically multiplicatives. Finally, the paper is
concluded with generalizations of some results in the Hochschild-Kamowitz cohomology theory.
47B48 Operators on Banach algebras
46H05 General theory of topological algebras
46J10 Banach algebras of continuous functions, function algebras
46H25 Normed modules and Banach modules, topological modules
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Lev Semenovich Pontryagin
Born: 3 September 1908 in Moscow, Russia
Died: 3 May 1988
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Lev Semenovich Pontryagin's father, Semen Akimovich Pontryagin was a civil servant. Pontryagin's mother, Tat'yana Andreevna Pontryagina, was 29 years old when he was born and she was a remarkable
woman who played a crucial role in his path to becoming a mathematician. Perhaps the description of 'civil servant', although accurate, gives the wrong impression that the family were reasonably well
off. In fact Semen Akimovich's job left the family without enough money to allow them to give their son a good education and Tat'yana Andreevna worked using her sewing skills to help out the family
Pontryagin attended the town school where the standard of education was well below that of the better schools but the family's poor circumstances put these well out of reach financially. At the age
of 14 years Pontryagin suffered an accident and an explosion left him blind. This might have meant an end to his education and career but his mother had other ideas and devoted herself to help him
succeed despite the almost impossible difficulties of being blind. The help that she gave Pontryagin is described in [1] and [2]:-
From this moment Tat'yana Andreevna assumed complete responsibility for ministering to the needs of her son in all aspects of his life. In spite of the great difficulties with which she had to
contend, she was so successful in her self-appointed task that she truly deserves the gratitude ... of science throughout the world. For many years she worked, in effect, as Pontryagin's
secretary, reading scientific works aloud to him, writing in the formulas in his manuscripts, correcting his work and so on. In order to do this she had, in particular, to learn to read foreign
languages. Tat'yana Andreevna helped Pontryagin in all other respects, seeing to his needs and taking very great care of him.
It is not unreasonable to pause for a moment and think about how Tat'yana Andreevna, with no mathematical training or knowledge, made by her determination and extreme efforts a major contribution to
mathematics by allowing Pontryagin to become a mathematician against all the odds. There must be many other non-mathematicians, perhaps many of whom are unrecorded by history, who have also by their
unselfish acts allowed mathematics to flourish. As we try to show in this archive, the development of mathematics depends on a wide number of influences other than the talents of the mathematicians
themselves: political influences, economic influences, social influences, and the acts of non-mathematicians like Tat'yana Andreevna.
But how does one read a mathematics paper without knowing any mathematics? Of course it is full of mysterious symbols and Tat'yana Andreevna, not knowing their mathematical meaning or name, could
only describe them by their appearance. For example an intersection sign became a 'tails down' while a union symbol became a 'tails up'. If she read 'A tails right B' then Pontryagin knew that A was
a subset of B!
Pontryagin entered the University of Moscow in 1925 and it quickly became apparent to his lecturers that he was an exceptional student. Of course that a blind student who could not make notes yet was
able to remember the most complicated manipulations with symbols was in itself truly remarkable. Even more remarkable was the fact that Pontryagin could 'see' (if you will excuse the bad pun) far
more clearly than any of his fellow students the depth of meaning in the topics presented to him. Of the advanced courses he took, Pontryagin felt less happy with Khinchin's analysis course but he
took a special liking to Aleksandrov's courses. Pontryagin was strongly influenced by Aleksandrov and the direction of Aleksandrov's research was to determine the area of Pontryagin's work for many
years. However this was as much to do with Aleksandrov himself as with his mathematics ([1] and [2]):-
Aleksandrov's personal charm, his attention and helpfulness influenced the formation of Pontryagin's scientific interests to a remarkable extent, as much in fact as the personal abilities and
inclinations of the young scholar himself.
The year 1927 was the year of the death of Pontryagin's father. By 1927, although he was still only 19 years old, Pontryagin had begun to produce important results on the Alexander duality theorem.
His main tool was to use link numbers which had been introduced by Brouwer and, by 1932, he had produced the most significant of these duality results when he proved the duality between the homology
groups of bounded closed sets in Euclidean space and the homology groups in the complement of the space.
Pontryagin graduated from the University of Moscow in 1929 and was appointed to the Mechanics and Mathematics Faculty. In 1934 he became a member of the Steklov Institute and in 1935 he became head
of the Department of Topology and Functional Analysis at the Institute.
Pontryagin worked on problems in topology and algebra. In fact his own description of this area that he worked on was:-
... problems where these two domains of mathematics come together.
The significance of this work of Pontryagin on duality ([1] and [2]):-
... lies not merely in its effect on the further development of topology; of equal significance is the fact that his theorem enabled him to construct a general theory of characters for
commutative topological groups. This theory, historically the first really exceptional achievement in a new branch of mathematics, that of topological algebra, was one of the most fundamental
advances in the whole of mathematics during the present century...
One of the 23 problems posed by Hilbert in 1900 was to prove his conjecture that any locally Euclidean topological group can be given the structure of an analytic manifold so as to become a Lie group
. This became known as Hilbert's Fifth Problem. In 1929 von Neumann, using integration on general compact groups which he had introduced, was able to solve Hilbert's Fifth Problem for compact groups.
In 1934 Pontryagin was able to prove Hilbert's Fifth Problem for abelian groups using the theory of characters on locally compact abelian groups which he had introduced.
Among Pontryagin's most important books on the above topics is topological groups (1938). The authors of [1] and [2] rightly assert:-
This book belongs to that rare category of mathematical works that can truly be called classical - book which retain their significance for decades and exert a formative influence on the
scientific outlook of whole generations of mathematicians.
In 1934 Cartan visited Moscow and lectured in the Mechanics and Mathematics Faculty. Pontryagin attended Cartan's lecture which was in French but Pontryagin did not understand French so he listened
to a whispered translation by Nina Bari who sat beside him. Cartan's lecture was based around the problem of calculating the homology groups of the classical compact Lie groups. Cartan had some ideas
how this might be achieved and he explained these in the lecture but, the following year, Pontryagin was able to solve the problem completely using a totally different approach to the one suggested
by Cartan. In fact Pontryagin used ideas introduced by Morse on equipotential surfaces.
Pontryagin's name is attached to many mathematical concepts. The essential tool of cobordism theory is the Pontryagin-Thom construction. A fundamental theorem concerning characteristic classes of a
manifold deals with special classes called the Pontryagin characteristic class of the manifold. One of the main problems of characteristic classes was not solved until Sergei Novikov proved their
topological invariance.
In 1952 Pontryagin changed the direction of his research completely. He began to study applied mathematics problems, in particular studying differential equations and control theory. In fact this
change of direction was not quite as sudden as it appeared. From the 1930s Pontryagin had been friendly with the physicist A A Andronov and had regularly discussed with him problems in the theory of
oscillations and the theory of automatic control on which Andronov was working. He published a paper with Andronov on dynamical systems in 1932 but the big shift in Pontryagin's work in 1952 occurred
around the time of Andronov's death.
In 1961 he published The Mathematical Theory of Optimal Processes with his students V G Boltyanskii, R V Gamrelidze and E F Mishchenko. The following year an English translation appeared and, also in
1962, Pontryagin received the Lenin prize for his book. He then produced a series of papers on differential games which extends his work on control theory. Pontryagin's work in control theory is
discussed in the historical survey [3].
Another book by Pontryagin Ordinary differential equations appeared in English translation, also in 1962.
Pontryagin received many honours for his work. He was elected to the Academy of Sciences in 1939, becoming a full member in 1959. In 1941 he was of one the first recipients of the Stalin prizes
(later called the State Prizes). He was honoured in 1970 by being elected Vice-President of the International Mathematical Union.
Article by: J J O'Connor and E F Robertson
Click on this link to see a list of the Glossary entries for this page
List of References (4 books/articles)
A Poster of Lev Pontryagin Mathematicians born in the same country
Honours awarded to Lev Pontryagin
(Click below for those honoured in this way)
LMS Honorary Member 1952
Speaker at International Congress 1958
Speaker at International Congress 1970
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History Topics Societies, honours, etc. Famous curves
Time lines Birthplace maps Chronology Search Form
Glossary index Quotations index Poster index
Mathematicians of the day Anniversaries for the year
JOC/EFR © January 1999 School of Mathematics and Statistics
Copyright information University of St Andrews, Scotland
The URL of this page is:
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For problems 1–5, simplify the expression. Answers written in decimal form will not be accepted. Each of these problems is worth 1 point. "v/" is a radical by the way ___ 1.v/96 ______ 2. 8 v/63x^5
___________ 3. v/128x^5y^2 ___ 4. ^3v/32 ________ 5. ^3v/56x^14
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So, I need them in radical form.
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Factor everything first and then apply your radical rules... for example \[\sqrt[3]{x^3} = x\]
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Problem is that I have no idea what you're saying. ._.
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Ok for example \[\sqrt[8]{256x^4}\] becomes, if you factor \[\sqrt[8]{2^{8}x^4}\] which then becomes \[2\sqrt[8]{x^4}=2 x^{\frac{4}{8}}=2x^{1/2}=2\sqrt{2}\]
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This uses the idea that \[\sqrt[a]{x} = x^{1/a}\]
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just wondering, would you mind using bigger font? I can't see the exponents. To do that, put what you are saying in the curly braces in \(\huge\text{}\.) but take out the . at the end.
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the ^ means exponents. like, 2^3 is 2 cubed.
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\(\Large\text{But again, I have no idea how to do this.}\)
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is ^3v/56x^14 supposed to be \[\bigg(\sqrt{56x^{14}}\bigg)^3\] then?
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So, \(\huge\sqrt[8]{256x^4}\) turns into \(\huge\sqrt[8]{2^{8}x^4}\) which turns into \(\huge2\sqrt[8]{x^4}=2 x^{\frac{4}{8}}=2x^{1/2}=2\sqrt{2}\) But how? \(\huge\text{:|}\)
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Sorry that should be \[2\sqrt{x}\]
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and no, it's supposed to be \[^{3}\sqrt{56x^{14}}\]
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ok the basic theorems that you need for these type of problems are that \[\sqrt[a]{x} = x^{\frac{1}{a}}\] So for example \[\large \sqrt[3]{x^{10}}\to x^{\frac{10}{3}}\to x^3x^{\frac{1}{3}}\to x^3
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I understand the first half of your example equation, but not the second half.
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So any complicated radical you are given you can convert to exponents, use the rules of exponents shamelessly and then convert back to radical. So \[\large \sqrt[3]{56x^{14}}\to \sqrt[3]{(7)(8)x^
{14}}\to 7^{1/3}8^{1/3}x^{14/3}\] This then becomes \[ 7^{1/3}8^{1/3}x^{14/3}\to 7^{1/3}(2^3)^{1/3}x^{12/3}x^{2/3}\] becomes \[7^{1/3}2x^{4}x^{2/3}\to 2x^4 7^{1/3}x^{1/3}\to 2x^4\sqrt[3]{7x}\]
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my brain is about to explode. e_o
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So you don't understand how \[x^{10/3} \to x^3x^{1/3}\]? This results from the fact that \[X^aX^b = X^{ab}\] and the reverse. So if you have \[x^{10/3}\] you have \[x^{9/3 + 1/3}\to x^{9/3}x^{1/
3}\to x^3x^{1/3}\]
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again, might I suggest the larger font size? Maybe Large would be a good replacement for huge though.
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The estimates on the energy functional of an elliptic system with Neumann boundary conditions
We consider an elliptic system of the form , in Ω with Neumann boundary conditions, where Ω is a domain in , f and g are nonlinearities having superlinear and subcritical growth at infinity. We
prove the existence of nonconstant positive solutions of the system, and estimate the energy functional on a configuration space by a different technique, which is an important step in the proof of
the solution’s concentrative property. We conclude that the least energy solutions of the system concentrate at the point of boundary, which maximizes the mean curvature of ∂Ω.
elliptic system; estimates; energy functional
1 Introduction
We are concerned with the following singularly perturbed system with Neumann conditions:
where is a small parameter, Ω is a bounded domain in ( ), . f and g are nonlinearities having superlinear and subcritical growth at infinity.
Problem (1.1) arises in many applied models concerning biological pattern formations. For example, the steady states in the Keller-Segel model, the Gierer-Meinhardt model, see [1,2] for more details.
Problem (1.1) has been studied extensively for last twenty years. The motivation for the study of such a problem goes back to the pioneering work of [2,3] concerning the scalar case (single
They proved a priori estimates, existence of least energy solutions and the concentrative properties of the solution. Furthermore, in [4,5], Ni and Takagi proved the existence of a nontrivial
solution to problem (1.2) for ε small enough. They showed that attains its maximum value at a point , and the subsequences of converge to P, which is the maximum point of mean curvature on ∂Ω.
The subject was studied by many authors for both Neumann and Dirichlet boundary conditions. There are many well-known results about (1.2). Del Pino and Felmer in [6] introduced shorter and more
elementary arguments with respect to those in [4,7]. Wang in [8] obtained multiple solutions of (1.2) by using Ljusternik-Schnirelman method. In [9], Grossi et al., obtained a solution of (1.2) with
k maxima points, k is a given positive integer. We refer the reader to [10-14] for further references.
As far as we know, Avial and Yang [15] were the first to approach the singularly perturbed system (1.1) with Neumann boundary conditions; they considered (1.1) with special nonlinearities , ( ). By
means of a dual variational formulation, they proved that there exist nontrivial positive solutions and in , which have global maximum point at different points.
A more direct approach was proposed in [16-18]. In these papers, the authors extend the idea, which is introduced by Del Pino and Felmer in [6], to system (1.1). In [18], Pistoia and Ramos proved the
least energy solutions of system (1.1) concentrate at a point of the boundary, which maximizes the mean curvature of the boundary of Ω. Pistoia and Ramos [19] consider system (1.1) with Dirichlet
boundary condition, they proved the existence of the least energy solutions. The solutions are concentrated, as ε goes to zero, at a point of Ω, which is maximized in distance to the boundary of Ω.
Let us recall the idea mentioned in [3-5], their proof based on the well-known result of Gidas et al.[20], that is, the uniqueness solution of the equation
here w is radially symmetric and is strictly decreasing, and for . But the uniqueness result for the following system corresponding to (1.1) is not known.
Besides, it is known that the underlying minimax theorem associated to ground-state level of (1.1) is an infinite-dimensional linking, this is in contrast with (1.2). We refer the reader to [18,21]
for more details on this.
In this paper, we prove the existence of nonconstant positive solutions of system (1.1), and estimate the energy functional of (1.1) on the configuration space (defined in Section 2) by a different
technique, which is compared with [18]. This estimation is an important step in the proof of , where denotes the mean curvature of ∂Ω at the boundary point P. We conclude the least energy solutions
of system (1.1), concentrated at the point of boundary, which maximizes the mean curvature of the boundary of Ω.
2 Statement of main results
The assumption to is a typical superlinear subcritical one, as in [18], we assume that the following holds.
(S[1]) , for . , . There exist two real numbers , such that
Remark 2.1 Examples of nonlinearities satisfying (S[1])-(S[3]) are
We should point out that (S[1])-(S[3]) are the natural extension of the assumptions for the scalar case (single equation). Let us recall the assumptions on single equation such as (1.2).
Assume that is continuous and satisfies the following structure assumptions.
(f[1]) for and near . as , for some if , and if .
(f[2]) There exists a constant such that for , in which .
(f[3]) The function is strictly increasing.
Remark 2.2 Assumption (S[1]) is the ‘system edition’ of (f[1]). (f[2]) is the famous Ambrosetti-Rabinowitz superlinear condition [22], which has appeared in most of studies for superlinear problems.
In fact, it implies that the super-quadratic condition on . It has been used in a crucial way not only in establishing the mountain-pass geometry of the functional, but also in obtaining bounds of
(PS) sequences. Assumption (S[2]) implies that , , which play a important roll in the proof of the existence of system’s solutions. So it is the ‘system edition’ of (f[2]).
Without loss of generally, we may assume that . By the following rescaling:
equation (1.1) becomes
To simplify the notations, we define . Associated with (2.1) is the energy functional
(2.2) is a functional defined over the Hilbert space .
We define the norm
It can be observed that the following orthogonal splitting holds: , here, , . We set , .
Theorem 2.3Assume (S[1])-(S[3]), then there exists , such that for any , system (1.1) has nonconstant positive solutions . If , the estimation of the energy functional on is
Remark 2.4 We point out that, in contrast with Theorem 2.3, only constant positive solutions are expected to exist for large values of ε[3,15].
Remark 2.5 The estimation (2.3) is an important step in the proof of
where denotes the mean curvature of ∂Ω at the boundary point P. So we can conclude the least energy solutions of system (1.1) concentrate at a point of the boundary, which maximizes the mean
curvature of the boundary of Ω.
3 Proof of Theorem 2.3
To prove Theorem 2.3, we need the following lemma. Let , be the solutions of (1.1), in order to simplify the notations, we define , ,
Lemma 3.1Supposing the assumptions in Theorem 2.3 hold, admits the following properties:
Proof Proof of (i)[1]. By the definition of , , then , where u and v are the solutions of (1.1), that follows .
Again by (3.1),
From (S[2]), we can deduce there exist , such that , then
The left side in last inequality is equal to , so . By the same way, . Then (3.2) can change to
So, we get the properties (i)[1].
To prove (ii)[1] is equal to show
By (S[1]), there exist C such that
As the same,
We prove (3.3) by contradiction. Assume that there exists a subsequence such that
that is,
Take in the last equality, then
For p is a number between 2 and , , we deduce . It contradicts with the original assumption of .
Now, we turn to the prove of (iii)[1]. By (3.1),
in the same way,
Proposition 3.2 (Theorem 1.1 in [18])
Under assumptions (H), there exists such that for any , problem (1.1) has nonconstant positive solutions . Moreover, both functions and attain their maximum value at some unique and common point . (
The assumption (H) is composed of (S[1]), (S[3]) and the following (3.5).)
Remark 3.3 We will compare our assumptions (S[1])-(S[3]) with the conditions (H) of Proposition 3.2 in the following proof of Theorem 2.3.
Proof of Theorem 2.3 The existence of solutions of (1.1) can follow the steps of Theorem 1.1 in [18]. They use some ideas introduced by Del Pino and Felmer [6], and differ from the method of Ni and
Takagi. It needs to be pointed out that (S[2]) implies the following conditions:
The assumption (H) in Proposition 3.2 is composed of (S[1]), (S[3]) and (3.5). By Proposition 3.2, the existence of solutions can be proved under (H). So, we can get the existence of solutions of
(1.1) under (S[1])-(S[3]).
The rest of the paper is devoted to the proof of (2.3). By the definition of space , we only need to prove for any , , the following holds,
Obviously . By (3.1), (3.6) is equal to
We prove (3.7) by contradiction, suppose that the maximum point of is , and .
Thus, by (3.8), we have
We claim that the function , defined in (3.9), has the following properties:
First, we prove (i)[2]. In fact, by (3.1) and (3.9),
Following from assumption (S[2]), we obtain .
Again by (3.9) and Lemma 3.1, and , so . Next, we want to compute . From (3.9),
Combined with (3.4), . By Lemma 3.1, , so .
Now, we turn to the proof of (ii)[2]. Let , by (3.11),
Then by (3.10),
By (S[3]), we get
then by (3.12), . We proved the property (ii)[2].
The proof of (iii)[2] is similar to (ii)[1]. Then we complete the proof of the claim.
Suppose that the maximum point of is , , then either or . If , , by (3.1), (3.9) and property (i)[2] of , , . By the property (iii)[2] of , there exist , such that
If , then . By Lemma 3.1, , , so there exist , such that , and . Thus, . , so there exist , such that
In fact, (3.13) and (3.14) is a contradiction to the nature (ii)[2] of , that is, for any , if , the value of must be smaller than 0.
Having reached a contradiction, this completes the proof of Theorem 2.3.□
Competing interests
There are no financial competing interests in this manuscript. There are no non-financial competing interests (political, personal, religious, ideological, academic, intellectual, commercial or any
other) to declare in relation to this manuscript.
Authors’ contributions
There is only one author in the manuscript. JZ designed the study. She carried out the studies, and drafted the manuscript. The author read and approved the final manuscript.
The author is supported by the project of ‘Youth Innovation,’ funded by the Department of Science and Technology of Fujian province (2011J05003), and supported by the Projects A of the Educational
Department of Fujian Province (JA11053).
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Posts by teal
Total # Posts: 10
14C has a half-life of 5,730 years. If we start with 70 grams of 14C at time zero how many grams of 14C will be left after three half-lives?
@damon physics
damon, does the car that weighs 500kg have more acceleration? if so, would e be correct then? here's the bottom three choices again: c. The two cars have the same acceleration due to the collision.
d. car at rest has the greater acceleration due to the collision. e. The ca...
@damon econ
okay . and for question # 1 you woud recommonded choosing none of the above correct ?
@damon physics
for question # 2 you woud say the only answer would be that the lighter car has a greater acceertion due to the collison?
@damon physics
some of the possible answers to question # 1 is 4000n,10n,40n, not enough info or none of the above, would you recommend none of the above ?
@damon physics
I was wondering if you can help me with these two questions. Thank you Question # 1 A horse is pulling a heavy cart up a hill. The cart has 400 kg of mass inside it. The horse is pulling the cart
with 10 N of force. What is the magnitude of the force on the horse due to the ca...
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Topic: Re: [mg4898] Transformation rule exercise
Replies: 0
Re: [mg4898] Transformation rule exercise
Posted: Oct 9, 1996 2:10 PM
Robert Hall wrote
Dear Abbey,
This has baffled me for two days now. It's exercise 4 from p. 113 of
Introduction to Programming with Mathematica, by Gaylord, Kamin &
The problem is to rewrite
g[x_] = x /. Plus[z___] -> Times[z]
so that
g[a + b + c]
a b c
The authors provide the following information:
"Hint: You need to maintain the lhs of the transformation rule
unevaluated for purposes of pattern-matching and the rhs of the rule
unevaluated until the rule is used."
Unfortunately, both my solutions violate both clauses of the hint.
g1[x_] := x /. Plus -> Times
g2[x_] := x /. Plus[y___, z___] -> Times[y, z]
I could replace Rule with RuleDelayed in either
function (replace -> with :>) but it isn't necessary. The authors
state that the lhs is evaluated before the rule is applied, and
Trace[] bears this out. I'm looking for a solution to which the
authors' hint applies, but for the life of me I can't find it.
Am I overlooking something obvious?
Baffled near Baltimore
Bob Hall | "Know thyself? Absurd direction!
rhall2@gl.umbc.edu | Bubbles bear no introspection." -Khushhal
Khan Khatak
A solution is
g[x_] := (x /. Literal[Plus[z___]] :> Times[z])
Frans Martens
Eindhoven University of Technology
The Netherlands
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