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History of the Quadratic Equation
Paper Rating: Word Count: 1938 Approx Pages: 8
Throughout the years, the history of mathematics has taken its fair share of changes. It has stretched across the world from the Far East, migrating into the Western Hemisphere. One of the most
fundamental and key principles of mathematics has been the quadratic formula. Having been used in several different cultures, the formula has been part of the base of mathematics theory. The general
equation has been derived from many different sources, most commonly: ax2 + bx + c = 0, with x being the variable and a, b, and c its respective constant terms. Though this is how modern mathematics
perceives the equation, different symbols and notations have been used to represent the formula.
Beginning in the "Before Christ era, the Babylonians were the first to have been recorded demonstrating the equation, circa 400 BC. The form most mathematics students use today is:
To solve a quadratic equation the Babylonians essentially used the standard formula, with the a term being included in the x2 variable. They considered two types of quadratic equations, namely:
Here b and c were positive but not necessarily integers. The form that their solutions took was, respectively:
x = [(b/2)2 + c] - (b/2) and x = [(b/2)2 + c] + (b/2).
Notice that in each case this is the positive root from the two roots of the quadratic and the one that will make sense in solving "real" problems. For example problems which led the Babylonians to
equations of this type often concerned the area of a rectangle. For example if the area is given and the amount by which the length exceeds the width is given, then the width satisfies a quadratic
equation and then they would apply the first version of the formula above (website one).
The efforts the Babylonians made at using this method were far from futile and, actually, served a very important purpose.
It was an important task for the rulers of Mesopotamia to dig canals and t... Continue Reading | {"url":"http://www.exampleessays.com/viewpaper/35214.html","timestamp":"2014-04-20T11:43:47Z","content_type":null,"content_length":"74823","record_id":"<urn:uuid:13240138-020e-4d66-857a-bed04c61bd6c>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00506-ip-10-147-4-33.ec2.internal.warc.gz"} |
Delivery: An Open-Source Model-Based Bayesian Seismic Inversion Program
Delivery: An Open-Source Model-Based Bayesian Seismic Inversion Program (2003)
Download Links
by James Gunning , Michael E. Glinsky
author = {James Gunning and Michael E. Glinsky},
title = {Delivery: An Open-Source Model-Based Bayesian Seismic Inversion Program},
year = {2003}
We introduce a new open-source toolkit for model-based Bayesian seismic inversion called Delivery. The prior model in Delivery is a trace--local layer stack, with rock physics information taken from
log analysis and layer times initialised from picks. We allow for uncertainty in both the fluid type and saturation in reservoir layers: variation in seismic responses due to fluid e#ects are taken
into account via Gassman's equation. Multiple stacks are supported, so the software implicitly performs a full AVO inversion using approximate Zoeppritz equations. The likelihood function is formed
from a convolutional model with specified wavelet(s) and noise level(s). Uncertainties and irresolvabilities in the inverted models are captured by the generation of multiple stochastic models from
the Bayesian posterior, all of which acceptably match the seismic data, log data, and rough initial picks of the horizons. Post-inversion analysis of the inverted stochastic models then facilitates
the answering of commercially useful questions, e.g. the probability of hydrocarbons, the expected reservoir volume and its uncertainty, and the distribution of net sand. Delivery is written in java,
and thus platform independent, but the SU data backbone makes the inversion particularly suited to Unix/Linux environments and cluster systems.
2017 Stegun eds., Handbook of mathematical functions with formulas, graphs, and mathematical tables - Abramowitz, A - 1972
382 Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces - Storn, Price - 1997
73 Bayesian model comparison via jump diffusions - Phillips, Smith - 1995
44 Hypothesis testing and model selection - Raftery - 1996
22 Markov chain Monte Carlo methods for conditioning a permeability field to pressure data - Oliver, Cunha, et al. - 1997
21 A modular system of algorithms for unconstrained minimization - Schnabel, Koontz, et al. - 1985
20 Model selection by MCMC computation - Andrieu, Djurić, et al. - 2001
19 The rock physics handbook - Mavko, Mukerji, et al. - 1999
15 et al. Bayesian Data Analysis - Gelman - 1995
9 On conditional simulation to inaccurate data - Oliver - 1996
7 Markov chain Monte Carlo and its application - Brooks - 1998
7 Offset dependent reflectivity- theory and practice of AVO analysis: Society of Exploration Geophysicists - Castagna, Backus - 1993
7 Reservoir Description From Static and Well-Test Data Using Efficient Gradient Methods,’’ paper - Chu, Reynolds, et al.
6 Petroleum Geostatistics - Omre, Tjelmeland - 1997
5 Prediction of reservoir variables based on seismic data and well observations - Eide, Omre, et al. - 2002
3 Seismic impedance and porosity: support effects - Abrahamsen - 1997
3 CWP/SU: Seismic Unix release 35: A Free Package for Seismic Research and Processing - Cohen, Stockwell - 2001
2 Seismic reservoir prediction using Bayesian integration of rock physics and Markov random fields: A North Sea example. The Leading Edge - Eidsvik
2 Constraining random field models to seismic data: getting the scale and the physics right - Gunning - 2000
2 Conditioning a fluvial reservoir on stacked seismic amplitudes - Huage, Skorstad, et al. - 1998
2 A promising approach to subsurface information integration - Leguijt - 2001
1 Bayesian AVO inversion - Buland, Omre - 2000 | {"url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.2061","timestamp":"2014-04-18T17:48:26Z","content_type":null,"content_length":"26109","record_id":"<urn:uuid:aa790f87-8abe-4dea-a042-0a4e08f27d85>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00206-ip-10-147-4-33.ec2.internal.warc.gz"} |
Architecture and Mathematics in the Pantheon by Giangiacomo Martines for the Nexus Network Journal vol.2 no.3 July 2000
The Relationship Between Architecture and Mathematics in the Pantheon
Giangiacomo Martines
Soprintendenza Archeologica di Roma
Piazza Santa Maria Nova, 53
00186 Rome, Italy
Fax: +39-06-67 87 689
Over the last fifteen years, publications, archaeological excavations and restoration work have changed our traditional view of the Pantheon. Under the direction of Paola Virgili, the Rome Town
Council Archaeology Office has recently excavated the piazza where the Pantheon is situated and the State Office for Monuments has begun restoration of the rotunda, under the direction of Mario Lolli
Ghetti and Giovanni Belardi. The l8th century restoration work on this monument, with its fine marbles and colours, is the subject of a book by Susanna Pasquali. Archaeological excavation and
restoration have provided objective data on which to base our observations.
In 1986 Godfrey and Hemsoll [6] put forward a new theory on the original function of the monument: "an imperial audience chamber". Subsequently, Hemsoll, Davies and Wilson Jones [8] published a
plausible hypothesis on the size of the pronaos in the original design. In his book, Das Pantheon in Rom, Abbild und Mass des Kosmos [14], published in 1999, Sperling analysed the whole monument in
terms of first century AD. mathematics and gnomonics. Although the most important works we have on the architecture and understanding of the Pantheon remain those by De Fine Licht in 1968 [4] and
MacDonald in 1976 [11], respectively, Sperling's book revolutionizes our knowledge of the monument in terms of its relation to neoplatonic mathematics. It is not easy to follow on from Sperling with
new arguments and his book requires careful and meditated reading.
Gert Sperling has carried on from where Herman Geertman left off in his celebrated 1980 study "AEDIFICIVM CELEBERRIMVM, studio sulla geometria del Pantheon" [5]. Sperling's conclusions also make use
of previous works by Jacobson, 1986 [9]; Alvegard, 1987 [1]; Haselberger, 1995 [7]; Williams, 1997 [15]. It is as if Sperling is taking a "peripatetic stroll" inside the monument in the company of
all these writers, with me tagging along behind, a little boy led by the hand, with the work I wrote in 1989 [12]. Sperling's book has several merits which I wish to underline:
• Almost all the mathematical relationships among the lines and points of the monument are visible to the naked eye.
• The floor on which we walk is a mathematical abacus, which we measure with our steps and, again with our eyes, appreciate its shapes and proportions, this time with a little help from Kim
• The monument is an allegory of the cosmos, but it must be seen in relation to another building in Rome: the Mausoleum of Augustus in the Campus Martius. Here we are indebted to Edmund Buchner,
who excavated the Horologium Solarium Augusti under the block of houses between Piazza del Parlamento and Piazza San Lorenzo in Lucina.
• The analemma, which is used in constructing sun dials, is applied by Sperling on the section of the Pantheon and, thus pinpoints some architectonic lines that look as though they have been
arranged according to the declination of the sun at the solstice and the equinox. We know of gnomonics applied to architecture in another geometrical monument, Castel del Monte, from the studies
of Aldo Tavolaro published in 1981.
I do not wish to list all those who have written on the Pantheon after De Fine Licht. I will mention only Howard Saalman [13] and William Loerke [10]. In effect a rational bibliography on the
Pantheon would be a very useful publication. It would certainly include Rodolfo Lanciani and especially Antonio Michetti and Fabrizio Esposito [3], respectively professor and disciple in the science
of construction, for breaking new ground in their study on the reconstruction of the proportionment of the dome structure by means of classical geometry, 1995-1996; unfortunately literary sources are
silent on this subject. We know from Hero of Alexandria about the statics of columns and architraves, but there is nothing from that period on vaults.
Many studies on the Pantheon are carried out far from Rome and so ideas on the monument cannot be checked easily or frequently. For this reason, together with a group of architects and archaeologists
working in Rome , I thought it might be a good idea to try and resolve some seemingly banal but still unanswered questions:
□ What is the exact orientation of the Pantheon?
□ What is its topographic position with respect to the Mausoleum of Augustus and to the preexisting buildings, adjacent to the Pantheon?
□ What path does the sun follow on the lacunaria at midday at the solstice and the equinox?
□ Have there been any variations in astronomical coordinations since the year 1 S 0 AD?
□ Which stars can be seen at night through the eye of the dome, during the course of the seasons?
In charge of this programme is Riccardo Migliari, professor of the Department of Representations and Reliefs at Rome University "La Sapienza" (Fax 06 49 91 88 84). Working in collaboration with him
are Mark Wilson Jones of Bath University, Matthias Bruno of Rome University "La Sapienza", Cinzia Conti of the Rome Archaeological Office and Giovanni Belardi, Director of the Pantheon. There will be
a web site for the data so that it may be utilized also by those far from Rome. I hope Prof. Migliari will be able to present these results here, at Nexus IV, next year. Gert Sperling has given his
full backing to the project, and has suggested taking weekly readings of the sun at the same time on the Rome meridian. All suggestions are welcome and this is why I have included in the text the fax
numbers of the Office and the Department.
A question that is often asked is: Could the inside of the Pantheon have been an astronomical observatory? The Golden House of Nero was another building which was architecturally inspired by the sun
and, according to Suetonius, "the main hall, a rotunda, went round according to the motion of the earth, with perpetual motion day and night"; but it was definitely not an astronomical observatory.
This aspect has been discussed by Cesare D'Onofrio in a chapter of the new edition of Gli Obelischi di Roma, 1992. However I think the eye of the Pantheon is too small to be an astronomical
observatory and too large to be a gnomonic hole; in San Petronio, Bologna, for example, the gnomonic hole of the meridian is only 27 mm wide. Moreover, in the history of gnomonica scientia, the
discovery of the use of a ray of light passing through a hole is attributed to Ibn Yunis, a Muslim astronomer active in Cairo and who died in 1009 AD. However D'Onofrio's theory is stimulating and
next year we hope to have some precise answers. Mathematical analysis of architecture is scientific by definition. The method is scientific because it is based on numbers and theorems, but the result
is not always scientific because we cannot repeat an experiment the way it was done by Galileo Galilei: we cannot always be certain that a mathematical observation we make corresponds to historical
fact, that is, the intention of constructing a building according to a mathematical rule that modern research may have discovered. We must be honest with historians working with Latin and Greek, who
put their trust in mathematicians, rnethods used in philology and logic being so similar. In order to clarify matters I would like to suggest adopting certain criteria, at least with regard to
ancient architecture, which is where my everyday work lies.
1. Mathematics can be used to describe any physical experience, but often the equation is not a law, like the laws of Galileo on the Pendulum and falling bodies, but only a mathematical
representation of a reality that escapes us. In our field this is true of proportions in plan and elevation.
2. Proportions and numbers should be verified on the monument itself, not on scale drawings. Autoptic inspection, that is, personal observation with the naked eye, is indispensable for evaluating
the state of measurement points, instrumental errors and construction material.
3. Studies which are applied to historical facts must make use of sources as documentary proof of an intention.
4. When there are no direct sources we must refer to treatises and scientific knowledge concerning the architecture we are observing.
5. In ancient architecture proportions and numbers must b e evident and easily perceived by the eye. I must mention here the work of Maria Teresa Bartoli [2] on the drawing of the lacunaria of the
Pantheon, 1995, and Mark Wilson Jones on the Arch of Constantine, 1995. Both Wilson Jones and I are indebted to Heinrich Bauer for the importance of this criteria of visibility of proportions in
ancient architecture. Our culture, in this respect, is still that of the Renaissance, as if we were all pupils of Luca Pacioli: it is a seed of the Renaissance that lives in our humanistic
civilisation and we must be aware of this when we study ancient architecture, which, precisely because of this, is Classical but not Renaissance.
After having formulated these criteria I feel like Lucian's Menippus, the cynic-philosopher, who, in the Dialogues of the Dead, desecrates the myths of Olympus and the behaviour of men. For this
reason people called him ton kuna ( tòn kùna), the dog that bites with irony.
Lucian, who was in Rome at the time of Hadrian, wrote famous descriptions of works of art; in fact he is considered the founder of art criticism and a particular literary genre, the ekfrasis (
ékphrasis), which, in fact, means description. One of these descriptions is entitled The Room and is about architecture and oratory. Lucian's room is not that of any particular building but a model
room: it is rectangular, apsidal, facing the east; "the relationship between length and width and between these and height is harmonious". Lucian speaks of "invincible pleasure to the eyes" and says
that "words decorate" architecture, and in so doing underlines the risks of art criticism. Moreover, Lucian maintains that architecture and oratory are aimed at "the common man" and "the cultured
man". Towards the end of the Classical world, similar criteria of aesthetics can be found in Procopius, who wrote On the Buildings of the Emperor Justinian. In these writers, architecture is a
complex organism, which unites matter and ideas in an aesthetic perception that is harmonious, not divided, as it is for us moderns, into humanistic perception and geometrical perception. On the
other hand, it is true that in the Classical world mathematics is aesthetics; mathematicians used similes taken from architecture: Nicomacus of Gerasa, a contemporary of Hadrian, uses bridges and
ladders to explain how mathematics leads to epistemology.
The exceptional flowering of mathematical studies on the Pantheon is due to its eminent architecture and the shapes of its interior space: it is the greatest example in the world, peri sfairas kai
kulindrou (perì sphaìras kài kulìndru), of Archimedes' sphere and cylinder, that is, one inscribed in the other. In antiquity, these shapes helped students learn Archimedes' universal law, 4/3 pr^3.
Archimedes had these shapes inscribed on his tomb, where Cicero was to see them; but man walks inside the sphere and cylinder of the Pantheon, like in the kosmw (kòsmo): a Greek word with three
meanings, "order", "beauty", "world". This is a summary of the work I wrote in 1992, but this is not my aim here. In studies on works of art, gnosiology always surpasses ontology; that is, the
problem of the knowledge of something surpasses its reality. We have more studies on the significance of the Pantheon than on its structure: for example, our knowledge of the dome is still based on
Piranesi's etchings and observations made by Alberto Terenzio in the thirties.
The boom in mathematical studies on the Pantheon is also due to a quality of mathematics: an important theorem generates numerous corollaries, not always known to the mathematician who advanced that
theorem. Most modern critics attribute the design of the Pantheon to one of the world's greatest architects, Appollodorus of Damascus, who was also a famous mechanical engineer: this is thanks to the
work of Wolf Dieter Heilmeyer, and the great archaeologists before him.
I would like to thank my English teacher, Fred Moffa of the British Institute of Rome, for his translation.
[1] L. Alvegard, The Pantheon Metrological System - a consistent, anthropometrical, time-calendar system based on golden section approximation ratios, Billdal, Chalmers University of Technology,1987.
[2] Maria Teresa Bartoli, "Scaenographia vitruviana: il disegno delle volte a lacunari tra rappresentazione e costruzione" in Disegnare, Rivisat semestrale del Dipartimento di Rappresentazione e
Rilievo, Università degli studi di Roma "La Sapienza", vol. V-VI, no. 9/10 (1994), pp. 51-62.
[3] Fabrizio Esposito and Antonio Michetti, "Il Pantheon: Teoria e tecnica della commodulatio" in "Disegnare, Rivisat semestrale del Dipartimento di Rappresentazione e Rilievo, Università degli studi
di Roma "La Sapienza", 13, pp. 69-80.
[4] Kjeld De Fine Licht, The Rotunda in Rome, Copenhagen, 1968.
[5] Herman Geertman, "AEDIFICIVM CELEBERRIMVM, studio sulla geometria del Pantheon", in Bulletin Antieke Beschaving, 55 (1980), pp. 203-229.
[6] P. Godfrey and D. Hemsoll, "The Pantheon: Temple or Rotunda?" in M. Henig and A. King, eds., Pagan Gods and Shrines of the Roman Empire, 1986, pp. 195-209.
[7] L. Haselberger, "Ein Leibelriss der Vorhalle des Pantheon. Die Werkrisse von dem Augustusmausoleum", Roemische Mitteilungen des Dt. Arch. Instituts, 101, Mainz, 1995, pp. 279-308.
[8] D. Hemsoll, P. Davies and M.Wilson Jones, "The Pantheon: Triumph of Rome or Triumph of Compromise", Art History, 10 (1987), 133-153.
[9] D.M. Jacobson, "Hadrianic Architecture and Geometry", in American Journal of Archaeology, 90 (1986).
[10] William C. Loerke, "A Rereading of the Interior Elevation of Hadrian's Rotunda" in Journal of the Society of Architectural Historians, 49, pp. 22-43.
[11] William L. MacDonald, The Pantheon: Design, Meaning and Progeny, London, 1976. To order this book from Amazon.com, click here.
[12] Giangiacomo Martines, "Argomenti di Geometria Antica a proposito della cupola del Pantheon" in Quaderni dell'Istituto di Storia dell'Architettura, 13, 1989.
[13] Howard Saalman, "The Pantheon Coffers: Pattern and Number", in Architectura, 18, (1988), pp. 122-123.
[14] Gert Sperling, Das Pantheon in Rom, Abbild und Mass des Kosmos, Neuried,1999.
[15] Kim Williams, "Il Panteon e la creazione dell'universo, Lettera Matematica Pristem, 24 (June1997), pp. 4-9.
History of Roman Architecture: The Pantheon
Hadrian's Temple
The Pantheon, from Platner, A Topological Dictionary of Ancient Rome
Great Buildings Online: The Pantheon
The Pantheon by Darlene Bishop and Diana Eggers
The Architecture of Hadrian
Giangiacomo Martines, architect with the Soprintendenza Archeologica of Rome, directed the restoration of Column of Trajan and the Column of Marcus Aurelius (1981-1988), about which he has published
several essays (including an forthcoming essay in La colonne Aurélienne, John Scheid and Valerie Huit, eds., Biblioteque des Hautes Etudes, in print). He is now responsible for the restoration of the
Flavian Amphitheatre. He studied art in Sicily under the direction of Federico Zevi, author of Storia dell'arte Italiana (Einaudi, 1980). He has also published on the Palazzo Ducale di Gubbio
(Convegno su Federico da Montefeltro, Urbino, 1982); the science of the gromatici (land surveyors) (in Misurare la terra, exhibit catalogue, Rome, 1985); the Pantheon in Rome (in Quaderni
dell'Istituto di Storia dell'Architettura, 1992). He participated in the Trattato di restauro architettonico, directed by Giovanni Carbonara (Utet, 1996). His current studies concern the poliorcetica
of Apollodorus of Damascus and the mechanics of Heron of Alexandria (in L'arte dell'assedio di Apollodoro di Damasco, Adriano La Regina, ed., Electa, 1999), and the Annali di architettura, 10-11,
The correct citation for this article is:
Giangiacomo Martines, "The Relationship Between Architecture and mathematics in the Pantheon", Nexus Network Journal, vol. 2, no. 3 (July 2000), http://www.nexusjournal.com/Martines.html
Copyright ©2000 Kim Williams
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Patent US7599979 - Apparatus for hybrid multiplier in GF(2m) and method thereof
This application claims the priority of Korean Patent Application No. 10-2004-0087044, filed on Oct. 29, 2004 in the Korean Intellectual Property Office, the disclosure of which is incorporated
herein in its entirety by reference.
1. Field of the Invention
The present invention relates to an apparatus for hybrid multiplication, and more particularly, to an apparatus for hybrid multiplication in finite field GF(2^m) capable of achieving trade-off
between the area and the performance of an apparatus for multiplication.
2. Description of the Related Art
A variety of operations in GF(2^m) are widely used in communications systems or public-key cryptosystems. The GF(2^m) operation in communication systems is used to enhance the reliability of
information and m is determined with respect to the amount of data to be guaranteed for reliability. The exponent m has close relation with the size of hardware for calculation. For communication
systems, m in a range from 8 to 32 is used, and a basic calculator for this, such as an adder, a multiplier, an inverse multiplier, is relatively easily implemented.
Meanwhile, in public-key cryptosystems, m is determined according to a guaranteed security, and in case of an elliptic curve cryptosystem (ECC), in order to guarantee sufficient security, m of 160 or
over is recommended. Thus, for large m, the area as well as the performance of hardware should be considered. In particular, in case of a multiplier taking a major part of public-key cryptosystem
calculations, the difference between the performance and the area can increase depending on the implementation method, and consequently, the difference of the performance of the entire system can
An apparatus for multiplication in GF(2^m) can be designed by a bit-serial method or a bit-parallel method. The bit-serial method has an advantage of hardware implementation with a small area, but
the operation should be repeatedly performed m or more times such that the operation time increases and the performance of the system can be lowered. Meanwhile, the bit-parallel method can be
expected to provide a high-speed operation performance, but with increasing m, the area of the hardware increases by a factor of 2 such that in case of a large system, there is difficult in
The present invention provides an apparatus and method for hybrid multiplication in finite field GF(2^m) capable of achieving trade-off between the area and the performance of an apparatus for
multiplication with optimizing the operation in finite field GF(2^m).
According to an aspect of the present invention, there is provided an apparatus for hybrid multiplication including: a matrix Z generation unit generating [m×k] matrix Z for performing a partial
multiplication of a(x) and b(x), by dividing b(x) by k bits (k≦┌m/2┐), when multiplication of m-bit multiplier a(x) and m-bit multiplicand b(x) is performed from [(m+k−1)×k] coefficient matrix of a
(x) in GF(2^m); a partial multiplication unit performing the partial multiplication ┌m/k┐k−1 times in units of rows of the matrix Z to calculate an (┌m/k┐k−1)-th partial multiplication value and a
final result value of the multiplication; and a reduction unit receiving the (┌m/k┐k−1)-th partial multiplication value fed back from the partial multiplication unit and performing reduction of the
value in order to obtain a partial multiplication value next to the (┌m/k┐k−1)-th partial multiplication value.
According to another aspect of the present invention, there is provided a hybrid multiplication method for multiplication of m-bit multiplier a(x) and m-bit multiplicand b(x) in GF(2^m) including:
generating [m×k] matrix Z for performing a partial multiplication of a(x) and b(x), by dividing b(x) by k bits (k≦┌m/2┐); by performing the partial multiplication ┌m/k┐k−1 times in units of rows of
the matrix Z, calculating an (┌m/k┐k−1)-th partial multiplication value and a final result value of the multiplication; and reducing the obtained (┌m/k┐k−1)-th partial multiplication value in order
to obtain a partial multiplication value next to the (┌m/k┐k−1)-th partial multiplication value.
The above and other features and advantages of the present invention will become more apparent by describing in detail exemplary embodiments thereof with reference to the attached drawings in which:
FIG. 1A is a diagram of the structure of a preferred embodiment of an apparatus of the present invention;
FIG. 1B is a flowchart of the operations performed by a preferred embodiment of a method of the present invention;
FIG. 2 is a diagram showing a detailed structure of register B shown in FIG. 1A;
FIG. 3 is a diagram showing a detailed structure of Z[n ]calculation unit shown in FIG. 1A;
FIG. 4 is a diagram showing a detailed structure of a matrix Z generation unit shown in FIG. 1A;
FIG. 5A is a diagram of the structure of a partial multiplication unit shown in FIG. 1A;
FIG. 5B is a flowchart of the operations performed in a partial multiplication operation shown in FIG. 1B;
FIG. 5C is a diagram showing a detailed structure of a bit multiplication unit shown in FIG. 5A;
FIG. 5D is a diagram showing a detailed structure of a bit addition unit shown in FIG. 5A;
FIG. 6 is a diagram of a detailed structure of a reduction unit shown in FIG. 1A; and
FIG. 7 is a diagram of the structure of another preferred embodiment of the present invention.
The attached drawings for illustrating preferred embodiments of the present invention are referred to in order to gain a sufficient understanding of the present invention, the merits thereof, and the
objectives accomplished by the implementation of the present invention. Hereinafter, the present invention will be described in detail by explaining preferred embodiments of the invention with
reference to the attached drawings. In the drawings, whenever the same element reappears in subsequent drawings, it is denoted by the same reference numeral.
The present invention is designed to solve the problem of the multiplication methods described above. The present invention provides an apparatus and method for hybrid multiplication in finite field
GF(2^m) capable of achieving trade-off between the area and the performance of an apparatus for multiplication with optimizing the operation in finite field GF(2^m).
For convenience of understanding, the implementation process of multiplication according to the present invention will now be explained considering the numerical formula aspect of the process.
Assuming that f(x)=x^m+x^n+1(1≦n≦m/2) is an irreducible polynomial in GF(2^m), an arbitrary element, a(x), in GF(2^m) can be expressed as
$a ( x ) = ∑ i = 0 m - 1 a i x i ,$
Here, the irreducible polynomial under condition n≧m/2 is a form of a irreducible polynomial recommended in the standards such as SEC, WTLS, ISO, NIST, and FIPS.
Assuming that
$b ( x ) = ∑ i = 0 k - 1 b i x i$
(k>m), if partial multiplication of an m-bit multiplier, a(x), and an arbitrary k-bit multiplicand, b(x), is considered, the partial multiplication d(x) of a(x) and b(x) can be expressed as the
following equation 1, and can also be expressed by using a [m+k−1]-row coefficient matrix M:
$d ( x ) = a ( x ) b ( x ) = ∑ i = 0 m + k - 2 d i x i [ d 0 d 1 ⋮ d k - 1 d k ⋮ d m - 1 d m ⋮ d m + k - 2 ] = [ a 0 0 0 ⋯ 0 0 a 1 a 0 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ a k - 1 a k - 2 a k - 3 ⋯ a 1
a 0 a k a k - 1 a k - 2 ⋯ a 2 a 1 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ a m - 1 a m - 2 a m - 3 ⋯ a m - k + 1 a m - k 0 a m - 1 a m - 2 ⋯ a m - k + 2 a m - k + 1 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ 0 a m - 1 ] [ b 0 b 1 b 2 ⋮ b k - 1 ] ( 1
Since d(x) includes terms having a higher order than [m−1], it can be reduced to the [m−1]-th order by using x^m=x^n+1. Assuming k≦┌m/2┐, the highest order term x^m+k−2 of d(x) satisfies the
following equation 2:
x ^m+k−2 =x ^k−2(x ^n+1)=x ^n+k−2 +x ^k−2(n+k−2≦m/2+┌m/2┐−2<m)(2)
That is, if k≦┌m/2┐, reduction is performed only once for each of terms of the m-th or higher order of d(x).
By this reduction, each row element d[m+j](0≧j≧k−2) of the matrix M is added once to d[n+j ]and d[j]. Assuming that Z denotes an [m×k] matrix obtained from matrix M after the reduction, matrix Z is
formed as the sum of three matrices X, T, and U, that is, Z=X+T+U. Here, matrix X is a matrix formed by obtaining 1st through m-th rows of matrix M and matrix T is a matrix formed by obtaining the
(m+1)-th and higher rows of matrix M and extending the remaining rows with 0's. Matrix U is formed by shifting matrix T down by n rows and filling 0's in the upper n rows. The three matrices are as
the following:
$X = [ a 0 0 0 ⋯ 0 0 a 1 a 0 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ a k - 1 a k - 2 a k - 3 ⋯ a 1 a 0 a k a k - 1 a k - 2 ⋯ a 2 a 1 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ a m - 1 a m - 2 a m - 3 ⋯ a m - k + 1 a m - k ] , T = [ 0 a m - 1 a m -
2 ⋯ a m - k + 2 a m - k + 1 0 0 a m - 1 ⋯ a m - k + 3 a m - k + 2 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ 0 a m - 1 0 0 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ 0 0 ]$ $U = [ 0 0 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ 0 0 0 a m - 1 a m - 2 ⋯ a
m - k + 2 a m - k + 1 0 0 a m - 1 ⋯ a m - k + 3 a m - k + 2 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ 0 a m - 1 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ 0 0 ] | 0 th row n - th row ( n + k - 2 ) - th row$
Due to predetermined regularity, an arbitrary i-th row Z[i ]of matrix Z can be obtained by signal line mapping without adding an additional logic gate. The n-th row Z[n ]of matrix Z is calculated as
the following equation 3:
Z [n]=(a [n] a [n−1 ] . . . a [0] a [m−1] a [m−2 ] . . . a [m−k+n+1])+(0a [m−1 ] . . . a [m−k+1]) if n<k
Z [n]=(a [n] a [n−1 ] . . . a [n−k+1])+(0a [m−1 ] . . . a [m−k+1]) if n≧k(3)
The partial multiplication of a(x) and b(x) is calculated as matrix multiplication of matrix Z and b(x). By using this partial multiplication and considering two m-bit multiplications, it is assumed
$a ( x ) = ∑ i = 0 m - 1 a i x i , and b ( x ) = ∑ i = 0 m - 1 b i x i .$
In order to calculate a(x)b(x) mod f(x)=c(x) that is the multiplication result of a(x) and b(x), reduction matrix Z is generated and then b(x) is divided into k bits according to the following
equation 4:
$b ( x ) = ∑ i = 0 k - 1 b i x i + ∑ i = k 2 k - 1 b i x i + … + ∑ i = ( ⌈ m / k ⌉ - 1 ) k m - 1 b i x i = ∑ i = 0 k - 1 b i x i + ∑ i = k 2 k - 1 b i x i + … + ∑ i = ( ⌈
m / k ⌉ - 1 ) k ⌈ m / k ⌉ k - 1 b i x i = T 0 + T 1 + … + T ⌈ m / k ⌉ - 1 T j = ∑ i = jk ( j + 1 ) k - 1 b i x i = x jk ∑ i = 0 k - 1 b i + jk x i$
Here, b[i]=0 (m≧i<┌m/k┐k).
Assuming s=┌m/k┐−1, by using matrix Z, it can be seen that
$a ( x ) T s = x sk a ( x ) ∑ i = 0 k - 1 b i + jk x i = x sk ∑ i = 0 m - 1 c i x i .$
As for a(x)T[s−1], k-bit reduction is performed as the following equation 5 and this is added when a(x)T[s−1 ]is calculated:
$a ( x ) T s = x ( s - 1 ) k ∑ i = 0 k - 1 c i x i + k = x ( s - 1 ) k ( ∑ i = k m - 1 c i - k x i + ∑ i = m m - 1 + k c i - k x i ) = x ( s - 1 ) k ( ∑ i = k m - 1 c i
- k x i + ∑ i = 0 k - 1 c i + m - k x i + m ) = x ( s - 1 ) k ( ( ∑ i = k m - 1 c i - k x i + ∑ i = 0 k - 1 c i + m - k x i + m ) + ∑ i = 0 k - 1 c i + m - k x i + n ) ( 5 )$
Since this reduction process can be performed in parallel when the partial multiplication of a(x)T[s−1 ]is performed, the calculation time is not delayed. As described above, the multiplication in
finite field GF(2^m) in the present invention is performed by repeatedly performing this k-bit partial multiplication and reduction from j=s to j=0.
The method for performing the multiplication described above will now be explained with reference to attached drawings.
FIG. 1A is a diagram of the structure of a preferred embodiment of the present apparatus invention and FIG. 1B is a flowchart of the operations performed by a preferred embodiment of the present
method invention.
As showing in FIG. 1A, the structure of the embodiment of the apparatus provided by the present invention includes: register A 10 storing m-bit multiplier a(x), register B 11 storing m-bit
multiplicand b(x), Z[n ]calculation unit 12 calculating the n-th row of matrix Z in advance, register Z[n ] 13 storing Z[n ]that is calculated in advance, a matrix Z generation unit 14 generating
matrix Z by using Z[n ]and a(x), a partial multiplication unit 15 performing partial multiplication of matrix Z and b(x), register C 16 storing the result value d(x) of partial multiplication, and a
reduction unit 17 performing reduction of the partial multiplication result value and adding to the next partial multiplication value.
The details of the multiplication method shown in FIGS. 1A and 1B will now be explained with reference to FIGS. 2 through 6.
Multiplier a(x) and multiplicand b(x) input are stored in register A 10 and register B 11, respectively. Register A 10 stores the m-bit input and outputs this as is and register B 11 stores the m-bit
input and outputs high-order k-bit values each time.
FIG. 2 is a diagram showing a detailed structure of register B shown in FIG. 1A. Register B 11 is formed with ┌m/k┐k registers 111 storing input values and when a multiplication is performed,
performing shift operations of stored values and multiplexers (MUX) 112 selecting between input values and shift register inputs. When an m-bit input value is stored in register B 11, according to
the equation 4, the input value is divided into k-bit units and stored, values between m and ┌m/k┐k are stored as 0's. After that time, if a multiplication begins, b[k ]values of stored high order k
bits are output and in registers 111 and 113, a shift operation is performed in units of k bits ┌m/k┐k−1 times.
When multiplier a(x) and multiplicand b(x) are stored in register A 10 and register B 11, Z[n], the n-th row value of matrix Z repeatedly used in multiplication is calculated in Z[n ]calculation unit
12 and stored in register Z[n ] 13.
FIG. 3 is a diagram showing a detailed structure of Z[n ]calculation unit 12 shown in FIG. 1A. Z[n ]calculation unit 12 is formed with [k−1] XOR operation units 123, and performs XOR operations, as
shown in equation 3, of input values 121 and 122 selected among multiplier input values according to n and k values. The calculation result value Z[n ]is stored in register Z[n ] 13 and is repeatedly
used in multiplication operations. Register Zn 13 is formed with registers having [k−1] bit inputs and outputs.
When necessary, without being stored in register Z[n ] 13, the calculation result value of the n-th row of matrix Z may be directly calculated from the output value of the Z[n ]calculation unit 12
and a multiplication can be performed. This has an advantage of reducing a hardware element required for the register Z[n ] 13 because the register is not needed. FIG. 7 is a diagram of the structure
of an embodiment of an apparatus for hybrid multiplication in GF(2^m) not using register Z[n ] 13. This structure is basically the same as that of the apparatus shown in FIG. 1, but the calculation
result value of the n-th row of matrix Z is input directly to the matrix Z generation unit 14 without being stored in register Z[n ] 13.
FIG. 4 is a diagram showing a detailed structure of the matrix Z generation unit shown in FIG. 1A.
Matrix Z is generated by using the calculation result value, Z[n], of the n-th row of matrix Z, and multiplier a(x) stored in register A 10 in operation S11. Due to predetermined regularity as
described above, matrix Z, as shown in FIG. 4, is implemented by signal line mapping without a separate logic. By repeatedly using the [k−1]-bit output of register Z[n ] 13 and the m-bit output of
register 10, each element value of the [m×k] bit matrix Z is generated.
FIG. 5A is a diagram of the structure of the partial multiplication unit shown in FIG. 1A, and FIG. 5B is a flowchart of the operations performed in the partial multiplication operation shown in FIG.
If each element value of matrix Z is generated, partial multiplication of this value and the output value, b[k], of register B 11 is performed in the partial multiplication unit 15 in operation S12.
The partial multiplication unit 15 is formed with a bit multiplication unit 151 performing bit multiplication of the [m×k] bit output of the matrix Z generation unit 14 and k-bit output b[k ]of
register B 11, and a bit addition unit 152 calculating partial multiplication result value partial_mul by performing bit addition of row elements of the bit multiplication result value and bit
addition of reduced values of the pre-calculated partial multiplication results.
FIG. 5C is a diagram showing a detailed structure of the bit multiplication unit 151 shown in FIG. 5A. The bit multiplication unit 151 is formed with [m×k] AND operation units 1511 and performs bit
multiplication of m rows of matrix Z by each bit of a k-bit output of register B 11 in operation S121. Bit addition of each row element of the bit multiplication result values, bit_mul, is performed
in the bit addition unit 152.
FIG. 5D is a diagram showing a detailed structure of the bit addition unit 152 shown in FIG. 5A.
The bit addition unit 152 is implemented by [m×k] XOR operation units 1521. In order to add each row element of the output value bit_mul of the bit multiplication unit 151, the bit addition unit 152
performs bit addition in units of k rows in operation S122, and at the same time, performs further addition of reduction calculation value c_reduct corresponding to each row. Reduction is performed
in the reduction unit 17.
FIG. 6 is a diagram of a detailed structure of a reduction unit shown in FIG. 1A.
The reduction unit 17 is implemented by k XOR operation units 171 and signal line mapping as shown in FIG. 6. The reduction unit 17 receives pre-calculated partial multiplication result value
partial_mul fed back, performs a reduction operation according to the equation 5, and outputs the reduction value c_reduct in operation S13. The partial multiplication result value partial_mul is
stored in register C 16 and in the next partial multiplication, is reduced and added. If this process is repeatedly performed ┌m/k┐k−1 times, the final multiplication result c(x) is stored in
register C 16 and the multiplication result is output.
If the present invention is used, in GF(2^m) defined as irreducible polynomial x^m+x^n+1(1≧n≧m/2), trade-off between the area and the operation speed of the multiplication apparatus can be achieved
such that efficient performing the multiplication is enabled. In particular, this enables an efficient implementation of an elliptic curve cryptosystem using a large m value.
While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it will be understood by those of ordinary skill in the art that various changes
in form and details may be made therein without departing from the spirit and scope of the present invention as defined by the following claims. The preferred embodiments should be considered in
descriptive sense only and not for purposes of limitation. Therefore, the scope of the invention is defined not by the detailed description of the invention but by the appended claims, and all
differences within the scope will be construed as being included in the present invention. | {"url":"http://www.google.com/patents/US7599979?ie=ISO-8859-1","timestamp":"2014-04-17T19:05:02Z","content_type":null,"content_length":"95312","record_id":"<urn:uuid:34ff141f-06f2-4b5e-a98f-26dcb2a46531>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00578-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hi Johnathon,
Use the definition:
limf(x)=L means for each e>0 there is a d>0 such that |x-a|<d ==> |f(x)-L|<e; that is,
lim(x - 1)=3 means for each e>0 there is a d>0 such that |x-2|<d ==> |(x - 1) -3|<e.
Now choose d in terms of e. Example: let d=g(e) but be specific like d=sqrt(e). You may have to
get creative. Then you need to "transform" the |x-2|<g(e) into |x^2-1-3|<e;that is, you need
to show that |x-2|<d implies that |x^2 - 4|<e.
That's the general procedure. But there are many functions that finding an appropriate d and
doing the "transformation" is nigh unto impossible. That's why they prove that products, sums,
differences, and quotients (excluding where the denominator may be or approach zero) are
continuous. This makes all polynomials continuous. Then x^2-1 is continuous since it is a product
and a difference of the continuous functions x and 1..
Then the nice little theorem that says if a function is continuous at a then the limit as x-->a is f(a).
makes the limit here as x-->2 turn out to be f(2)=2^2 - 1 = 3.
Good luck!
By the way, what would it mean to say that L is NOT the limit of f(x) as x approches a?
Last edited by noelevans (2012-08-17 06:16:43)
Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make. | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=228789","timestamp":"2014-04-20T21:21:46Z","content_type":null,"content_length":"11088","record_id":"<urn:uuid:0fff5576-1385-4544-bff5-b1f2a673193c>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00317-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bounded Quantification is Undecidable
Results 1 - 10 of 82
, 1994
"... The design of a module system for constructing and main- taining large programs is a difficult task that raises a number of theoretical and practical issues. A fundamental issue is the
management of the flow of information between program units at compile time via the notion of an interface. Experie ..."
Cited by 268 (23 self)
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The design of a module system for constructing and main- taining large programs is a difficult task that raises a number of theoretical and practical issues. A fundamental issue is the management of
the flow of information between program units at compile time via the notion of an interface. Experience has shown that fully opaque interfaces are awkward to use in practice since too much
information is hidden, and that fully transparent interfaces lead to excessive interdependencies, creating problems for maintenance and separate compilation. The "sharing" specifications of Standard
ML address this issue by allowing the programmer to specify equational relationships between types in separate modules, but are not expressive enough to allow the programmer com- plete control over
the propagation of type information be- tween modules.
, 1994
"... This paper presents a variant of the SML module system that introduces a strict distinction between abstract types and manifest types (types whose de nitions are part of the module speci
cation), while retaining most of the expressive power of the SML module system. The resulting module system pro ..."
Cited by 224 (8 self)
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This paper presents a variant of the SML module system that introduces a strict distinction between abstract types and manifest types (types whose de nitions are part of the module speci cation),
while retaining most of the expressive power of the SML module system. The resulting module system provides much better support for separate compilation. 1
- The Computer Science and Engineering Handbook , 1997
"... This paper presents an overview of the programming language Modula-3, and a more detailed description of its type system. 1 ..."
, 1995
"... PolyTOIL is a new statically-typed polymorphic object-oriented programming language that is provably type-safe. By separating the de nitions of subtyping and inheritance, providing a name for
the type of self, and carefully de ning the type-checking rules, we have obtained a language that is ve ..."
Cited by 137 (10 self)
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PolyTOIL is a new statically-typed polymorphic object-oriented programming language that is provably type-safe. By separating the de nitions of subtyping and inheritance, providing a name for the
type of self, and carefully de ning the type-checking rules, we have obtained a language that is very expressive while supporting modular type-checking of classes. The matching relation on types,
which is related to F-bounded quanti cation, is used both in stating type-checking rules and expressing the bounds on type parameters for polymorphism. The design of PolyTOIL is based on a careful
formal de nition of type-checking rules and semantics.
- Journal of Functional Programming, 16:375 – 414 , 2006
"... Recent years have seen the development of several foundational models for statically typed object-oriented programming. But despite their intuitive similarity, di erences in the technical
machinery used to formulate the various proposals have made them di cult to compare. Using the typed lambda-calc ..."
Cited by 119 (3 self)
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Recent years have seen the development of several foundational models for statically typed object-oriented programming. But despite their intuitive similarity, di erences in the technical machinery
used to formulate the various proposals have made them di cult to compare. Using the typed lambda-calculus F! as a common basis, we nowo er a detailed comparison of four models: (1) a
recursive-record encoding similar to the ones used by Cardelli [Car84],
- In Proc. 29th Int’l Coll. Automata, Languages, and Programming, volume 2380 of LNCS , 2002
"... Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M
is typable. For example, suppose there is a derivation in S ending with the judgement A M : # meanin ..."
Cited by 86 (12 self)
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Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M is
typable. For example, suppose there is a derivation in S ending with the judgement A M : # meaning that M has result type # when assuming the types of free variables are given by A. Then (A, #) is a
typing for M .
, 1993
"... We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML. This family comprises all levels of the stratification of the second-order
lambda-calculus by "rank" of types. We show that typability is an undecidable problem at every rank k >= 3 o ..."
Cited by 78 (14 self)
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We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML. This family comprises all levels of the stratification of the second-order
lambda-calculus by "rank" of types. We show that typability is an undecidable problem at every rank k >= 3 of this stratification. While it was already known that typability is decidable at rank 2,
no direct and easy-to-implement algorithm was available. To design such an algorithm, we develop a new notion of reduction and show howto use it to reduce the problem of typability at rank 2 to the
problem of acyclic semi-unification. A by-product of our analysis is the publication of a simple solution procedure for acyclic semi-unification.
, 2000
"... The need for subtyping in type-systems with dependent types has been realized for some years. But it is hard to prove that systems combining the two features have fundamental properties such as
subject reduction. Here we investigate a subtyping extension of the system *P, which is an abstract versio ..."
Cited by 70 (6 self)
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The need for subtyping in type-systems with dependent types has been realized for some years. But it is hard to prove that systems combining the two features have fundamental properties such as
subject reduction. Here we investigate a subtyping extension of the system *P, which is an abstract version of the type system of the Edinburgh Logical Framework LF. By using an equivalent
formulation, we establish some important properties of the new system *P^, including subject reduction. Our analysis culminates in a complete and terminating algorithm which establishes the
decidability of type-checking.
, 1997
"... The ease of understanding, maintaining, and developing a large program depends crucially on how it is divided up into modules. The possible ways a program can be divided are constrained by the
available modular programming facilities ("module system") of the programming language being used. Experien ..."
Cited by 58 (0 self)
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The ease of understanding, maintaining, and developing a large program depends crucially on how it is divided up into modules. The possible ways a program can be divided are constrained by the
available modular programming facilities ("module system") of the programming language being used. Experience with the Standard-ML module system has shown the usefulness of functions mapping modules
to modules and modules with module subcomponents. For example, functions over modules permit abstract data types (ADTs) to be parameterized by other ADTs, and submodules permit modules to be
organized hierarchically. Module systems with such facilities are called higher-order, by analogy with higher-order functions. Previous higher-order module systems can be classified as either opaque
or transparent. Opaque systems totally obscure information about the identity of type components of modules, often resulting in overly abstract types. This loss of type identities precludes most
interesting uses of hi...
- In Conf. Rec. POPL ’99: 26th ACM Symp. Princ. of Prog. Langs , 1999
"... Principality of typings is the property that for each typable term, there is a typing from which all other typings are obtained via some set of operations. Type inference is the problem of
finding a typing for a given term, if possible. We define an intersection type system which has principal typin ..."
Cited by 52 (17 self)
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Principality of typings is the property that for each typable term, there is a typing from which all other typings are obtained via some set of operations. Type inference is the problem of finding a
typing for a given term, if possible. We define an intersection type system which has principal typings and types exactly the strongly normalizable -terms. More interestingly, every finite-rank
restriction of this system (using Leivant's first notion of rank) has principal typings and also has decidable type inference. This is in contrast to System F where the finite rank restriction for
every finite rank at 3 and above has neither principal typings nor decidable type inference. This is also in contrast to earlier presentations of intersection types where the status (decidable or
undecidable) of these properties is unknown for the finiterank restrictions at 3 and above. Furthermore, the notion of principal typings for our system involves only one operation, substitution,
rather than severa... | {"url":"http://citeseerx.ist.psu.edu/showciting?doi=10.1.1.39.7516","timestamp":"2014-04-16T06:14:09Z","content_type":null,"content_length":"36106","record_id":"<urn:uuid:b74509c9-4ba5-40b7-b14b-e94dff28772b>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00577-ip-10-147-4-33.ec2.internal.warc.gz"} |
Patent US6466566 - Low complexity adaptive interference mitigating CDMA detector
The present invention relates to the field of Code Division Multiple Access (CDMA) communications with direct sequence/spread spectrum (DS/SS) modulated signals. It is concerned in particular with a
novel adaptive signal receiver for CDMA communication systems.
A CDMA communication system transmits information from a number of users over a common channel using different code sequences referred to as signatures. In these systems, the transmitters send the
information independently from each other. Therefore, signals from different users arrive asynchronously at the receiver. Because the cross correlations between the signatures for signals from
different users are non zero, the received signal from a user is sensitive to nearby interfering signals from other users.
This sensitivity to interference from other users contrains the number of active users operating at a specified system bit error rate (BER), and therefore CDMA systems are said to bear a modest
overall capacity in terms of number of users/bandwidth.
Some attention has already been focused on the Multiple Access Interference (MAI) and in the last decade there has been an explosion of interest on multiuser interference rejection and multiuser
detection (MUD). The purpose of MUD is to achieve the joint demodulation of all user data streams. One of the major outputs of the extensive research activity, inspired by the discovery of MUD, was
the recognition that the sensitivity to MAI of CDMA is not inherent to the access method itself, rather it is essentially related to the incorrect utilization, in a multiple-access environment, of
the single-user correlation receiver that presently represents the optimum scheme for processing a single DC/SS signal in an additive white Gaussian noise (AWGN) channel (see “Optimum Multiuser
Asymptatic Efficiency” by S. Verdú, IEEE Transactions on Communications, September 1986, pp. 890-897).
The implementation of a multiuser receiver is not straightforward: firstly due to the high computational complexity for the optimum scheme and secondly because even sub-optimum lower-complexity
schemes call for accurate estimation of the code timing, the carrier phase and/or the signal power of all the active network users. The latest step towards a practical implementation of a
MAI-resilient detector is represented by the class of single-user adaptive receiver an example of which is disclosed for instance in “MMSE Interference Suppression for Direct-Sequence Spread-Spectrum
CDMA”, by Upamanyu Madhow et Michael L. Honig, IEEE Transactions on Communications, Vol. 42, No. 12, December 1994, pp. 3178-3188. This type of receiver of capable of eliminating a substantial
portion of the multiuser interference, but it needs to be adjusted by a proper training sequence that must be known a priori, before data transmission can occur.
A robust and simple receiver of this kind that can operate without need for a training sequence is disclosed in “Blind Adaptive Multiuser Detection”, by Michael Honig, Upamanyu Madhow, Sergio Verdú,
IEEE Transactions on Information Theory, Vol. 41, No. 4, July 1995, pp. 944-960. The simulation results presented therein are relative to the case of BPSK DS/SS synchronous signals with rectangular
chips, ideal coherent detection and a low AWGN level. The ideal coherent detection is in fact a strong assumption that largely reduces the applicability of blind adaptive interference-mitigating
detection (BAID) scheme to practical systems. The use of the BAID detector requires the prior knowledge of the useful channel carrier frequency and phase that is generally unknown to the receiver and
that can not be accurately estimated prior to DS/SS signal detection.
In particular, the authors of said publication envisage application of BAID to the initial receiver acquisition phase only, just to bring the signal-to-noise ratio after detection in the vicinity of
0 dB.
In such conditions, a decision-aided detector can directly take over without the need of a special training sequence. Such decision-aided detector is disclosed in “Adaptive Receiver Structures for
Asynchronous CDMA Systems”, by Predag B. Rapajic and Branka S. Vucetic, IEEE Journal on Selected Areas in Communications, Vol. 12, No. 4, May 1994, pp. 685-697. Although optimal in performance, a
data-aided detector requires the prior knowledge of the transmitted symbols. Assuming a slowly variant channel as compared to the symbol duration, it is possible to insert a preamble of known data in
the data stream at regular intervals to help the detector convergence.
In this case, however, there is a penalty due to the increased data rate and the need for fast converging algorithms like Recursive Least Square that drastically increases the algorithm complexity
and presents numerical stability problems. While the decision-aided approach does not require the prior knowledge of a training sequence, it has however a is major drawback in its poor performance
under low signal-to-noise ratio conditions whereby symbol estimates become unreliable and detector performance rapidly degrades. This behavior is particularly harmful when the signal amplitude
significantly fades, as is the case in many practical (mobile) systems or when path diversity is exploited.
In this case, equalization has to be performed on each Rake receiver finger at lower input signal-to-noise ratio. Furthermore, the adaptive detection approach in this known detector is not
insensitive to the unknown phase of the useful signal and the useful channel carrier frequency error.
It is known that the data-aided MMSE algorithm provides intrinsic phase estimation on top of interference mitigation. This is an apparent advantage as it relates the algorithm convergence speed for
interference mitigation to the carrier phase tracking. Over fading channels the algorithm performs poorly unless the useful signal phase is pre-corrected before entering the adaptive MMSE detector
(see A. N. Barbosa, S. L. Miller “Adaptive Detection of DS/CDMA Signals in Fading Channels” IEEE Trans. on Comm., Vol. 46, No. 1, January 1998 whereby a solution based on a complicated phase
estimates to remove phase errors at the detector input is proposed). The proposed solution being phase transparent does not suffer from this important problem and allows to effectively separate the
useful signal phase estimation from CDMA interference mitigation.
The present invention deals with an extension of the blind adaptive interference mitigating detection (BAID) scheme to the general case of asynchronous CDMA signals with arbitrary interferer timing
and frequency offset in order to enhance the performance of a DS-CDMA receiver with affordable complexity increase.
It is an object of this invention to provide a blind adaptive receiver for CDMA signals which minimises the detrimental effect of multiple access interference (MAI) on the bit-error rate (BER)
performance and which does not require using a known training sequence.
A further object of this invention is to provide a low-complexity blind adaptive receiver for CDMA signals that is robust to asynchronous MAI.
Yet another object of this invention is to provide a blind adaptive receiver scheme that is robust to the interferers frequency error and is invariant to a possible carrier phase offset.
These and other objects of the invention are attained by an adaptive receiver for CDMA signals which comprises means arranged to perform the symbol detection in the incoming data stream using a user
signature sequence comprised of a first fixed term and a second complex adaptive part having a predetermined length that extends over a number of samples. Means are provided to update automatically
and at successive regular intervals encompassing the duration of one symbol or a plurality of symbols.
The second complex adaptive part is determined starting from the error signal of the carrier frequency measured on the signal obtained after detection.
The blind adaptive detector incorporated in the CDMA receiver of the invention can be implemented in modular form well suited for a compact ASIC digital implementation on a user terminal. The
embodiment includes unobvious techniques to avoid the impact of a definite number of bits being used by DSP/ASIC to represent the signals.
The solutions proposed herein are particularly suited for application to satellite communication systems, but they can be adapted for use in terrestrial communication systems as well. Also, the
invention can be adapted for use in communication systems having a variable transmission rate as proposed for the third-generation standards for wireless CDMA communication.
The main advantages of the invention can be summarized as follows:
1. There is no need for training sequences to help algorithm convergence nor knowledge of interferer parameters. The only parameters required are the useful channel signature sequence and chip timing
information as for a conventional detector.
2. Robustness to asynchronous CDMA interference even for large interferer power unbalance and carrier frequency offsets.
3. The detector is insensitive to the unknown phase of the useful signal and can adopt conventional BPSK symbol rate phase estimators.
4. Increased robustness to the residual basehand useful channel carrier frequency error as compared to conventional data-aided Minimum Mean Square Equalizer (DA-MMSE).
5. System performance is very close to the optimum DA-MMSE linear detector in the SNR region of practical interest for a coded system.
6. The adaptive detector can operate at very low SNR typical of a coded system and supports more spectral efficient Dual-BPSK Spread Spectrum signal formats (D-BPSK/SS).
7. It is well suited for a compact ASIC digital implementation on a user terminal.
The invention can find practical applications in the following domains:
Fixed and mobile satellite communication networks.
CDMA-based positioning systems.
Terrestrial CDMA networks.
FIG. 1 is a block diagram of a prior art blind adaptive receiver;
FIG. 2 is a block diagram of an adaptive receiver in accordance with the invention;
FIG. 3 is a functional block diagram of the detection unit shown in the receiver of FIG. 2;
FIG. 4 illustrates a variation of the adaptive receiver according to the invention;
FIGS. 5a and 5 b are functional block diagrams of two implementations of the detection unit incorporated in the receiver of FIG. 4;
FIG. 6 is a functional block diagram of a variation of the adaptive receiver of FIG. 4;
FIGS. 7a-7 c are functional block diagrams of two implementations of the detection unit incorporated in the receiver of FIG. 6;
FIG. 8 shows an optimized architecture for a variation of the detection unit according to the invention.
FIG. 9 is a diagram showing the BER performance of a system incorporating the invention.
FIG. 10 is a functional block diagram of a further variation of the adaptive receiver according to the invention.
FIG. 11 illustrates the adaptation convergence of the adaptive receiver of the invention;
FIG. 12 shows the variation of BER vs the adaptation factor Gamma for various detector types;
FIG. 13 shows the variation of BER vs Δ·f[k]·T[s ]for various detector types;
FIG. 14 shows the variation of VER vs the number of interferers for various detector types;
FIG. 15 shows the PDF vs factor ρ for various detector types;
FIG. 16 shows the variation of BER vs the ratio E[o]/N[o ]for various detector types;
FIG. 17 illustrates the simulation results showing BER vs the ratio E[o]/N[o ]for various detector types;
FIG. 18 is a functional block diagram of another variation of the adaptive receiver of the invention;
FIG. 19 is a block diagram of a multi-rate detector of the invention for the highest rate in a network;
The CDMA Signal Format
The signal format introduced here is quite general and corresponds to DS/Spread Spectrum (DS/SS) with two-dimensional modulation. In the most general case the incoming binary data stream at rate R[b
]for the k-th user is split between the two Phase-Quadrature (P-Q) rails by means of a serial-to-parallel converter. The resulting symbols a[k,p](u), a[k,q](u) ε{−1,1} are independently spread by the
P-Q signature sequences c[k,p](l), c[k,q](l) and filtered prior P-Q carrier modulation. The resulting complex k-th signal is given by $e k ( t ) = Dp k ∑ U = - ∝ ∝ [ a k , p ( u ) s k , p
( t - uT s - τ k ) + ja k , q ( u ) s k , q ( t - uT s - τ k ) ] · exp [ j ( 2 π Δ f k t + φ k ) ]$$s k , h ( t ) = ∑ l = 1 L c k , h ( l ) g T ( t - lT c ) , h =
p , q$
where D is an amplitude factor related to the signal modulation dimensionality (see Table 1), P[k ]is the k-th signal power, L is the period for both spreading sequences, T[c ]is the chip time, T[s]=
1/(2R[b])=Lt[c ]is the symbol time, Δf[k ]is the k-th carrier frequency offset with respect to the nominal frequency f[o], φ[k ]is the k-th user carrier phase, g[T](t) is the impulse response of the
chip shaping filter, and τ[k ]represents the k-th user signal delay. Without loss of generality we also assume 0≦τ[k]<T[s].
The signature sequences c[k,h](l) may be “compound” sequences such as Walsh Hadamard (WH) functions overlaid by extended pseudo-noise (PN) sequences having the same period and start epoch [1].
Compound sequences are necessary in multi-beam, multi-satellite systems, or cellular terrestrial systems wherein the overlay PN is beam/sector unique, and different WH signature sequences are
assigned to each different user within the same beam/sector. Notice that we also assumed short codes, i.e., T[s]=Lt[c ]in order for the EC-BAID to be applicable. In this case, the code length L is
also coincident with the spreading factor T[s]/T[c].
Equation (1) represents a variety of modulation and spreading formats as summarized in Table 1. For conventional detectors with symbol length spreading sequences d-BPSK and QPSK-RS doubles the
available codebook size for orthogonal sequences with no bandwidth penalty. In case of carrier phase errors, d-BPSK provides increased robustness with respect to QPSK-RS [1].
The (baseband equivalent of the) received signal r(t) is a multiplex of K different signals in the form (1), plus (the baseband equivalent of) the AWGN complex process v(t) with two-sided power
spectral density N[o]. Thus $r ( t ) = ∑ k = 1 K e k ( t ) + v ( t )$
Assuming now, for notation simplicity, that channel k=1 is the wanted channel we can re-write eqn. (2) as
r(t)=e [1](t)+J(t)+v(t)
TABLE 1
Signal formats
Scheme Modulation Spreading D T[s]/T[b] Properties
BPSK-RS BPSK real (RS) 2 1 a[k,q ]= c[k,q ]= 0
BPSK-CS BPSK complex 1 1 a[k,p ]= a[k,q]
d-BPSK 2xBPSK (CS) 1 2 c[k,p ]≠ c[k,q]
QPSK-RS QPSK 2xreal 1 2 c[k,p ]= c[k,q]
real (RS)
with J(t)^ΔΣ[k=2 ]e[k](t) representing the MAI term. The sampled chip matched filter (CMF) output can be expressed as:
y(τ)=τ(t){circle around (X)}g [R](t)|[t=rTc]
being g[R](t) the CMF impulse response.
Referring now to FIG. 1, there is shown a prior art blind adaptive receiver as disclosed in the publication “Blind Adaptive Multiuser Detection” mentioned herein before. After being translate into
the baseband, the signal y(m) is received in a baseband filter 11, e.g. a Nyquist square-root raised cosine chip matched filter, the function of which is to limit the noise bandwidth without
affecting the useful signal. The signal is then sampled at intervals Tc in sampler 12. Each sampling interval Tc is a sub-multiple of the symbol duration Ts. The complex samples are thereafter
applied to a detector 13 for being processed to deliver the demodulated symbols at the symbol rate. To this end, the detector 13 receives the user code sequence as generated by a replica code
generator 14. A replica code clock acquisition unit 15 is provided as it is in each conventional spread-spectrum demodulator. A frequency and phase estimator 16 serves to eliminate the frequency
offset and carrier phase errors.
As mentioned above herein, the receiver according to the invention is designed and arranged in such a way as to cancel the multiuser interferences in a communication channel without need for a known
training sequence prior to data transmission, nor any other information related to the interfering signals. For this purpose, the receiver performs a detection processing and incorporates means to
detect the useful channel bit stream so as to minimize the mean square error (MSE) between the actual detector output and the correct output we would get in the absence of noise and MAI.
Broadly stated, the adaptive detection processing is an extension of the one disclosed in the above-mentioned publication “Blind Adaptive Multiuser Detection”, the content of which is incorporated
herein by way of reference. The detection processing of the invention differs from the Michael Honig et al approach in that detection is made by performing a correlation on a number L of samples Y
using a modified user signature sequence thereby to minimize the effect of interfering signals while despreading the signal without loss of performance.
In accordance with the invention, the user signature sequence is composed of two components: a fixed term c[1 ](the anchor) and a complex adaptive part x[1 ](auxiliary code) that is updated at least
symbol per symbol using a recursive algorithm.
For the most simple C-VAID embodiment, this can be expressed as:
h [1](r)=c [1] +x [1](r)
c [1] =[c [1](1), . . . c [1](L)]^T
where h[1](r) is the signature sequence.
The auxiliary code has a length that encompasses the duration of at least two interfering symbols and it is automatically computed for each symbol or after several symbols using an adaptation rule
that can expressed by the following relations: $b 1 ′ ( r ) = 1 L C 1 - Y ( r )$$x 1 ( r ) = x 1 ( r - 1 ) - γ b 1 ( r ) [ y ( r ) - Y ( r ) T · c i L · C 1 ]$
b[1 ](r) is the detector output symbol,
γ is the update step to be set as a compromise
between acquisition speed and steady-state
Contrary to the prior art, the adaptation coefficients of the response vector h[1](r) are complex valued as well as the detector output symbol b[1](r).
This novel approach in accordance with this invention provides a number of advantages. First, the detector is rotationally phase invariant. This allows to exploit two (four in case of Dual BPSK
format) projections of the received signal on the two (four) dimensions of the signal space, which are in general required to perform optimum coherent signal demodulation in the presence of carrier
phase errors.
Second, the detector is resistant to non co-frequency MAI and this has big relevance to practical systems where large frequency offsets among different carriers occur with respect to the digital
signaling rate.
A third advantage resides in that the complex blind adaptive receiver of the invention can be used with D-BPSK DS/SS signals, which minimizes the standard mean square error. This feature is very
important for systems using short spreading sequences to exploit cyclo-stationary properties of the code. This is disclosed in “Two Different Philosophies in CDMA-A Comparison”, by S. Ventú and A. J.
Viterbi, IEEE Vehiculor Technology Conference, Atlanta, Ga., Apr. 28-May 1, 1996.
The receiver of the invention can operate with any spreading sequence that meets with the specified condition. In many practical systems (satellite or terrestrial), use is made of sequences composed
of a unique internal sequence for each channel within a sector or beam (sequence Walsh-Hadamard or Gold) and an external sequence having the same timing and the same length as those of the internal
sequence (pseudo-noise). Using a second slow external sequence with a pulse duration equal to the symbol duration permits to solve very large delay differences necessary for instance in case of
combination of signals from different satellites. The length of this sequence is an integer multiple of the symbol duration.
The complex response vector h[1 ]is designed so as to minimize the mean square error (MSE) between the actual detector output and the output in the absence of noise and interference (MAI). The
detector is made adaptive through a simple stochastic gradient algorithm thereby to find the solution to the minimization problem stated above. More sophisticated adaptation algorithms such as the
so-called Recursive Least Squares (RLS) or variations thereof can be used to enhance speed convergence with the penalty of higher complexity.
FIG. 2 shows an example of possible implementation for the adaptive receiver structure according to the invention Dual BPSK/SS signals. After baseband translation, the received signal y(m) enters a
chip matched filter (CMF) 21 and is then sampled every Tc seconds in sampler 22. The complex baseband samples enter then the P-Q C-BAID detectors 23. Each of them processes one of the two D-BPSK/SS
signal components and delivers the complex baseband samples at symbol rate on lines 201 and 202 respectively. Processing in the detectors 23 occurs using a user replica code sequence generated by a
replica code generator 24. Phase and frequency error detectors 25 and 26 operate at the rate 2/Tc to estimate the phase and frequency to remove the frequency offset and phase residual error prior to
final P-Q soft sample delivery.
Thanks to the rotationally phase invariance of the C-BAID detector 23 of the invention, frequency and phase errors can be estimated after despreading and adaptation, thus at a SNR typical for digital
communication systems. The implementation shown in FIG. 2 is provided for coherent detection.
The rotationally phase invariance allows the demodulator to be simplified by providing a simple differential symbol detector at the output of the C-BAID detectors 23.
Despite the C-BAID rotationally phase invariance, the carrier frequency estimation needs to be quite accurate. However, this is not a major drawback as accurate frequency estimate is required anyway
for initial signal acquisition. In order to avoid that the frequency error at the C-BAID input exceeds the tolerable range, the receiver includes a long frequency control loop 203 that receives the
frequency error signal Δω at 27 and uses it to control the downconverter oscillator 30. The phase error signal Δφ at an output 28 of the detectors 26 is applied to phase rotators 29 for the samples
of channels P and Q. The in-phase modulated symbols b^P(1) and the in-quadrature modulated symbols b^Q(2) are delivered at outputs 210 and 220, respectively.
As mentioned earlier herein, the correlation processing according to the invention uses a signature sequence comprised of a fixed term (replica code) and an adaptive part (auxiliary code) that is
updated at least symbol per symbol. This updating and the auxiliary code despreading can be performed in a simple digital circuit comprising a feedback loop with a sampling clock equal to the signal
clock. FIG. 3 shows a functional block diagram of a possible implementation for the C-BAID detector 23. The correlation is performed in two blocks 31 and 32 that perform multiply and add operations:
the first block 31 performs the correlation using the replica code C[1 ]while the second block 32 performs the correlation using the auxiliary code X[1](r). Updating the auxiliary code requires a
correlation loop 301 that is easily implementable by simple digital devices such as multiplier, adder, delay circuit, shift register etc. Of particular interest is the way the adaptive auxiliary code
is generated by a simple complex shift register 39 and an adder 37. The blocks denoted by the reference numerals 34, 36 and 38 represent delay circuits (known per se).
The outputs 310 and 320 form blocks 31 and 32, respectively, are combined in adder 33, the output of which delivers the detected symbol b[1]. It is to be noted in this regard that the correlation
processing involves an inherent delay since it is performed on L samples. As a result, while the detector 23 is detecting the symbol b[1](r), its output delivers the preceding symbol b[1](r−1).
A further interesting feature of the invention is that the response vector h[1](r) of the detector can be extended over an enlarged observation window spanning over the duration of several
interfering symbols. Tis time span increases the detector performance so as to better cope with the asynchronous CDMA interference effects. A three symbol window represents a good trade-off between
complexity increase and performance advantage. A shorter window (two symbols) may provide similar performance under certain circumstances, but does not substantially simplify the demodulation
hardware described in the following.
FIG. 4 represents a block diagram of an adaptive receiver for Direct-Sequence (DS) CDMA signals. This receiver comprises three identical detector (EC-BAID) units 43. Each of them operate over an
observation window having a three symbol length (3L) with the window being delayed relative to each other and overlapping each other. The diagram of FIG. 4 is pretty similar to that of FIG. 2 except
that each detector unit is arranged to perform a three-symbol correlation using an auxiliary code having a 3L length and that the adaptation coefficients are updated after three symbols. The blocks
41 and 42 represent digital delay lines to shift the observation windows. Block 44 represents a time multiplexer running at frequency 1/Ts. Block 45 represents a PSK demodulator running at the symbol
rate. The frequency error signal Δω is generated in the demodulator and applied to the frequency control loop 303. The reference numerals 21, 24, 25 and 30 denote a base band filter, a replica code
generator, a replica code acquisition unit and a down-converter oscillator as shown in the diagram of FIG. 2.
FIG. 5a illustrates the structure of a detector unit (EC-BAID I) similar to that of FIG. 3. On FIG. 5b there is illustrated another structure for the detector unit (EC-BAID II) that is also similar
to the structure shown in FIG. 3, except that it includes a so-called gating function 51 for multiplying the replica code C[1 ]under control of the symbol clock (runing at frequency 1/LT[c]) and the
super-symbol clock (running at frequency 1/3LT[c]).
In the diagram of FIG. 4, the detector units 43 are independent from each other, that is each of them generates a respective frequency error signal. However, the invention permits to implement a
receiver in which the error signals generated by the different detector units are combined with each other thereby to increase the detection performance. To this end, the arrangements described
before herein can be modified as illustrated in FIGS. 6 and 7. On FIG. 7a there is shown a structure for the EC-BAID I detector unit; and FIG. 7b shows a structure for the EC-BAID II detector unit.
In this case the adaptive receiver comprises a unique external auxiliary code generator 64 that is completely updated after every symbol interval. The frequency error signal on line 203 for the
generation of the auxiliary code is produced by coherent combination of the three individual error signals delivered at the output of the digital delay lines 65, 66 and 67. The reference numerals 41,
42, 61 and 62 also denote digital delay lines.
The fact that the EC-BAID I embodiment (see FIG. 5a) imposes the orthogonality condition for the auxiliary code {overscore (x)}[e ]with respect to the anchor {overscore (c)}[1 ]is due to the fact
that in this way the demodulator is robust vis-a-vis non perfect randomness of the modulating data pattern. It can be seen that the EC-BAID II performance is degraded when the data pattern is partly
non random. The EC-BAID II does not won at all in case of an unmodulated ODMA signal whilst EC-BAID I wons perfectly in all conditions with insignificant performance loss compared to EC-BAID II even
for pure random data.
The Select and Add Architecture
Another possible EC-BAID option has been dubbed Select and Add. As depicted in FIG. 8, the correlations y(r)·c[1 ]and x^e [1]·c^e [1], respectively, are evaluated in blocks 81 and 82 to yield the b
[1](r) output at symbol rate. The vector x^e [1 ]is stored in memory 86 and each of its 3L elements is updated every T[c]/3; in particular, during the i-th chip period within the r-th symbol period,
the coefficients of x[1 ]relevant to the i-th chip of y(r=1), y(r) and y(r+1) are updated. The most recent 3L input chips are stored in memory 87. Multiplexers 85 and 88 properly re-align internal
dataflow, while multiplexer 83 selects the desired EC-BAID algorithm version between I and II. More precisely for EC-BAID II the switch is set to zero for the external chuncks of the error signal as
the anchor orthogonality condition shall only be imposed for the central (useful) symbol. The AGC 84 on the feedback loop is needed in order to keep the b[1 ]value constant for the error signal
generation at different SNIR operating conditions.
The main difference between S&A and O&A resides in the evaluation of the error signals and in the relevant update of the adaptive vector x[1]: in the O&A groups of three error contributions are
summed every 3T[s ]while in the S&A each error contribution is summed separately every T[s ]and this way numerical results and convergence speed are almost equal.
The S&A architecture is characterized by a considerable hardware complexity saving in terms of both arithmetical elements and memory cells at the expenses of an increased clock rate.
In fact, for O&A proper operations three distinct arithmetical units are required and moreover, in addition to the memory capacity for x[1], some extra delays (memory elements) are needed to provide
proper timing between signals of the various circuit branches.
On the other hand, in the S&A version only one arithmetical unit is needed due to hardware multiplexing, and moreover only vector x[1 ]and 3L input (y^c) samples have to be stored, so that the extra
delay blocks previously mentioned are no longer needed.
These modifications, without affecting overall performances, allow for nearly 50% gate complexity saving and for nearly 70% RAM capacity saving at the expenses of a three times higher clock rate. The
latter may limit the S&A applicability to very high chip rate applications.
Error Signal Truncation Effects
The S&A fixed-point ASIC implementation inevitably introduces some truncation errors with respect to theory which is based on floating-point arithmetic. For this particular adaptive architecture,
which is based on a feedback loop, these quantization errors may have dramatic effects on the overall algorithm convergence. In particular, because of this quantizations, the error contributions
which are used to generate the adaptation vector x[1 ]may be not perfectly orthogonal to the code sequence vector; this situation has to be avoided because the algorithm is able to maintain a stable
steady-state value only for the x[1 ]component which is orthogonal to c^e [1 ](it is referred here to the EC-BAID II, but the same considerations are valid for the EC-BAID I, taking into account that
in that case the algorithm controls the x[1,ω. ]components orthogonal to c^e [1]). On the other hand, if the finite arithmetic effects generate a x^e [1 ]component which is not orthogonal to c^e [1],
this one may indefinitely increase without being controlled by the algorithm, and thus causing failure of the whole system.
In order to avoid this situation it is sufficient to calculate the error signal e[1 ](based on the quantized y^e and b[1 ]values as shown here below with full-precision arithmetic. $E 1 Δ b 1 [ y
c… ( τ ) - y c . ( τ ) T · c 1 L c 1 ] ;$
The error signal generated this way is perfectly orthogonal to c[1 ]thus preventing the aforesaid problem. This means that, starting from quantized y^e and b[1 ]values, the processing relevant to e[1
](and so X^e [1]) is performed with an internal word-length suitable for whole signal dynamics, so that no further truncation is introduced. FIG. 9 shows that the BER system performances (for L=128,
N=64, E[b]/N[o]=6 dB and C/1 =−6 db) when no further truncation is introduced in the evaluation of e^e [1 ](and so x^e [1]) as well as when there is just 1-bit error in the internal word-length
Finite arithmetic effects on all the other S&A circuit internal signals can be regarded as additional noise without causing any problem to the algorithm convergence towards the steady-state vector x^
e [1,opt]. Although this solution has been exemplified for the S&A case its applicability can be more general.
The Linear Combiner Architecture
The linear combiner (LC) architecture represents a hybrid solution between the Baseline and the OA/SA. The main feature is that while there are three separate auxiliary sequences for each EC-BAID
detector. Differently from the baseline EC-BAID, the error signals are linearly combined so that the three detectors are not anymore disjoint. By doing so the convergence speed is improved compared
to the baseline although the complexity advantage shown by the S&A is not achieved. Details about the combination law are provided in next Section herein. The LC EC-BAID top level architecture is
shown in FIG. 10.
EC-BAID Architectures Summary
Baseline Architecture
As shown in the previous paragraphs, the EC-BAID I and EC-BAID II versions formerly proposed require three separate units, each with its own local copy of x[1]; the first unit processes symbol
periods (r−1)-th, r-th and (r+1)-th to produce the output b[1](τ), the second one processes symbol periods (r)-th, (r+1)-th and (r+2)-th to produce the output b[1](r+1), and the third unit operates
likewise to produce b[1](r+2). This way, the generic n unit includes its own x^e,n [1 ]local vector, to be updated every 3T with the contribution (error signal) of its own output. The equations for
this baseline architecture are summarized in Table 1.
TABLE 1
Equation EC-BAID-I and EC-BAID-II for the
baseline architecture
EC-BAID-I and EC-BAID-II, Baseline output construction:
$b 1 ( 3 s + n - 1 ) = 1 L h 1 ( s ) T · y c ( 3 s + n - 1 ) with h 1 ( s ) = x 1 ( s ) + c 1$
with n EC-BAID detector index
EC-BAID-I, updating of vectors x[1]
x[1,w](s + 1) = x[1,w](s) − γe[1,w](s) w = −1, 0, 1
$ e 1 , w ( s ) Δ b 1 ( 3 s + n - 1 ) [ y w ( 3 s + n - 1 ) - Y w ( 3 s + n - 1 ) T · c 1 L C 1 ] w = - 1 , 0 , 1$
with s super-symbol index
EC-BAID-II, updating of vectors x[1]
x[1 ](s + 1) = x[1 ](s) − γe[1 ](s)
$e 1 ( s ) Δ b 1 ( 3 s + n - 1 ) [ y en ( 3 s + n - 1 ) - y en ( 3 s + n - 1 ) T · c 1 L C 1 ]$
with s super-symbol index
$c 1 Δ 0 c 1 0 , x 1 ( s ) Δ x 1 , - 1 ( s ) x 1 , 0 ( s ) x 1 , + 1 ( s ) , e 1 ( s ) Δ e 1 , - 1 ( s ) e 1 , 0 ( s ) e 1 , 1 ( s ) $
“Overlap and Add” Architecture
The EC-BAID-I and EC-BAID-II versions named “Overlap and Add” (O&A), bring about an improvement: the three units still process input data symbols producing in turn the desired output b[1], this time
using a unique vector x[1]. This vector is now updated with the sum of the three unit contributions (sum of the three error signals), still every 3T[s].
The advantage obtained from this architecture is twofold: from the area saving point of view now only one vector x[1 ]rather than 3 need to be stored, while from the speed performance point of view
the x1 updating is three times faster, since every 3T, three contributions are summed together instead of one only. The equations for the O&A architecture are summarized in Table 2.
TABLE 2
Equations EC-BAID-I and EC-BAID-II for the
“Overlap and Add” architecture
EC-BAID-I and EC-BAID-II, Overlap and Add-output construction:
$b 1 ( 3 s + n - 1 ) = 1 L h 1 ( s ) T · y c ( 3 s + n - 1 ) with h 1 ( s ) = x 1 ( s ) + c 1$
with n EC-BAID detector index
EC-BAID-I Overlap and Add, updating of vectors x[1]
x[1,w](s + 1) = x[1,w](s) − γ[e[1,w](s − 1) + e[1,w](s − 1)] w = −1, 0, 1
$e 1 , w ( s ) Δ b 1 ( 3 s + n - 1 ) [ y w ( 3 s + n - 1 ) - y w ( 3 s + n - 1 ) T · c 1 L C 1 ] w = - 1 , 0 , 1$
with s super-symbol index
EC-BAID-II Overlap and Add, updating of vectors x[1]
x[1](s + 1) = x[1](s) − γ[e[1 ](s − 1) + e[1 ](s − 1) + e[1 ](s − 1)]
$e 1 ( s ) Δ b 1 ( 3 s + n - 1 ) [ y en ( 3 s + n - 1 ) - y en ( 3 s + n - 1 ) T · c 1 L C 1 ]$
with s super-symbol index
$c 1 Δ 0 c 1 0 , x 1 ( s ) Δ x 1 , - 1 ( s ) x 1 , 0 ( s ) x 1 , + 1 ( s ) , e 1 ( s ) Δ e 1 , - 1 ( s ) e 1 , 0 ( s ) e 1 , 1 ( s )$
“Select and Add” Architecture
The final architecture proposed to implement EC-BAID-I and EC-BAID-II, named “Select and Add” (S&A) provides the same bit error rate and convergence speed performances of the O&A one but it allows a
considerable circuit complexity reduction.
Its top level block diagram shown in FIG. 2, where the detector unit is now the one sketched in FIG. 8.
The S&A architecture exploits the possibility of using a clock of period T[c]/3 so that it is possible to use the arithmetical part of the circuit three times for each T[c]; this allows to calculate
the output b[1](τ) in one period T[s], getting the entire product x^e [1](r)^T·c^e [1 ]and to update, in the same period T[s], the whole vector x^e [1 ]with the 3L error signal coefficients. The
equations for the S&A architecture are shown in Table 3.
The Linear Combiner Architecture
The linear combiner (LC) architecture represents a hybrid solution between the Baseline and the OA/SA. The main feature is that while there are three separate auxiliary sequences for each EC-BAID
detector they interact through a linear combination of the respective error signals as it has been discussed before.
The functioning of the LC EC-BAID is regulated by the set of equations shown in Table 4:
TABLE 3
Equations EC-BAID-I and EC-BAID-II for the
“Select and Add” architecture
EC-BAID-I and EC-BAID-II, Select and Add-output construction:
$b 1 ( r ) = 1 L h 1 ( r ) T · y c ( r ) with h 1 ( r ) = x 1 ( r ) + c 1$
EC-BAID-I Select and Add, updating of vectors x[1]
x[1,w](r + 1) = x[1,w](r) − γ e[1,w](r − 1) w = −1, 0, 1
$e 1 , w ( r ) Δ b 1 ( r ) [ y w ( r ) - y w ( r ) T · c 1 L C 1 ] w = - 1 , 0 , 1$
with r symbol index
EC-BAID-II Select and Add, updating of vectors x[1]
x[1](r + 1) = x[1](s) − γ e[1 ](r − 1)
$e 1 ( r ) Δ b 1 ( r ) [ y en ( r ) - y en ( τ ) T · c 1 L C 1 ]$
with r symbol index
$c 1 Δ 0 c 1 0 , x 1 ( r ) Δ x 1 , - 1 ( r ) x 1 , 0 ( r ) x 1 , + 1 ( r ) , e 1 ( r ) Δ e 1 , - 1 ( r ) e 1 , 0 ( r ) e 1 , 1 ( r )$
TABLE 4
Equations EC-BAID-I and EC-BAID-II for the
“Linear Combiner” architecture (n = 1, 2, 3)
EC-BAID-I and EC-BAID-II, Overlap and Add-output construction:
$b 1 ( 3 s + n - 1 ) = 1 L h 1 ( s ) T · y c ( 3 s + n - 1 ) with h 1 ( s ) = x 1 ( s ) + c 1$
with n EC-BAID detector index
EC-BAID-I Linear Combiner, updating of vectors x[1]
x[1,w](s + 1) = x[1,w](s) − γ[e[1,w](s) + e[1,w](s) + e[1,w](s)]
x[1,w](s + 1) = x[1,w](s) − γ[e[1,w](s + 1) + e[1,w](s) + e[1,w](s)] w = −1, 0, 1
x[1,w](s + 1) = x[1,w](s) − γ[e[1,w](s + 1) + e[1,w](s + 1) + e[1,w](s)]
$ e 1 , w ( s ) Δ b 1 ( 3 s + n - 1 ) [ y w ( 3 s + n - 1 ) - y w ( 3 s + n - 1 ) T · c 1 L C 1 ] w = - 1 , 0 , 1$
with s super-symbol index
EC-BAID-II Linear Combiner, updating of vectors x[1]
x[1 ](s + 1) = x[1 ](s) − γ[e[1 ](s) + e[1 ](s) + e[1 ](s)]
x[1 ](s + 1) = x[1 ](s) − γ]e[1 ](s + 1) + e[1 ](s) + e[1 ](s)]
x[1 ](s + 1) = x[1 ](s) − γ]e[1 ](s + 1) + e[1 ](s + 1) + e[1 ](s)]
$e 1 ( s ) Δ b 1 ( 3 s + n - 1 ) [ y en ( 3 s + n - 1 ) - y en ( 3 s + n - 1 ) T · c 1 L C 1 ]$
with s super-symbol index
$c 1 Δ 0 c 1 0 , x 1 ( s ) Δ x 1 , - 1 ( s ) x 1 , 0 ( s ) x 1 , + 1 ( s ) , e 1 ( s ) Δ e 1 , - 1 ( s ) e 1 , 0 ( s ) e 1 , 1 ( s )$
Numerical results, both analytical and computer simulated, show excellent and fast convergence of the adaptation algorithm and further show an independence of the mean square error to the phase
offset of interfering signals. This convergence of the adaptation algorithm is illustrated by the diagram of FIG. 11 that shows the convergence of the real and imaginary parts of the adaptation
coefficients in the conditions specified.
It should also be mentioned that the blind adaptive receiver of the invention shows an excellent robustness to the residual frequency errors of the useful signal at the input of the detector unit.
This result is obtained thanks to the use of complex coefficients for the auxiliary code as outlined earlier herein. This behaviour of the receiver of the invention is illustrated by the diagram of
FIG. 12 showing the variation of the binary error rate BER versus the adaptation factor γ for various detector types.
One of the strong points in favor of the (E)C-BAID characterized by complex-valued coefficients, is related to its performance independence to interferers frequency offset. Such same frequency offset
can be quite large compared to the symbol rate in a Low- or Medium Earth Orbiting (LEO-MEO) satellite constellation. This important (E)C-BAID feature is testified by FIG. 13.
The diagram of FIG. 13 shows that the receiver according to this invention is more robust to frequency error than a known DA-MMSE detector. More particularly, the robustness to frequency error of the
receiver of the invention is three orders of magnitude larger than that of the DA-MMSE schema. It is thus compatible to the typical standard deviation of the frequency estimator 25. In addition,
cancellation of MAI expressed in terms of BER is comparable to the performance of the data-aided adaptive processing.
With regard to the number of interfering users, the binary error rate (BER) of the detector of the invention reveals that the extended version of the detector makes it possible to increase
substantially the number of interfering users even in cases where the useful signal power is substantially lower than the interfering signal power. This is exemplified in the case of a satellite
communication system in which the power control is slow because of the propagation time. This performance is comparable to that of the conventional data-aided adaptive detector (EDA-MSE) as shown by
FIG. 14 which represents the BER versus the number of interfering signals for various adaptivbe detector types with different values of the ratio E[b]/N[o ]between the average received energy of the
quaternary constellation symbols at the edge of the beam and the one-sided AWGN (additional white gaussian noise) power spectral density N[o].
The impact of the different types of detectors is well evidentiated looking at the statistical distribution of the total signal-to-noise plus interference ratio at the detector output ρ.
FIG. 15 shows the probability density function (PDF) of ρ for the CR, the C-BAID and the EC-BAID, with random interferers' carrier phases and code delays. It is fairly evident that the values of ρ at
the output of the CR show a relatively large dispersion, due to the different delay and carrier phase assignments on the active channels. The effect of the interference-mitigating detectors is
twofold: first, they obviously increase the mean value of ρ so as to improve the mean quality of the link; second, they reduce the dispersion of ρ around its mean value, and this has the effect of
reducing the outage probability of the link. CDMA with long codes (sometimes referred to as Random CDMA) is not easily amenable to interference mitigation, but the use of long codes causes a sort of
symbol-by-symbol randomization of MAI to pursue the same goal of outage probability reduction. This attractive feature of the EC-BAID is further enhanced by the use of d-BPSK or QPSK-RS signal
formats. FIG. 11b shows simulation results for the pdf of ρ in the case of BPSK, d-BPSK and QPSK. The occupied bandwidth being equal, d-BPSK/QPSK-RS allow the use of signature sequences having a
repetition period which is twice that of the codes of the BPSK-RS or BPSK-CS cases, thus doubling the codebook size (i.e., the maximum number of sequences available) and yielding a twofold capacity.
Also, looking at FIG. 11b it can be observed that, though BPSK and d-BPSK have similar average value of ρ, the variance is reduced for the latter. Compared to d-BPSK, QPSK-RS provides a further
advantage in terms of average ρ increase and σ[τ] reduction due to the fact that each signal employs one and not two spreading sequences as for d-BPSK. Although immaterial to the CR, this feature
reduces the number of space dimensions occupied by the CDMA multiplex, thus enhancing EC-BAID interference mitigation capabilities. The QPSK-RS advantage versus BPSK-RS and d-BPSK in terms of BER for
a typical (fixed) interferer delay distribution is shown in FIG. 16. Monte Carlo BER simulations results utilizing the adaptive LMS version of the EC-BAID described earlier closely follows
theoretical predictions.
Fading Performance
Thanks to the previously described (E)C-BAID rotationally phase invariance the proposed detector is capable to provide remarkable performance also over flat fading channels where both useful signal
and interferers ar affected by independent Rician fading with Rice factor K and the user speed ranges from 5 to 80 mph (assuming a 2 GHz carrier frequency). As the simulation results in FIG. 17
indicate, the EC-BAID provides a remarkable advantage over conventional CDMA correlation receivers even other fast fading channels. In all simulations the useful signal carrier phase has been
(ideally) estimated after the EC-BAID. Although results in FIG. 17 assume ideal channel estimation, similar performance have been achieved using pilot-aided channel estimation techniques.
It was observed that simulation results remain practically unchanged by removing the fading on the CDMA interferers even in the case of fast fading. This fact demonstrates that the EC-BAID is
intrinsically robust with respect to MAI flat fading fluctuations (both in amplitude and phase).
Satellite Path Diversity
The proposed (E)C-BAID detector is also well suited for mobile satellite systems whereby satellite path diversity is adopted. In this case typically a different spreading sequence is used for each
forward link satellite signal although the information bits transmitted are the same. In this case the FIG. 18 block diagram is modified as shown in the following FIG. 15 whereby a same EC-BAID
possible implementation is exemplified. Assuming that up to N[D ]diversity order is implemented in the system, then the demodulator is composed by N[D ]EC-BAID each of them dealing with the signal
coming from a different satellite. Because of the different satellite geometry independent fine frequency and time/phase adjustment is required on each finger. In case the differential delay exceeds
the symbol duration (as it is often the case), the delay ambiguity shall be resolved by means of external and as frame unique word not shown here for simplicity.
In any case, the symbols of the same data stream transmitted by different satellites shall be spread by a satellite and user unique sequence and staggered in time so that there is no possibility that
the same symbol (±1 symbol in case of EC-BAID) overlaps with another diversity path coming from a different satellite.
This staggering technique is illustrated as follows:
T[s ]= LTo
Satellite d[1] d[2] d[3] Transmitted
#1 d[4 ]symbols
Satellite d[1] d[2]
(E)C-BAID Maximum Guard
window Differen- time
(BLTc) tial
Actual diversity
signals time offset
This modulator symbol staggering is then recovered at the demodulator side by means of the delay adjustment block shown in FIG. 18. This is a very important technique to avoid possible destructive
interaction between useful and diversity path within each EC-BAID demodulator.
EC-BAID Application to Multi-rate CDMA
All of the third-generation standards for wireless CDMA, be it terrestrial or satellite, encompass a multi-rate access capability to accommodate best multimedia services. All rates are integer
sub-multiples of a maximum rate R[M ](e.g., 2048 or 384 kb/s) which depends on the kind of terminal (fixed, indoor, fully mobile etc.) and network (indoor, outdoor terrestrial, satellite etc.).
Usually, but not necessarily, the supported bit rates are in the form R[ck]=R[M]/2^k just for easier implementation and coordination, while of course the chip rate R[c ]is invariably the same on
every allocated channel.
A recently proposed approach to deal with multi-rate CDMA makes use of the so-called OVSF codes. The OVSF codes family is just a re-labeled version of the popular set of Walsh-Hadamard codes class,
wherein the re-labeling is the value-added feature. Specifically, the OVSF are a re-organization of the Walsh code in layers. The codes on each layer have twice the length of the codes in the layer
Also, the codes are organized in a tree, wherein any two “children” codes on the layer underneath a “parent” code are generated by repetition and repetition with sign change, respectively. The
peculiarity of the tree is that a pair of codes is not only orthogonal within each layer (each layer is just the complete set of the Walsh codes of the corresponding length), but they are also
orthogonal between layers, provided that the shorter one is not an ancestor of the longer.
FIG. 19 illustrates an example of two user signals with different bit rates, namely, rate R[o ]for the upper signal and rate 4R[o ]for the lower one. The former uses the length-8 code C[3](3), while
the latter employs the length-2 code C[1](1). The two codes are orthogonal since the longer one does not have C[1](1) as ancestor at layer 1. Therefore, a conventional correlation receiver suffers
from no MAI either when used on signal 1 or on signal 2.
Let us first examine the demodulator for the “slow” signal # 1. It can be said that the interference generated by signal # 2, when observed on a “long” symbol time T[c]=1/R[c ]bears a time-varying
spreading code. Assuming a[2](1) as the “reference” symbol, signal #2 can be fictitiously seen as a rate R[o ]DS-SS signal transmitting just a[2](1), and with a spreading code equal to
[C [1](1)a [2](1)a [2](2)C[1](1)a [2](1)a [2](3) C[1](1)a [2](1)a [1](1)a [2](4)C [1](1)]
Of course, if the signals are synchronous, they stay orthogonal by virtue of the OVSF codes properties, and the time-variance described above is immaterial. An issue may arise when the multi-rate MAI
is asynchronous to the desired user (i.e., inter-cell or inter-beam interference in the forward link). Assume now we have to demodulate the generic user signal #m in a CDMA network with asynchronous
interference (either intra-cell/beam MAI in the uplink or inter-cell/beam MAI in the downlink), and assume also that it has the highest rate allowed in the network. The relation of all interfering
signal to signal #m will be similar to that of signal #1 to signal #2, all of them will bear a longer code than user #m. The latter will therefore see a set of cyclically varying spreading codes on
the interfering signal, whose repetition period will be in general the ratio M between the highest and the lowest bit rate allowed in the network. The interference-mitigating detector for user #m can
therefore be designed exploiting this cyclical regularity. It will be made of a bank of M conventional EC-BAIDs which are cyclically operated every M symbol periods, in such a way that everyone
always sees the same (sub-)code on the interfering signals. The outputs of such detectors are then “demultiplexed” so as to give a symbol-rate stream for subsequent processing.
The situation is different if user #m has the lowest rate in the network (basic rate). In such condition, it sees a random-varying set of codes, according to the remarks in sect. 2. The solution to
this case can be found as follows. What we should have to do now is to split the symbol interval into M sub-intervals and to use again M different EC-BAIDs each matched to the relevant sub-code, and
each yielding a different soft-valued output. With this partitioning, each detector would see now the same code from symbol interval to symbol interval, and would follow a conventional design. The M
soft outputs of such detectors would then be combined to yield the final decision variable. This sort of multi-layer linear detector is fully equivalent to a linear detector for the whole length of
the symbol. Therefore, the conclusion is that a conventional EC-BAID with no modifications can be used in the case of a lower-rate signal.
Of course, if user #m has an intermediate bit-rate, the only parameter that matters is the ratio between its actual rate and the maximum rate in the net to build-up a multi-rate architecture as in
FIG. 3. This figure represents the functional block diagram for a user operating at a bit rate M times the basic system bit rate 1/T[h]. As it is apparent from the previous discussion, the basic
EC-BAID detectors can be reused although each of the M parallel detectors operates on a disjoint short symbol. The M detectors are sequentially activated with periodicity T[h] ^slow and a duty cycle
T[h] ^slow/M. This can be easily achieved distributing the clocks with a selector similar to the one distributing the input samples to the different EC-BAID detectors. The output soft samples at rate
1/T[h] ^fast M/T[h] ^slow are sequentially obtained from detectors 1 to M. The selection procedure is then repeated at the end of each period T[h] ^slow. It is also evident that the number of EC-BAID
detectors required to a specific demodulator operating in a CDMA environment employing OVSF multi-rate techniques is proportional to the ratio between the maximum rate supported with respect to the
basic network rate. The additional demodulator complexity is therefore only impacting users transmitting at higher rates then the basic one.
EC-BAID Applicability to Quasi-Random CDMA
One of the major disadvantages of symbol length sequences is that in case of a stationary system geometry the cross-correlation between two signals will repeat on a symbol-by-symbol basis thus
penalizing users experiencing worst-case cross-correlation. A technique commonly adopted to overcome this problem in many practical systems consists of overlaying the user specific channelization
sequence with a PN scrambling sequence having duration equal to an integer multiple (m) of the symbol duration. By doing so the cross-correlation is changing on a symbol-by-symbol basis and repeats
after m symbols. This is the approach that has been adopted for the CDMA cellular Telecommunication Industry Association (TIA) standard IS-95 and the third generation (IMT-2000) Wideband CDMA
proposed by several countries to the International Telecommunication Union (ITU).
It is easy to see that this quasi-random CDMA (a true random CDMA system will have a PN sequence never repeating) can be reconducted to the multi-rate case whereby a bank of (E)C-BAID detectors
working in parallel but with duty cycle staggered in time, will make possible to cope with the symbol-by-symbol varying spreading sequence. It is easy to show that the random CDMA performances can be
closely achieved with a relative little value for m (e.g. 4, 8) thus with an affordable demodulator complexity.
EC-BAID Applicability to Frequency Selective Channels
In case of terrestrial mobile communications (especially in urban areas), the radio channel can not be considered frequency-flat any longer, due to the large number of propagation paths generated by
multiple signal reflections on buildings that cause the propagation channel to be frequency-selective. As a consequence, in this case, the BAID algorithm, that was designed for a non
frequency-distorting channel, may not work properly anymore. Therefore, the BAID algorithm must be suitably modified in order to retain its effectiveness in combating the MAI also in this harsh
propagation scenario. For the sake of simplicity, in the following we will refer to a two-ray only propagation channel (i.e. direct plus reflected path), but the extension of the relevant concepts to
a more involved case of multipath propagation with more than two rays is rather straightforward. Under these assumptions, the complex array of the received signal samples w(r) can be expressed as
y(r) is the complex array of the signal samples we would receive in the absence of multipath propagation (direct path);
y′(r) is the complex array of the delayed signal samples (reflected path);
α is a complex coefficient representing the attenuation of the delayed path.
From the detector standpoint, the useful (i.e. the information-bearing) part of the received signal appears as it be spread by a modified signature code s[1 ]that can be expressed as follows:
s [1] =c [1] +αc [1],
where assuming for simplicity of illustration that the multipath is delayed by an integer number of chips we can write: $c 1 = [ 0 , 0 , … , 0 , c 1 , 1 m , c 1 , 2 , … , c 1 , 4 - m L = m ]$
represents the delayed (truncated) replica of the spreading code, and m is the time delay, normalized with respect to the chip interval T[t]. This fact causes a sort of ‘mismatching’ between the
actual received signature s[i ]and the expected nominal code c[1]. As matter of fact, the BAID algorithm updates the adaptive part of the detector coefficients x[1](r) on the basis of the ‘anchoring’
condition c[l]·x[1](r)=0 that leads to the cancellation of every signal belonging to the subspace orthogonal to c[1]. Unfortunately, in the case of multipath propagation (or any other mismatch
condition caused for example by incorrect input signal sampling), the a-priori anchor c[1 ]does not match anymore the input signal, and it is thus causes useful signal partial cancellation. This
leads to a waste of useful signal power, that definitely causes the algorithm to become less efficient as the contribution of the delayed path gets larger (i.e. for increasing α).
This inconvenience can be circumvented by resorting to a simple modification of the original BAID algorithm. In particular, the coefficients of the detector h[1](r) shall now be expressed as:
h [1](r)=s [1] +x [1](r),
and the ‘anchoring’ condition shall be modified accordingly: s[1]·x[1](r)=0. The new version of the algorithm thus becomes:
x [1,1](r)=x [1,1](r−1)−γb [1](r−2 )
y[i](r−2)−b [1](r−2)s [1,i]]
where $b 1 ( r ) = h T · y ( r ) s 1 2 , b 1 ( r ) = ( s 1 ) · y ( r ) s 1 2$
and x[1,1](r), y[1](r), s[1,1 ]are the i-th elements of x[1](r), y(r) and s[1], respectively. The above modified algorithm allows to not only avoid the destructive multipath effects on the EC-BAID
detector but also to coherently combine the multipath energy avoiding the use of the rake demodulator typically used in conventional CDMA detectors operating over frequency selective fading channels.
Essentially, in the modified algorithm, the nominal signature code c[1 ]shall be replaced by the actual received signature code (or anchor) s[1], which takes into account for the multipath
propagation phenomena. It should be observed that this modified anchor s[1 ]is not anymore a binary code, as it was the case for the flat fading case, but a complex-valued sequence. The information
concerning the channel delay profile, can be obtained from the pilot channel, which in the forward link is a DS/SS-CDMA signal broadcast with very low-rate (or without) data modulation, at a power
level typically higher than the traffic channels, by every network's Base Station (BS) as a synchronization aid for the Mobile Terminals (Mts) within the cell's coverage range. Channel estimation can
also be easily obtained using pilot symbol periodically inserted in the traffic channel to allow for channel estimation in the presence of smart directive antennas.
In the reverse link the use of a code division or time multiplexed pilot symbols makes possible to perform data-aided channel estimation similarly to the forward link.
Numerical results confirm the interference-mitigation capability of the modified BAID algorithm operating in the presence of multipath channel, even with strong reflected signal component (i.e. |α|=
1), provided that a reliable channel delay profile is made available.
It is to be understood that the embodiments shown in the drawings are only a few exemplary implementations given to illustrate the way of carrying out the adaptive correlation processing in
accordance with the invention. | {"url":"http://www.google.com/patents/US6466566?dq=5726663","timestamp":"2014-04-16T13:47:29Z","content_type":null,"content_length":"196581","record_id":"<urn:uuid:fffacc08-e652-46ff-8f31-f57a068e04cd>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00402-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Total # Posts: 71
math please help!!!!!!!!!
no, it would be a decimal. im so confused. and im late! its not b, what is it?
math please help!!!!!!!!!
ummmm 4?
math please help!!!!!!!!!
Express the fractions 1/2, 3/16, and 7/8 with an LCD. A. 1/4, 3/4, 7/4 B. 1/32, 3/32, 7/32 C. 4/8, 6/8, 14/8 D. 8/16, 3/16, 14/16 I think its b, but im really bad at math. can someone please help me
out? I have to finish this by 12:15 so I can be on time to volunteer at the h...
In the following sentence, identify the part of speech of the italicized word. Catching fish is one of the oldest pastimes. the italicized word is is. A. Adverb B. Conjunction C. Preposition D. Verb
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Maths-can someone explain something to me???
well i'll check out the website but im homeschooled so facetime isn't gonnna happen.
Maths-can someone explain something to me???
hey, can someone explain vectors to me? im soooooo lost, I don't understand them at ALL. please please please help me. :(
Maths- very very simple
thanks yall :)
Maths- very very simple
can someone tell me what the title of this proof is? a^2+b^2=c^2 im writing a paper on the Pythagorean theorem and this is the proof I would use to explain it to someone but idk the name of it.
PLEASE help me out here.
Find the lengths of the missing sides in the triangle. Write your answers as integers or as decimals rounded to the nearest tenth. The diagram is not drawn to scale. triangle has sides 7, Y and X and
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thanks anyways Damon :) and thanks Steve
so the answer is 89.22? sorry I hope I don't sound dumb but I don't understand trig AT ALL.
maths- preferably Steve
can someone help me?
find the missing value to the nearest hundredth Tan ___ = 73 I just don't get these!
Ms. Sue
alright thanks
Ms. Sue
can you do me a favor and look at my science question? please?
someone plz plz plz plz plz plz plz plz plz help me!
List the five conditions that can disturb genetic equilibrium in a population I think Infinite population size, no mutation, no selection against a certain homozygous gene, random mating, no
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to what point are you referring?
To Corie
English; Help/Check.
I think its really good. but you need to change "my father love me" to "my father loves me" in the 4th line from the bottom. other than that I think its amazing.
English; Help/Check.
Whats your question?
thanks. could you show me how to do another one?
A courtyard is shaped like a square with 250-ft-long sides. What is the distance from one corner of the courtyard to the opposite corner? Round to the nearest tenth. 353.6 ft 500 ft 433.0 ft 176.8 ft
can someone show me how to do this?
2*10=20 stickers +6 single stickers=26 stickers +5*10 stickers=50=76 stickers +2 stickers= 78 stickers theres how I got it.
78 stickers.
life oriantation
pollution, deforestation, and im not sure about the third one. maybe salmonella.
The measures of the angles of a triangle are in the extended ratio 2 : 2: 8. What is the measure of the smallest angle? can someone pretty please show me how to do this?
Bokgoni technical secondary school
4. The wind was like a lovers soul because it embraced me.
thanks so much!!!
if x+3/3=y+2/2, then x/3=? whats the answer? I think its y/2. but im probly wrong. please explain to me.
Biology....someone who can explain something to me
thank you Devron
Biology....someone who can explain something to me
can someone explain the Miller-Urey Experiment to me? I've read it again and again and I just don't really get it. Please help me out here.
Geometry-yes or no
10/4=18/x would you cross multiply then divide to solve this proportion?
24 gallons over 7 minutes
mechanics of materials-SPAM
seriously dude, stop spamming. its annoying and uncalled for. and you are just trying to get people to do your homework for you. post a question with your answer and someone will let you know if you
are correct or not.
Social Studies
could someone please tell me if im right or not?
Social Studies
According to Sigmund Freud, human behavior was? A)ruled by two related emotions: love and pity. B)determined by the physical makeup of the brain, and could not be altered. C)the one factor that made
human beings superior to all other animals. D)strongly determined by repressed...
Geometry- Steve
Hey Steve, in the proportion 9/x=15/25 would you cross multiply 9 and 25 and x and 15???
Geometry-Solve for x
THANK YOU SOOOOOOOOOO MUCH STEVE!!!!! YOU ARE A LIFE SAVER!!!
Geometry-Solve for x
ok so I have to solve for x in this problem. and I have tried and tried to figure it out but I cant. 12(4x-2)=9(3x+2) I distributed them but after that when I try to simplify im lost. can someone
please help???
1 C2H4(g) + 3 O2(g) ---> 2 CO2(g) + 2 H2O(g) 0.821 moles C2H4 x (3 moles O2 / 1 mole C2H4) = 2.46 moles O2 Therefore, the answer is 3
ill check out the site :) thanks ms sue
can someone check my answer?
Biology-Ms. Sue?
Ms. Sue can you check my answer plz?
How do scientists calibrate a molecular clock? A)They determine the number of mutations accumulated in a gene of a species of known age. B)They compare the ticking of the molecular clock with that of
the atomic clock. C)They use the clock that ticks at the appropriate rate for...
#4 is an isosceles triangle bc a isosceles triangle is a triangle with (at least) two equal sides
thanks :) I don't have one....
I think what it is saying there implies that its C. is that right? :)
What is a molecular clock? A)a tool scientists use to date fossils B)a tool scientists use to determine how many mutations a species accumulates over time C)a tool scientists use to determine how
long two species have been evolving independently D)a tool scientists use to dete...
lolz your welcome. sorry I wasn't exactly accurate. :)
Phoenix I think.
are you looking for a specific city in Arizona? bc Southern Arizona is pretty warm in winter but if you need it to be more specific I can look. I know off the top of my head thought that southern
Arizona is pretty warm in winter.
Miss Prism is saying that she's attracted to Dr. Chasuble, she wants to marry him, and she ll continue to be attracted to him once they re married. This is her earnest feeling.
Penn Foster
A :)
consumer math
so whats your question?
sorry sandy idk how to do your problem. I did try though :( could you please take a look at mine right below yours though and see if you could tell me how to do it?
ok im gonna go with Reiny's answer. thanks for breaking it down so I could see how you got the answer :)
thanks Dylan. but howd you get that?
please someone tell me how to solve this. im very confused and im homeschooled so my teachers are no help and my mom cant do math. please please help!!!
What is the solution to the proportion 3y-8/12=y/5? can someone help me do this?
Posted by Writeacher on Friday, January 11, 2013 at 8:48am. anang, John, Prya, Sergey, Dylanprof, Amelia, kkr, kiril, Lena, c ~ Dumping ALL your homework questions in any one place (such as this
website) is considered spamming. I've removed your collective 50+ posts except...
check my answers? 1. What is the difference between Darwin's theory of evolution and Lamarck's theory of evolution? A)Darwin believed that organisms changed gradually over time, but Lamarck thought
they changed suddenly. B)Lamarck believed that organisms could acquire ...
ENGLISH HELP!!!
thanks :)
ENGLISH HELP!!!
b or c I think
ENGLISH HELP!!!
A feature article about a plane crash would be more likely than a hard news article to include ? A)the date, time, and location of the crash. B)a profile of one of the survivors of the crash. C)
statistics about other crashes involving the same airline. D)comments from top exec...
ENGLISH PLEASE HELP!!! URGENT!!!
I think its a unique tradition that brings together a lot of Chinese culture. And my high school English teacher said it was a good thesis statement.
ENGLISH PLEASE HELP!!! URGENT!!!
The Lantern Festival is a unique Chinese cultural tradition that includes aspects such as fancy lanterns, watching the Lion Dance, and eating tangyuan. I wrote this thesis statement at the beginning
of a paper I wrote about the Chinese Lantern Festival. but I have to restate i...
how did you get the 360? and then where did 24 come from??? im confused but I have the same math problem. | {"url":"http://www.jiskha.com/members/profile/posts.cgi?name=Corie","timestamp":"2014-04-16T13:37:43Z","content_type":null,"content_length":"18939","record_id":"<urn:uuid:749908d5-877a-44f3-af7c-a1881e80f851>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00453-ip-10-147-4-33.ec2.internal.warc.gz"} |
Percent change: Know the formula
Here’s a question I posed to some college students recently:
Let’s say you cover the Town of East Middleburgtown. The mayor announces that this year’s town budget comes in at $12.6 million. Last year’s budget was $11.4 million. What is the percent change?
Better yet, what’s the formula for figuring it out?
If you don’t know the answer, or how to obtain it, you’re not alone. This kind of problem — which is in my son’s 7th grade math textbook — routinely stumps most journalists in most of the newsrooms
across America.
I’ll avoid the temptation to moralize here. If you’re a journalist — if you have a pulse — you need to know this very basic operation. With it, you’ll have the power to analyze all kinds of data and
even double-check the mayor’s math.
Here it is:
(the_new_number - the_original_number) / the_original_number
or, in the case of East Middletownburg:
Remember (and you learned this in fifth grade) that operations in parentheses come first. That gives you this:
1.2 / 11.4 = .105 = 10.5%
So, the mayor’s new budget is a 10.5% increase over last year’s. Now you have something to write about!
my math-savvy wife was just explaining this to me the other day, now i can bookmark your post so I’ll never have to ask her again. great site Tony, added to my google reader!
Social comments and analytics for this post…
This post was mentioned on Twitter by seattlesketcher: A math formula every journalist should know (via @TonyDB) http://bit.ly/1KNNyq…
It amazes me that kids come through 12 to 16 years of academics with no understanding of statistics at all, which greatly hampers your ability to make sense of the world. I take every opportunity to
use and explain things like “right edge of the bell curve” and “two standard deviations” and go over ways to interpret raw data with the kids… Invaluable tools mental modeling and turning “knowledge”
into “wisdom.”
I’m glad I remembered how to do this type of problem!
Alternate approach:
Divide larger number by the smaller smaller number, subtract 1 and move the decimal point two positions to the right.
Or . . . in the case of Middletownburg: 12.6/11.4=1.105
I.e . . . the new budget is 110.5% of the old budget . . . or 10.5% larger. I find this easier . . . but . . . different strokes.
What REALLY freaks folks out is that having one’s salary move from $100K to 150K a year is a 50% increase . . . but if the boss takes back the $50k, it’s only a 33% decrease.
Ahhhh . . . a 17% raise? (50-33?) . . . that costs the boss . . . nothing !
Yes, math is funny that way — different routes can lead to the same destination. As long as it’s accurate! That’s always the main thing. | {"url":"http://www.anthonydebarros.com/2009/10/14/percent-change-the-formula/","timestamp":"2014-04-19T19:33:43Z","content_type":null,"content_length":"22559","record_id":"<urn:uuid:34f4fa5e-b686-4c29-a498-95eb2a2faa9c>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00493-ip-10-147-4-33.ec2.internal.warc.gz"} |
Power Tip 57: Design a flyback primary switch snubber | EE Times
Power Tips
Power Tip 57: Design a flyback primary switch snubber
How to best control the voltage stress on the primary switch in a single-ended flyback converter (shown in Figure 1) is a multi-faceted problem. You have to solve a combination of technical issues
while still keeping an eye on the overall cost. You have to:
• Limit the MOSFET voltage stress to an acceptable level
• Discharge the leakage inductance very quickly to maintain good efficiency (see Power Tip 17)
• Minimize circuit losses due to adding the snubber
• Avoid impacting the power supply dynamics
The lowest cost approach to solve these issues is shown in
Figure 1
of Power Tip 17 and consists of a standard recovery diode, capacitor and loading resistor. The circuit works by transferring excessive transformer leakage energy onto the snubber capacitor and
dissipating it over the switching period. Unfortunately in this approach there is always energy dissipated in the snubber resistor, regardless of output power. In each switching cycle, the voltage on
the capacitor always will be recharged to at least the reflected output voltage. This degrades the efficiency, particularly, at light loads.
Figure 1
of this power tip presents an alternative circuit approach, which replaces the resistor/capacitor with a resistor (R1) and zener diode (D1). When the FET turns off, the drain voltage rises to the
point that the diodes conduct to discharge the leakage inductance of the transformer. The rate at which the current discharges is set by the difference between the reflected output voltage and the
clamp voltage. Note that for best efficiency, as Power Tip 17 points out, it is critical to discharge the leakage inductance energy as quickly as possible. In choosing values, first consider the
MOSFET voltage rating and derating criterion to determine a suitable maximum voltage stress on the MOSFET. First choose the zener voltage to be above the reflected output voltage so that it does not
continue to conduct after the leakage inductance has been reset. Next size the resistor/zener combination so that you do not exceed the allowed MOSFET voltage stress at high-line and maximum current.
Click on image to enlarge.
Figure 1: This FET clamp provides good light load efficiency.
Now trade circuit ringing for efficiency. In
Figure 2
, resistor R1 has been shorted so that the zener solely sets the voltage stress on the MOSFET. At turn-off, the drain voltage flies up and the leakage inductance current is discharged with a constant
voltage, which provides the fastest discharge and best efficiency. However, once the leakage inductance is discharged, the drain voltage rings around the reflected output plus input voltage, which
creates a couple of concerns. Obviously one concern is electromagnetic interference (EMI), as this 4 MHz ringing creates common-mode currents in the power transformer and increases the power line
filtering need. The second issue is related to the choice of controllers. There are a number of integrated circuits (ICs) that eliminate secondary-side measurement of the output voltage and rely on
the primary bias winding voltage to provide a representative sample of the output. With this type of controller, the ringing can result in poor output voltage regulation accuracy.
Click on image to enlarge.
Figure 2: High voltage zener clamp discharges leakage inductance quickly to improve efficiency.
If the ringing is an issue, reduce the zener voltage to approximately the reflected output voltage and add series resistance to increase the peak drain voltage.
Figure 3
shows the waveforms from the circuit shown in
Figure 1
. The yellow trace is the drain voltage and the red is the voltage at the junction of D3 and R1. The difference between the two voltages is proportional to the leakage inductor current. The drain
voltage starts at a high voltage and reduces the differential voltage and, hence, leakage inductance current to zero. So when the diode turns off, there is little voltage difference between the drain
voltage and the reflected output voltage. Consequently, there is little ringing. Unfortunately, with this approach, you pay an efficiency penalty. In this case it was about two percent. As was
pointed out in Power Tip 17, the longer it takes to discharge the leakage inductance, the worse the efficiency will be. In
Figure 2
, the leakage was discharged in 70 nS while it took 160 nS in
Figure 3
Click on image to enlarge.
Figure 3: Series resistance reduces EMI.
To summarize, RCD clamps are the simplest way to snub a flyback. However, with an RCD clamp, the light-load losses suffer from continuous power dissipation. If light-load loss is an issue, consider a
snubber with a Zener, which only dissipates power when it is needed. An abrupt zener provides the best efficiency; but it can cause unacceptable ringing. The best trade-off may be using a reduced
zener voltage along with a series resistance.
Please join us next month when we take a look at some classic power supply layout mistakes.
For more information about this and other power solutions, visit: www.ti.com/power-ca.
About the author
Robert Kollman
is a senior applications manager and distinguished member of technical staff at Texas Instruments. He has more than 30 years of experience in the power electronics business and has designed magnetics
for power electronics ranging from sub-watt to sub-megawatt with operating frequencies into the megahertz range. Robert earned a BSEE from Texas A&M University and an MSEE from Southern Methodist | {"url":"http://www.eetimes.com/author.asp?section_id=183&doc_id=1280601","timestamp":"2014-04-17T04:05:16Z","content_type":null,"content_length":"132987","record_id":"<urn:uuid:0484e36c-3e97-4a9e-8a44-41429d1a4992>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00183-ip-10-147-4-33.ec2.internal.warc.gz"} |
Computer Algorithms II
They are not heaps, of course, because it makes no sense to talk of a heap of numbers, and yet the numbers 1 through 5 form a perfectly acceptable set - {1, 2, 3, 4, 5}. Still less are heaps
sets. A heap of sand does not yield a set of sand. The heaps belong off with the mounds, the piles, the stacks, the masses, and the batches; the sets are otherwise—remote, detached, abstract.
David Berlinski, The Advent of the Algorithm
This is the second of two courses on computer algorithms and data structures; see the CS 306 entry in the course catalog for the formal description of the course.
The prerequisite for this class is CS 305. In particular, from the textbook for this class, you are assumed to be familiar with the contents of chapters 1 through 8, 11, sections 12.1 through 12.6,
and appendices C and D.
The course is divided into three three-week sections and one six-week section; see the syllabus for details.
The class meets in Howard Hall C1 (third floor) on Tuesdays and Thursdays, 6:00 p.m. to 7:50 p.m. There's no class on Thanksgiving, Thursday, 22 November.
The course objective is to extend and exercise your algorithm design and implementation knowledge in three areas:
• algorithm characterization (How fast is it? Is it correct?),
• designing algorithms (What approach best solves a problem?), and
• data structures (What best represents the problem and solution?).
R. Clayton, rclayton@monmouth.edu. Office hours are from 5 to 6 p.m. on Tuesdays and Thursdays in HH C1; see the schedule for details. The usual grade ranges are in effect:
95 ≤ A
90 ≤ A- < 95
86.6 ≤ B+ < 90
83.3 ≤ B < 86.6
80 ≤ B- < 83.3
76.6 ≤ C+ < 80
73.3 ≤ C < 76.6
70 ≤ C- < 73.3
60 ≤ D < 70
F < 60
All grades are kept with one digit of precision to the right of the decimal point and 0.05 rounded up. No grades are adjusted to a curve; that means, for example, that 89.9 is always a B+, never an
A-. The final grade is the weighted sum of the test-grade average and the assignment-grade average with the weights
45% test grades
45% assignment grades
10% presentation grade
The test- and assignment-grade averages are straight, unweighted averages.
Mid-term grades are the straight, unweighted averages of whatever assignment and quiz grades have accumulated by the mid-term grade deadline (Tuesday, 23 October). A presentation grade, assuming it
exists by the deadline, provides 10% of the mid-term grade with the other grades providing the rest; otherwise the other grades provide up the full amount.
There are four tests, one test for each of the sections; see the syllabus for the schedule. Tests are given in class, and are closed book with no notes; calculators and computers will not be
necessary. The tests are cumulative, covering everything taught up to and including the class before the test. Tests should take no more than an hour to complete, and are given in the first hour of
class. Test answers will be made available off the syllabus. There are no mid-terms or finals. There are four programming assignments, one for each of the sections; see the syllabus for details. Each
person in class will give a talk about a problem they've solved; see the problem-talk page for more details. The textbook for this class is ADTs, Data Structures, and Problem Solving with C++ by
Larry Nyhoff Prentice-Hall, 2004. An errata is available off Nyhoff's web page for the book. Feel free to send e-mail to rclayton@monmouth.edu . Unless I warn you beforehand, I'll usually respond
within a couple of hours during the usual work days; if I don't respond within a day, resend the message.
Mail relevant to the class are stored in a hyper-mail archive. If your message is of general interest to the class, I'll store it, suitably stripped of identification and along with my answer, in the
If you're reading this on paper, you can find the class home page at http://www.monmouth.edu/rclayton/web-pages/f07-306/index.html. I'll make the class notes, assignments, and tests available off the
syllabus; you should get in the habit of checking the syllabus regularly. People who need assistance or accommodations above and beyond what is usually provided in class should contact the
University's ADA/504 coordinator to get those needs met. See the Disability Services page for more details. I have no class attendance policy; you may attend class or not as you see fit. However, I
hold you responsible for knowing everything that goes on in class; "I wasn't in class for that." is not an acceptable excuse for a wrong answer, or for giving no answer at all.
My attendance policy applies only to lecture attendance; it does not apply to other kinds of attendance which may be required for the course. Repeated failures to meet the attendance expectations set
for tests, meetings, projects, labs or other forms of course work will have a bad influence on your grade.
Monmouth University does have a class attendance policy, which you can find in the Academic Information chapter of the Student Handbook. To the extent that I need to keep the record straight, I will
take attendance. Attendance lists, however, are entirely for the University's benefit; I will make no use of them in grading.
I deal with suspected cheating by failing first and asking questions later. Although cheating has many forms, I generally consider cheating to be any attempt to claim someone else's work as your own;
also, I consider both the provider and the user of the work guilty of cheating. See the chapters on Academic Information and the Student Code of Conduct in the Student Handbook for more details. I
recognize and encourage a student's sacred right to complain about their grade. There are, however, a few rules under which such complaining should take place, and those students who don't follow the
rules will be less successful in their complaints than those students who do follow the rules.
First, the only complaint that matters is that something got marked wrong when it was actually right. When you come to complain, be prepared to present, in explicit detail, what it is you did and why
you think it's right.
Second, complaints about a particular test or assignment are only valid until the next test or assignment is due; after that point the book is permanently closed on all previous test or assignment
grades. Assignments must be turned in by their due date; assignments turned in after their due date are late. You should contact me as soon as possible if you need to negotiate a due-date extension.
The longer you wait to negotiate, the less likely it is you'll be successful; in particular, you have almost no chance of getting an extension if you try for one the day before the due date, and you
have no chance of getting an extention on the due date.
A late assignment is penalized five points a day for each day it's late. I use a 24-hour clock running from midnight to midnight to measure days; note this means that an assignment handed in the day
after it's due is penalized ten points: five for the day it was due and five for the next day.
There may occasionally be a conflict between taking a test and doing something else, particularly among those working full time. If you're going to be out of town, or on jury duty, or whatever, on a
test day, let me know beforehand and we'll discuss a make-up test.
A make-up test must be scheduled to be taken by the date of the test following the missed test (or the final exam if you miss the last test). If a missed test is not made up by the time of the next
test, you get a zero for the missed test.
There will be only one make up given per missed test. If more than one person misses the same test, those people will have to coordinate among themselves to pick a mutually agreeable date for the
make up.
The NIST dictionary of algorithms and data structures; the Stony Brook repository of algorithms and data structures, with the associated book.
FreeTechBooks' list of on-line data structures and algorithm books.
Softpanorama's old but wide ranging link page for data structures and algorithms.
Algosort's link page to algorithm pages.
Algorithm search engines (as opposed to search-engine algorithms) in general (seems borken, 30 Aug 07), for Haskell, and in detail.
This page last modified on 7 November 2007. | {"url":"http://bluehawk.monmouth.edu/~rclayton/web-pages/f07-306/index.html","timestamp":"2014-04-19T14:29:24Z","content_type":null,"content_length":"16792","record_id":"<urn:uuid:518a5b30-96c6-4017-ab93-1b0a6caaee2c>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00200-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hanover Park Algebra 2 Tutor
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Deterministic system
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In mathematics and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system.^[1] A deterministic model will thus always produce
the same output from a given starting condition or initial state.^[2]
Physical laws that are described by differential equations represent deterministic systems, even though the state of the system at a given point in time may be difficult to describe explicitly.
The systems studied in chaos theory are deterministic. If the initial state were known exactly, then the future state of such a system could be predicted. However, in practice, knowledge about the
future state is limited by the precision with which the initial state can be measured.
Markov chains and other random walks are not deterministic systems, because their development depends on random choices.
A pseudorandom number generator is a deterministic algorithm, although its evolution is deliberately made hard to predict; a hardware random number generator, however, may be non-deterministic.
See also | {"url":"http://psychology.wikia.com/wiki/Deterministic_system","timestamp":"2014-04-20T04:03:14Z","content_type":null,"content_length":"58691","record_id":"<urn:uuid:1ee30586-d2f8-4ba8-b22e-b15082049b70>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00164-ip-10-147-4-33.ec2.internal.warc.gz"} |
Selfish Unsplittable Flow
Results 1 - 10 of 48
, 2005
"... Abstract Selfish routing is a classical mathematical model of how self-interested users might route traffic through a congested network. The outcome of selfish routing is generally inefficient,
in that it fails to optimize natural objective functions. The price of anarchy is a quantitative measure o ..."
Cited by 175 (12 self)
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Abstract Selfish routing is a classical mathematical model of how self-interested users might route traffic through a congested network. The outcome of selfish routing is generally inefficient, in
that it fails to optimize natural objective functions. The price of anarchy is a quantitative measure of this inefficiency. We survey recent work that analyzes the price of anarchy of selfish
routing. We also describe related results on bounding the worst-possible severity of a phenomenon called Braess's Paradox, and on three techniques for reducing the price of anarchy of selfish
routing. This survey concentrates on the contributions of the author's PhD thesis, but also discusses several more recent results in the area.
- In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC , 2005
"... Abstract We consider the price of anarchy of pure Nash equilibria in congestion games with linearlatency functions. For asymmetric games, the price of anarchy of maximum social cost is \Theta (p
N),where N is the number of players. For all other cases of symmetric or asymmetric games andfor both max ..."
Cited by 122 (7 self)
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Abstract We consider the price of anarchy of pure Nash equilibria in congestion games with linearlatency functions. For asymmetric games, the price of anarchy of maximum social cost is \Theta (p
N),where N is the number of players. For all other cases of symmetric or asymmetric games andfor both maximum and average social cost, the price of anarchy is 5 /2. We extend the results tolatency
functions that are polynomials of bounded degree. We also extend some of the results to mixed Nash equilibria.
- In Proc. 37th Symp. Theory of Computing (STOC , 2005
"... The essence of the routing problem in real networks is that the traffic demand from a source to destination must be satisfied by choosing a single path between source and destination. The
splittable version of this problem is when demand can be satisfied by many paths, namely a flow from source to d ..."
Cited by 106 (4 self)
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The essence of the routing problem in real networks is that the traffic demand from a source to destination must be satisfied by choosing a single path between source and destination. The splittable
version of this problem is when demand can be satisfied by many paths, namely a flow from source to destination. The unsplittable, or discrete version of the problem is more realistic yet is more
complex from the algorithmic point of view; in some settings optimizing such unsplittable traffic flow is computationally intractable. In this paper, we assume this more realistic unsplittable model,
and investigate the ”price of anarchy”, or deterioration of network performance measured in total traffic latency under the selfish user behavior. We show that for linear edge latency functions the
price of anarchy is exactly 2.618 for weighted demand and exactly 2.5 for unweighted demand. These results are easily extended to (weighted or unweighted) atomic ”congestion games”, where paths are
replaced by general subsets. We also show that for polynomials of degree d edge latency functions the price of anarchy is dΘ(d). Our results hold also for mixed strategies. Previous results of
Roughgarden and Tardos showed that for linear edge latency functions the price of anarchy is exactly 4 3 under the assumption that each user controls only a negligible fraction of the overall traffic
(this result also holds for the splittable case). Note that under the assumption of negligible traffic pure and mixed strategies are equivalent and also splittable and unsplittable models are
equivalent. 1
- IN FOCS , 2005
"... We introduce the concept of a sink equilibrium. A sink equilibrium is a strongly connected component with no outgoing arcs in the strategy profile graph associated with a game. The strategy
profile graph has a vertex set induced by the set of pure strategy profiles; its arc set corresponds to transi ..."
Cited by 68 (11 self)
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We introduce the concept of a sink equilibrium. A sink equilibrium is a strongly connected component with no outgoing arcs in the strategy profile graph associated with a game. The strategy profile
graph has a vertex set induced by the set of pure strategy profiles; its arc set corresponds to transitions between strategy profiles that occur with nonzero probability. (Here our focus will just be
on the special case in which the strategy profile graph is actually a best response graph; that is, its arc set corresponds exactly to best response moves that result from myopic or greedy
behaviour.) We argue that there is a natural convergence process to sink equilibria in games where agents use pure strategies. This leads to an alternative measure of the social cost of a lack of
coordination, the price of sinking, which
- Algorithmica , 2004
"... Abstract We revisit a classical load balancing problem in the modern context of decentralized systems andself-interested clients. In particular, there is a set of clients, each of whom must
choose a server from ..."
Cited by 52 (3 self)
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Abstract We revisit a classical load balancing problem in the modern context of decentralized systems andself-interested clients. In particular, there is a set of clients, each of whom must choose a
server from
, 2006
"... We study the speed of convergence to approximately optimal states in two classes of potential games. We provide bounds in terms of the number of rounds, where a round consists of a sequence of
movements, with each player appearing at least once in each round. We model the sequential interaction betw ..."
Cited by 29 (2 self)
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We study the speed of convergence to approximately optimal states in two classes of potential games. We provide bounds in terms of the number of rounds, where a round consists of a sequence of
movements, with each player appearing at least once in each round. We model the sequential interaction between players by a best-response walk in the state graph, where every transition in the walk
corresponds to a best response of a player. Our goal is to bound the social value of the states at the end of such walks. In this paper, we focus on two classes of potential games: selfish routing
games, and cut games (or party affiliation games [7]).
- ICALP 2006. LNCS , 2006
"... Abstract. We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client.
We consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it ..."
Cited by 28 (5 self)
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Abstract. We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client. We
consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it selects to run its job to the server among its permissible servers having the smallest latency
given the assignments of the jobs of other clients to servers. In online load balancing, clients appear online and, when a client appears, it has to make an irrevocable decision and assign its job to
one of its permissible servers. Here, we assume that the clients aim to optimize some global criterion but in an online fashion. A natural local optimization criterion that can be used by each client
when making its decision is to assign its job to that server that gives the minimum increase of the global objective. This gives rise to greedy online solutions. The aim of this paper is to determine
how much the quality of load balancing is affected by selfishness and greediness. We characterize almost completely the impact of selfishness and greediness in load balancing by presenting new and
improved, tight or almost tight bounds on the price of anarchy and price of stability of selfish load balancing as well as on the competitiveness of the greedy algorithm for online load balancing
when the objective is to minimize the total latency of all clients on servers with linear latency functions. 1
, 2006
"... We show exact values for the price of anarchy of weighted and unweighted congestion games with polynomial latency functions. The given values also hold for weighted and unweighted network
congestion games. ..."
Cited by 25 (4 self)
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We show exact values for the price of anarchy of weighted and unweighted congestion games with polynomial latency functions. The given values also hold for weighted and unweighted network congestion
, 2001
"... We study the number of steps required to reach a pure Nash equilibrium in a load balancing scenario where each job behaves selfishly and attempts to migrate to a machine which will minimize its
cost. We consider a variety of load balancing models, including identical, restricted, related and unrelat ..."
Cited by 22 (4 self)
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We study the number of steps required to reach a pure Nash equilibrium in a load balancing scenario where each job behaves selfishly and attempts to migrate to a machine which will minimize its cost.
We consider a variety of load balancing models, including identical, restricted, related and unrelated machines. Our results have a crucial dependence on the weights assigned to jobs. We consider
arbitrary weights, integer weights, K distinct weights and identical (unit) weights. We look both at an arbitrary schedule (where the only restriction is that a job migrates to a machine which lowers
its cost) and specific efficient schedulers (such as allowing the largest weight job to move first). A by product of our results is establishing a connection between the various scheduling models and
the game theoretic notion of potential games. We show that load balancing in unrelated machines is a generalized ordinal potential game, load balancing in related machines is a weighted potential
game, and load balancing in related machines and unit weight jobs is an exact potential game. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=103087","timestamp":"2014-04-20T19:55:02Z","content_type":null,"content_length":"37351","record_id":"<urn:uuid:16bea1d6-4e70-4925-a0c2-1958d73424e5>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00482-ip-10-147-4-33.ec2.internal.warc.gz"} |
Using R to answer a football question
Last Monday night, I was watching the Ravens playing the Packers at Green Bay. Mostly, I was watching penalties. This game featured an astounding number of penalty calls: 23 calls, 310 yards. At
first glance, you'd think that penalties were a bad thing; teams for which officials call more penalties should perform worse than teams for which officials call fewer penalties.
However, I wondered if this was really true. Penalties could be a sign that a team is playing passionately. Or they could be completely random and meaningless. I decided to take a look at this
question. Here's what I figured out in 15 minutes, with a little help from R.
The NFL web site has a good statistics section. I decided to take a look at 2008 team offense statistics and 2008 team defense statistics. With these stats, I could look at the correlation between
penalties and points. If penalties really affected games, you'd expect a negative correlation.
I copied these data tables from my web browser (Safari 4 for Mac OS X) to Microsoft Excel, then saved the data as a set of text files.
Now, I was ready to import the data into R, using read.delim. I started with defensive stats:
> def.08 <- read.delim(file="~/Desktop/nfldef2008.txt")
Before analyzing the data, I took a quick look at the penalty data to make sure that everything looked OK.
> names(def.08)
[1] "Rk" "Team" "G" "Pts.G" "TotPts"
[6] "Scrm.Plys" "Yds.G" "Yds.P" "X1st.G" "X3rd.Md"
[11] "X3rd.Att" "X3rd.Pct" "X4th.Md" "X4th.Att" "X4th.Pct"
[16] "Pen" "Pen.Yds" "ToP.G" "FUM" "Lost"
> def.08$Pen.Yds
[1] 801 792 593 639 866 1,002 750 601 660 636 543
[12] 772 869 540 615 663 691 736 816 721 827 659
[23] 637 854 708 770 633 654 738 671 588 753
32 Levels: 1,002 540 543 588 593 601 615 633 636 637 639 654 ... 869
This reveals one small problem: R loaded the numeric fields as text fields, converting them to factors. The problem was that "1,002" value; R didn't correctly parse that field. I corrected the
underlying data and tried again:
> def.08 <- read.delim(file="~/Desktop/nfldef2008.txt")
> fivenum(def.08$TotPts)
[1] 223 313 350 390 517
> fivenum(def.08$Pen.Yds)
[1] 540.0 636.5 699.5 782.0 1002.0
Let's plot total points vs total penalty yards:
> plot(TotPts~Pen.Yds,data=def.08.n)
Here's what the chart looks like:
I didn't see any obvious correlation between penalty yards and points allowed, but I decided to check for correlation using the cor.test function in R:
> cor.test(def.08.n$Pen.Yds,def.08.n$TotPts)
Pearson's product-moment correlation
data: def.08.n$Pen.Yds and def.08.n$TotPts
t = -0.4091, df = 29, p-value = 0.6855
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.4188408 0.2862818
sample estimates:
As you can see, the p value shows that there was no statistically significant correlation for defensive statistics. The story is much the same for offensive stats:
> off.08 <- read.delim(file="~/Desktop/nfloff2008.txt")
> plot(TotPts~Pen.Yds,data=off.08.n)
Here is a plot of points allowed vs penalty yards for offenses in 2008:
And here is the result of the correlation test:
> cor.test(off.08.n$Pen.Yds,off.08.n$TotPts)
Pearson's product-moment correlation
data: off.08.n$Pen.Yds and off.08.n$TotPts
t = 0.8927, df = 30, p-value = 0.3791
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.1989928 0.4824930
sample estimates:
As with defensive statistics, the correlation isn't statistically significant, though it is slightly stronger.
7 Comments
You analyzed an entire season. Take a single week of play and I think you will probably find a greater correlation. When you take a teams entire body of work you pretty much average out any loss in
points which could be associated with high penalty yardage.
Yeah, agree with Chris' comment. Was gonna say the same. It's probably a wash over the course of a season, but at a per-game level it's probably significant. Neat idea btw!
Chris and Andy:
You're right that you'd see more significance in single week. Taken to the extreme, it's obvious that there are single penalties that can affect the result of games (or seasons, or players' or
cocahces careers). However, many football commentators comment on a teams' cumulative penalty yards, implying that this statistic is important.
However, the big reason I wrote this post was to show how to pull this data into R and take a look at it by yourself. Let me know if you find anything interesting!
Wouldn't it be more meaningful to look for a correlation between penalties incurred and games lost? A high number of total points could hide negative consequences of penalties incurred if the other
teams scored more points.
Interesting question. Ultimately, what matters most is games won and lost.
However, there are a bunch of reasons why W/L statistics aren't the best measure here. First, there are only 16 games in an NFL season, so it's really tough to find a statistically significant
effect. Secondly, wins and losses are a function of lots of different effects. Among other things, quality of opponents has a big effect.
Mostly, the point of the exercise was the show that there are both good and bad teams that get large and small numbers of penalties. Oh, and to show how to get the data and play with it in R.
Actually you've got a bug in the way you read the data. Notice that the penalty yards should be in the range 540 to 1002, but on your graph they're in the range 1 to 32. That's because you converted
the factor to numeric by using its underlying factor-level-number and there are 32 teams. So no wonder there was no correlation. =)
Here's the way I read my data in, which gets rid of the commas in the first place (that's what was tripping up the read). Note that it's specific to OS X:
Still not much correlation in the data, but the effect is somewhat stronger when the correct data is used.
Good catch Ken; I updated the example to fix the issue. Looks like R got stuck on the value "1,002" in the original data. | {"url":"http://broadcast.oreilly.com/2009/12/using-r-to-answer-a-football-q.html","timestamp":"2014-04-18T16:11:18Z","content_type":null,"content_length":"44199","record_id":"<urn:uuid:8a7f9586-a9ed-40a1-b830-005c6cf76280>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00189-ip-10-147-4-33.ec2.internal.warc.gz"} |
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On the Distribution of a Product of Two Random Variables
Accession Number : AD0736166
Title : On the Distribution of a Product of Two Random Variables
Corporate Author : CLEMSON UNIV SC DEPT OF MATHEMATICAL SCIENCES
Personal Author(s) : Alam, Khursheed
PDF Url : AD0736166
Report Date : SEP 1971
Pagination or Media Count : 14
Abstract : Let W=UV where U and V are independent random variables. It is shown that if V is distributed according to the non-central Chi-square distribution, then W is distributed according to the
Chi-square distribution if and only if U=1 with probability 1. If V is normally distributed, it is shown that W is normally distributed if and only if the distribution of U is a two-point
Descriptors : *STATISTICAL DISTRIBUTIONS, MULTIVARIATE ANALYSIS, STATISTICAL TESTS, THEOREMS, RANDOM VARIABLES, PROBABILITY DENSITY FUNCTIONS
Subject Categories : Statistics and Probability
Distribution Statement : APPROVED FOR PUBLIC RELEASE | {"url":"http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=AD0736166","timestamp":"2014-04-19T06:51:49Z","content_type":null,"content_length":"3031","record_id":"<urn:uuid:f179872a-5039-4697-bd74-771020493f44>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00129-ip-10-147-4-33.ec2.internal.warc.gz"} |
Variance Problem
April 7th 2008, 07:26 AM #1
Apr 2008
Variance Problem
Hi, I haven't done used variances in a couple of years and I have this econ problem:
VIEW BMP
Basically i have a variance in a variance, one is j and the other i. What can i do, if i plug the log Y bar into the equation I can simplify to 0, any ideas?
Last edited by murdoc; April 7th 2008 at 08:44 AM.
I am sorry
but if you need help you should learn LaTeX...because that is completely hard to follow...if you can put it in a more reader friendly format I would love to help!
Look at the math man's post three down...learn LaTeX code and see how clear and lucid your questions become....Mathstud
April 7th 2008, 08:30 AM #2 | {"url":"http://mathhelpforum.com/advanced-statistics/33526-variance-problem.html","timestamp":"2014-04-19T19:52:43Z","content_type":null,"content_length":"33147","record_id":"<urn:uuid:97dd5a16-c8cb-4189-bdd3-cecd6581660c>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00436-ip-10-147-4-33.ec2.internal.warc.gz"} |
Exton Statistics Tutor
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Results 1 - 10 of 21
, 1998
"... . In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and
introduce some special systems which exhibit some interesting behavior. We then apply these results to the set o ..."
Cited by 30 (9 self)
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. In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce
some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the
Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries. 1
Research supported by NSF Grant DMS-9502952. AMS(MOS) subject classifications(1980). Primary 28A80; Secondary 58F08, 58F11, 28A78 Key words and phrases. Iterated function systems, continued
fractions, Hausdorff dimension, Hausdorff and packing measures, arithmetic densities. Typeset by A M S-T E X Mauldin and Urba'nski Page 1 x1. Introduction: Setting and Notation Let I be a nonempty
subset of N , the se...
, 2007
"... We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to
do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the ..."
Cited by 28 (6 self)
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We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We
also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals,
are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often
much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, we obtain an Ω±-result for the variation of the error term in the prime number
theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to
Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n)
and ΛR(n) can be employed in diverse problems concerning primes.
- ALGORITHMIC NUMBER THEORY , 2008
"... ..."
- Adv. in Math , 1981
"... ABSTRACT. Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x):::: (c + o ( I)) logx loglogx
log log log 10gx(loglog logx)-2, where c = eY and y is Euler's constant. Here, this result is shown with ..."
Cited by 11 (3 self)
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ABSTRACT. Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x):::: (c + o ( I)) logx loglogx log
log log 10gx(loglog logx)-2, where c = eY and y is Euler's constant. Here, this result is shown with c = coe Y where Co = 1.31256... is the solution of the equation 4 / Co- e-4/co = 3. The principal
new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus. 1.
- Israel Journal of Mathematics , 1962
"... This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1
Cited by 9 (1 self)
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This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1<j<co be an infinite sequence of numbers on the unit circle. Put
"... Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some
corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of ex ..."
Cited by 8 (2 self)
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Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some
corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→ ∞ (qn+1 − qn) ≤ 26. If an appropriate
generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6. 1.
"... ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for
any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between ..."
Cited by 7 (3 self)
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ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any
η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.
- Proc. Symp. Pure Math. (Analytic Number Theory , 1972
"... Introduction. This talk is about the interplay between computers and theoretical research, as experienced by someone who is not a computer expert. The story involves, among other things, a
measure of good luck. Several instances of this will emerge in due course, but one example now may give the ide ..."
Cited by 6 (0 self)
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Introduction. This talk is about the interplay between computers and theoretical research, as experienced by someone who is not a computer expert. The story involves, among other things, a measure of
good luck. Several instances of this will emerge in due course, but one example now may give the idea: The speaker and his co-worker, Douglas Hensley,
"... We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the
Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. E ..."
Cited by 6 (1 self)
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We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the
Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance
apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, pn+1 − pn lim inf =0. n→ ∞ log pn We will
quantify this result further in a later paper (see (1.9) below). | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2130988","timestamp":"2014-04-19T19:59:14Z","content_type":null,"content_length":"33623","record_id":"<urn:uuid:0a8d8f78-7bce-4101-a4d8-65eeddcc37ad>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00470-ip-10-147-4-33.ec2.internal.warc.gz"} |
[SciPy-user] fast max() on sparse matrices
Peter Skomoroch peter.skomoroch@gmail....
Mon Jan 5 19:04:18 CST 2009
You said:
"... some matrices permit duplicate entries.
Currently, we implicitly sum duplicate values together (e.g. when
computing sparse matrix-vector products) and when converting to other
Could you elaborate on that a bit? I'm trying to track down a nasty bug
right now where the result of a sparse matrix-matrix product (A_sparse *
B_dense) does not agree with the corresponding dense product (A_dense *
On Mon, Jan 5, 2009 at 5:40 PM, Nathan Bell <wnbell@gmail.com> wrote:
> On Mon, Jan 5, 2009 at 4:43 PM, nicky van foreest <vanforeest@gmail.com>
> wrote:
> >
> > A few days ago I encountered just the same problem, and solved by
> > taking the max of the values(), just as suggested below. However, it
> > took me some minutes to fiugre this out, and I first, of course, tried
> > the max() function. Thus, I suggest that the max function will be
> > added to the sparse class. Is there a reason not to do so?
> >
> Hi Nicky,
> It should be added, but it's not as straightforward as you might think.
> For conformity with dense matrices, max() should return zero if the
> nonzero entries of the matrix are all negative and there is at least
> one missing value in the matrix. This might surprise people who
> expect the largest nonzero value instead. For instance,
> csr_matrix([[0,-1]]).max() should be 0.
> Another minor problem is that some matrices permit duplicate entries.
> Currently, we implicitly sum duplicate values together (e.g. when
> computing sparse matrix-vector products) and when converting to other
> formats. We'd probably want to make max() and min() agree with this
> behavior.
> --
> Nathan Bell wnbell@gmail.com
> http://graphics.cs.uiuc.edu/~wnbell/<http://graphics.cs.uiuc.edu/%7Ewnbell/>
> _______________________________________________
> SciPy-user mailing list
> SciPy-user@scipy.org
> http://projects.scipy.org/mailman/listinfo/scipy-user
Peter N. Skomoroch
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Sample questions
Next: Algebra Up: Optimization/Numerical Linear Algebra Previous: Outline
1. Compare and contrast the line search and trust region methods of globalizing a quasi-Newton method. Your discussion should touch on the following points:
(a) The cost of taking a step (that is, of solving the line search subproblem and trust region subproblem), and the complexity of the algorithms.
(b) Dealing with nonconvexity.
(c) Convergence theorems.
2. Derive the BFGS update used for approximating the Hessian of the objective function in an unconstrained minimization problem. Explain the rationale for the steps in the derivation.
(a) State carefully and prove the first-order necessary condition for x=x^*.
(b) Give an example to show that the first-order condition is only necessary, not sufficient.
(c) State carefully and prove the second-order necessary condition for x=x^*.
(d) Give an example to show that the second-order condition is only necessary, not sufficient.
(a) Let
Define ``
(b) Let f is locally q-quadratically convergent. State carefully and prove this theorem.
5. Suppose
(a) State and prove a theorem indicating that the usual first-order necessary condition is, in this case, a sufficient condition.
(b) Prove that every local minimum of f is, in fact, a global minimum.
6. Consider the equality-constrained problem
7. Recall that the
Derive the formula for the corresponding induced matrix norm of
(a) What is the condition number of a matrix
(b) Explain how this condition number is related to the problem of computing the solution x to Ax=b, where b is regarded as the data of the problem, and A is regarded as being known exactly.
9. Consider the least-squares problem Ax=b, where m>n, and x is to be determined.
(a) Assume that A has full rank. Explain how to solve the least-squares problem using:
i. the normal equations;
ii. the QR factorization of A;
the SVD of A.
In each case, your explanation must include a justification that the algorithm leads to the solution of the least-squares problem (e.g. explain why the solution of the normal equations is the
solution of the least-squares problem).
(b) Discuss the advantages and disadvantages of each of the above methods. Which is the method of choice for the full rank least-squares problem?
(a) Give a simple example to show that Gaussian elimination without partial pivoting is unstable in finite-precision arithmetic. (Hint: The example can be as small as
(b) Using the concept of backward error analysis, explain the conditions under which Gaussian elimination with partial pivoting can be unstable in finite-precision arithmetic. (Note: This
question does not ask you to perform a backward error analysis. Rather, you can quote standard results in your explanation.)
(c) Give an example to show that Gaussian elimination with partial pivoting can be unstable.
(a) Suppose that
Explain how to perform the power method, and under what conditions it converges to an eigenvalue.
(b) Explain the idea of simultaneous iteration.
(c) Explain the QR algorithm and its relationship to simultaneous iteration.
12. Suppose that B is an estimate of A^-1, and AB = I+E. Show that the relative error in B is bounded by
13. Show that if A is symmetric positive definite and banded, say a[ij] = 0 for |i-j| > p, then the Cholesky factor B of A satisfies b[ij] = 0 for j > i or j < i-p.
14. Suppose that Gaussian elimination (without partial pivoting) is applied to a symmetric positive definite matrix A. Write
where each E[j] is an elementary (lower triangular) matrix (left-multiplication by E[j] accomplishes the jth step of Gaussian elimination). None of the E[j]s is a permutation matrix, that is, no
row interchanges are performed. The purpose of this exercise is to prove that this is possible (i.e. that Gaussian elimination can be applied without row interchanges) and to prove the following
Do this by proving the following three lemmas:
(a) Let B be a symmetric positive definite matrix. Then b[ii] > 0for B (in magnitude) occurs on the diagonal.
(b) Let A be a symmetric positive definite matrix, and suppose one step of Gaussian elimination is applied to A to obtain
(c) Using the notation of the previous lemma,
Now complete the proof by induction. (Note that this result both proves that no partial pivoting is required for a symmetric positive definite matrix, and also that Gaussian elimination is
perfectly stable when applied to such a matrix.)
15. Let A=USV^T, where S is diagonal (S[ij]=0 if S^(k) by
and define A^(k) by
What is
Next: Algebra Up: Optimization/Numerical Linear Algebra Previous: Outline Math Dept Webmaster | {"url":"http://www.math.mtu.edu/graduate/comp/node12.html","timestamp":"2014-04-18T10:34:36Z","content_type":null,"content_length":"17183","record_id":"<urn:uuid:53ca127b-0359-4049-9553-dee3367da10e>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00083-ip-10-147-4-33.ec2.internal.warc.gz"} |
Seasonal trades in stocks
Readers of this blog have seen my discussions of various seasonal trades in commodities futures (e.g. see this
). Recently,
Mark Hulbert
of the NYTimes drew our attention to a seasonal trade in stocks. The strategy is very simple: each month, buy a number of stocks that performed the best in the same month a year earlier, and short
the same number of stocks that performed poorest in that month a year earlier. The average annual return is more than 13% before transaction costs, and since it is market neutral, this already
considerable return can be leveraged to 2 or 3 times higher. Also, since it turns over the stocks only once a month, transaction costs should not be a major problem. The strategy was developed by
Profs. Steven Heston and Ronnie Sadka, and details can be found online
. Besides its simplicity, the strategy is not as affected by survivorship bias in the data set as a mean-reverting strategy, since survivorship bias would tend to lower its backtest performance by
excluding very poorly performing stocks that we would short. All in all, it seems to be a market neutral strategy made for retail trading!
17 comments:
FYI I consider this analyst the authority in seasonal trading, at least in Canada. He combines technical, seasonal, and fundamental trading analysis. http://market-minute.dvtechtalk.com/
Dear Ernie
This is a very interesting trading idea. Thank you for posting about it on your blog.
First, I’m really not sure to what extent this would in fact be a profitable trading strategy after allowing for costs. I note that the authors themselves doubt that it would be a profitable
stand-alone strategy once costs are allowed for. The monthly pre-cost profitability is little more than one percent per month. Round trip costs per trade of 50 bps – not too unreasonable an
estimate allowing for commission, spread and financing costs - would account for all of the ‘profitability’ (keeping in mind that this is 50 bps per side i.e. 100 bps for the long and short-side
Second, I note that the study uses data up-to 2002, no later. Many strategies have been observed to exhibit declining profitability in recent years (you have noted this yourself in a number of
posts and it is discussed in the Loe et al article that you have cited previously).
To see whether the seasonality phenomenon identified by Heston and Sadka might persist in more recent years, I tested the idea on data from 2002 to June 2007 (the sample included data from 2001
in order to determine which stocks to buy and sell in 2002). I looked at US large caps only (contemporaneous market cap greater than $2bn). The authors find that the seasonality effect applies
throughout the size distribution (though I have not read the article closely enough to see whether it is particularly prnounced at certain points of the size distribution). I chose large caps
simply because I currently happen to have reliable data for large caps in memory (including for delisted companies).
The results are consistent with the Heston and Sadka findings, though the effect is more modest. The authors rank stocks according to their performance in the same calendar month a year earlier
and buy the stocks in the top performing decile and short the stocks in the worst performing decile. For 2002 onwards, the top decile out-performs the bottom decile by 35 bps per month (4.2% per
It would be interesting to repeat this exercise for mid and small cap stocks as well as for other, arguably less efficient markets. I have also not read the Heston and Sadka paper in great
detail, so I may well have missed out an important part of the calculations (apologies in advance). However, given trading costs, I am not convinced that there is much to be gained from this
particular seasonal trading strategy, at least as implemented above.
But there may be variations on this strategy that may be worth pursuing. For starters, selecting the top and bottom 5% rather than 10% of winners and loosers should in theory yield higher returns
(though at the expense of greater noise). This is indeed the case, but the improvement in returns is extremely slight. The returns increase from 35 to 39 bps per month on average.
Heston and Sadka find the seasonablity effect to be particularly pronounced in the months of January, October and December. So, a possible strategy might involve restricting trading of this
strategy to these months. The evidence suggests that this aproach seems far more promising. The strategy based on the decile groups produces an average monthly return of 189 bps and the strategy
based on the 5% groups produces a an average monthly return of 231 bps (results apply to 2002-2006 i.e. completed years). These results are extremely impressive. However, the average monthly
returns appear to decline over time and yield a negative return in 2006. It will be interesting to see the effect for the whole of 2007, though January 2007 (the only month for which I have data)
produces a negative return.
Thus, I believe there may be a trading strategy in this but I am not yet convinced. If the results for January, October and December 2007 together generate decent returns, then this trading idea
might merit some consideration for January, October and December 2008.
Many strategies building on market inefficiencies have been found to persist longer in mid cap compared to large cap stocks (see, for example, the article by Loe et al, 2007). I intend to repeat
the above exercise on mid cap stocks. It will be especially interesting to observe the results for 2006 (and 2007 when available).
Dear JR,
Thank you for publishing your detailed analysis here! Actually, I just did a backtest myself using SP500 stocks (this data has survivorship bias as I use the current constituents of SPX). I found
that the returns were negative even without transaction costs from 20050102 - present. In the earlier years, where my data is less reliable because of survivorship bias, a small positive return
can be found, but with high volatility, such that the Sharpe ratio was never greater than 1.
I agree with your conclusion: it is not a strategy I would personally engage in, whether in conjunction with other strategies or not.
Thank you for your backtest. A possible explanation for bad results in the last years is the very low volatility. Low volatility is the worst enemy of every algorithmic strategy.
In my opinion if a strategy has proven robustness in many many years (and isn't overfitted) one/two years of bad results wouldn't change our opinion about its 'goodness'.
Otherwise, do you know strategy that works every single years for many many years?
I think it's impossible.
Dear statarb,
I agree that it is not easy to find strategies that keep working year after year going forward. However, there are many strategies that I know of which have worked every year in backtest. If a
strategy hasn't worked for a year or two in the recent past, I would not trade it. As a stat arb trader, the goal is to achieve a maximum drawdown duration that is shorter than 1 month.
a) strategies which have worked every year in backtest. The question is: how many years? Five, six? Is it enough? In accademic literature (scientific method) i haven't read nothing which use less
then five economic cycle of data (5 years).
b) can you tell approximately which strategies are you talking about? So we can do our backtest on more history/market.
c) Drawdown shorter then 1 month. My problem is: how you can know first the length of your future Drawdown? It's impossible to know ex-ante. So the best thing that statistic can do for us is to
give a positive mathematical expectancy, not for the next month, but for the next years. Do you agree with this?
1) Yes, I believe 5 years is a good number. Financial market is not stationary, so longer history is not necessarily better.
2) I am referring to high frequency mean-reverting strategies.
3) It is true that one won't know what one's drawdown is in the future, but it is safe to say that the future drawdown is going to be longer than the drawdown in backtest. Thus if the dd is
already longer than 1 year in backtest, there is no hope for a desirable future dd.
Ok, however there is a difference beetwin high frequency trading and almost position strategies. In this seasonal strategies you are always IN the market.
The high frequency trading strategies you are talking about is what you give in premium section of your blog? Is it a long/short strategy on single stocks?
Dear statarb,
The high frequencies strategies include those that have been disclosed to me in confidence, as well as those that I am currently actively pursuing. As a result, I have not publicized them in any
What could one imagine under these secret high-frequency strategies. Is it cointegration /pairs trading in a much smaller time frame? Momentum strategies? Maybe you can gibe a hint ;-) or a link
to some ressources?
This comment has been removed by a blog administrator.
I am reading your book, and i can see that your seasonal pattern on natural gas has disappeared in the last few years. Isn t it difficult to establish a seasonal rules or make statistically
significant when test sample is only 20 years for instance
Actually, what you said applies to any trading strategy: practically no trading strategy works for ever!
Ernie, the link to the seminar that you provided no longer seems to work. Is there another source for that file?
A google search of the authors turned up this link:
Thanks, yes I had found that paper, but I was looking for the seminar which you linked to, which I assumed was something different. Never mind...
This is a great strategy AND there is a great Seasonal Stock Trading Service at http://www.blashing.com that provides Stock Seasonality information for every stock, ETF, and mutual fund. It also
provides a Seasonal Stock Scanner that gives you the best equities going Up and Down for every Month and every Week of the year! | {"url":"http://epchan.blogspot.com/2007/11/seasonal-trades-in-stocks.html","timestamp":"2014-04-19T04:49:44Z","content_type":null,"content_length":"107760","record_id":"<urn:uuid:e1a7f2cf-8b94-4373-a8cc-57df12789e3f>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00504-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math failures – haven’t we heard this before?
Roberta M. Eisenberg is chair of the UFT Math Teachers Committee.
As controversies rage about the best way to teach math and whether students should be allowed to use calculators — incidentally, the State Education Department on Dec. 1 declared that calculators
will now be considered teaching materials, like textbooks, and schools must provide them to students — the real question is why children in this country are not better at learning math. Is it the
curriculum? Is it the equipment? Is it the tests? And, haven’t we heard all this before?
In 1957, the Russians sent up Sputnik, stealing a march in the space race, and the United States decided that something had to be done, in a hurry, about math and science instruction in this country.
Thus were born National Science Foundation grants to teachers of math and science so that they might get master’s degrees in their subjects rather than in education. A generation of teachers
excitedly brought their advanced knowledge back to their classrooms.
Also in the early ’60s, the so-called New Math was influencing curricula across the country. The result was an emphasis on concepts to the detriment of the basics. Naturally, there was an eventual
backlash when parents could no longer understand their children’s homework.
By the ’70s, teachers in middle and high schools were noticing that students were getting weaker on their recall of times tables and other basics. This could not then be blamed on calculators because
there were no calculators yet in general use.
In the ’90s there was growing concern that lack of math skills by American kids would reduce us to a third-world economy. A few weeks ago, an article in The New York Times said essentially the same
thing. In “As Math Scores Lag, a New Push for the Basics” (Nov. 14), Tamar Lewin stated, “For the second time in a generation, education officials are rethinking the teaching of math in American
This was not the second time nor was it only in one generation. Changing the curriculum has been going on for at least 150 years. At one time, the math skills needed by the citizenry were mainly
arithmetic and practical geometry. Carpenters knew about rectangles and squares in order to produce cabinets with right angles in the right places. Very few people went to college, and therefore very
few needed to know algebra and more advanced math.
It is instructive — and funny — to read some of the old tirades against slide rules and typewriters. They were blamed for students’ loss of ability to do times tables, and it was even claimed that
students would no longer be able to write legibly.
Every advance in technology has brought about changes in curriculum. The State of New York has been very slow in permitting and then requiring calculators on high school Regents exams — finally
allowing basic calculators. In 1989, graphing calculators began to appear. As teachers got excited by the new technology and began to change the way mathematics is taught, they also began to push the
SED to require these calculators on math exams. Finally, for the past few years, they have been allowed on the Math A and required on the Math B Regents.
Now New York math standards are changing again in reaction to outcries from parents and teachers after disastrous results on the Math A Regents a few years ago. Math A and B are being replaced after
a short-lived, unsuccessful life. No one knows what the new Regents in algebra, geometry and intermediate algebra and trigonometry will look like.
So why have student skills gradually deteriorated over the decades?
Is it the fault of the curriculum? There is no national curriculum in any subject. In New York State, since the introduction of Math A and B, we have the completely illogical situation of standards
and assessments without any curriculum. The UFT and NYSUT have followed the AFT’s call for a grade-by-grade curriculum. Teachers need to know exactly what to teach and in how much depth. Students and
parents must know what is required on each assessment (as exams are now called).
Is it the fault of calculators and other technology? Students learn much more exciting mathematics and can literally see things with graphing calculators that were never really seen before, not even
by authors of calculus textbooks. The types of questions asked have necessarily changed as the technology has improved. In fact, with a graphing calculator — and simpler calculators at lower levels —
the math that students do is much harder than without them.
So why are students not learning math and comparing unfavorably with students in other countries on international tests? Around 1990, Al Shanker cited a statistic that was shocking to hear but
realistic upon reflection. He said, “Sixty-five percent of high school students in our country never do any homework. Never do any.”
Parents and the public don’t expect excellence to occur, nor even passable skills, in sports and music without lots of practice and repetition. How can they expect less from academic subjects?
Related to this is a public belief that it’s OK not to be able to do math. Parents often tell their kids that they themselves could “never do math” either. As Nicholas D. Kristof stated in a Times
Op-Ed piece (“Watching the Jobs Go By,” Feb. 11, 2004), “The broader problem is not just in schools but society as a whole: There’s a tendency in U.S. intellectual circles to value the humanities but
not the sciences. Anyone who doesn’t nod sagely at the mention of Plato’s cave is dismissed as barely civilized, while it’s no blemish to be ignorant of statistics, probability and genetics.”
He concluded, “In 1957, the Soviet launching of Sputnik frightened America into substantially improving math and science education. I’m hoping that the loss of jobs in medicine and computers to India
and elsewhere will again jolt us into bolstering our own teaching of math and science.”
The hard part is not the teaching but the changing of attitudes in a country.
6 Comments:
• Math failures – haven’t we heard this before?…
Roberta M. Eisenberg:As controversies rage about the best way to teach math and whether students should be allowed to use calculators — incidentally, the State Education Department on Dec. 1
declared that calculators will now be considered teaching m…
• Thanks for addressing an issue in that is of concern to virtually every elementary teach I meet, and every math teacher on the secondary level. The question of why our students can’t do math
absolutely needs to be asked, and you seem to give two tentative answers of your own, with which I agree with.
First, you say, “Parents and the public don’t expect excellence to occur, nor even passable skills, in sports and music without lots of practice and repetition. How can they expect less from
academic subjects?”
I agree. And it seems that every rank-and-file teacher I talk to agrees with you too. In fact, when it comes to Math, they have agreed for years. And for years they have been yelling about the
need for students to master the basics – from times tables to basic math algorithms – but have been derided as old-fashioned. They were forced – as they have been forced in other subjects – to
teach against what they knew would be best. On the elementary level it was a different math concept everyday – five miles wide and one inch deep. In high school, every math teach I knew was
skeptical about the change from the algebra-geometry-trig sequence. No one listened, and now – after ten bad years – it’s coming back.
Since no one but we teachers seem to care what teachers say – I offer you as support an article published in American Educator in Spring 2006 by Daniel Willingham, titled, How Knowledge Helps.
Willingham, who is a professor of cognitive psychology, explains how important it is for students to acquire background knowledge, and shows how this applies to math as well.( http://
www.nychold.com/talk-ocken-051002.doc) I am probably about to distort and simplify what Willingham says, and I suggest all our members read the article. But put simply,Willingham explains how
working memory can only juggle a limited amount of discreet bits of information. To juggle more, it has to rely on chunking. Random letters, for example, are hard to remember and take up a lot of
space in our short term memories; but NCLB takes up less because our familiarity/recall allows us to see them as a single unit. Willingham offers a better example.
“Consider, for example, the plight of the algebra student who has not mastered the distributive property. Every time he faces a problem with a(b + c), he must stop and plug in easy numbers to
figure out whether he should write a(b) + c or a + b(c) or a(b) + a(c). The best possible outcome is that he will eventually finish the problem—but he will have taken much longer than the
students who know the distributive property well (and, therefore, have chunked it as just one step in solving the problem). The more likely outcome is that his working memory will become
overwhelmed and he either won’t finish the problem or he’ll get it wrong.”
So, mastery of basics has the blessings of the teachers, the cognitive scientists – and finally the National Council of the Teachers of Mathematics, whose fuzzy notions about math have driven
education here in NY, much to the dismay of classroom teachers. This Fall NCTM finally reversed itself (something it won’t admit), and perhaps now math teachers can all get some relief.
I agree also with your second point: there is no curriculum. As you point out, and as the UFT and AFT have pointed out, there is no clear idea of what children should know, and when they should
know it. Our voices have been in line with some of NY’s top educators, which of course have been ignored. For example, Stanley Ocken of CUNY delivered a paper in October of 2005 entitled,
Mathematics Education Reform: Toward a Coherent K-12 Curriculum. ( http://www.aft.org/pubs-reports/american_educator/issues/spring06/willingham.htm) In it, he pointed out that under Harold Levy,
the Board of Education and CUNY convened a Math Commission charged with setting directions for NYC K-12 math education, which resulted in the Goldstein Report. Ockem tells us:
“A principal recommendation of the resulting Goldstein Report was to focus on K-16 education in New York as a “seamless system,” with co-ordination between CUNY mathematics departments and K-12
educators. The word “seamless” was used to indicate proper alignment of mathematics requirements from elementary school through college. That was a great idea. Had it been implemented, following
the California model, we could by now have been well on the way to establishing a K-16 mathematics curriculum that is both seamless and coherent……Unfortunately, that goal seems rather distant.”
Ocken goes on to tell us how he and other CUNY and NYU math scholars were shut out of the discussions about Math once Klein took over. In fact, they even sent a letter of warning about the
programs Klein was adopted, but as Ocken says, “That letter never received the courtesy of a reply.” Instead Klein relied on the NCTM standards that came up short on basic math skills, in favor
of concepts.
So, it is not for lack of voices that math has suffered in this country, and especially in New York. It’s for lack of ears.
• Thanks for writing this, Bobbi.
It is the attitude in this country that is a problem, but not the only problem.
We can look far back, and things before were not rosy either. How many students used to really “get” algebra? 60%? Geometry? 30%?
So we made lots of changes to help reach the kids who weren’t getting math, and made things worse for the kids who were. I am as nervous about the new regents as you are, but I am glad that at
least the names of the exams now make sense. We need to take some steps backwards, and that is one.
At the same time, there is still the problem of reaching the kids who never got math. Just because A and B were complete disasters does not remove our obligation to teach as much math as students
are able to learn, and to keep trying to increase the amount they are able to learn.
• Over six years ago the Bronx Superintendant told us that every high school in the Bronx would change their math programs to IMP or Math Connections (two constructivist programs). Math teachers
griped, and the UFT stepped up.
Our District Rep at the time, Dave Schulman, organized a committee of us to file a district-wide request for professional conciliation under Article 24 of our contract.
About two dozen teachers met on and off over the course of a year to prepare. There were maybe five or six core teachers, and I was one of them.
We did good work. We had our hearing with the Supe and his deputy present. And we won.
Why mention this now? Because along the way our core group became very familiar with other, national aspects of the Math Wars. And part of that education was being exposed to what I would
characterize as extremists: those who were using mathematics as a political battleground. As a group, we were not comfortable with either side.
Here’s part of what I wrote to Dave, on the evening before our hearing: “Let’s start with the “Math Wars.” It makes me damned nervous to be on the same side as what I would call right-wing kooks.
It started as a California thing: “Back to basics” vs. “Constructivists” along roughly the same fault lines as the anti-Bilingual and the anti-Affirmative Action fights there. … I like to think
of us as taking the reasonable center against the Ed nuts on one side, but then holding it against hte back to basics cretins who will certainly be emboldened enough to start making real noise…”
(June 14, 2001)
NYCHOLD is very much part of that back-to-basics extreme. And as far as the CUNY math chairs being excluded, and mind you, I like these guys, but they do not even have a consistent curriculum
campus to campus. They were right that Diana Lam and Joel Klein’s decisions stunk, but they did not have a better proposal, and do not; they are not pedagogues. It is easier to be a critic, which
is a suitable role for them, than to actually make the curricular decisions.
• I agree with you Jonathan- extremes on either side are bound to fail. And if you think Math is politically fueled — well, you should see reading. At least the content in math classes is not
politically charged, but in English it is(even at the elementary level), and I think the wars there are even fiercer than they are in Math.
I think teachers tend to know how to avoid the extremes and the politics — their focus is on real teaching, not theory. That’s the problem with the Kleinists. They say their programs are
balanced, but the teachers I speak to (and my own experience as a high school English teacher) tell me they are not. Teachers find themselves compelled to enact someone else’s theoretical and
political agenda, in programs that lean too heavily to constructivism in math and whole language in English. In reading, writing, and math in the elementary schools either you teach the Klein
way, or, well, you teach the Klein way. Curriculum is de-emphasized, and pedagogy is not permitted to grow intrinsically from the content .Then in high school we are left to pick up the pieces.
You were lucky to be able to have that kind of collaboration in your district, and that sounds like the heart and soul of what unions and DR’s ought to do with their members. Six years ago sounds
pre Klein. Was it?
• 6 years ago was indeed Pre-Klein. And all of our effort (tremendous, and successful) was wiped out two years later by the Lam-adoptions…
However, the sense of control and professional input that we gained was immense, even if the results were bureaucratically over-ridden later on.
And I have heard a bit about reading. It is often the same people arguing whole language vs phonics as argue traditional vs. constructivist math. | {"url":"http://www.edwize.org/math-failures-havent-we-heard-this-before","timestamp":"2014-04-20T10:46:37Z","content_type":null,"content_length":"55883","record_id":"<urn:uuid:60b1d6c3-1935-4bf6-b2e9-e8f2573957ca>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00595-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: Industrial Compiler Optimization Features Survey
"Christopher Glaeser" <cdg@nullstone.com>
28 Jan 2002 01:05:08 -0500
From comp.compilers
| List of all articles for this month |
From: "Christopher Glaeser" <cdg@nullstone.com>
Newsgroups: comp.compilers
Date: 28 Jan 2002 01:05:08 -0500
Organization: Concentric Internet Services
References: 02-01-123
Keywords: optimize
Posted-Date: 28 Jan 2002 01:05:08 EST
> I am working on a survey on the optimization features of industrial
> compilers, especially those machine independent optimizations. In
> literature, there are numerous optimization/transformation
> documented, I am interested to know what subset of those are
> considered important and got implemented in the industrial
> compilers.
Most commercial compilers are promoted with marketing literature using
the industry standard laundry list of optimizations. That is, most of
these compilers perform constant propagation, common subexpression
elimination, strength reduction, instruction scheduling, and so on and
so on. The hard part is determining the sophistication of each of
these optimizers. For the relatively weak optimizers, some of the
optimizations performed are best characterized as an existence proof,
which is to say, there exists at least one program fragment for which
the optimizer will perform that optimization. In contrast, the more
powerful optimizers can perform that optimization for many different
program fragments.
Let me give a simple example. Consider the following code fragment and
common subexpression elimination.
x = a + b;
y = a + b;
Weak optimizers will only perform this optimization if and only if
- x, y, a, and b are local or register
- x, y, a, and b must be int
- the operator must be addition or subtraction or possibly multiplication
- the two statements must occur in the same basic block
Sophisticated optimizers can perform this optimization when
- x, y, a, and b can be local, register, static, or extern
- x, y, a, and b can be char, short, int, long, float, double
- CSE is performed for all operators such as and, or, not, left shift,
right shift, etc, etc
- the two statements can be in the same basic block, extended basic block,
or more complex control flow
This is just one simple example, but the concept holds true for all other
optimizations. Something to consider as you collect information for your
survey. Hope that helps.
Christopher Glaeser cdg@nullstone.com
Nullstone Corporation http://www.nullstone.com
Post a followup to this message
Return to the comp.compilers page.
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Wilsons Theorem First Proof
Wilsons Theorem First Proof
A selection of articles related to wilsons theorem first proof.
Original articles from our library related to the Wilsons Theorem First Proof. See Table of Contents for further available material (downloadable resources) on Wilsons Theorem First Proof.
Wilsons Theorem First Proof is described in multiple online sources, as addition to our editors' articles, see section below for printable documents, Wilsons Theorem First Proof books and related
Suggested Pdf Resources
Fermat's Little Theorem and Wilson's Theorem are two of the most famous and .
Frequently, in Wilson's Theorem, only the if part is stated. The Case n is a Composite. We are going to reach the proof of this theorem in stages.
will cover the necessary algebra, a proof of Wilson's Theorem, and a proof of .
In order to prove this, we'll first prove the following. Theorem 3. ..
first proving Wilson's Theorem : (p–1)! ≡ p–1 mod p . What follows are two more direct proofs of Fermat's Little Theorem.
Suggested Web Resources
If p is composite, then its positive divisors are among the integers 1, 2, 3, 4, … , p − 1 and it is clear that gcd((p − 1)!
Aug 16, 2005 We first show that, if $p$ is a prime, then $(p-1)! This is version 5 of proof of Wilson's theorem, born on 2002-01-05, modified 2005-10-11.
Mar 3, 2011 1 Theorem; 2 Proof This proof was attributed to John Wilson by Edward Waring in his 1770 It was first stated by Ibn al-Haytham ("Alhazen").
Wilson's Theorem Number Theory discussion. I am having trouble getting this proof started. Can you please give me some direction?
Frequently, in Wilson's Theorem, only the if part is stated. The Case n is a Composite. We are going to reach the proof of this theorem in stages.
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. | {"url":"http://www.realmagick.com/wilsons-theorem-first-proof/","timestamp":"2014-04-20T03:36:32Z","content_type":null,"content_length":"28576","record_id":"<urn:uuid:16f0f349-3459-4188-8745-0e331309c301>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00251-ip-10-147-4-33.ec2.internal.warc.gz"} |
[ODE] center of mass
Guillaume Jouffroy oxymore at tele2.fr
Wed Oct 19 20:50:22 MST 2005
Sorry for second posting, the question has been in another subject,
don't know why.
Hope this time it will work
I have a problem to clearly understand the center of mass position. So I
will ask these
questions to try to have it limpid :-/
1. Imagine I want to build an object composed of 2 cylinders (2 geoms),
the cylinders
are linked to each other at one extremity, and form a 90 degree angle.
Where will be the rigidbody if I create this geom structure if I attach
these 2 geoms to the rigidbody ?
2. Is the center of mass at the "middle" of a geom the default ?
3. how to compute the center of mass of the whole object ? (should the
rigidbody and the center of mass
coincide ?)
4. Actually I don't really understand this part of the test_boxstack
example :
// move all encapsulated objects so that the center of mass is (0,0,0)
for (k=0; k<2; k++) {
dGeomSetPosition (g2[k],
dMassTranslate (&m,-m.c[0],-m.c[1],-m.c[2]);
Just above this code the mass of the geoms are already translated and
rotated like the geoms are.
So ??? @@@#"~""é"'é~#{
Please enlight me :'(
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NAG Library
NAG Library Routine Document
1 Purpose
G02BYF computes a partial correlation/variance-covariance matrix from a correlation or variance-covariance matrix computed by
2 Specification
SUBROUTINE G02BYF ( M, NY, NX, ISZ, R, LDR, P, LDP, WK, IFAIL)
INTEGER M, NY, NX, ISZ(M), LDR, LDP, IFAIL
REAL (KIND=nag_wp) R(LDR,M), P(LDP,NY), WK(NY*NX+NX*(NX+1)/2)
3 Description
Partial correlation can be used to explore the association between pairs of random variables in the presence of other variables. For three variables,
, the partial correlation coefficient between
is computed as:
$r12-r13r23 1-r1321-r232 ,$
is the product-moment correlation coefficient between variables with subscripts
. The partial correlation coefficient is a measure of the linear association between
having eliminated the effect due to both
being linearly associated with
. That is, it is a measure of association between
conditional upon fixed values of
. Like the full correlation coefficients the partial correlation coefficient takes a value in the range (
) with the value
indicating no association.
In general, let a set of variables be partitioned into two groups
variables in
variables in
and let the variance-covariance matrix of all
variables be partitioned into,
The variance-covariance of
conditional on fixed values of the
variables is given by:
The partial correlation matrix is then computed by standardizing
${\Sigma }_{y\mid x}$
$diagΣy∣x -12Σy∣xdiagΣy∣x -12.$
To test the hypothesis that a partial correlation is zero under the assumption that the data has an approximately Normal distribution a test similar to the test for the full correlation coefficient
can be used. If
is the computed partial correlation coefficient then the appropriate
statistic is
which has approximately a Student's
-distribution with
degrees of freedom, where
is the number of observations from which the full correlation coefficients were computed.
4 References
Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press
Morrison D F (1967) Multivariate Statistical Methods McGraw–Hill
Osborn J F (1979) Statistical Exercises in Medical Research Blackwell
Snedecor G W and Cochran W G (1967) Statistical Methods Iowa State University Press
5 Parameters
1: M – INTEGERInput
On entry
: the number of variables in the variance-covariance/correlation matrix given in
Constraint: ${\mathbf{M}}\ge 3$.
2: NY – INTEGERInput
On entry: the number of $Y$ variables, ${n}_{y}$, for which partial correlation coefficients are to be computed.
Constraint: ${\mathbf{NY}}\ge 2$.
3: NX – INTEGERInput
On entry: the number of $X$ variables, ${n}_{x}$, which are to be considered as fixed.
□ ${\mathbf{NX}}\ge 1$;
□ ${\mathbf{NY}}+{\mathbf{NX}}\le {\mathbf{M}}$.
4: ISZ(M) – INTEGER arrayInput
On entry
: indicates which variables belong to set
The $\mathit{i}$th variable is a $Y$ variable, for $\mathit{i}=1,2,\dots ,{\mathbf{M}}$.
The $i$th variable is a $X$ variable.
The $i$th variable is not included in the computations.
□ exactly NY elements of ISZ must be $\text{}<0$;
□ exactly NX elements of ISZ must be $\text{}>0$.
5: R(LDR,M) – REAL (KIND=nag_wp) arrayInput
On entry
: the variance-covariance or correlation matrix for the
variables as given by
. Only the upper triangle need be given.
Note: the matrix must be a full rank variance-covariance or correlation matrix and so be positive definite. This condition is not directly checked by the routine.
6: LDR – INTEGERInput
On entry
: the first dimension of the array
as declared in the (sub)program from which G02BYF is called.
Constraint: ${\mathbf{LDR}}\ge {\mathbf{M}}$.
7: P(LDP,NY) – REAL (KIND=nag_wp) arrayOutput
On exit
: the strict upper triangle of
contains the strict upper triangular part of the
partial correlation matrix. The lower triangle contains the lower triangle of the
partial variance-covariance matrix if the matrix given in
is a variance-covariance matrix. If the matrix given in
is a partial correlation matrix then the variance-covariance matrix is for standardized variables.
8: LDP – INTEGERInput
On entry
: the first dimension of the array
as declared in the (sub)program from which G02BYF is called.
Constraint: ${\mathbf{LDP}}\ge {\mathbf{NY}}$.
9: WK(${\mathbf{NY}}×{\mathbf{NX}}+{\mathbf{NX}}×\left({\mathbf{NX}}+1\right)/2$) – REAL (KIND=nag_wp) arrayWorkspace
10: IFAIL – INTEGERInput/Output
On entry
must be set to
$-1\text{ or }1$
. If you are unfamiliar with this parameter you should refer to
Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$-1\text{ or }1$
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit
unless the routine detects an error or a warning has been flagged (see
Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by
Errors or warnings detected by the routine:
On entry, ${\mathbf{M}}<3$,
or ${\mathbf{NY}}<2$,
or ${\mathbf{NX}}<1$,
or ${\mathbf{NY}}+{\mathbf{NX}}>{\mathbf{M}}$,
or ${\mathbf{LDR}}<{\mathbf{M}}$,
or ${\mathbf{LDP}}<{\mathbf{NY}}$.
On entry, there are not exactly NY elements of ${\mathbf{ISZ}}<0$,
or there are not exactly NX elements of ${\mathbf{ISZ}}>0$.
On entry, the variance-covariance/correlation matrix of the $X$ variables, ${\Sigma }_{xx}$, is not of full rank. Try removing some of the $X$ variables by setting the appropriate element of ${\
Either a diagonal element of the partial variance-covariance matrix,
${\Sigma }_{y\mid x}$
, is zero and/or a computed partial correlation coefficient is greater than one. Both indicate that the matrix input in
was not positive definite.
7 Accuracy
G02BYF computes the partial variance-covariance matrix,
${\Sigma }_{y\mid x}$
, by computing the Cholesky factorization of
${\Sigma }_{xx}$
. If
${\Sigma }_{xx}$
is not of full rank the computation will fail. For a statement on the accuracy of the Cholesky factorization see
F07GDF (DPPTRF)
Models that represent the linear associations given by partial correlations can be fitted using the multiple regression routine
9 Example
Data, given by
Osborn (1979)
, on the number of deaths, smoke (
) and sulphur dioxide (parts/million) during an intense period of fog is input. The correlations are computed using
and the partial correlation between deaths and smoke given sulphur dioxide is computed using G02BYF. Both correlation matrices are printed using the routine
9.1 Program Text
9.2 Program Data
9.3 Program Results | {"url":"http://www.nag.com/numeric/fl/nagdoc_fl24/html/G02/g02byf.html","timestamp":"2014-04-19T22:19:43Z","content_type":null,"content_length":"27127","record_id":"<urn:uuid:963fb603-f5d7-4664-94a8-4fa0ef9f2499>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00292-ip-10-147-4-33.ec2.internal.warc.gz"} |
Naming or Defining types of Growth.
Naming or Defining types of Growth.
I was in a discussion with a friend. I understand that Geometric growth and Exponential growth often amount to the same thing as mentioned in Wikipedia under 'exponential growth'. However, I thought
'exponential growth' is sometimes also called 'logarithmic growth' and in Wikipedia under 'bacterial growth' there is an associated discussion of exponential growth during a 'log phase'. Can
population growth sometimes be synonymously called: geometric growth, exponential growth, and logarithmic growth? I just registered on "Purple Math" and like math but have found advanced math
difficult. Thanks.
"Exponential" growth is usually meant to be increasingly increasing; that is, it gets bigger faster and faster. Also, exponential growth generally has a fixed minimum, a horizontal line that it is
never below. You can see a sampling of typical graphs at the bottom of
this page
"Logarithmic" growth is usually meant to be decreasingly increasing; that is, it gets bigger, but slower and slower. Also, logarithmic growth generally has a restricted domain, a vertical line that
it never crosses. You can see a sampling of typical graphs at the bottom of
this page
Re: Naming or Defining types of Growth.
Thank you for the help. The Wikipedia article for "Bacterial Growth" does have a discussion that parallels what you just showed as well as said. Even if a growth curve has a rapidly increasing
population 'log phase', that does not mean the rapidly doubling population is increasingly increasing -- or does it? Must there be an inflection point somewhere in a bacterial growth curve where (1)
there is a single bacterium; (2) there is a stage of exponential growth with much doublings of the population; (3) there then begins a stage of logarithmic growth with gradual slowing due to
limitations of growth perhaps from food supply or space; (4) then there may be a steady phase; and, (5) maybe even a declining phase.
Perhaps the Wikipedia article on "Bacterial Growth" was confusing one part of a curve which consisted of both exponential growth and logarithmic growth and calling it by one name. If the rate of
doublings of the population are growing shorter and shorter, then do we have exponential growth? And, if the rate of doublings of the population are growing longer and longer, then do we have
logarithmic growth?
Do the so-called 'sigmoid growth curves' consist of an exponential phase and a logarithmic phase? I hope this is an okay question to ask about in the "pre-calculus area." Or, is there a better name
than "sigmoid"? Thanks again.
Re: Naming or Defining types of Growth.
Growth curves are probably "logistic" but the growth part may be both exponential and logarithmic and the decline part could be exponential and logarithmic too. Maybe with a logistic curve there is
no inflection point. ??? I guess there is need for me to take a trip to the library or check further on the internet. Thanks. It is much clearer. | {"url":"http://www.purplemath.com/learning/viewtopic.php?t=1916","timestamp":"2014-04-20T04:03:23Z","content_type":null,"content_length":"25966","record_id":"<urn:uuid:46068b18-02c2-4817-a42f-fc3619c2a29d>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00258-ip-10-147-4-33.ec2.internal.warc.gz"} |
Back to Basics: Thermowell protection
Thermowells, while protecting temperature sensors from a process fluid, can undergo tremendous stresses. A new standard is designed to lower risk of failures.
Thermowells protect temperature sensors from direct contact with a process fluid. But once inserted into the process, the thermowell can obstruct flow around it, leading to a drop in pressure. This
phenomenon creates low-pressure vortices downstream of the thermowell (the same principle underlying vortex flowmeters). Vortices can occur at one side of the thermowell and then the other, which is
known as alternating vortex shedding. This effect can be seen in the example of a flagpole rippling a flag in the wind.
The result is that thermowells experience a combination of stresses: the flow pushing on the thermowell (drag forces) and the vortex shedding (lift forces). Instrument engineers should evaluate the
thermowell to see if it can withstand these stresses as they can cause mechanical failure. The industry standard for this evaluation is ASME PTC 19.3 TW-2010, which, in 2010, superseded ASME PTC 19.3
1974. Motivation for the new standard followed some catastrophic failures of thermowells in nonsteam service. These thermowells had passed the criteria laid out in 1974. The 2010 standard includes
significant advances in the knowledge of thermowell behavior, increasing from four pages in 1974 to over 40 pages in 2010. The recent standard evaluates thermowell suitability with new and improved
calculations including:
• Various thermowell designs including stepped thermowells
• Thermowell material properties
• Detailed process information
• Review of the acceptable limit for frequency ratio
• More accurate evaluation of stresses that affect thermowells.
Forces on a thermowell
As mentioned earlier, flow passing the thermowell creates alternating vortices downstream known as shedding vortices. These shedding vortices cause the thermowell to vibrate. If this vortex shedding
rate (fs) matches the natural frequency (fnc) of the thermowell, resonance occurs, and dynamic bending stress on the thermowell greatly increases.
Forces created by the fluid in the Y plane (in-line with flow) are called drag, and forces created in the X plane (transverse to flow) are called lift, as shown in Figure 2. The vortex shedding rate
for the drag and lift must be calculated. The in-line forces (parallel to flow) are approximately 2x the transverse forces.
Low velocities
If the fluid is flowing at a very low velocity, the forces exerted on the thermowell are small, which greatly reduces the risk of thermowell failure. The new standard states that the natural
frequency, frequency limit, steady-state stress, and dynamic stress do not need to be calculated if all the following conditions are met:
• The process velocity, V, is less than 0.64 m/s [2.1 ft/s]
• Root diameter minus bore diameter (A – d) ≥ 9.5 mm [0.376 in.]
• Unsupported length, L ≤ 0.61 m [24 in.]
• Root diameter, A ≥ 12.7 mm [0.5 in.]
• Tip diameter, B ≥ 12.7 mm [0.5 in.]
• Maximum allowable working stress, S ≥ 69 MPa [10 ksi]
• Fatigue endurance limit, Sf ≥ 21 MPa [3 ksi]
• The thermowell material is not subject to corrosion or embrittlement
Although the risk of thermowell failure is small if these conditions are met, in-line resonance can still be excited at low velocities, which may lead to sensor failure.
Quantitative criteria
There are four evaluations to be carried out on a thermowell at each set of process conditions to determine the suitability.
Frequency ratio—the forces on the thermowell due to the process conditions—shall not allow the thermowell to vibrate at the critical resonance. See the Frequency ratio limit section below for details
on this criterion.
• Steady stress—combines radial, axial, and tangential stresses due to external pressure with stress caused by drag at the design velocity. This information is then used in the Von Mises criterion,
which must be less than 1.5 times the maximum allowable working stress for the thermowell material.
• Dynamic stress—the dynamic predicted stress (including drag and lift forces) must not exceed the fatigue stress limit for the thermowell
• Pressure stress—the stress put on the thermowell due to the process pressure must not be more than the thermowell is rated to. This evaluation is carried out at the shank and at the tip; both
must be rated higher than the process pressure.
Frequency ratio limit
The frequency ratio (fs/fnc) is the ratio between the vortex shedding rate and the installed natural frequency. In the old standard, the frequency ratio limit was set to 0.8. This was to avoid the
critical resonance caused by the transverse (lift) forces. Figure 3 shows the transverse resonance band above the 0.8 limit. Following the inclusion of the in-line (drag) forces, a second resonance
band (shown in black) also needs to be avoided.
The frequency limit ratio is set at either 0.4 or 0.8. The criteria for which limit to use is defined in ASME PTC 19.3 TW-2010, and the theory is simplified in Figure 4. This is the theory used in
the calculation and should not be estimated without carrying out the full evaluation.
Improvements to design
If a thermowell fails the evaluation, the design can be changed in the following ways:
• Shorten the thermowell to reduce the unsupported length
• Increase the thickness of the thermowell (A and B)
A velocity collar can be added to reduce the unsupported length, although this is not generally recommended. A velocity collar is used to provide a rigid support to the thermowell and will work only
if there is an interference fit between the standoff wall and the collar. Care must be taken to ensure the collar meets the standoff wall at installation and is not affected by corrosion. If a
velocity collar is the only viable solution, it is the responsibility of the operator to ensure there is an interference fit between the standoff wall and the velocity collar.
- Jennifer Wilson attended the University of Nottingham, achieving a BEng Hons in Chemical Engineering. She has been working for ABB for 5 years following completion of the ABB graduate scheme and
has been involved in engineering and design for temperature and differential pressure (DP) products.
Figures 2 and 3 were reprinted from ASME PTC 3 TW-2010, by permission of The American Society of Mechanical Engineers. All rights reserved. No further copies can be made without written permission
from ASME. | {"url":"http://www.plantengineering.com/industry-news/automation-news/single-article/back-to-basics-thermowell-protection/3c5bf2adeae5931923d8c3d1d457d280.html","timestamp":"2014-04-20T12:08:53Z","content_type":null,"content_length":"64142","record_id":"<urn:uuid:0d32b271-fb1f-4157-9cae-ccf93157028a>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00158-ip-10-147-4-33.ec2.internal.warc.gz"} |
Checking Taylor Expansions
August 24th 2010, 12:11 AM #1
Junior Member
Jul 2010
Checking Taylor Expansions
Hi, I just want to check that I have the Taylor expansions correct for these choices.
I cannot find any examples on the net. So would be great if you could give me these.
Thanks in advance
I think we're missing an awful lot of information here. What is $y?$ Where are your $x_{i}$'s?
Ok, so it looks like you're trying to construct an iterative sequence out of the Taylor series expansion for a function. Is that correct? If so, where do you want to expand the Taylor series? It
matters where you're trying to expand the series.
The actual question I am asked is:
Find the principle local truncation error and the order of Quade's method
And the idea is to use the Taylor series and cancel out to get the error. So I assume it is around x=0.
Couple comments:
1. I assume the last term on the LHS is meant to be lower-case.
2. In googling Quade's method (which I've never heard of before), I saw at least one definition of it that has different indices than yours. Here's one such example. I don't know if it's correct
or not. There seem to be multiple definitions of the method out there.
I'm out of my league here. CB, what do you think?
August 24th 2010, 02:30 AM #2
August 24th 2010, 02:39 AM #3
Grand Panjandrum
Nov 2005
August 24th 2010, 04:50 AM #4
Junior Member
Jul 2010
August 24th 2010, 05:00 AM #5
August 24th 2010, 05:14 AM #6
Junior Member
Jul 2010
August 24th 2010, 06:53 AM #7 | {"url":"http://mathhelpforum.com/advanced-applied-math/154291-checking-taylor-expansions.html","timestamp":"2014-04-21T03:17:11Z","content_type":null,"content_length":"54578","record_id":"<urn:uuid:42727606-49b0-463b-8e70-14683e36fe6e>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00548-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wilsons Theorem First Proof
Wilsons Theorem First Proof
A selection of articles related to wilsons theorem first proof.
Original articles from our library related to the Wilsons Theorem First Proof. See Table of Contents for further available material (downloadable resources) on Wilsons Theorem First Proof.
Wilsons Theorem First Proof is described in multiple online sources, as addition to our editors' articles, see section below for printable documents, Wilsons Theorem First Proof books and related
Suggested Pdf Resources
Fermat's Little Theorem and Wilson's Theorem are two of the most famous and .
Frequently, in Wilson's Theorem, only the if part is stated. The Case n is a Composite. We are going to reach the proof of this theorem in stages.
will cover the necessary algebra, a proof of Wilson's Theorem, and a proof of .
In order to prove this, we'll first prove the following. Theorem 3. ..
first proving Wilson's Theorem : (p–1)! ≡ p–1 mod p . What follows are two more direct proofs of Fermat's Little Theorem.
Suggested Web Resources
If p is composite, then its positive divisors are among the integers 1, 2, 3, 4, … , p − 1 and it is clear that gcd((p − 1)!
Aug 16, 2005 We first show that, if $p$ is a prime, then $(p-1)! This is version 5 of proof of Wilson's theorem, born on 2002-01-05, modified 2005-10-11.
Mar 3, 2011 1 Theorem; 2 Proof This proof was attributed to John Wilson by Edward Waring in his 1770 It was first stated by Ibn al-Haytham ("Alhazen").
Wilson's Theorem Number Theory discussion. I am having trouble getting this proof started. Can you please give me some direction?
Frequently, in Wilson's Theorem, only the if part is stated. The Case n is a Composite. We are going to reach the proof of this theorem in stages.
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. | {"url":"http://www.realmagick.com/wilsons-theorem-first-proof/","timestamp":"2014-04-20T03:36:32Z","content_type":null,"content_length":"28576","record_id":"<urn:uuid:16f0f349-3459-4188-8745-0e331309c301>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00251-ip-10-147-4-33.ec2.internal.warc.gz"} |
Suppose That A Rocket Is Lauched Straight Up From ... | Chegg.com
Suppose that a rocket is lauched straight up from the surface of the earth with initial velocity v0 = sqrt(2gR) Where R is the radiuus of the earth. Neglect air resistance.
Find the time required for the rocket to go 240000mi. Assume that R = 4000mi
Electrical Engineering | {"url":"http://www.chegg.com/homework-help/questions-and-answers/suppose-rocket-lauched-straight-surface-earth-initial-velocity-v0-sqrt-2gr-r-radiuus-earth-q2860638","timestamp":"2014-04-18T20:03:51Z","content_type":null,"content_length":"20547","record_id":"<urn:uuid:706645ca-7321-4276-976d-e82f2175e3be>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00656-ip-10-147-4-33.ec2.internal.warc.gz"} |
Four Identical Particles Of Mass 0.589 Kg Each ... | Chegg.com
Four identical particles of mass 0.589 kg each are placed at thevertices of a 2.39 m x 2.39 m square and held there by fourmassless rods, which form the sides of the square. What is therotational
inertia of this rigid body about an axis that(a) passes through the midpoints of opposite sidesand lies in the plane of the square, (b) passesthrough the midpoint of one of the sides and is
perpendicular tothe plane of the square, and (c) lies in the planeof the square and passes through two diagonally oppositeparticles? | {"url":"http://www.chegg.com/homework-help/questions-and-answers/four-identical-particles-mass-0589-kg-placed-thevertices-239-m-x-239-m-square-held-fourmas-q411825","timestamp":"2014-04-19T17:56:48Z","content_type":null,"content_length":"21460","record_id":"<urn:uuid:867439c2-f1b7-4a87-b6ea-66dc3f1158b8>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00507-ip-10-147-4-33.ec2.internal.warc.gz"} |
Binary Numbers
Date: 12/25/95 at 22:42:18
From: Bruce Zaktzer
Subject: Binary Number Systems
Dear Dr. Math:
My name is Courtney Zaktzer, I am in 9th grade and I am doing a
project on Binary Numbers and Systems.
I am having a difficult time finding information about this topic
and would ask for your help to find information about binary
Thank You,
Courtney Zaktzer
Date: 3/18/96 at 2:9:9
From: Doctor Jodi
Subject: Re: Binary Number Systems
Hi there!
I think what you're talking about is base two. Different bases
are REALLY cool.
One base we use every day is base 60 - can you think of where we
use it? This is a tradition that the Babylonians left us...
I believe that base 10 was popularized by the Arabs (who gave us
the numerals currently in use).
Base two is best known for its use in computing: 0 and 1 can
represent off and on, yes and no, etc.
Any numbers can be written in binary. Have you learned about this
Well, here's a brief intro, just in case...
Here's what Dr. Steve said:
We have ten symbols for counting (0,1,2,3,4,5,6,7,8,9). So what do
we do when we need to use numbers higher than 9? We make different
places in our numbers and know that each place has a different
meaning. The first place is the "ones" place. The second is the
"tens". The third is the "hundreds" and so on. Each place is ten
times greater than the one to its right.
So the number 159 means: 1 hundred + 5 tens + 9 ones
notice: 10^2 10^1 10^0 (aka 1)
in base 2, the places are similar:
Instead of being 100, 10, 1, they are 4(2^2), 2 (2^1), 1 (2^0),
So, what does
Anyway, write us back with more questions or if you want to know
-Doctor Jodi, The Math Forum
Date: 3/18/96 at 9:49:50
From: Bruce Zaktzer
Subject: Re: Binary Number Systems
Thank you for the information. It arrived just in time to complete
the project! | {"url":"http://mathforum.org/library/drmath/view/55770.html","timestamp":"2014-04-16T11:24:54Z","content_type":null,"content_length":"6710","record_id":"<urn:uuid:16e83138-77a9-4d5a-8f6b-456b1b534952>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00127-ip-10-147-4-33.ec2.internal.warc.gz"} |
Lizette decides that starting in January she will deposit $50 into her bank account at the start of each month. Her account earns 0.25% interest per month. Interest is calculated at the end of each
month. Truncate answers to two decimal places.
(a) In the middle of February, how much money is in Lizette's account?
(b) In the middle of March, how much money is in Lizette's account?
(c) In the middle of the nth month (where January is the 1st month, February is the 2nd month, etc.), how much money is in Lizette's account? Give your answer in closed form.
(d) In the middle of December, how much money is in Lizette's account? | {"url":"http://www.shmoop.com/series/word-problem-exercises-6.html","timestamp":"2014-04-18T00:41:35Z","content_type":null,"content_length":"30551","record_id":"<urn:uuid:f74dbeed-0d04-4077-b936-53573e9bb006>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00226-ip-10-147-4-33.ec2.internal.warc.gz"} |
Using the Rayleigh Quotient
May 4th 2010, 12:45 PM #1
Using the Rayleigh Quotient
Boundary conditions are:
$\alpha_1\phi(a) + \alpha_2\phi'(a) = 0$
$\beta_1\phi(b) = \beta_2\phi'(b) = 0$
Prove that, if u(x) satisfies the BC (but not necessarily the DE), then
$\lambda_1^2 = \min_i{\lambda_i^2} \leq \min_u{RQ(u)}$
Where RQ(u) is the Rayleigh Quotient where $\phi = u$
by expanding u(x) in eigenfunctions: $u \approx \sum_{n=1}^{\infty} c_n\phi_n$
and using the operator: $L[\phi] \equiv \frac{d}{dx}\left(p(x)\frac{d\phi}{dx}\right) + q(x)\phi$
My professor told us that the first eigenvalue is the smallest eigenvalue, but did not show us how to begin to prove it with the stuff provided. Any help would be appreciated.
Follow Math Help Forum on Facebook and Google+ | {"url":"http://mathhelpforum.com/differential-equations/143046-using-rayleigh-quotient.html","timestamp":"2014-04-19T05:49:41Z","content_type":null,"content_length":"31509","record_id":"<urn:uuid:c766023f-9954-4a77-be50-be9f9f246ef1>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00471-ip-10-147-4-33.ec2.internal.warc.gz"} |
Weyl's theorem on complete reducibility
The purpose of this post is to show that the category of finite-dimensional representations of a semismple Lie algebra is a semisimple category; there is thus an analogy with Maschke’s theorem,
except in this case the proofs are more complicated. They can be simplified somewhat if one uses the cohomology of Lie algebras (i.e., appropriate Ext groups), which I may talk more about, but most
likely only later. Here we will give the proofs based on linear algebra.
The first step is to construct certain central elements in the enveloping algebra.
Casimir elements
Let ${B}$ be a nondegenerate invariant bilinear form on the Lie algebra ${\mathfrak{g}}$. (E.g. ${\mathfrak{g}}$ could be semisimple and ${B}$ the Killing form.) Given a basis ${e_i \in \mathfrak{g}}
$, we can consider the dual basis ${f_j}$ with respect to it, i.e. such that ${B(e_i, f_j) = \delta_{ij}}$. Consider the Casimir element
$\displaystyle C := \sum e_i f_i \in U \mathfrak{g}.$
I claim that ${C}$ is independent of the choice of ${e_i}$ and is in the center of the enveloping algebra. First off, consider the isomorphisms of ${\mathfrak{g}}$-modules,
$\displaystyle \hom_k( \mathfrak{g}, \mathfrak{g}) \simeq \mathfrak{g} \otimes \mathfrak{g}^{\vee} \simeq \mathfrak{g} \otimes \mathfrak{g} .$
The last one is given by the form ${B}$.
Now the identity, an invariant element of ${\hom_k( \mathfrak{g}, \mathfrak{g})}$, is sent to ${\sum e_i \otimes f_i \in \mathfrak{g} \otimes \mathfrak{g}}$. Since there is a well-defined
homomorphism of vector spaces,
$\displaystyle f: \mathfrak{g} \otimes \mathfrak{g} \rightarrow U \mathfrak{g}, \ a \otimes b \rightarrow ab$
we see that ${C}$ is unique. Moreover, we can make ${U\mathfrak{g}}$ into a ${\mathfrak{g}}$-module by the adjoint—or, equivalently, commutator—mapping, i.e. ${x \cdot a := xa - ax}$ for ${a \in U\
mathfrak{g}, x \in \mathfrak{g}}$. Then ${f}$ becomes a ${\mathfrak{g}}$-homomorphism, because
$\displaystyle x \cdot f(a \otimes b) = xab - abx$
$\displaystyle f( [x,a] \otimes b + a \otimes [x.b]) = xab - axb + axb - abx.$
So ${C}$ is then an invariant element under this action of ${\mathfrak{g}}$ on ${U\mathfrak{g}}$, which means that ${C}$ is in the center of ${U\mathfrak{g}}$.
Complete reducibility
Lemma 1 Let ${\mathfrak{g} \subset gl(V)}$ be a semisimple subalgebra. Let ${B_V}$ be the bilinear form via ${x,y \mapsto \mathrm{Tr}(xy)}$. Then ${B_V}$ is nondegenerate on ${\mathfrak{g}}$.
Now ${B_V}$ is the form associated to the representation ${V}$ so is invariant. The kernel of the form is thus an ideal ${\mathfrak{z}}$, and by the second version of Cartan’s solvability criterion,
${\mathfrak{z}}$ is solvable. This proves the lemma. One half the proof of Cartan’s semisimplicity criterion can be generalized, as it shows.
Fix a semisimple Lie algebra ${\mathfrak{g}}$ over a field ${k}$ of characteristic zero. Let ${M}$ be a simple ${\mathfrak{g}}$-representation, i.e. containing no proper subrepresentations besides
Lemma 2 (Raising of invariants) Consider an exact sequence of ${\mathfrak{g}}$-modules$\displaystyle 0 \rightarrow V \rightarrow W \rightarrow k \rightarrow 0$
where ${k}$ is acted upon trivially. Then it splits.
We prove this by induction on ${\dim V}$. If ${V}$ is not simple, then we can take a smaller submodule ${X}$ and consider the sequence
$\displaystyle 0 \rightarrow V/X \rightarrow W/X \rightarrow k \rightarrow 0.$
We get by the inductive hypothesis a ${\mathfrak{g}}$-section ${k \rightarrow W/X}$, whose image is a one-dimensional submodule ${\tilde{X}/X \subset W/X}$. Then ${\tilde{X} + V = W}$.
There is another exact sequence
$\displaystyle 0 \rightarrow X \rightarrow \tilde{X} \rightarrow k \rightarrow 0$
from which we can take a section ${k \rightarrow \tilde{X}}$, whose image ${Y}$ satisfies ${Y+X = \tilde{X}}$. In particular, ${Y+V = Y+ V+X = \tilde{X} + V =W}$.
So we may assume ${V}$ simple, and work with our exact sequence ${0 \rightarrow V \rightarrow W \rightarrow k \rightarrow 0}$ as before.
We may make one further reduction, namely to assume that the action of ${\mathfrak{g}}$ on ${V}$ is faithful. If ${\mathfrak{n}}$ is the kernel of this action, it is an ideal—and, as a direct summand
in ${\mathfrak{g}}$, semisimple itself (as a Lie algebra). I claim that this ideal acts trivially on ${W}$. Now, ${\mathfrak{g}}$ sends ${ W}$ into ${V}$ because it acts trivially on ${k}$. Since $
{[\mathfrak{n}, \mathfrak{n}]=\mathfrak{n}}$, it follows that ${\mathfrak{n}}$ must act trivially on ${W}$. So we can treat our sequence as a sequence of ${\mathfrak{g}/\mathfrak{n}}$-modules, where
this quotient is a semisimple Lie algebra (as $\mathfrak{n}$ was a direct factor).
We have now made the reduction of the previous lemma to the following claim:
Lemma 3 (Special case) Let ${V}$ be a faithful, simple representation of the semisimple Lie algebra ${\mathfrak{g}}$ and consider an exact sequence$\displaystyle 0 \rightarrow V \rightarrow W \
rightarrow k \rightarrow 0 ;$
this splits.
We have the nondegenerate bilinear form ${B_V}$ and the corresponding Casimir element ${C}$. Then ${C}$ acts trivially on ${k}$; so ${CW \subset V}$. Moreover, I claim ${CV = V}$. Indeed, ${C}$ is a
${\mathfrak{g}}$-endomorphism of ${V}$, and once we prove it is nonzero, it will follow that it is an isomorphism. But
$\displaystyle \mathrm{Tr}_V(C) = \sum \mathrm{Tr}_V( e_i f_i) = \dim V eq 0$
since we are in characteristic zero.
As a result, we take a nonzero vector ${v}$ annihilated by ${C}$ as the splitting. Then ${\mathfrak{g}v = C \mathfrak{g} v = \mathfrak{g} Cv = 0}$, so ${v}$ is invariant.
Finally, we can do the general case:
Theorem 4 (Weyl) Let ${W \subset V}$ be a ${\mathfrak{g}}$-submodule of the finite-dimensional representation ${V}$ of the semisimple Lie algebra ${\mathfrak{g}}$. Then ${W}$ has a ${\mathfrak
(I.e. the category ${Rep(\mathfrak{g})}$ is semisimple.)
We can assume that ${W}$ is simple (i.e. contains no proper subrepresentations). It then follows by induction that every finite-dimensional ${U\mathfrak{g}}$-module is a semisimple module, which
implies the theorem.
So, there is an exact sequence of ${\mathfrak{g}}$-modules
$\displaystyle \hom_{k}(V,W) \rightarrow \hom_k(W,W) \rightarrow 0.$
We consider the submodule ${\hom_{\mathfrak{g}}(W,W)}$ of ${\hom_k(W,W)}$ consisting of multiples of the identity and the inverse image ${A \subset \hom_k(V,W)}$; this is then a ${\mathfrak{g}}$
-submodule as well. There is an exact sequence
$\displaystyle A \rightarrow \hom_{\mathfrak{g}}(W,W) \rightarrow 0.$
By the previous result on raising invariants, we can find a 1-dimensional ${\mathfrak{g}}$-submodule of ${A}$ which maps isomorphically onto ${\hom_{\mathfrak{g}}(W,W)}$. Note that any 1-dimensional
${\mathfrak{g}}$-module is the trivial (action by zero) one because ${\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]}$. This 1-dimensional module is generated by some ${\mathfrak{g}}$-invariant ${f: V \
rightarrow W}$ which restricts to a nonzero multiple of the identity on ${W}$. An appropriate multiple gives a ${\mathfrak{g}}$-invariant projection ${V \rightarrow W}$ and thus a complement.
This result is hugely important, as we will see in the future.
January 31, 2010 at 11:39 am
[...] Climbing Mount Bourbaki Thoughts on mathematics « Weyl’s theorem on complete reducibility [...]
January 10, 2011 at 11:48 am
I’ve got some questions here, and a few typos. Otherwise, thanks for this nice review!
- Shortly before Lemma 1, the $axa$ in http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle+f%28+%5Bx%2Ca%5D+%5Cotimes+b+%2B+a+%5Cotimes+%5Bx.b%5D%29+%3D+xab+-+axa+%2B+axb+-+abx.&bg=ffffff&fg=
000000&s=0 should be $axb$, and the $[x.b]$ should be $[x,b]$ (though it’s maybe a problem with rendering precision).
- In Lemma 1, the $\to$ arrow should be $\mapsto$.
- First sentence after Lemma 1: “representation of $V$” should either be “representation on $V$” or “representation $V$”. $V$ is not an algebra.
- Proof of Lemma 2: “We may one further reduction, that the action of $\mathfrak g$ on $V$ is faithful.” This sentence needs two more words: “We may DO one further reduction, ASSUMING that the action
of $\mathfrak g$ on $V$ is faithful.” or something like that.
- Shortly after that: “and, as a direct summand in $\mathfrak g$, semisimple itself”. “Semisimple” may mean “semisimple as $\mathfrak g$-module” or “semisimple as Lie algebra”. You mean the latter
here, I think.
- Before you can redduce Lemma 2 to Lemma 3, you need to ensure that $\mathfrak g / \mathfrak n$ is semisimple. This is, of course, easy since $\mathfrak g / \mathfrak n = \mathfrak n^{\perp}$.
- Shortly before Theorem 4, you write $\mathfrak g v = C \mathfrak g v = \mathfrak g C v = 0$. The first equality sign should be an isomorphism sign, I think.
- The proof of Theorem 4 is obscure to me (but I know a different one from Weibel, so this is not important anyway). Why can we assume that $W$ is simple? (Here is where I would use the long exact
sequence in Lie algebra cohomology…) Why is your exact sequence of $hom$’s exact? How do you know that the submodule $hom_{\mathfrak g} (W,W)$ consists of multiples of the identity? (This kind of
Schur needs $k$ to be algebraically closed).
January 10, 2011 at 9:25 pm
Thanks for the corrections! I’ve made the changes (apparently I didn’t pay too much attention when writing this post — I think I was on some kind of blogging spree).
To be honest, I haven’t thought about this in a while and can’t remember the argument well, but I may have been assuming that the field was algebraically closed at the beginning. The restriction
to $W$ simple is reasonable because every object in this category is artinian, so contains a simple subobject; then keep splitting off chunks (so any object decomposes into a sum of simple ones
since the category is artinian).
January 10, 2011 at 6:06 pm
I think I know how to prove Theorem 4 now. The idea comes from: C. A. Weibel, “An Introduction to Homological Algebra”, proof of Theorem 7.8.11. Weibel uses $H^1$ rsp. $\operatorname*{Ext}$ groups to
do this proof, but this all can be rewritten in elementary terms:
Let $U = V / W$, so we have a short exact sequence $0 \to W \to V \to U \to 0$. Let $E$ be the subspace of $\operatorname*{hom}_k (W, W)$ generated by $\operatorname*{id}$, and let $F$ be the
preimage of this subspace $E$ under the canonical map $\operatorname*{hom}_k (V, W) \to $\operatorname*{hom}_k (W, W)$ (which is induced by the inclusion map $W\to V$). Then, the image of the map $\
operatorname*{hom}_k (U, W) \to \operatorname*{hom}_k (V, W)$ (which is induced by the epimorphism $V \to U$) is contained in $F$, and the sequence
$0 \to \operatorname*{hom}_k (U, W) \to F \to E \to 0$
is a short exact sequence of $\mathfrak g$-modules (this is easy to prove). Since $E\cong k$ as $\mathfrak g$-module (this is obvious), we can replace $E$ by $k$ in this sequence, and thus Lemma 2
yields that this sequence splits. Thus, the element $\operatorname*{id} \in E$ can be lifted by a $\mathfrak g$-module homomorphism $E \to F$ to an element of $F$. This element of $F$ will be $\
mathfrak g$-invariant and thus a $\mathfrak g$-module homomorphism from $V$ to $W$. And being a lift of $\operatorname*{id}$, this homomorphism must be a section for the short exact sequence $0 \to W
\to V \to U \to 0$. So this short exact sequence splits, i. e. the submodule $W$ of $V$ has a $\mathfrak g$-complement. Done.
January 10, 2011 at 9:28 pm
Right. OK, so I suspect this is kind of the same argument (it’s due to Weyl, I think? — there’s a nicer one if you allow Lie groups, because then the Lie algebra corresponds to a compact group
and you can use the usual averaging trick as in Maschke’s theorem, with integration w.r.t. the Haar measure replacing averaging) but I am not really sure. I will get back to you after I actually
flip through a book on Lie algebras so that I refresh my memory on this! | {"url":"http://amathew.wordpress.com/2010/01/31/weyls-theorem-on-complete-reducibility/","timestamp":"2014-04-21T02:01:28Z","content_type":null,"content_length":"84083","record_id":"<urn:uuid:1403d59b-c672-46e6-bde4-61a126535092>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00540-ip-10-147-4-33.ec2.internal.warc.gz"} |
Obey the Law
Statistical sampling works
because of the law of large numbers.
Think of this as the statistician's answer to cost over-runs. The law of large numbers says that as the size of the sample increases, so does the chance it accurately reflects the whole.
If you don't think a random sample can represent a much larger group, look at the situation backwards. If you selected 1,600 people at random in the U.S. population, how likely is it that they would
not represent the whole?
If the law is so great, why would you ever want to increase the sample size beyond 1,000 or so? (I'm thinking of polls and studies with thousands, or even hundreds of thousands, of respondents...).
• You want to know about subgroups within the sample (women, or women smokers, or women smokers in New York City or Topeka, Kansas, for example). Remember, every time the sample size decreases, the
accuracy does as well. Thus if we had only 25 women smokers in a sample of 1,600 people, the margin of error for that subgroup would be 20%, far below the 2.5% margin for the whole study.
• you were studying something rare, like the incidence of a rare disease, or whether a vaccine really works.
• The poll is an excuse for something unrelated to legitimate public opinion research, like making publicity for a product, organization, or cause. In these cases, the larger the sample, the
Then again, sometimes the opposite of the law is true.
Reader Beware
A legitimate political poll should come with some information to help you assess it: the number of people contacted, when the poll was conducted, and the margin of error. The margin is typically
phrased as "accurate to plus or minus 3 percentage points," for a true range of uncertainty of 6 percent.
That's All There Is To It?
Sorry. We need to discuss some limitations. First of all, one time in 20, the results can be outside the margin of error. So if you read a lot of polls, some of them will be wrong. It's part of the
statistical game of chance.
More important, remember that the margin of error is only valid if the poll was perfectly designed and perfectly executed. That means no errors in writing the questionnaire. And it means making sure
everybody was treated equally.
Want to read about statistical issues in medical epidemiology?
We've located the ultimate in randomness. Wanna check it out? | {"url":"http://whyfiles.org/009poll/math_primer2.html","timestamp":"2014-04-16T07:14:40Z","content_type":null,"content_length":"3545","record_id":"<urn:uuid:58c10be4-2adf-48d0-ac81-8e178cc467cc>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00113-ip-10-147-4-33.ec2.internal.warc.gz"} |
Operations Research Applications and Algorithms 4th Edition Chapter 23.11 Solutions | Chegg.com
Summary report shows that a bid between $11,000 and $13,000 will give maximum profit. Expected profit is $825. Values of Bid amount is changed by changing the formula
By running the simulation result again we get the following report.
From the summary report we find that maximum expected profit is $855 with a bid amount of $12,000. | {"url":"http://www.chegg.com/homework-help/operations-research-applications-and-algorithms-4th-edition-chapter-23.11-solutions-9780534380588","timestamp":"2014-04-18T03:30:10Z","content_type":null,"content_length":"32193","record_id":"<urn:uuid:5afe8ef8-d94f-4a9b-b5fb-b95800b9b5f2>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00372-ip-10-147-4-33.ec2.internal.warc.gz"} |
Find the Percent of a Number
5.4: Find the Percent of a Number
Difficulty Level:
Created by: CK-12
Practice Percent of a Number
Do you like to ride roller coasters? Take a look at this dilemma.
At an amusement park there is an average of 30,000 visits daily. Over 75% of those people ride on the biggest roller coasters. About how many people ride the roller coasters?
To figure this out, you will need to find the percent of a number. Pay attention and you will know how to solve this problem by the end of the Concept.
We said that percent are very useful. Part of the usefulness is being able to find percentages of a number. In other words, if you are planning a barbecue and the butcher tells you that 30% of the
people want chicken…well, how many people is that? For how many people should you buy chicken?
The key words here are “of a number” this means that you multiply to solve the problem. We can convert the percent to a decimal and multiply or convert it to a fraction and multiply.
Let’s suppose that you invited 58 people to the barbecue. If 30% prefer chicken, then we need to know how many people that is.
First, convert 30% to either a decimal or a fraction
$30\% = .30 \ or \ 30\% = \frac{30}{100} = \frac{3}{10}$
Using the decimal or fraction, you now multiply by the number of people you invited: $.30 \times 58 = 17.4$
$\frac{3}{10} \times 58 = \frac{174}{10} = 17.4$
This is the answer.
Solve each problem.
Example A
What is 20% of 18?
Solution: 3.6
Example B
What is 25% of 40?
Solution: 10
Example C
What is 5% of 80?
Solution: 4
Now back to the dilemma from the beginning of the Concept.
First, let's write a statement about the problem.
We need to figure out 75% of 30,000.
To do this, first convert 75% to a decimal.
$75\% = .75$
Next, multiply.
$.75 \times 30,000 = 22,500$
22,500 people ride the roller coasters.
a comparison between two quantities.
a ratio that is being compared to the quantity of 100. Percent means out of 100.
a part of a whole written using a numerator and a denominator.
a part of a whole written in base ten place value.
two equal ratios form a proportion.
Guided Practice
Here is one for you to try on your own.
A survey stated that 15% of the people surveyed like olives. Out of the 500 people surveyed, about how many like olives?
First, let's write a statement describing the problem.
15% of 500
Now change the percent to a decimal.
$15\% = .15$
Next, multiply.
$.15 \times 500 = 75$
75 of the people like olives.
Video Review
Find the percent of the given number by converting the percent to a decimal. You may round if necessary.
1. 30% of 90
2. 3% of 12
3. 14% of 900
4. 33% of 99
5. 18% of 100
6. 8% of 72
7. 11% of 50
8. 14.5% of 30
9. 12% of 80
10. 2% of 800
11. 150% of 21
12. 45% of 60
Read the following situations carefully and answer the questions accordingly.
13. For every paycheck you receive, your employer pays 6% to social security. Write this percent as a ratio with a denominator of 100.
14. Jimmy’s height is 1.78m. Write his height as a percent of a meter.
15. A store did a survey and found that $\frac{4}{5}$
Files can only be attached to the latest version of Modality | {"url":"http://www.ck12.org/book/CK-12-Middle-School-Math-Concepts-Grade-8/r12/section/5.4/","timestamp":"2014-04-18T09:51:56Z","content_type":null,"content_length":"126115","record_id":"<urn:uuid:30f75e6e-b9e0-4715-84bc-d0e299a6f2fc>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00082-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Seek a Predictable Regular Return: The Autocall
Over the years financial product designers and marketing types in the UK have constantly tinkered with the basic zero structure, with perhaps the most important innovation being the autocall (or kick
out plan).
The basic idea behind an autocall is nearly as simple as the zero – an annual return in exchange for taking some risk:
1. You hand over, say, £100 per share or unit, potentially for a period of up to five years. The level of the underlying or reference index – in this case, the S&P 500 – is noted at issue.
2. The issuer looks back at that underlying index at the end of year one to see whether it increased (any increase, no matter how small, is relevant) or remained stable. As long as the underlying
index hasn’t dropped in value, the structure calls: that is, it pays out an agreed upfront return of 5 per cent for the year.
3. Your £100 is now £105 and the autocall matures – that is, closes or winds up.
What happens, you may ask, if the underlying index doesn’t advance but falls back? Well, the barrier on a typical autocall is usually set at 50 per cent (at which point your capital is at risk), but
if the index has fallen by, say, 10 per cent you don’t need to panic.
Your autocall simply rolls on to the next year, to the second anniversary of the fund, when again you see whether the underlying index has advanced from the initial level. If it has, you receive a
return for both years, in other words, 10 per cent or £110.
Every year over the next five or six years you’re presented with a chance to receive an annual income, as long as the underlying index stays stable or advances.
If the index doesn’t advance at all over the full five years (based on the initial level of the S&P 500) you simply receive back your initial investment of £100 as long as the barrier hasn’t been
breached (that is, the S&P 500 hasn’t fallen by more than 50 per cent).
Returns from autocalls are, like zeros, regarded as capital gains, which can be advantageous for certain investors. Autocalls also carry with them the same set of risks as a zero: counterparty risk
of the bank issuer going bust; the barrier being breached in a nasty market downturn; and the opportunity cost of markets rising and of dividends not being paid.
Yet the intrepid investor also receives some equally obvious upsides including a potentially steady 5 per cent return plus a capital gain.
Autocalls have also been tweaked over the years, with a defensive version perhaps the most noteworthy. The defensive autocall is structured exactly like a conventional autocall with annual call dates
and a barrier, but with one key difference – you can receive an annual call return even if the market you’re tracking has fallen in that year.
Imagine for one moment that the FTSE 100 is at 6,000 when a defensive autocall is issued with a 10 per cent annual return. The defensive version may still make that 10 per cent payment in one year’s
time even if the underlying index falls by no more than 10 per cent.
Therefore, as long as the FTSE 100 index is at 5,400 or more, you get a 10 per cent annual return; that is, you may make a 10 per cent positive return even if the stock market falls by 10 per cent
over the same period.
The call level used to trigger annual payments may in fact keep on falling every year over the next five years, perhaps settling at just 50 per cent of the initial level of 6,000 for the FTSE 100;
that is, it pays out a positive return of 10 per cent per annum even if the FTSE 100 index is at 3,001. | {"url":"http://www.dummies.com/how-to/content/seek-a-predictable-regular-return-the-autocall.html?cid=RSS_DUMMIES2_CONTENT","timestamp":"2014-04-17T13:42:27Z","content_type":null,"content_length":"98685","record_id":"<urn:uuid:97208553-3b6a-4d0d-b5cb-e0a8ffca0755>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00553-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Tutor] Use of sqrt() from math module
Matt Smith matt at mattanddawn.orangehome.co.uk
Sat Dec 1 16:46:03 CET 2007
Michael H.Goldwasser wrote:
> After using "import math" you will need to use the qualified name
> math.sqrt(blah) to call the square root function. That explains the
> NameError when trying to use the unqualified name, sqrt.
> As to your first message, the ValueError that you are reporting with
> the usage math.sqrt is likely due to an attempt to take the square
> root of a negative number (presumably because your (ypos - 384 * 160)
> factor is negative.
Thanks Michael and Ziyad, it seems I just had my brackets in the wrong place
leading to trying to square root a number less than 0.
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Impossible Questions
Being in a casino in a vacuum in a vacuum with no money to play with.
Does the set of all sets contain itself?
Re: Impossible Questions
Set-ception??? Haha, I suppose it would contain itself. And inside the contained set of all sets would be another set of all sets inside of which is another set of all sets inside of which is another
set of all sets inside of which is another set of all sets inside of which is another set of all sets inside of which is another set of all sets inside of which is another set of all sets inside of
which is another set of all sets inside of which is another set of all sets inside of which is another set of all sets inside of which is another set of all sets inside of which is another set of
sets inside of which is another set of all sets inside of which is another set of all sets inside of which is another set of all sets inside of which is another set of all sets inside of which is
another set of all sets.................
It reminds me of other infinite paradoxes, one specifically coming to mind is the Infinite Regress Argument made by skeptics questioning the premises of any argument.
It goes as such:
Premise 1. Any given piece of knowledge (a) can must be supported by stating another piece of knowledge (b) in order to be true.
Premise 2. That second piece of knowledge (b) would require a third piece of knowledge (c) to be proven true.
Premise 3. That third piece of knowledge (c) would require a third piece of knowledge (d) to be proven true.
Premise 4. Because there can never be any "first" proof of an argument, whatever happens to be the first premise (in this case, I only went to (d), but as you can tell it will go on indefinitely)
will never be 100% known to be true.
CONCLUSION: Nothing is known to be absolutely true.
Premise 1 seems pretty logical because every piece of evidence in a court case, every mathematical equation, and most pieces of knowledge we encounter come with some support.
However, the paradox that arises if Premise 1 is originally believed to be true, then the conclusion will be held true. The problem is, how can we know that Premise 1 is true if the conclusion that
follows is that nothing can be known to be absolutely true? Therefore Premise 1 is not necessarily 100% true, which goes against what was originally believed. So here is my impossible question: Is
Premise 1 true?
Follow up question: Can we ever know anything with absolute certainty?
Extra Credit: Is it possible to know something without proving that it is true? If so, what do you know, and why do you not need to prove it?
Re: Impossible Questions
I know with absolute certainty that I do NOT know the answer.
Re: Impossible Questions
schmittd wrote:So here is my impossible question: Is Premise 1 true?
Follow up question: Can we ever know anything with absolute certainty?
Extra Credit: Is it possible to know something without proving that it is true? If so, what do you know, and why do you not need to prove it?
Question #1: It is neither true nor untrue. It is a paradox like you stated. Good thing paradoxes never happen. If Pinocchio said "My nose will grow longer." what would happen? Chainsaw mode?
Nothing? Explosion? If he said that, who knows what would happen, so good thing there aren't puppets with lie-detecting noses.
Question #2: We can in fact know some things with certainty. I have thought about this a lot, and I have come up with only two things that you can know for sure. The first is that the universe
exists. You are constantly observing it with your five senses, and therefore something exists. It may not be at all what we say it is, but something exists. The only other thing I can think of is
that certain mathematical principles are true. The only reason I can prove that certain mathematical principles are true is because they aren't real things, only figments of our imagination with
perfect laws to govern them. For example, if one of something is there and another one joins it, you know there are two. Or to get more complex, pi will always be 3.1415... no matter what your vision
of the universe is. It's just a simple ratio.
Of course, remember that pi may not necessarily be 3.1415... it could also be 3.131313131... or perhaps the square root of 10. Because the only reason you think pi is 3.1415... is because you have
been told that. Have you ever gone out and measured a giant circle yourself to get exact measurements yourself? Even if you have, your measuring tape could have been off. If you used a calculator,
your calculator could be wonky. If you used a pencil and paper, you could have made mental errors. We do not know pi exactly, we only know that the ratio exists. Or do we? To know that we have to
know that perfect circles exist, but they don't. No perfect circle exists in nature, not even an electron or a quark or the circumference of a string. Pi, circles, and math itself are all figments of
our imagination, but we know they exist because we are the observers of the universe.
I therefore conclude that the universe is only as we see it. A tree only falls if someone witnesses it and thinks it fell. In fact, the universe only exists as long as there is life there to witness
its existence, yet it does not let life itself know anything except that the universe exists and a few mathematical principles.
Question #3: It is not possible to know something for sure unless you prove it, though some people would say differently, commonly religious people, who know their religion is true though they have
no tangible evidence. Some say that knowledge can just kind of appear, but it doesn't make sense to me. Then again, things just kind of "appear" all the time with no apparent cause. That is what dark
energy does, it creates something from nothing. It creates space, time, gravity, matter, everything you need in a universe from nothing. But dark energy isn't everywhere, according to astronomers, so
it could be part of the stuff from the big bang. But then again, the big bang kind of came from nothing as well, according to some astronomers. Some also say it came from the tattered remains of an
older universe, or came through another dimension of space from another universe. But where did everything start? Who knows. Maybe it didn't have a start, maybe some things are just there, in which
case you do not always need proof to show that something exists. I don't think question #3 is answerable, but I tried, eh? Extra credit?
My question, back to the pi idea:
Why is pi the number that it is?
Re: Impossible Questions
Maximirobbes wrote:
schmittd wrote:So here is my impossible question: Is Premise 1 true?
Follow up question: Can we ever know anything with absolute certainty?
Extra Credit: Is it possible to know something without proving that it is true? If so, what do you know, and why do you not need to prove it?
Question #1: It is neither true nor untrue. It is a paradox like you stated. Good thing paradoxes never happen. If Pinocchio said "My nose will grow longer." what would happen? Chainsaw mode?
Nothing? Explosion? If he said that, who knows what would happen, so good thing there aren't puppets with lie-detecting noses.
Question #2: We can in fact know some things with certainty. I have thought about this a lot, and I have come up with only two things that you can know for sure. The first is that the universe
exists. You are constantly observing it with your five senses, and therefore something exists. It may not be at all what we say it is, but something exists. The only other thing I can think of is
that certain mathematical principles are true. The only reason I can prove that certain mathematical principles are true is because they aren't real things, only figments of our imagination with
perfect laws to govern them. For example, if one of something is there and another one joins it, you know there are two. Or to get more complex, pi will always be 3.1415... no matter what your
vision of the universe is. It's just a simple ratio.
Of course, remember that pi may not necessarily be 3.1415... it could also be 3.131313131... or perhaps the square root of 10. Because the only reason you think pi is 3.1415... is because you
have been told that. Have you ever gone out and measured a giant circle yourself to get exact measurements yourself? Even if you have, your measuring tape could have been off. If you used a
calculator, your calculator could be wonky. If you used a pencil and paper, you could have made mental errors. We do not know pi exactly, we only know that the ratio exists. Or do we? To know
that we have to know that perfect circles exist, but they don't. No perfect circle exists in nature, not even an electron or a quark or the circumference of a string. Pi, circles, and math itself
are all figments of our imagination, but we know they exist because we are the observers of the universe.
I therefore conclude that the universe is only as we see it. A tree only falls if someone witnesses it and thinks it fell. In fact, the universe only exists as long as there is life there to
witness its existence, yet it does not let life itself know anything except that the universe exists and a few mathematical principles.
Question #3: It is not possible to know something for sure unless you prove it, though some people would say differently, commonly religious people, who know their religion is true though they
have no tangible evidence. Some say that knowledge can just kind of appear, but it doesn't make sense to me. Then again, things just kind of "appear" all the time with no apparent cause. That is
what dark energy does, it creates something from nothing. It creates space, time, gravity, matter, everything you need in a universe from nothing. But dark energy isn't everywhere, according to
astronomers, so it could be part of the stuff from the big bang. But then again, the big bang kind of came from nothing as well, according to some astronomers. Some also say it came from the
tattered remains of an older universe, or came through another dimension of space from another universe. But where did everything start? Who knows. Maybe it didn't have a start, maybe some things
are just there, in which case you do not always need proof to show that something exists. I don't think question #3 is answerable, but I tried, eh? Extra credit?
My question, back to the pi idea:
Why is pi the number that it is?
pi is the number of times the radius of a circle can fit around its circumference...
edit: half of its circumference, my bad
Re: Impossible Questions
Ouch! That would be 2Pi, or Tau.
Edit: gotcha
Re: Impossible Questions
marjo wrote:Ouch! That would be 2Pi, or Tau.
Edit: gotcha
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geoid (geology) :: Ellipsoidal era
Article Free Pass
Ellipsoidal era
The period from Eratosthenes to Picard can be called the spherical era of geodesy. A new ellipsoidal era was begun by Newton and the Dutch mathematician and scientist Christiaan Huygens. In Ptolemaic
astronomy it had seemed natural to assume that Earth was an exact sphere with a centre that, in turn, all too easily became regarded as the centre of the entire universe. However, with growing
conviction that the Copernican system is true—Earth moves around the Sun and rotates about its own axis—and with the advance in mechanical knowledge due chiefly to Newton and Huygens, it seemed
natural to conceive of Earth as an oblate spheroid. In one of the many brilliant analyses in his Principia, published in 1687, Newton deduced Earth’s shape theoretically and found that the equatorial
semiaxis would be ^1/[230] longer than the polar semiaxis (true value about ^1/[300]).
Experimental evidence supporting this idea emerged in 1672 as the result of a French expedition to Guiana. The members of the expedition found that a pendulum clock that kept accurate time in Paris
lost 2^1/[2] minutes a day at Cayenne near the Equator. At that time no one knew how to interpret the observation, but Newton’s theory that gravity must be stronger at the poles (because of closer
proximity to Earth’s centre) than at the Equator was a logical explanation.
It is possible to determine whether or not Earth is an oblate spheroid by measuring the length of an arc corresponding to a geodetic latitude difference at two places along the meridian (the ellipse
passing through the poles) at different latitudes, which means at different distances from the Equator. This can be seen from the figure, in which the geodetic latitude at any point (P) is
represented by the angle made between a line perpendicular to the ellipsoidal surface at the point P and the equatorial plane. This angle differs from the geocentric latitude that is determined by a
line directed from the point P toward Earth’s centre. Such measurements of arc were made by the astronomer Gian Domenico Cassini and his son Jacques Cassini in France by continuing the arc of Picard
north to Dunkirk and south to the boundary of Spain. Surprisingly, the result of that experiment (published in 1720) showed the length of a meridian degree north of Paris to be 111,017 metres, or 265
metres shorter than one south of Paris (111,282 metres). This suggested that Earth is a prolate spheroid, not flattened at the poles but elongated, with the equatorial axis shorter than the polar
axis. This was completely at odds with Newton’s conclusions.
In order to settle the controversy caused by Newton’s theoretical derivations and the measurements of Cassini, the French Academy of Sciences sent two expeditions, one to Peru led by Pierre Bouguer
and Charles-Marie de La Condamine to measure the length of a meridian degree in 1735 and another to Lapland in 1736 under Pierre-Louis Moreau de Maupertuis to make similar measurements. Both parties
determined the length of the arcs by using the method of triangulation. Only one baseline, 14.3 kilometres long, was measured in Lapland, and two baselines, 12.2 and 10.3 kilometres long, were used
in Peru. Astronomical observations for latitude determinations from which the size of the angles was computed were made by using the zenith sectors having radii up to four metres. The expedition to
Lapland returned in 1737, and Maupertuis reported that the length of one degree of the meridian in Lapland was 57,437.9 toises. (The toise was an old unit of length equal to 1.949 metres.) This
result, when compared with the corresponding value of 57,060 toises near Paris, proved that Earth was flattened at the poles. Later, large errors were found in the measurements, but they were in the
“right direction.”
After the expedition returned from Peru in 1743, Bouguer and La Condamine could not agree on one common interpretation of the observations, mainly because of the use of two baselines and the lack of
suitable computing techniques. The mean values of the two lengths calculated by them gave the length of the degree as 56,753 toises, which confirmed the earlier finding of the flattening of Earth. As
a combined result of both expeditions, these values have been reported in the literature: semimajor axis, a = 6,397,300 metres; flattening, f = ^1/[216.8].
Almost simultaneously with the observations in South America, the French mathematical physicist Alexis-Claude Clairaut deduced the relationship between the variation in gravity between the Equator
and the poles and the flattening. Clairaut’s ideal Earth contained no lateral variations in density and was covered by an ocean, so that the external shape was an equipotential of its own attraction
and rotational acceleration. Under these assumptions, gravity at sea level can be written as a function of latitude ϕ in the form
The expression deduced by Clairaut is
where m = centrifugal acceleration at Equator / attraction at Equator.
The quantity m is on the same order of magnitude as f; it can be obtained more precisely by calculation than by measurement. Clairaut’s result is accurate only to the first order in f, but it shows
clearly the relationship between the variation of gravity at sea level and the flattening. Later workers, particularly Friedrich R. Helmert of Germany, extended the expression to include higher-order
terms, and gravimetric methods of determining f continued to be used, along with arc methods, up to the time when Earth-orbiting satellites were employed to make precise measurements.
Historical determinations of the Earth’s radius and flattening
author year method equatorial radius (in metres) l/f*
P. Bouguer and P.-L. M. de Maupertuis 1735–43 arc 6,397,300 216.80
G.B. Airy 1830 arc 6,376,542 299.30
A.R. Clarke 1866 arc 6,378,206 295.00
F.R. Helmert 1884 gravimetric 299.25
J.F. Hayford 1906 arc 6,378,283 297.80
W.A. Heiskanen 1928 gravimetric 297.00
H. Jeffreys 1948 arc 6,378,099 297.10
*Flattening denoted by f.
Numerous arc measurements were subsequently made, one of which was the historic French measurement used for definition of a unit of length. In 1791 the French National Assembly adopted the new length
unit, called the metre and defined as 1:10,000,000 part of the meridian quadrant from the Equator to the pole along the meridian that runs through Paris. For this purpose a new and more accurate arc
measurement was carried out between Dunkirk and Barcelona in 1792–98. These measurements combined with those from the Peruvian expedition yielded a value of 6,376,428 metres for the semimajor axis
and ^1/[311.5] for the flattening, which made the metre 0.02 percent “too short” from the intended definition.
The length of the semimajor axis, a, and flattening, f, continued to be determined by the arc method but with modification for the next 200 years. Gradually instruments and methods improved, and the
results became more accurate. Interpretation was made easier through introduction of the statistical method of least squares.
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Two Enumeration Questions
January 12th 2011, 03:58 PM #1
Junior Member
Nov 2010
Two Enumeration Questions
1. Find an enumeration of the set of all odd integers
I know x/2 when x is even and -(x+1)/2 when x is odd represent natural numbers I just cannot turn this into odd numbers only.
when x is even the numbers I need are {2,6,10,14...}
when x is odd the numbers are {1,5,9,13...}
each are separated by 4 but when put together {1,2,5,6,9,10,13,14,...} I cannot find something that works like this.
2. Find an enumeration of the set of all rational numbers that have a terminating decimal representation.
(E.g. 17=10 = 1:7 and 17=8 = 2:125 are in this set, but 17=9 = 1:88888 : : : is
I'm not sure how to do this one
1. Find an enumeration of the set of all odd integers
I know x/2 when x is even and -(x+1)/2 when x is odd represent natural numbers I just cannot turn this into odd numbers only.
when x is even the numbers I need are {2,6,10,14...}
when x is odd the numbers are {1,5,9,13...}
each are separated by 4 but when put together {1,2,5,6,9,10,13,14,...} I cannot find something that works like this.
Do you know an enumeration for all integers?
Do you know a bijection from integers to odd integers?
Compose these two and you have an enumeration for odd integers.
2. Find an enumeration of the set of all rational numbers that have a terminating decimal representation.
(E.g. 17=10 = 1:7 and 17=8 = 2:125 are in this set, but 17=9 = 1:88888 : : : is
I'm not sure how to do this one
Do you know the enumeration for pairs of natural numbers $\mathbb N \times \mathbb N$?
Find the bijection between paris of natural numbers and terminating decimals.
Again, compose
1. Find an enumeration of the set of all odd integers
Does this mean to find a surjection from $\mathbb{N}$ to $\{n\in\mathbb{Z}\mid\text{n is even}\}$? Does $f$ have to be an injection?
I know x/2 when x is even and -(x+1)/2 when x is odd represent natural numbers
I am not sure why you have - before (x+1)/2. The number -(x+1)/2 is not always natural when x is an odd integer, and it is never natural when x is an odd natural number.
when x is even the numbers I need are {2,6,10,14...}
when x is odd the numbers are {1,5,9,13...}
each are separated by 4 but when put together {1,2,5,6,9,10,13,14,...}
What does {1,2,5,6,9,10,13,14,...} have to do with odd integers, for which you need to find an enumeration? Why do you "need" {2,6,10,14...} and {1,5,9,13...}?
2. Find an enumeration of the set of all rational numbers that have a terminating decimal representation.
(E.g. 17=10 = 1:7 and 17=8 = 2:125 are in this set
It's pretty hard to understand this. 17 is not equal to 10 and neither is 10 to 1:7. Also, 1/7 gives a nonterminating decimal representation 0.142857142...
Hint from Wikipedia: "Terminating decimals represent rational numbers of the form $k/(2^n5^m)$."
Last edited by DrSteve; January 13th 2011 at 12:05 PM.
so how do I 'compose' the enumeration and bijection? The enumeration is x/2 if x is even and -(x+1)/2 if x is odd and the bijection is 2x+1... I just don't get how to describe the nth term of the
When you substitute x/2 into 2x+1, you get 2(x/2)+1=x+1. When you substitute -(x+1)/2 into 2x+1 you get 2[-(x+1)/2+1=-x-1+1=-x
ah ok this makes sense... so my nth term would be x+1 if x is even and -x if x is odd and this is the correct final answer yes?
I do no know the enumeration for pairs of natural numbers so I'm somewhat stuck here.... I guess I need some further explanation
Yes - that's correct.
I think the idea behind the way to do the second one being suggested is the following:
You can think of any positive rational number that terminates as a.b where a and b are natural numbers. Now use snowpea's idea, together with the usual trick (as we've been discussing) to get the
negatives as well.
(I don't remember off the top of my head an explicit bijection between natural numbers and pairs of natural numbers, but you should be able to find one easily with an internet search for "pairing
Be aware that these solutions are far from unique. There are lots of ways to do enumeration problems.
By the way, we've all been giving you hints to produce explicit functions that give bijections. It's not clear from the way you've written the questions that this is necessary. If you only have
to "show the list" (as I did in my first post), then we can answer these questions much more quickly and easily.
so if I'm correct I should have some function f(x,y) = x.y ... or should it be (x/2).y when x is even and [-(x+1)/2].y when x is odd
Let $\Phi:\mathbb{Z}^+\times\mathbb{Z}^+ \to\mathbb{Z}^+$ defined by $\Phi(m,n)=2^{m-1}(2n-1)$.
That is a bijection.
Easily extended to all rationals.
That's the idea but not quite. Your final function will be a function of one variable. The domain is the set of natural numbers and the range is a subset of the rational numbers. You're going to
compose as follows:
$\mathbb{N}\rightarrow \mathbb{N}\times \mathbb{N}\rightarrow \mathbb{Z}\times \mathbb{N}\rightarrow \mathbb{Q}$.
The first is the pairing function. The second is the map $(a,b)\rightarrow (f(a),b)$ where f is the function we've previously discussed, and the third is the map $(a,b)\rightarrow a.b$.
I am asked to find the enumeration but I have to specify the nth term of the enumeration
so is f(a) the a/2 if a is even and -(a+1)/2 if a is odd? Then how do I map these to a.b?
January 12th 2011, 07:51 PM #2
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January 13th 2011, 02:50 AM #3
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Essays on Mathematics
Links to a few choice essays on mathematics, teaching math, and the philosophy of math can be found below.
If you are interested in these and other writers, check out our Math News and Media page. If you have a suggestion to add to this page, please contact us.
The opinions expressed in external websites are those of the authors of those sites and do not necessarily reflect the positions of the City University of New York or the CUNYMath Oversight Committee
• David Aldous
• Keith Devlin
• Martin Gardner
• Douglas Hofstadter
□ selections from the essay collection Metamagical Themas (1996) are available on Google Books
• Morris Kline
• John Allen Paulos
• Eugene Wigner
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Re: Coordinate Systems used in Autodesk Robot Stru
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Coordinate Systems used in Autodesk Robot Structural Analysis
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235 Views, 7 Replies
07-08-2012 07:47 AM
What is the difference between the local bar coordinate system and the global bar coordinate system ?
Message 1 of 8 (235 Views)
Re: Coordinate Systems used in Autodesk Robot Structural Analysis
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07-09-2012 12:07 AM
in reply to: Robot12
The local is determined by the position of the bar in the 3D space (X along a bar) and its rotation about its own X axis (Y and Z).
If you find your post answered press the Accept as Solution button please. This will help other users to find solutions much faster. Thank you.
Artur Kosakowski
Message 2 of 8 (214 Views)
Re: Coordinate Systems used in Autodesk Robot Structural Analysis
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I am analyzing a building using a 3D frame. Would the displacement experienced by each node used in the frame be in the local coordinate system or in the global coordinate system ?
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Re: Coordinate Systems used in Autodesk Robot Structural Analysis
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07-09-2012 11:38 PM
in reply to: Robot12
In the global coordinate system but deflections of bars in the local ones.
If you find your post answered press the Accept as Solution button please. This will help other users to find solutions much faster. Thank you.
Artur Kosakowski
Message 4 of 8 (180 Views)
Re: Coordinate Systems used in Autodesk Robot Structural Analysis
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In your answer you mention bars. Would the displacement of the nodes used in the 3D frame be in the global coordinate system and the deflection of the nodes used in the frame be in the local
coordinate system ?
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Re: Coordinate Systems used in Autodesk Robot Structural Analysis
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07-10-2012 07:39 AM
in reply to: Robot12
The deflection does not refer to a node; you can only see a displacement of a node regardless of an object it belongs to.
See also http://forums.autodesk.com/t5/Autodesk-Robot-Structural/RESULTS-DISPLACEMENT-vs-DEFLECTION-Definitio...
Artur Kosakowski
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Re: Coordinate Systems used in Autodesk Robot Structural Analysis
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Would the values of RX, RY and RZ that appear in the displacement results table be in the global coordinate system ?
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Re: Coordinate Systems used in Autodesk Robot Structural Analysis
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07-12-2012 02:58 AM
in reply to: Robot12
Artur Kosakowski
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Competitive niche: Way of population regulation
Based on the theory of robust coexistence by Meszéna et al. (TPB, 69:68, 2006), we establish a clear meaning of ecological niche assuming that the intended use of the word is the precise formulation
of Gause's principle. In line with Gause’s principle the expected content of a ``niche theory'' can be vaguely summarized as:
-Species partition an abstract “niche space” of the ecological possibilities among themselves.
-They are able to coexist, i.e. to avoid competitive exclusion, exactly because they partition that space.
Our central notion is the regulating variable, a generalization of the resource concentration. An environmental variable is regulating, if it is involved in the population regulation feedback loop.
The niche space is identified with the set of the regulating variables. This is the set, which should be partitioned between the coexisting populations to reduce competition. This “space” is not
necessarily a continuous entity. In case of two distinct resources it is a two-element set. Hutchinson’s niche axes/variables represent continua of regulating variables. The concentrations of the
infinitely many different seed-sizes are infinitely many different regulating variables, distinguished by the niche variable, i.e. the seed-size.
Regulating variables at different locations, distinguished by different environmental conditions, are to be considered as separate ones. Therefore, conditions, like temperature, can serve as niche
variables. Successional niche-segregation in a metapopulation is a separation along the patch-age, as niche axis. Here the regulating variables are the frequencies of the patches of different ages.
Mathematical implementation of these cases of spatio-temporal heterogeneity employs the theory of structured (meta)populations.
The niche of a species is defined by its way of being regulated. Two species may coexist because their growth is regulated differently enough. Quantitatively, a species’ niche is given by the
strength of its impact on, and sensitivity towards, the regulating variables. Specifying the pair of impact and sensitivity is the measurable alternative to the intuitive notion of resource
utilization. The strength of competition between two species can be calculated from the overlap between the impact of one species and the sensitivity of the other. Robust coexistence requires
sufficient differentiation in the impact as well as in the requirement niches; increasing similarity decreases robustness. | {"url":"http://eco.confex.com/eco/2008/techprogram/P11515.HTM","timestamp":"2014-04-16T19:15:49Z","content_type":null,"content_length":"4502","record_id":"<urn:uuid:5f497c52-4d6f-434d-98d7-0f29e30c6746>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00561-ip-10-147-4-33.ec2.internal.warc.gz"} |
Special quadrangle with 135° and 90°
February 13th 2013, 11:50 AM #1
Feb 2013
Special quadrangle with 135° and 90°
I hope you guys can help me with the following problem:
I have a quadrangle ABCD. <ADC = 135°, <ABC = 90°, [AB] = [BC]
That's all I know, but now I have to prove that [BD] = [AB] = [BC]
I tried some things with equations but it doesn't really work.
Thanks for your help!
Btw: I'm from Austria and I'm not a native speaker, so please feel free to correct my linguistic mistakes.
Last edited by jalt; February 13th 2013 at 11:59 AM.
Re: Special quadrangle with 135° and 90°
Re: Special quadrangle with 135° and 90°
thanks a lot for your reply and your solution. I know this might sound strange but my teacher told me that there was a solution without any trigonometry.
Unfortunately he isn't gonna tell me until I find it out.
I think I tried everything but it seems to be impossible to me. Do you have another idea or can you at least tell me whether it's possible or not?
Re: Special quadrangle with 135° and 90°
your teacher is right
you do not need all the complicated answers..after all the beuty of Mathematics lies on one word SIMPLICITY.
Anyway the quadrelateral you mention is part of a regular octagone with B as center of the circle... check it.....as simple as such
Re: Special quadrangle with 135° and 90°
I'm afraid I disagree with the previous answer. A regular octagon has all of its sides equal, but there's no restriction on your quadrilateral that AD = DC.
Back to the problem. Since the result is true, the circumcenter of triangle ACD mus be B; i.e. the intersection of the perpendicular bisectors of AD and DC must be B. I can see no reason why this
is true. There's probably a simple geometric argument, but I can't see it.
Re: Special quadrangle with 135° and 90°
construct a regular octagone with B as the center of circle.and BA=BD=BC as the radii of the circle .The sides AD=DC .The angle ABC=90 degrees and angle abd=135 degrees.
it is very simple try it.
Re: Special quadrangle with 135° and 90°
What you say about constructing a regular octagon is certainly true, but it does not answer the original question. The sides AD and DC are not necessarily equal. See the figure in my initial
Re: Special quadrangle with 135° and 90°
construct a right angle at B with equal legs BA and BC.Swing an arc from C to A.Mark a point D anywhere on the arc. Prove that the angle ADC is always 135 deg It is simple
Re: Special quadrangle with 135° and 90°
Construct a regular octagone with B as the center of circle and BA,BC BD RADII . THE QUADRILATERAL ABCD IS ONE OF THE TRILLIONS YOU CAN GET WITH THE SAME PROPERTY ONCE YOU SLIDE THE VERTEX D OVER
THE ARC AD. TRY IT IT IS SIMPLE. I AGREE WITH BJHOPPER IT IS SIMPLE.
Re: Special quadrangle with 135° and 90°
I agree with bjhopper that his construction always produces a quadrilateral as specified originally -- it's nothing more than an application of the inscribed angle theorem. I'm not being obtuse
on purpose, but why is every quadrilateral, as originally described, an instance of this construction? It seems to me that you need to know that length BD is the same as BA and BC (of course this
is true, but this was the original question).
minoanman: I think my source of confusion with you is that you keep saying regular octagon. As defined everywhere, a regular polygon must have sides of equal length. In particular, a regular
octagon has equal sides.
Re: Special quadrangle with 135° and 90°
Geometry fascinates me, but I've always known my skills at geometry are poor. The missing item in my information bank was:
Given an obtuse triangle ADC with obtuse angle $\theta$ at D, the circumcenter of ADC is the unique point P on the perpendicular bisector of AC which satisfies:
1. P is in that half plane determined by line AC which does not contain D.
2. The angle APC is $2\pi-2\theta$.
So the proof as suggested by bjhopper is absolutely correct. There's nothing magic about 135 degrees, it can be replaced by any $\theta$ and the angle ABC being $2\pi-2\theta$.
MY ignorance caused all the hubbub. Sorry.
Re: Special quadrangle with 135° and 90°
I am sorry, I was on vacation the last week.
Unfortunately I still haven't understood your solution.
I have an arc with the center B and the radius AB and now I claim that <ADC is always 135° if D is on the arc.
Well that's true but how can I prove it?
@johng What angle is pi?
Re: Special quadrangle with 135° and 90°
Hi jalt,
See post 8.
Triangle ABD and DBC are isosceles.Sides are radii
Angle BAD=BDA =a Angle BDC =BCD =b Property of Isosceles triangles
360 =90 +135 +2a +2b Property of quads
270 = 2a+2b
a+b = 135 angle D.D can be anywhere on the arc CA
Re: Special quadrangle with 135° and 90°
Thanks a lot!
I am still just a little confused, because you proved that D can be anywhere on the arc CA, but you didn't prove that D has to be on the arc.
Why isn't it possible that there is a point outside of the arc that makes <ADC = 135°?
There's probably a very easy solution but I don't see it.
February 14th 2013, 07:41 AM #2
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Fundamentals of Algebra
Chapter 13, Lesson 9
13-9.A understand the direction of a graph
For which coordinates will points P and Q determine a line that is falling?
I P(—5, 0) and Q(0, 5)
II P(—3, 5) and Q(3, 5)
III P(0, 5) and Q(5, 0)
(a) I only (b) II only (c) III only (d) I and III only
13-9.B identify possible values for the slope of a line
Which statement is true about the slope of a line?
I When a line and the positive portion of the x-axis form an acute angle, that line has a positive slope.
II When a line and the positive portion of the x-axis form an obtuse angle, that line has a negative slope.
III A horizontal line has a slope of 0.
(a) I, II, and III only (b) I and II only (c) I and III only (d) II and III only | {"url":"http://www.sadlier-oxford.com/math/mc_prerequisite.cfm?sp=family&grade=7&id=1973","timestamp":"2014-04-18T10:34:05Z","content_type":null,"content_length":"15772","record_id":"<urn:uuid:a2213b4a-afd2-4b17-a21e-98cda486966d>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00207-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Card Games Home Page | Other Invented Games
Contributed by Geoff Klassen (geoffk1@shaw.ca), who writes: "A few years back, my older brother taught me a card game called Snitch. At least that's what he was told its name was. I'm not sure where
he got it from, but I was wondering if you knew of any other names for the game or any similar games."
Editor's Note. This game is reminiscent of the Iranian game known as Gharat (Loot). It's not quite the same though - in Gharat you are dealt four cards and do not draw any more until they are all
played. There are no wild cards, and the scores are slightly different. If anyone can shed more light on the origin of Snitch and its relationship to Gharat, please let me know - John McLeod.
Players: 3 or more
Number of Decks: 2 (with Jokers); 108 cards total
Card Rank (from low to high): 3-4-5-6-7-8-9-10-J-Q-K-A-2*-Joker*
*2s and Jokers are wild
Object: To gain points by creating sets and stealing other players' sets.
Play: Each player is dealt 3 cards. The remaining stock pile is placed in the middle, from which a card is drawn to begin the discard pile. The player left of the dealer begins play, after which play
moves clockwise. On a turn, a player may do one of the following:
1. Discard a card to the discard pile and pick up a card from the stock pile.
2. Create a 'set' by combining two like-cards from his hand (e.g. 5-5, K-K, 7-2, A-Joker, etc.) and placing them down in front of him, on top of any sets he currently has if necessary. (A
combination of 2-2 or 2-Joker etc. is not allowed.)
3. Create a set by combining one card from his hand with the top card on the discard pile.
4. Challenge an opponent's topmost set with a similar card or wild (e.g. challenge a 4-4 with another 4, a 2, or a Joker) by placing his challenge card down somewhere (in the vicinity of the
challenged player) and declaring the challenge.
Upon being challenged, a player may do one of the following:
1. 'Defend' the set by also placing a similar card or wild down somewhere and declaring the defence.
2. Give up the set, regardless of whether he can defend them.
A challenge need not end after a single phase of challenge-defend. If he so wishes, the challenger may continue the challenge by placing another similar card or wild down and restating the challenge,
after which the defendant may defend in a likewise manner. Obviously, this process can only occur up to three times, since the players begin the turn with only three cards each. Whoever wins the
challenge adds the played cards to the challenged set and, if necessary, makes the set his by placing it on the top of his 'stack'.
After a player has made his turn, if he holds fewer than three cards, he draws until he has three, after which the defendant (if there is one) replenishes his hand the same way.
Example: It is Player 1's turn. He has a hand of 7-K-2 and a set of Aces in front of him. Player 2 has a set of 7s, on top of which she has a set of Kings. The top card on the discard pile is a
9. He may choose to discard & draw, combine the top 9 with his 9 to make a set & place it on top of his aces, place a 9-2 (or K-2) set on top of his aces, or challenge Player 2's Kings with his
King or his 2. He may not challenge Player 2's 7s with his 2, because the 7s are covered by the Ks.
Play continues normally until the stock pile is empty, at which point players may no longer use the discard pile to make sets. Once no cards remain in the stock pile, players may only:
1. Discard;
2. Challenge (and defend); or
3. Create sets using only cards from their hand.
Once all players have no cards remaining in their hands, play is over, and counting can begin.
Each player gathers the cards in his sets and counts his points.
Point Values:
• 3-9: 5 points
• 10-K: 10 points
• A: 15 points
• 2: 20 points
• Joker: 50 points
The deal now shifts to the left; extensive shuffling is encouraged.
A winner is determined by either:
• The player with the most points after a previously determined number of rounds; or
• The first player to reach 1000 points.
Return to Index of Invented Card Games Last updated 20th November 2003 | {"url":"http://www.pagat.com/invented/snitch.html","timestamp":"2014-04-16T07:13:46Z","content_type":null,"content_length":"8863","record_id":"<urn:uuid:4bb7912a-5514-438c-98ea-e133a93532de>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00251-ip-10-147-4-33.ec2.internal.warc.gz"} |
Popular Science
I can’t score this book more than 3 stars because it’s not really popular maths, but it does what it sets out to do rather well, so it should be seen in this context. As Richard Elwes points out in
his introduction ‘I was never any good at maths,’ is
The approach is not as heavy as a textbook, though occasionally I did get the feel of a slight older, fussy teacher at work. (It’s notable that the precise expression we’re told Elwes has heard from
‘a thousand different people’ is ‘I was never any good at mathematics.’ Hardly anyone would say ‘mathematics’ rather than ‘maths’. Now it’s possible he was trying to avoid the UK/US maths/math split
– but it still fits that slightly fussy precision we meet on a regular basis through the text.)
I really can’t fault the step-by-step progress, starting with basic arithmetic, taking us on to fractions and powers, roots and logs, percentages, algebra, geometry and even a brief intro to
probability and statistics. Each of the sections is quite short, easily digested, well laid out and illustrated and finished off with a little quiz that’s not too taxing but helps reinforce the
message. I suppose the only question is whether it’s best to arrange such an introduction by the structure of maths itself (as this book is) or by application, taking the reader through typical
mathematical chores from checking a shopping bill to calculating odds at a bookies. That way you could cover the same ground but perhaps make it seem more real world. However, Elwes doesn’t resort to
an excess of mathematical jargon, keeping the focus simple – and at least by structuring the book on the maths itself it can have the most logical progression of experience.
As I mentioned at the start, this isn’t popular maths. A popular maths book is not a tutorial in how to use it, with tests, but an exploration of some aspect of maths, the people involved, the
history and its significance. This is much more a practical book. I would it see it being particularly useful to an adult learner who had trouble with maths at school and now wants to come back to it
and take it on. It is a lot less condescending than most modern maths textbooks and would appeal more to a mature reader. So for this particular audience it is definitely an option well worth
considering – and it’s excellent value, priced like cheap paperback but actually a good size and well-made book. Just not really for someone wanting a voyage of discovery about the history or nature
of mathematics.
Review by Brian Clegg
Maths 1001 [Mathematics 1001] – Richard Elwes ***
Like its sister title Science 1001, this book takes on an enormous task: telling us ‘everything we need to know about mathematics in 1001 bite-sized explanations’.
In a sense, then, this is a mini-encyclopaedia of maths, though arranged by subject, rather than alphabetically. I had mixed feelings about the science entry in the series and those feelings are more
extravagantly mixed than ever here. There is no doubt whatsoever that this is a useful book. A good marker of this is that, unlike many of the books that come into the review pile, I intend to keep
this one. I think I will come back to it time and again to brush up on what some specific aspect of maths is. (As it is, really, a reference book, it would have been more helpful if the topics were
alphabetic, but hey, what do you expect from a mathematician?)
However, as a popular science book to read from cover it has a number of deep flaws. Firstly it’s much too broken up into tiny segments. There is a bit of a flow, brought in by the way the topics are
organized, but it’s very weak, and certainly doesn’t make for casual reading matter.
Secondly, far too much of the book is definitions. Time after time, a topic consists of defining what a mathematical term means. I feel a bit like Richard Feynman, who was told in a biology class,
when explaining what the various bits of a cat were called, that everyone would be expected to memorise these. He said something to the effect of ‘no wonder this course takes so long’ – he didn’t see
why people need to keep all those definitions in memory, and I rather feel the same about maths.
Then there’s the difficulty that the structure has in terms of dealing with some of the essentials of maths. Time after time, the author refers to the number e, without telling us what it is until
over 200 pages after it is first mentioned. The assumption for a reader who hasn’t come across it might be that e is just a placeholder, the way j is used elsewhere – although many definitions here
aren’t necessary, explaining what something like e is, and why it’s important, is pretty crucial.
As someone with a physics background, I particularly struggle to understand why there’s a whole section in here called ‘mathematical physics.’ No, it’s just physics. Newton’s laws don’t belong in a
book on maths – there’s much too much to get your head around already without straying into a different subject.
And to top it all, I think the approach taken is often wrong. Popular science/maths, as opposed to textbooks, adds in explanation and context, not just the theory. By being so strong on definitions,
there doesn’t seem to be room for this here. We find very little out about all the fascinating people involved. But even if you decide the format doesn’t allow for context and history, there is still
far too little explanation. Two example out of literally hundreds: we are told ‘Up until the early 20th century, 1 was classed as prime, but no longer.’ Why? There are good reasons for this, but it
is totally counter-intuitive. The number 1 seems like a prime. After all, it is only divisible by 1 and itself. We need explanation, not statement from authority. Another example is the topic on
Bayes’ theorem. This is fascinating in its application, but the explanation is almost unreadable, being mostly equations, and there is nothing about its application in that section (a later one does
make use of it, but doesn’t mention it is doing so). Highly frustrating.
Overall then, this is a very useful book if you dip into maths and need a quick reminder of what various things mean. It really is a great resource as a reference book. But it just doesn’t work as
popular maths.
Review by Brian Clegg
Richard Elwes – Four Way Interview
Richard Elwes is a writer, teacher and researcher in Mathematics and a visiting fellow at the University of Leeds. Dr Elwes is passionate about the public understanding of maths, which he promotes at
talks and on the radio. His more recent book is Maths 1001.
Why maths?
I don’t know anything else!
I have always enjoyed the subject, and the more I have studied, the more I have realised how incredibly deep it goes, and just how much there is to know. At the same time, I am aware of the gulf
between how most people see maths (a horrendous mix of tedious equations and incomprehensible jargon), and how I see it, which is as a whole other world, packed full of amazingly cool, interlocking
ideas. So, as well as enjoying studying maths myself, I suppose I have a drive to try to close this gap.
Why this book?
There are two answers, both true.
The first is that I don’t think a book like this has ever been attempted before. Of course, there are plenty of excellent books discussing various mathematical topics for a general audience. But I
don’t believe any have tried to be as comprehensive as this. It’s ambitious, there’s no doubt about it, and I was excited by the challenge.
At the same time, there seems to be a gap between ‘popular’ books on one hand, which take a completely equation-free, discursive approach to a mathematical subject, and ’technical’ volumes or
textbooks on the other, which go fully into all the gory details. My book treads a middle path. I didn’t want to sex things up too much, I wanted the mathematics to speak for itself, and for the book
to work as a reference volume. At the same time, some of the material is undoubtedly difficult and unfamiliar, and people need a way in, to understand what fundamental questions are being addressed.
I wanted it to be enjoyable to read, and for people genuinely to learn from it. In some ways, I suppose I wanted to write the book that I would like to have read aged 17.
The second answer is… someone offered me money to write it.
What’s next?
I’m pleased to say that I have a couple of projects in the pipeline. In Spring 2011 I have a book called “How to build a brain (and 34 other really interesting uses of mathematics)” coming out, which
has been a fun one to write. It covers some of the same areas as Mathematics 1001, but in a much more light-hearted and less technical style. Perhaps you could guess that from the title.
There are other things in the works too… but it is probably still too early to go into details. I can say that I am looking forward to working on them though!
What’s exciting you at the moment?
Maths 1001 is my first book, and it’s just come out. I’m quite excited about that, to be honest!
Otherwise, I find that the internet makes a wonderful blackboard, these days. There are so many people out there talking about maths, from primary school teachers discussing games kids can play to
start to enjoy numbers, right up to Fields medallists presenting their latest research. I follow several mathematical and scientific blogs (I’ve got my own too, may I plug it? www.richardelwes.co.uk/
blog Thank you!). It is just fun to be a part of that huge conversation.
In terms of mathematics itself, I have been thinking about recent work by the logician Harvey Friedman, which I find very exciting. It’s a sort of sequel to Kurt Gödel’s famous work. I think it will
turn out to be important. I am getting quite interested in ideas from logic to do with provability, computability, and randomness, and how they relate. My background is not in exactly this type of
logic, but I do find it fascinating. | {"url":"http://www.popularscience.co.uk/?tag=richard-elwes","timestamp":"2014-04-18T10:34:24Z","content_type":null,"content_length":"69425","record_id":"<urn:uuid:004e283e-af52-4434-861b-d42ed98fbfc8>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00272-ip-10-147-4-33.ec2.internal.warc.gz"} |
Early Career Profiles
Si Ken Long
• Undergraduate Institution: Swarthmore College '06
• Position: Analyst
• Company: Analysis Group
• Industry Sector: Consulting
What he does:
• Ken works with Analysis Group in Denver, Colorado, in the field of economic and financial consulting. The cases he works on concern litigation support or transfer pricing.
Math on the job:
• Ken works with statistics more than mathematics. Most of the material he studied in mathematics classes in college is not directly relevant, but the thought process he developed during his
classes is invaluable.
Ken's background:
• Ken double majored in Economics, and Mathematics with an emphasis on Statistics. He has always preferred applied mathematics over pure mathematics. Before college, Ken spent six years learning
math in Mandarin and five years learning math in Malay in Malaysia.
Ken's advice to students:
• Having any slight interest in mathematics is a valid reason to major in mathematics. For those interested in using mathematics in other fields, taking classes in other disciplines will give you
ideas and options on where this interest in mathematics can take you. Plan your classes at college well, but do not hesitate to try new things - during and after college. | {"url":"http://www.swarthmore.edu/NatSci/math_stat/AMS/SiKenLong.html","timestamp":"2014-04-20T23:40:31Z","content_type":null,"content_length":"3924","record_id":"<urn:uuid:82ded186-8769-4dd7-9b76-10df8ced8577>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00161-ip-10-147-4-33.ec2.internal.warc.gz"} |
How to solve the function fg(2x)
March 15th 2012, 11:13 AM
How to solve the function fg(2x)
I've become a bit confused on a question that is:
let $f(x)=x^2-1$ and $g(x)=2x+1$
Calculate $fg(2x)$
I'm not too sure. Is it:
$f(2(2x)+1)$ or $f(2(2x+1))$?
Thanks very much!
March 15th 2012, 11:16 AM
Re: How to solve the function fg(2x)
I think fg should be 2x^3 + x^2 - 2x - 1 and then you just plug in 2x in place of x to get your ans.
March 15th 2012, 11:23 AM
Re: How to solve the function fg(2x)
March 15th 2012, 11:32 AM
Re: How to solve the function fg(2x)
Ok thanks very much for the help! Much appreciated.
March 15th 2012, 11:37 AM
Re: How to solve the function fg(2x)
March 15th 2012, 12:02 PM
Re: How to solve the function fg(2x)
Oh right. I originaly got 16x^2 + 8x though although I wasn't sure of it...? | {"url":"http://mathhelpforum.com/pre-calculus/196004-how-solve-function-fg-2x-print.html","timestamp":"2014-04-17T04:09:45Z","content_type":null,"content_length":"8860","record_id":"<urn:uuid:dc486172-1512-421e-bb83-679729225cbb>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00240-ip-10-147-4-33.ec2.internal.warc.gz"} |
sketch graph of quadratic equation
January 12th 2009, 10:24 AM #1
Super Member
Sep 2008
sketch graph of quadratic equation
sketch the graph of the following equation
$y= 6-10x-4x^2$
I rearranged the equation to $y= -4x^2 -10x +6$
but i am stuck here cause can't seem to factor the equation, and i am not meant to use the quadratic equation or completing the square.
how do i facto this?
i tried $(2x-3) ( -2x-2) =0$ but that does not work
So can you use the discriminant ?
$\Delta=100+4 \times 4 \times 6=14^2$
And it factorises to $2(x+3)(2x-1)$
So you have the two x-intercepts.
Now you need the vertex (where the derivative is 0) to draw the graph !
You should find the maxima/minima and intersections with x-axis. You can rewrite the function:
And the final :-)
So there is minus on the right side of the equation so it is parabola inverted by x-axis. Its maximum is in point
Intersection points you'll find by discriminant.
Since this is in the pre-algebra/algebra section, lets assume the OP doesn't know how to find derivatives (i hardly think s/he would have problems graphing parabolas were that the case).
for a quadratic of the form $y = ax^2 + bx + c$ the vertex occurs where $x = \frac {-b}{2a}$. of course you can find the corresponding y-value by plugging in the x-value for the vertex
Since this is in the pre-algebra/algebra section, lets assume the OP doesn't know how to find derivatives (i hardly think s/he would have problems graphing parabolas were that the case).
for a quadratic of the form $y = ax^2 + bx + c$ the vertex occurs where $x = \frac {-b}{2a}$. of course you can find the corresponding y-value by plugging in the x-value for the vertex
I am not sure what everyone here is suggesting, I have not been taught all that stuff yet.
basically I am meant to sketch the graph by firstly putting y=0 to find the x-axis crossing points coordinates and than put x=0 to find the y crossing points coordinates.
and cause $b^2>4ac$ and $a<0$ there are two different roots.
but i can't daw the graph as i am not able to factorise the eqution to solve for x ?
does anybody understand what i am meant to do now?
I am not sure what everyone here is suggesting, I have not been taught all that stuff yet.
basically I am meant to sketch the graph by firstly putting y=0 to find the x-axis crossing points coordinates and than put x=0 to find the y crossing points coordinates.
and cause $b^2>4ac$ and $a<0$ there are two different roots.
but i can't daw the graph as i am not able to factorise the eqution to solve for x ?
does anybody understand what i am meant to do now?
$b^2-4ac$ is called the discriminant.
$-4x^2-10x+6=0 \implies 2x^2+5x-3=0$
$b^2-4ac=5^2+4 \times 3 \times 2=49=7^2$
And we know that the roots are $x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}=\frac{-5 \pm 7}{4}=\left\{\frac 12,-3\right\}$
sketch the graph of the following equation
$y= 6-10x-4x^2$
I rearranged the equation to $y= -4x^2 -10x +6$
but i am stuck here cause can't seem to factor the equation, and i am not meant to use the quadratic equation or completing the square.
how do i facto this?
i tried $(2x-3) ( -2x-2) =0$ but that does not work
The first thing i would do is complete the square to see where the vertex is.
(I'm sure you know how to do that)
I get
$y= -4(x+\frac{5}{4})^2 + \frac{49}{4}$
$V=(-\frac{5}{4},\frac{49}{4})<br />$
Next I would make a table of values
Ex take the x values -4,-3,-2,-1,0,1,2 and plug it into the equation to solve for the different y values. (there will not be the exact vertex co-ordinates, although it will be very close.) Then
simply plot the points and you're done!
January 12th 2009, 10:35 AM #2
January 12th 2009, 10:40 AM #3
Jan 2009
January 12th 2009, 10:46 AM #4
January 12th 2009, 11:49 AM #5
Super Member
Sep 2008
January 12th 2009, 11:58 AM #6
January 12th 2009, 01:01 PM #7 | {"url":"http://mathhelpforum.com/algebra/67880-sketch-graph-quadratic-equation.html","timestamp":"2014-04-18T17:52:26Z","content_type":null,"content_length":"58066","record_id":"<urn:uuid:fedcd43c-e225-4522-ab83-a09d1ae1e5cb>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00538-ip-10-147-4-33.ec2.internal.warc.gz"} |
Algebra - Solving Equations
Date: 11/03/97 at 23:36:50
From: AVI MANTHA
Subject: ALGEBRA
Hi, Dr. Math,
I am in an enriched Algebra class, and I'm having a hard time solving
some of my math problems. Please help me:
4 - 3X = 5 - 6X - 7
7X = 5(X - 12)
Please explain the solution step by step.
Thank you very much for your help.
Date: 11/10/97 at 09:28:19
From: Doctor Allan
Subject: Re: ALGEBRA
Hello Avi, and thanks for writing to Dr. Math!
You do not tell me exactly where you have problems, so I am not sure
whether you have problems with these types of equations generally, or
if it is just the two problems you mention here.
What I will do is give you techniques to use when solving these
equations, and then help you with the first of your problems - then
you can check out whether that helps you solving the second problem
and getting the general picture.
When you want to solve an equation you want to find values such that
if you substitute these values for x in your equation, you will get
something that is true. Let me give you an example:
x + 5 = 8
What you need to do is find a value to substitute for x such that you
get 8. If you have five apples and you want to have eight apples how
many more apples should you buy?
That's basically what an equation is all about.
So the smart thing to do is to isolate x on one side of the equation
and that will tell you the exact value to be substituted.
In order to do this you have to remember some rules. The most
important one is this:
Whatever you do to one side of the equation you should do to
the other side.
That means if you for instance add 3x to one side you should add 3x to
the other side.
Furthermore, there is the distributive rule:
x(a+b) = (a+b)x = xa+xb
This means that when you multiply terms within a parenthesis by x, you
multiply x by each term of the parenthesis.
That should cover the basics. Now let's go to the first of your
4 - 3x = 5 -6x -7
First notice that 5 - 7 = -2 so you get
4 - 3x = -2 -6x
Now you want to isolate x on one side of the equation, so let's
subtract 4 from each side of the equation. We get
4 - 3x - 4 = -2 - 6x -4
That is
-3x = -6 -6x
Since you want the x on one side the thing to do now is to add 6x:
-3x + 6x = -6 - 6x + 6x
That is
3x = -6
So all you need to do now to isolate x is to divide by 3 on each side:
(3x)/3 = (-6)/3
So you get
x = -2
I hope this helped you. If you have further questions, please write
again and I will try to help you out.
-Doctor Allan, The Math Forum
Check out our web site! http://mathforum.org/dr.math/ | {"url":"http://mathforum.org/library/drmath/view/57440.html","timestamp":"2014-04-21T02:57:57Z","content_type":null,"content_length":"7534","record_id":"<urn:uuid:59d5dc5b-06ac-4e33-823f-9be6fc9a1069>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00649-ip-10-147-4-33.ec2.internal.warc.gz"} |
Simplify: I get y = (10-2x)/5 but book says y = (1/5)(10-2x)
I was wondering what is the correct answer or the simplest form of 2x + 5y=10?
I was asked to solve for y
I answered this: $y=\frac{10-2x}{5}$
The book says $\frac {1}{5}(10-2x)$ is correct. Which is the same though it looks simpler to me.
Wouldn't y = 2-.4x be even simpler?
I hope it displays correctly. It doesn't on the preview.
Re: Simplify
"Ah, now I see!" said the blind man.
$y=\frac15(10-2x)$is what the book says is correct.
I say this is as correct $y=\frac{10-2x}{5}$ but isn't this: y=2-.4x, simpler yet?
I like the decimal form of the answer much better as I'm very used to converting fractions to decimals in my work. To me the decimal is the simplest form and so the book should accept that as a
correct answer, well, in my world anyway!
diaste wrote:"Ah, now I see!" said the blind man.
You got it! (I've edited the original post to insert the missing formatting tags.)
diaste wrote:$y=\frac15(10-2x)$is what the book says is correct.
I say this is as correct $y=\frac{10-2x}{5}$
They're exactly the same thing, just arranged a little differently. Since multiplying by one-fifth is the same as dividing by five, either form should be fine.
diaste wrote:but isn't this: y=2-.4x, simpler yet?
Generally, books and teachers prefer the fractional form. But your decimal form should be acceptable since, as you point out, it is the same.
Re: Simplify: I get y = (10-2x)/5 but book says y = (1/5)(10-2x)
Sometimes it can be annoying, when you want to write the answer one way but the book has it another way. I haven't figured out what rules or whatever they use to pick which way they do it. You just
have to get good at seeing when they're really the same thing.
Re: Simplify: I get y = (10-2x)/5 but book says y = (1/5)(10-2x)
Hey Little Dragon,
I know, isn't that annoying? In Linear equations they want the answer to look like $y=\frac15(10-2x)$ but now that we're graphing, the very next chapter, they want the answer to look like y = 2-.4x.
And they didn't say they wanted it in the simpler form. I'm still counting it as a right answer! | {"url":"http://www.purplemath.com/learning/viewtopic.php?p=206","timestamp":"2014-04-19T10:06:01Z","content_type":null,"content_length":"25311","record_id":"<urn:uuid:0ad63da8-70d9-48b8-b8e2-9cb4f86689b8>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00511-ip-10-147-4-33.ec2.internal.warc.gz"} |
Archives of the Caml mailing list > Message from skaller
Re: [Caml-list] Wikipedia
Date: -- (:)
From: skaller <skaller@u...>
Subject: Re: [Caml-list] Wikipedia
On Mon, 2005-11-07 at 11:14 +0100, Andreas Rossberg wrote:
> "skaller" <skaller@users.sourceforge.net>:
> >
> > In fact there is a lot of misleading or missing stuff in Wikipoedia,
> > and there are also some excellent articles. Try
> >
> > [...]
> > Referential Transparency (totally wrong)
> That article could well be more accurate and complete, but I don't see at
> all how you arrive at judging it "totally wrong".
It confuses purity with transparency. Purity is a semantic property
of functions. Transparency is a property of expressions.
Particularly, it is applications which are transparent NOT functions.
A function is pure if it depends only on its arguments.
A expression is transparent if its evaluation does not vary
with time or place (more or less).
As I pointed out in the commentary, 'functions' are necessarily
and trivially transparent, since ALL constant terms are transparent.
The article obscures an important theorem: given 'suitable conditions',
an expression is transparent if the function part of all applications
in the expressions are pure.
In particular, if ALL functions are pure (Haskell or Clean), then
ALL expressions are transparent.
I might add -- I know about this because at the moment,
Felix has the rule that functions may not have side effects.
But still, not all expressions are transparent because functions
can depend on variables .. so it matters when you evaluate them.
This cannot happen in Haskell because it has no variables.
I don't know what to call this kind of 'function' other
than 'impure', nor what to call the resultant expressions,
but I am calling them 'semi-transparent'.
What is actually interesting is that applications
of impure functions are "transparent within a
particular space/time locus".
The Felix optimiser uses this. For example:
var x = 1;
fun f(y:int)=> x + y; // impure
val a = f 1; // a = 2 at this point
val b = a + a; // b = 4
x = 2; // modify x
val c = b; // should be 4 not 6!
In this code, the optimiser knows it can inline to obtain
val b = f 1 + f 1;
however the substitution
val c = f 1 + f 1;
will give the wrong answer, since x has been modified --
so the expression f 1 is "transparent" in a certain part
of the control flow graph. Felix can follow only linear
sequence of val bindings, it gives up on branches,
assignments, and procedure calls: it is using simple
pattern matching, not a proper data flow analysis.
Anyhow, I hope my objection is clear: it is a serious
category error. 'transparency' is a property of terms
of the language -- bits of syntax. Purity is a property
of functions, not the terms which denote the function.
The coupling is vital, in the sense that transparency
allows you to examine a call such as:
f (g (y) )
and know for sure without looking at ANY other code that
IF the expressions and subexpressions are transparent,
THEN you can rewrite the expression like, for example:
let a = g (y) in f (a)
(or conversely) and numerous other rewrite rules,
without changing the semantics. To establish this
transparency in Ocaml you know to examine the
functions f and g to see if they're pure,
and it remains only to examine 'a' to see if it is
transparent (assuming f,g are function constants).
And here .. f and g are *of course* transparent,
when considered as expressions themselves -- whether
or not they're pure: all constants are transparent.
Transparent and pure are distinct concepts applying
to distinct categories of entity. The article is
therefore *totally* wrong. Purity and transparency
are nominally independent, they have to be connected
by a (language dependent) theorem. If you mess up
the meaning of these important words, there would
be no way to state this theorem, let alone prove it.
John Skaller <skaller at users dot sf dot net>
Felix, successor to C++: http://felix.sf.net | {"url":"http://caml.inria.fr/pub/ml-archives/caml-list/2005/11/8082d8ff4dd603164d5cecc424a7c9e0.en.html","timestamp":"2014-04-21T02:31:19Z","content_type":null,"content_length":"9672","record_id":"<urn:uuid:06b1a3df-fac2-4c58-8c91-3f4870678aab>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00608-ip-10-147-4-33.ec2.internal.warc.gz"} |
aterial for CSE 773, Spring 2003
L. Morris's material for CSE 773, Spring 2003
My e-mail address is lockwood@ecs.syr.edu
Purpose of this directory
Notes on some basic tactics in HOL
Notes on some additional HOL tactics
Notes for Jan. 28 lecture
Notes for Feb. 10 lecture
More examples of the use of lists
Generalized counters (See also the link below to LCM_chcScript.sml)
Notes for Mar. 27 lecture
Notes for April 10 lecture
Use of Hol_defn for making non-standard recursive definitions
Some tactics, conversions, etc. written originally for my own convenience in using HOL, but of possible interest and use to CSE 773 students: description (PostScript, 14 pp., index on page 14);
ante_allTacs.sml ML source (8 pp.);
ante_allTacs.sig ML signature (2 pp.).
The ML files and their compiled forms are located in, and loadable from, directory /home/ecefac/cse773.
A mathematical theory of LCM, Iter, period, and characteristic abstracted from gen_count.txt (concerning Prof. Stabler's generalized notion of a counter) which has in consequence become quite short.
Also compiled in /home/ecefac/cse773: LCM_chcScript.sml
A theory of breaking natural numbers and lists into as nearly equal halves as possible, largely avoiding struggles with DIV. Provides facilities for induction and, in effect, recursive function
definition by decomposition into (nearly) equal halves: halvesScript.sml
A theory specifying addition for natural numbers represented as bool lists, least significant bit first, and defining (with the facilities provided by halvesTheory) a carry-lookahead adder for any
fixed word length, and proving the implementation correct: adder_halvesScript.sml | {"url":"http://www.cis.syr.edu/~lockwood/html/index_cse773.html","timestamp":"2014-04-21T08:14:13Z","content_type":null,"content_length":"2886","record_id":"<urn:uuid:1057d621-8034-4081-84c1-b92a4a80eddc>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00455-ip-10-147-4-33.ec2.internal.warc.gz"} |
• integrability in thermodynamics, rge11x
• Interaction of gravitational waves with matter, Paul Colby
• Re: Action at a Distance vs. Entanglement, Anon E. Mouse
• tests of Einstein's equation in matter, Laconicus
• boxy galaxies, Laconicus
• Re: fast neutrinos?, Dr BDO Adams
• follow up to superluminal neutrinos?, clifford wright
• Re: Generic WIMPs Ruled Out, raymond
• Re: superluminal Neutrinos ?, Dr BDO Adams
• On Average Each Month 16 Close Gamma Bursts ??????, herbertglazier0
• Re: Feynman vs. Uncertainty Principle, herbertglazier0
• Five-dimensional quantum electrodynamics, Andreas Moser
• classical_limit, Ulf Klein
• Re: Are the Concepts of Mass in Quantum Theory and General Relativity, ANS
• Re: Joy Christian's Work on Bell's Inequality, Hendrik van Hees
• did string theory predict the Higgs-boson mass?, Jonathan Thornburg [remove -animal to reply] | {"url":"http://sci.tech-archive.net/Archive/sci.physics.research/2012-01/","timestamp":"2014-04-20T05:49:49Z","content_type":null,"content_length":"20168","record_id":"<urn:uuid:0a55b896-c4b7-49f3-8765-1bd8f132b3ac>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00036-ip-10-147-4-33.ec2.internal.warc.gz"} |
kg/m3 to KN/m3 - OnlineConversion Forums
Originally Posted by Unregistered
Anyone can help me? What is the reinforment/concrete ratio design (kg/m3) for a double storey terrace house?
Mind reply to this e-mail:
ratio reinforcement is the percentage of armature in concrete section so,
in each time it depends on the element
for coloumns is between 1%-3% of the surface of section
for beams u can increase this value
and for the slabs normaly is about 1% of the thickness multiply by widthness | {"url":"http://forum.onlineconversion.com/showthread.php?t=504","timestamp":"2014-04-20T23:38:51Z","content_type":null,"content_length":"57349","record_id":"<urn:uuid:6fcfabce-3c56-4143-a837-5d7d68abec43>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00535-ip-10-147-4-33.ec2.internal.warc.gz"} |
How should one present calculations?
up vote 17 down vote favorite
It is often necessary to present calculations, or at least their outlines, in a proof. However, when I need to do it, if the calculation takes more than one line (or perhaps two at the most), I feel
a bit uneasy, and have wrestled with trying to present it in a better way. On friday, I was browsing a journal, and came across a paper, in a subject that I was interested in, which contained a
number of calculations which were at least a half a page of nothing but formulas. This made me shudder. Ultimately the reason why we should write papers is to foster insight, and I think that such
presentations fail that test.
So, how do people deal with this problem?
soft-question exposition
3 You are lucky to work in a subject where most calculations take only a couple of lines. – Victor Protsak May 31 '10 at 7:14
Actually, calculations often do take up more than a couple of lines. However, that's the threshold at which I believe the calculations need to be broken up into conceptual pieces. – Victor Miller
May 31 '10 at 19:24
Conceptual blocks, then. I wholeheartedly agree with the spirit of your comment, but some fields are luckier than others in this regard. – Victor Protsak Jun 1 '10 at 5:40
add comment
6 Answers
active oldest votes
The 1970 paper of Halmos entitled "how to write mathematics" is a bit old, preposterous to the TeX era, but I think that many of his advices still make sense today. To summarize what he
says, "use well chosen words instead of plethora of symbols.".
"Think about the alphabet". And don't hesitate to associate a meaningful symbol to a piece of formula that makes sense by itself, so as to keep your calculation as compact as possible. In
particular, when there is some constant at the end of the computation, that comes from an agregation of many constants appearing during the computation, just call it C through the whole
calculation, and at the end, give its actual value. "The value of the constant C is..." (I think I read that trick in some paper by Krantz).
up vote 15 "Use words correctly". Explain what is going on. Irrelevancy should be avoided. Writing "Now applying the Cauchy-Schwarz inequality leads to...", and just giving the result, may be better
down vote than actually applying it in the middle of the computation without mentioning it.
Beware of too heavy use of formulae notations. Something like "Now applying (32), we get ..." is less helpful than "Let us apply the upper bound on the curvature that was obtained with the
help of the Gauss-Bonnet theorem." Give meaningful names to important formulas in your paper instead of refering to them through numbers. The section entitled "resist symbols" in the paper
of Halmos gives a few other tricks to replace heavy formalism by well worded sentences.
add comment
I think that in all writing, and especially in mathematical writing, signposting is extremely important. So a calculation-heavy proof should include a short summary, and then the
calculations should be very clearly marked so that the reader knows that "here are the nitty gritties of the calculations, and here's where they stop, so you can skip them and pick up at the
up vote For short papers, I like the following format. Begin with a very short introduction, setting the paper in context and giving a couple-sentence outline of the paper. In the second section,
9 down state all definitions and theorems precisely, but do not provide any proofs that are more than one or two sentences. Then the rest of the document, in as many sections as necessary, provides
vote the detailed proofs, with all non-immediate calculations. The point is that most readers can read sections 1 and 2 and get everything out of the paper, and the only people who will read
sections 3+ are: the reviewers (one hopes), and anyone trying to generalize the actual proof to a different setting.
add comment
Here are a few words of wisdom from the French poet Nicolas Boileau...
up vote 7 down vote Ce que l'on conçoit bien s'énonce clairement et les mots pour le dire arrivent aisément.
3 Translation: What is well conceived can be stated clearly and the words to say it come easily. – François G. Dorais♦ May 30 '10 at 14:55
add comment
Why not put long calculations in an appendix?
up vote 7 down vote
add comment
Isn't the standard wisdom to omit those things that should be a routine check for a reader familiar enough with the topic? This reminds me of Serre's statement about a Bourbaki proof
up vote 5 versus a proof. (The Bourbaki proof is for non-experts and the proof is for experts...) One should be writing to someone...and the level of detail should reflect who that someone is.
down vote
add comment
Often, the arXiv version contains more calculations than the journal version.
up vote 3 down vote
add comment
Not the answer you're looking for? Browse other questions tagged soft-question exposition or ask your own question. | {"url":"http://mathoverflow.net/questions/26452/how-should-one-present-calculations/26468","timestamp":"2014-04-18T08:50:31Z","content_type":null,"content_length":"72514","record_id":"<urn:uuid:438c8c7f-c3fe-4bad-b999-09dced6f5337>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00650-ip-10-147-4-33.ec2.internal.warc.gz"} |
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And CCENT Practice Exam
The answers:
1. Hex value aC4 into decimal
This means we have...
"a" units of 256
"c" units of 16
"4" units of 1
In hex, "a" is 10 and "c" = 12. Therefore, we have ....
10 units of 256 = 2560
12 units of 16 = 192
4 units of 1 = 1
Add them up and you get 2756.
By the way, get used to doing these conversions and additions by hand -- you can't use the Windows calculator in the exam room.
2. Decimal 1092 into hex
First question: Are there units of 256 in this value? Sure, there are actually four of them...
4 x 256 = 1024
Subtract that from 1092 and we have 68 left. Are there units of 16 in 68? Sure, there are 4 of those as well.
4 x 16 = 64
That leaves us 4, which is four units of 1.
Our hex value: 444
3. Binary string 00110010 into decimal
Going from left to right, the following bits are set to one -- 32, 16, and 2. Add them up and you have 50.
4. Binary string 11001101 into hex
A little more work is needed here, but not much. ;)
These bits are set to one: 128, 64, 8, 4, and 1. Add them up and you have 205.
Converting to hex, we obviously don't have any units of 256 in there. We have plenty of 16s, though - twelve of them.
16 x 12 = 192 (12 = "c"
We have 13 remaining, which in hex is "d" - so the resulting hex value is "cd".
5. Hex value C8 into binary
Converting that to decimal, we have 200 -- and you can probably convert that in your sleep at this point.
But just in case you're not sleeping, that binary string would have the 128, 64, and 8 bits set, resulting in:
For over 100 additional exercises designed to help you master binary and hex conversions, look for my first Kindle book, coming in April 2011 ...
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Open Source 'Sage' Takes Aim at High End Math Software
416541 story
Posted by
from the that'll-take-awhile dept.
"A new open source mathematics program is looking to push aside commercial software commonly used in mathematics education, in large government laboratories and in math-intensive research. The
program's backers say the software, called Sage, can do anything from mapping a 12-dimensional object to calculating rainfall patterns under global warming."
This discussion has been archived. No new comments can be posted.
• Very Nice (Score:3, Interesting)
by lansirill (244071) on Saturday December 08, 2007 @10:30AM (#21623711)
I haven't had a chance to play around with this yet, but if it's as good a replacement for Mat* as R is for S+ and SAS, I'm quite happy to see it. I'm sad that I'll probably never be able to
touch it unless I change my job as I've been told it would, quite literally, require an act of Congress to allow us to use anything other than SAS for our work. It will still be great to have
access to a (hopefully) well documented library of algorithms that I can tear into, instead of trying to cobble together things that seem good to me at the time. Huzzah, hip hip, and all those
fun things.
• Re:This makes me think..... (Score:2, Interesting)
by Anonymous Coward on Saturday December 08, 2007 @10:51AM (#21623863)
The thing is...programs are algorithms at their core, but to the user...well, a UI makes all the difference. And often the UI is a product of creativity. I mean, take Google, for example.
Visiting their page auto-focuses you to the search box. I find this *tiny* feature to be incredibly useful as I can leave my hands on the keyboard and continue typing, yet that's not something a
computer program would have generated. The part that is exposed to the user requires creativity to conceptualize and implement. I don't think that should be completely free of any copyright
• Re:another one bites the dust (Score:2, Interesting)
by Goaway (82658) on Saturday December 08, 2007 @10:51AM (#21623865) Homepage
Decent graphics software, apparently.
• FINALLY! (Score:3, Interesting)
by yamamushi (903955) <yamamushi.gmail@com> on Saturday December 08, 2007 @11:03AM (#21623943) Homepage
I think a lot of us can agree that open source software like this should have been developed YEARS ago, so I'm glad to finally see a good alternative to MATLAB and Mathematica out, I was getting
kind of tired of pirating my Mathematica software. Plus with the added benefit of being scripted in Python, I'm sure this project is going to take off like wildfire.
• Maxima vs Mathematica (Score:5, Interesting)
by hweimer (709734) on Saturday December 08, 2007 @11:32AM (#21624101) Homepage
But I use Mathematica because it is full of functionality, fairly reliable, and has a very elegant programming paradigm. Also, as a student, it'll cost me $100-150, depending on where I live, for
the lifetime of my studentship, assuming no site license; the kinds of business that run this software commercially really don't care too much about a $2500 license fee.
Free software isn't about price -- it is about freedom. One of the research groups at my university cannot use Mathematica since a few weeks because the license expired, and neither renewing the
license nor contacting tech support has so far brought a solution.
Another no-go is that Mathematica 6 notebooks are not compatible with Mathematica 5 notebooks. Also, the unwillingness of Wolfram to timely fix bugs leading to wrong results is unacceptable. I
could go on ranting like this, but recently I have completely switched to Maxima [osreviews.net] and have not regretted it.
• The Sage Notebook (Score:2, Interesting)
by mhansen444 (1200253) on Saturday December 08, 2007 @12:34PM (#21624531)
One nice feature that Sage has is its web-based interface -- the Sage Notebook. This inteface was designed with the Google documents interface in mind in terms of sharing and collaborating on
worksheets. The Sage notebook also provides a web-based interface to most every piece of math software out there (so long as you have it installed on your computer): Maple, Magma, Mathematica,
Matlab, Axiom, Maxima, Octave, Macauly2, Singular, etc.. Or, in one workshhet, one can have one cell be a Mathematica cell while the next one be a Maple cell. This interface does not depend at
all on the math functionality of Sage.
This is one area which could use some help from a web developers familiar with Python and AJAX -- a background in math is not needed at all. Eventually, we'd like to split off the interface into
its own project since it pretty useful on its own.
• Re:another one bites the dust (Score:3, Interesting)
by module0000 (882745) on Saturday December 08, 2007 @12:39PM (#21624577)
Sadly yes, there is a type of very expensive commercial software who's market is unable to be challenged by free software.
That market is custom database design: it's where your company pays $10,000 per license of some "cutting-edge" VB6.0 front-end to a MS Access database file because it has been completely
customized to their business model. They are rampant with bugs, bag programming procedures, and hidden [usually annual] costs.
Doesn't look like it's going anywhere either, until corporate purchasing mindsets evolve from "price = value".
• Matlab (Score:4, Interesting)
by Liquid Len (739188) on Saturday December 08, 2007 @12:42PM (#21624601)
I work in Europe, as a researcher, and two and three years ago, the Mathworks (the company behind Matlab) decided we weren't eligible to research/education prices anymore. They did the same with
a bunch of other institutes (in Europe, I don't know about the US). We operate an experimental reactor, whose control is largely based on Matlab programs. Some of these were developed a long time
and people left, or retired. There's a lot to be said about the way this was handled by our management, but that's the way it is. So, we had to admit we were screwed, having to pay the price. We
met with the Mathworks representatives, and I have to say all I saw a bunch of arrogant jerks.
Anyway, since then, we've renewed our licences every year, and we've been looking for an alternative. We even tried to migrate the whole lab to Scilab [scilab.org] but that didn't work out
(mostly because of the limited capabilities of Scilab in scientific plotting and GUIs). Some of us use Python + Matplotlib (I'm a big fan), some (often the same people) use Octave. Although we've
converted some individuals, we weren't able to find a software which could be used by everyone in the lab as a substitute to Matlab. This is frustrating, as the vast majority of people here use
only a fraction of the capabilities of Matlab.
I for one, would be really happy if we had something to replace Matlab, be it Sage or whatever else...
• Commercial vs Free (Score:3, Interesting)
by sjbe (173966) on Saturday December 08, 2007 @12:47PM (#21624647)
Is there any category of commercial software that can't
be challenged by free software?
Theoretically no, but in reality probably yes.
There are some applications that are simply very difficult to make work in an open source or free software model. CAD software comes to mind immediately. Creating a CAD system is highly
specialized, requires serious math skillz, and the end application is large and complicated (on par with operating systems or top tier database software) so a good team is required. There also
are likely to be patent issues to work around as well. From a user's perspective changing CAD systems has VERY VERY high switching costs, require a LOT of training, and the user bases are quite
small. Sure there are a few free/open-source CAD packages out there but they are toys compared with CATIA or ProEngineer or even AutoCAD. Don't get me wrong, lots of firms would love to not have
to spend huge $ on an expensive 3D modeling package like CATIA. It costs a bloody fortune. But there just aren't enough programmers out there with the right skills and the itch to create a CAD
package that will replace the commercial stuff any time soon.
Games seem to be another area where free software struggles to challenge commercial offerings. High development costs, small group of available programmers, requires artistic/creative skills not
widely possessed by programmers, and other reasons besides.
Basically, the more specialized the software or the more artistic content required, the more difficult it seems to be to develop under a free model. Not impossible mind you, just more difficult;
sometimes to the point where it is not practical even if it is theoretically possible.
• Re:SAGE is an interesting project (Score:2, Interesting)
by slawekk (919270) on Saturday December 08, 2007 @01:41PM (#21625075) Homepage
we need to integrate formal proof software concepts
I am very happy to hear that. Maybe this will result in a decent authoring and presentation tool for Isabelle. Mathematics is not about calculations, it is about theorems and proofs. I would not
call any software "math software" if I can not do math with it (i.e. write a proof and verify it). This is not to criticize Sage which I consider an awesome piece of software engineering, mostly
because of using existing excellent tools rather than inventing its own.
• Re:SAGE is an interesting project (Score:2, Interesting)
by mhansen444 (1200253) on Saturday December 08, 2007 @01:54PM (#21625195)
Math software, like Sage, is incredibly useful for coming up with and testing conjectures. Before you can prove something, you need to know what you want to prove. For example, in some of my
research, the direction we went was primarily driven by computational results, which led to conjectures, which in turn led to theorems and proofs. I've looked at Isabelle, and it looks to be a
long way off from being able to help with the math that I'm interested in.
• Re:What about other math software? (Score:2, Interesting)
by xtracto (837672) on Saturday December 08, 2007 @01:54PM (#21625197) Journal
Sage provides much more functionality than existing FLOSS projects. One of the ways it does this is by making use of those project. For example, Sage comes with Maxima and uses it as an engine to
do symbolic calculus type computations. Axiom can be used from within Sage if it is installed as well. Sage also includes GAP, which is the open-source package for doing abstract algebra
computations. One of the main reasons for starting a new project was to take advantage of existing projects and tie them together.
So, is there something that Sage does that can not be achieved by a BASH shell?
• Re:SAGE is an interesting project (Score:4, Interesting)
by ortholattice (175065) on Saturday December 08, 2007 @02:06PM (#21625303)
It may be argued that computers are not really an appropriate tool when truly "correct" mathematics must be relied upon. My response to that is that as problems of interest become ever more
complex, limitations both of the human mind and the human life span will ultimately limit the problems we can solve unaided. The task for us now is to create a system we CAN trust to solve
problems correctly, because someday we will have to trust it to solve problems we cannot handle.
There is a mathematical proof verification language, Metamath [wikipedia.org], whose rigor and/or correctness (meaning freedom from bugs) are probably near the top, if only because (1) the proof
language is trivially simple and (2) as a result half a dozen independently written proof verifiers have been coded, in C, Haskell, Python (300 lines of code), Java, Lisp, and Lua, so the
likelihood they all have the same bug is pretty small. It stands in contrast to some other proof verifiers or theorem provers that embed complex internal algorithms and tend to be very large
programs that would be hard to formally verify for correctness - and in some cases are closed source (like Mizar [wikipedia.org], which BTW probably has the largest body of mathematical knowledge
developed for it).
A problem with Metamath is that it is very labor-intensive to develop proofs. The proof of 2 + 2 = 4 [metamath.org] has 23,000 steps from ZF set theory axioms, and the computation of cosine of 2
[metamath.org] to one decimal place has some 75,000 steps that take several seconds for the verifier to verify. All of these steps were entered by hand (although once a collection of theorems are
developed they can be reused, so proofs become easier as a body of knowledge is developed). All of these steps are absolutely, rigorously correct - assuming that at least one of the independent
verifiers has no bugs. Unlike a 75,000 line computer program, there is no such thing a a bug in the proof - a proof is either right or wrong (i.e. not a proof).
• Re:FLOSS misses the point again (Score:2, Interesting)
by MonaLisa (190059) on Saturday December 08, 2007 @02:12PM (#21625355)
Mathematica is quite good at linear algebra, actually. Not such a great group theory tool, but I have used it for a lot of number theory projects. I paid $75 for a Mathematica license (as a grad
student) and it's definitely worth that much to me. It is a very nice tool for a lot things, but not everything. When I need to do a group theory calculation, I use GAP. When I need to do some
complicated commutative algebra calculation, I use Macaulay2. I think SAGE is cool and all, but they are all just tools to do research, and I'll use whatever I can get my hands on to get the job
done as fast as possible. Also, Mathematica as a programming language is really more of a pure functional language that is every bit as flexible as your "real" programming languages. If I need to
use a "real" programming language, it is for speed, in which case C++ is much faster than Python and all those "real" programming languages.
• Re:FLOSS misses the point again (Score:5, Interesting)
by mhansen444 (1200253) on Saturday December 08, 2007 @02:45PM (#21625645)
When I meant not good at linear algebra, I meant that it is slow. For example, Sage is over 30x faster at computing the characteristic polynomial of a matrix over the integers. Regarding number
theory, there isn't really any support in Mathematica for working with number fields, modular forms, or elliptic curves. What I meant by "real" programming language was that there is a lot of
software out there that can be taken advantage of. Say for instance I need to work with data stored in an relational database. How easy is that to do with Mathematica? It is trivial with Sage
since Sage uses Python. When Sage needs to do things fast, it uses Cython ( http://www.cython.org/ [cython.org] ) which is almost a superset of Python and compiles down to C.
• Re:another one bites the dust (Score:4, Interesting)
by 99BottlesOfBeerInMyF (813746) on Saturday December 08, 2007 @04:15PM (#21626363)
Blender is a user interface nightmare.
Blender is a UI for advanced users. It has very poor learnability, but I've heard it is a very good UI once you are used to it. I haven't seen any usability studies though, so it is just hearsay.
GIMP's no good for commercial artwork (Pantone swatches and CYMYK and whatnot)
I have used GIMP for commercial work for years and it has been the best tool on the market for certain uses, especially large automated batch jobs that are beyond Graphics Converter. More
recently, Pixelmator may have taken the title away from them, but to call GIMP "no good" in a commercial environment is just wrong. It is used a lot in certain segments, although it can't compete
with Photoshop for one off photo touch-ups and that sort of thing.
I can't comment on Inkscape.
Inkscape is pretty decent and a reasonable Illustrator replacement for many projects. The main drawbacks I have with it is for Visio type work it is not well suited, and support on the Mac (where
realistically most pro graphic artists work) is very weak.
They're more "challenged" than a challenge to commercial programs.
I disagree. Most of them are focused on different parts of the market than commercial competitors, but all of this software is probably the best for some uses.
• Re:User interface and documentation (Score:3, Interesting)
by William Stein (259724) <wstein@gmail.com> on Saturday December 08, 2007 @04:24PM (#21626443) Homepage
> The success of Sage won't be determined by how powerful it is.
The success of Sage with research mathematicians may be determined by how
powerful Sage is, but you're right -- the success for 99% of users won't be
determined by that.
> As others have observed, it is largely a mashup of existing stuff.
> Its success will be determined by how easy it is to use. If someone
> can put together some decent documentation
We have many people in the development team who are really very interested
in writing good documentation (and who write published mathematics books as
part of our jobs). For example, the author of "Adventures in Group Theory:
Rubik's Cube, Merlin's Machine, and Other Mathematical Toys" is
one of the main Sage developers (he's coming out with a new version of the
book that uses Sage soon).
> and a semi-intuitive UI, it will take off.
From the start we've had many undergraduates with a software engineering
background involved in the project and they have helped immensely with
the browser-based GUI (which one can use locally -- no need to be online!).
Also, us "professional mathematicians" -- even the ones that use mainly FOSS --
really do greatly value having a nice GUI. You might be able to try
out the GUI right now here:
https://sage.math.washington.edu:8101/ [washington.edu]
that is, if it hasn't been slashdotted into oblivion already!
-- Willam
• Re:another one bites the dust (Score:1, Interesting)
by Anonymous Coward on Saturday December 08, 2007 @04:53PM (#21626675)
There is F/OSS software being used in spaceflight. RTEMS is an Open Source RTOS that's being used in flight applications, for instance.
There ARE some practical issues, though..
1) flight software applications are usually not "ground up creations from scratch", but, rather, inherit a LOT of their code from previous go-arounds (it works, don't break it). That "known good
code" might well be closed source
2) flight software might be subject to export controls and ITAR (i.e. it's a munition). The same software that might control a science probe to the moon could also control a re-entering ICBM
warhead. (this is probably the biggest issue with releasing code)
3) In a schedule critical development, one might not to wait for a F/OSS developer to need to scratch that particular itch. So you have to pay a developer or contracting firm to develop the
software. That firm may wish to use their (closed source) libraries, or might see some downstream profit potential from the development in other markets. That closed source profit potential might
be orders of magnitude greater than what a space mission is willing to pay for the software.
4) Those F/OSS toilers in the bazaar might not want to work in the rigorous schedule and configuration control environment needed for flight-critical code development. Your Mars launch
opportunity only comes once every 2 years or so.. no time to follow the latest interesting branch.
• Re:Added benefit (Score:1, Interesting)
by Anonymous Coward on Saturday December 08, 2007 @06:55PM (#21627409)
This isn't a comment about William in particular, but I find the packaging of SAGE to be rather arrogant and self-important. At first glance it looks like SAGE is millions of lines of source
code. On closer inspection I find that SAGE is really just several dozen open source mathematics packages bundled together in a tarball with the SAGE name slapped on it. On even closer inspection
I find that there is actually SAGE code that appears very worthwhile, additional functionality is provided, a consistent interface, etc. However, I'm not going to use it seriously because I can't
'apt-get install sage-math'. SAGE wants me to download and install more than 200MB of stuff that isn't going to be handled by my OS package management, and it duplicates many of the components I
already have installed that are handled by my OS package management. Ten years ago I may have put up with the headaches this is bound to cause, but not today. Get SAGE packaged if you really want
people to use it. Don't make me use your own forked and patched versions of Pari or GAP. I can 'apt-get install pari-gp gap' today(and I already have). I'm not going to install your alternative
versions and deal with any inconsistancies between them. I'm not throwing out the ease of use that OS level package management provides to get SAGE. I know many other people who aren't going to
do it either.
• Re:Added benefit (Score:1, Interesting)
by Anonymous Coward on Monday December 10, 2007 @09:39PM (#21650491)
...Sage is a volunteer project for which most developers are naturally mathematicians. We simply don't have the time to maintain Debianizing dozens of packages.
You are taking packages that have already been Debianized and are Sage-ing them. The work to Debianize GAP, PARI/GP, and R has already been done. These packages are already maintained in Debian
and on other operating systems. I would think you would want to leverage that work. It looks like you are underestimating or overlooking the benefits and time savings that integration with the
distributions can provide. In short, I think you are making things hard not only on your users but also on all of the Sage developers. I don't understand what would motivate you to take on the
responsibility of building and distributing millions of lines of source code, most of which is already distributed via other means.
Related Links Top of the: day, week, month. | {"url":"http://science.slashdot.org/story/07/12/08/1350258/open-source-sage-takes-aim-at-high-end-math-software/interesting-comments","timestamp":"2014-04-20T13:25:55Z","content_type":null,"content_length":"144875","record_id":"<urn:uuid:c1fd01b8-26f7-4e28-acb5-b6c3c4e5f2f9>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00302-ip-10-147-4-33.ec2.internal.warc.gz"} |
Amesos_Scalapack: A serial and parallel dense solver. For now, we implement only the unsymmetric ScaLAPACK solver.
Amesos_Scalapack, an object-oriented wrapper for LAPACK and ScaLAPACK, will solve a linear systems of equations: A X = B using Epetra objects and the ScaLAPACK library, where A is an Epetra_RowMatrix
and X and B are Epetra_MultiVector objects.
Amesos_Scalapack can be competitive for matrices that are not particularly sparse. ScaLAPACK solves matrices for which the fill-in is roughly 10% to 20% of the matrix size in time comparable to that
achieve by other Amesos classes. Amesos_Scalapack scales well and hence its performance advantage will be largest when large number of processes are involved.
Amesos_Scalapack uses the ScaLAPACK functions PDGETRF and PDGETRS if more than one process is used. If only one process is used, Amesos_ScaLAPACK uses the LAPACK function PDGETRF and PDGETRS.
AmesosScaLAPACK uses full partial pivoting and will therefore provide answers that are at least as accurate as any direct sparse solver.
AmesosScalapack makes sense under the following circumstances:
Common control parameters :
Amesos_Scalapack supports the following parameters which are common to across multiple Amesos solvers:
• ParamList.set("MaxProcs", int MaximumProcessesToUse );
By default, this is set to -1, which causes Amesos_Scalapack to use a heuristic to determine how many processes to use. If set to a postive value, MaximumProcessesToUse, Amesos_Scalapack will use
MaximumProcessesToUse provided that there are that many processes available. Testing should be performed with MaximumProcessesToUse set to some value larger than one to force parallel execution.
• ParamList.set("PrintTiming", bool );
• ParamList.set("PrintStatus", bool );
• ParamList.set("ComputeVectorNorms", bool );
• ParamList.set("ComputeTrueResidual", bool );
• ParamList.set("OutputLevel", int );
• ParamList.set("DebugLevel", int );
• ParamList.set("ComputeTrueResidual", bool );
Amesos_Scalapack supports the following parameters specific to Amesos_Scalapack.
Teuchos::ParameterList ScalapackParams = ParameterList.sublist("Scalapack") ;
• ScalapackParams.set("2D distribution", bool );
By default this is set "true". In general, because a two dimensional data distribution generally produces faster results. However, in some cases, a one dimensional data distribution may provide
faster execution time. The code for the one dimensional data distribution uses a different data redistribution algorithm and uses the transpose of the matrix internally (all of which is
transparent to the user).
• ScalapackParams.set("grid_nb", bool );
By default this is set to 32. On some machines, it may be possible to improve performance by up to 10% by changing the value of grid_nb. (16,24,48,64 or 128) are reasonable values to try. For
testing on small matrices, small values of grid_nb will (if "MaxProcs" is set to a value greater than 1) force the code to execute in parallel.
None of the following limitations would be particularly difficult to remove.
The present implementation limits the number of right hand sides to the number of rows assigned to each process. i.e. nrhs < n/p.
The present implementation does not take advantage of symmetric or symmetric positive definite matrices, although ScaLAPACK has separate routines to take advantages of such matrices.
Definition at line 146 of file Amesos_Scalapack.h. | {"url":"http://trilinos.sandia.gov/packages/docs/r11.0/packages/amesos/browser/doc/html/classAmesos__Scalapack.html","timestamp":"2014-04-24T02:05:14Z","content_type":null,"content_length":"95303","record_id":"<urn:uuid:ad6c9472-57ac-4b58-91d3-d913114fc7b2>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00046-ip-10-147-4-33.ec2.internal.warc.gz"} |
Braingle: 'Difficult Danny' Brain Teaser
Difficult Danny
Math brain teasers require computations to solve.
Puzzle ID: #49976
Category: Math
Submitted By: eighsse
Difficult Danny has acquired his nickname (and some other, less polite nicknames) due to the fact that, whenever he buys anything, he insists upon paying the exact amount, with precisely 100 pieces
of currency. Danny walks into McArnold's and orders an iced coffee for $1.99. He pours 100 coins onto the counter, using some number (possibly zero) of each of the following: pennies, nickels,
dimes, quarters. There are equal, non-zero quantities of dimes and quarters. What is the breakdown of coins that Difficult Danny used to pay for his iced coffee?
(See hint for currency values.)
Show Hint
Show Answer
What Next? | {"url":"http://www.braingle.com/brainteasers/teaser.php?id=49976&comm=0","timestamp":"2014-04-17T04:01:37Z","content_type":null,"content_length":"23708","record_id":"<urn:uuid:ba1e056d-29b6-4f95-9ee0-77401eef1454>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00623-ip-10-147-4-33.ec2.internal.warc.gz"} |
Center for Philosophy of Science ::: huggett_04-11-08
(Again) A Philosopher Looks at String Theory
Nicholas Huggett
Abstract: According to Brandenberger and Vafa (1989), because of the 'T-duality' between string theories of with compactified dimensions with radius R and radius 1/R, ‘there is no physical experiment
which tells us whether today we live in a Universe of size [10^10 light years], or in a tiny Universe of size [10^-85m]’. This talk sketches the physics behind this remarkable-sounding claim (and
others of Greene, 2003), and explains what it means -- it doesn't seem to be the kind of thing we could be mistaken about taken literally, and indeed it involves a kind of pun. The talk will conclude
with some remarks about the underdetermination of geometry by experiment in this context (which is rather different from more familiar cases) and about the relation of T-duality to the emergence of
spacetime in quantum gravity. | {"url":"http://www.pitt.edu/~pittcntr/Events/All/Conferences/others/other_conf_2007-08/stringtime_in_pgh_04-11-08/abstracts/huggett.htm","timestamp":"2014-04-16T19:33:42Z","content_type":null,"content_length":"8459","record_id":"<urn:uuid:309c0d73-0484-423e-a8d9-ecc63450730d>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00103-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Error Bounds
September 10th 2007, 06:09 PM #1
Junior Member
Sep 2007
Error Bounds
How large should n be to guarantee that the Simpson's Rule approximation to the integral of 0 to 1 of e^(x^2) is accurate to within 0.00001?
so I took the function up to it's 4th derivative.
how am i supposed to find k?
stupid teacher did not clearly tell us how to find it and this book (Stewart's) i heard was not clear either.
You may wish to lose this attitude. It is possible both the teacher and the author want you to learn how to think.
The error term for Simpson's Rule is $\frac{1}{90}f^{4}(c)h^{5}$. Your task is to find where this expression MUSt be less than 0.00001.
Well, $\frac{1}{90}f^{4}(c)h^{5} < 0.00001$
$f^{4}(c)h^{5} < 0.00090$
It's a little tricky, right here. This requires some judgment. Where on your interval is the fourth derivative GREATEST? If we find where it is greatest, we can guarantee the error will be as
desired. Find the value of 'c' that maximizes the fourth derivative. Evaluate that derivative. Substitute in the inequality and finish solving for 'h'. This can be translated into the appropriate
value of 'n' that is required.
Let's see what you get.
September 10th 2007, 07:29 PM #2
MHF Contributor
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st: Defining variables for the ml procedure
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st: Defining variables for the ml procedure
From bpersson@bu.edu
To statalist@hsphsun2.harvard.edu
Subject st: Defining variables for the ml procedure
Date Sun, 9 Oct 2005 12:59:34 -0400
Dear Statalist,
I am trying to estimate a nonlinear regression equation using the ml
routine, and I am having problems defining one of my input variables.
Specifically, I would like to use the recursively defined variable
Z(it) = c*Z(it-1) + X(it)
as an independent variable in my regression equation. If the
coefficient "c" is known, this variable can be generated as follows:
gen Z = X
bysort group (time): replace Z = c * Z[_n-1] + X if _n >= 2
My problem is that I would like to estimate the coefficient "c" (along
with the other regression coefficients) and therefore I need to define
the variable Z(it) "within" the program itself.
It seems, however, as if the "by" command and the "if" command are not
allowed when specifying variables in the ml programs. I would be most
grateful if anyone would have any ideas on how to do this.
Thank you for your help.
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
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completing the square
September 18th 2009, 02:29 PM #1
Jan 2006
completing the square
by completing the square the expression that is x^2+14 x +156 equals (x+A)^2+B
so the question is what would A and B equal
this is my last question...i am stumped on a few of these..
Last edited by mr fantastic; September 18th 2009 at 03:38 PM. Reason: Restored original question and merged with another post (also originally deleted)
I'll put you out of you're misery. Given an equation $x^2+bx+c$, You want to get something of the form $x^2+bx+\left(\frac{b}{2}\right)^2+k$, for some constant $k$. (Note that $k+\left(\frac{b}
{2}\right)^2=c$.) When factored, this will become $\left(x+\frac{b}{2}\right)^2+k$.
So in this case, $\frac{b}{2}=7$ so $\left(\frac{b}{2}\right)^2=49$.
Break up the equation: $x^2+14x+156=x^2+14x+49+107$
Factor and you're done: $x^2+14x+49+107=(x+7)^2+107$.
So $A=7$ and $B=107$.
EDIT: You've already figured it out I see. Well, at least you have something to check it against now.
thanks a lot, but i figured that one out.
Last edited by mr fantastic; September 18th 2009 at 03:36 PM. Reason: Moved a question to another thread
September 18th 2009, 03:22 PM #2
September 18th 2009, 03:28 PM #3
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Fundamental domain for group generated by reflections in -2 curves
up vote 3 down vote favorite
Given a proper algebraic surface $S$, the Picard group $Pic(S)$ is endowed with the (symmetric) intersection form. We can therefore talk about reflections in the classes of $-2$-curves. These will
preserve the intersection form and therefore the positive cone inside the lattice $Pic(S)$. Many times I have come across the claim that if $S$ is a smooth K3, then the action of this group generated
by reflections is properly discontinuous on the positive cone inside $Pic(S) \otimes \mathbb{R}$ and has a (maybe infinite) polyhedral fundamental domain with faces defined by the hyperplanes
orthogonal to the classes of effective $-2$-classes. Why is this? And is there a good reference for the proof of this fact?
ag.algebraic-geometry quadratic-forms
Did you already have a look at Chapter VIII of "Compact Complex Surfaces" by Barth, Hulek, Peters, van de Ven; as well as S. Kovacs' paper "The cone of curves of a K3 surface"? – Christian Liedtke
Aug 8 '11 at 14:52
I just took a look at both sources, which don't seem to address this question. Any other suggestions? – A. Pascal Aug 8 '11 at 15:23
add comment
1 Answer
active oldest votes
This is a general fact about groups generated by reflections acting on lattices. The reflections in $-2$ curves generates an action on $H^2(S;\mathbb{Z})$ by reflections, with quadratic form
given by the cup product. Restricting this to $H^{1,1}(S)\cap H^2(S;\mathbb{Z})$, one gets an action on an integral lattice of signature $(1,19)$ (check out McMullen). For any group
preserving a Lorentzian lattice, the subgroup generated by reflections will have fundamental domain bounded by hyperplanes fixed by a subset of reflections. The point is that the set of all
hyperplanes which are fixed by reflections is equivariant with respect to the group action, since the conjugate of a reflection is a reflection. Thus, they cut out polyhedra in the positive
up vote cone (the hyperplanes must be locally finite since the group is discrete as it preserves a lattice). The fact that the quadratic form is Lorentzian is used here, since otherwise none of the
5 down complementary pieces of the null cone are convex. Given any two points in the cone, one takes a generic path between them, which will cross finitely many hyperplanes. The sequence of
vote reflections fixing the sequence of hyperplanes crossed by the path will send the polyhedron containing one point to the polyhedron containing the other. Thus, the polyhedra are all
accepted equivalent by the group action, and therefore form a fundamental domain for the subgroup generated by reflections (one must also check that no reflection fixes one of the polyhedra, which is
not hard to show geometrically as well). In fact, Vinberg has a nice algorithm which will compute the fundamental domain inductively, and I believe this kind of argument is discussed in his
I'd like to accept this as the answer, but don't completely understand yet. In the above argument, do we really use the Lorentzian condition? This is similar to another question I asked
about locally finite collections of root hyperplanes for an arbitrary even symmetric bilinear form. Also, which book of Vinberg might you have in mind? I took a look at the linked paper,
which gives an algorithm, but he doesn't really explain the existence of the chamber decomposition. – A. Pascal Aug 9 '11 at 5:27
A. Pascal: I put a link to the Mathscinet review of his book,although I don't have access to it now so I haven't checked if it contains the argument. In his papers, he just says that the
argument is exactly as in Coxeter's proof of the existence of a fundamental domain for a group generated by reflections in the Euclidean case. The fact that the quadratic form is
Lorentzian is used because the positive cone is convex - this is not true in other signatures. In some cases, one can find a convex fundamental domain for a reflection group preserving a
quadratic form of other signatures(Tits cone). – Ian Agol Aug 9 '11 at 16:59
Thanks. This cleared up some things and the Shvartsman-Vinberg book is very useful. – A. Pascal Aug 10 '11 at 8:41
Great answer, I needed this exactly. – Philip Engel Jun 5 '13 at 4:01
add comment
Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry quadratic-forms or ask your own question. | {"url":"http://mathoverflow.net/questions/72344/fundamental-domain-for-group-generated-by-reflections-in-2-curves","timestamp":"2014-04-20T11:11:54Z","content_type":null,"content_length":"58657","record_id":"<urn:uuid:77df521b-aadd-4f93-96c3-13ebf4687386>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00391-ip-10-147-4-33.ec2.internal.warc.gz"} |
what is cube, cuboid, cylinder and sphere
Cube as a Geometric Solid. A regular hexahedron, the cube is a six-sided geometric solid, also called a block. It is one of the ideal solids, which include the only five solids with congruent,
polyhedral angles and faces that are congruent, regular polygons. The others are the regular tetrahedron, regular octahedron, regular dodecahedron, and regular icosahedron, having 4, 8, 12, and 20
sides respectively. The cube is the only one of these five to have four edges per face: the tetrahedron, octahedron, and icosahedron – formed with equilateral triangle faces - have three; and the
dodecahedron - formed with regular pentagon faces - has five.
In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing incompatible definitions of a cuboid in the mathematical literature. In the more
general definition of a cuboid, the only additional requirement is that these six faces each be a quadrilateral, and that the undirected graph formed by the vertices and edges of the polyhedron
should be isomorphic to the graph of a cube.[1] Alternatively, the word “cuboid” is sometimes used to refer to a shape of this type in which each of the faces is a rectangle, and in which each pair
of adjacent faces meets in a right angle; this more restrictive type of cuboid is also known as a right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this
surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity.
In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whose cross section is an ellipse, parabola, or
hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder.
A sphere (from Greek sfa??a—sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in three dimensions, a perfect
sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The maximum
straight distance through the sphere is known as the diameter of the sphere. It passes through the center and is thus twice the radius.
In higher mathematics, a careful distinction is made between the sphere (a two-dimensional spherical surface embedded in three-dimensional Euclidean space) and the ball (the three-dimensional shape
consisting of a sphere and its interior).
Added 4/13/2010 1:05:47 PM | {"url":"http://weegy.com/Home.aspx?ConversationId=27B18FF8","timestamp":"2014-04-20T13:19:26Z","content_type":null,"content_length":"33149","record_id":"<urn:uuid:fb3fbf1b-12bd-4249-9d47-a09c6619c273>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00009-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sterling, VA Algebra 2 Tutor
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December 12th, 2003, 04:01 AM
Hi Gurus,
Its somewhat frustrating question.
Is there any way to compare two numbers without using any conditional operator like (<,>) ?
December 12th, 2003, 04:18 AM
why cant you use the '<','>', etc. operators ?
December 12th, 2003, 04:18 AM
What do you mean under "to compare two numbers" ?
December 12th, 2003, 05:32 AM
Suppose a = 10 and b = 20. are two numbers.
i want to know which is greater out of them. For this purpose we can use '<','>' operators. but i don't want to use them to achieve the result as 20.
Is there any way out ?
December 12th, 2003, 05:37 AM
You dont want to use:
PHP Code:
int a = 10 ;
int b = 20 ;
// do something
// do something
// do something
December 12th, 2003, 05:39 AM
Yah, inequality operators are terrible for determining order.
I mean what does one have to do with the other.
December 12th, 2003, 05:43 AM
yes, TheRogue
I really don't want to use these operators.
I am also thinking that we can' t do this.. but still i wan't to raise on CODEGURU coz there are really good ppls who can answer it well.
December 12th, 2003, 05:47 AM
Well, I am just curious. Please tell us why you don't want to use
December 12th, 2003, 05:52 AM
Pretty please..with sugar on top.:)
December 12th, 2003, 05:52 AM
Hi souldog,
Thanx for being curios with the question.
I want to write an assembly language code for one specific microprocessor which doesn't contain any cmp ('compare' ), jl('jump if less'), jg (jump if greater) instructions. so i am thinking how
it is possible to find greatest number out of A and B.
May be my approach is wrong. isn't it ?
please suggest.
December 12th, 2003, 05:56 AM
If it has a check for zero,
could you decrement both variables until one of them is zero ?
December 12th, 2003, 06:02 AM
Well if you can look at the bits in the numbers starting with
the MSB and find the first place they are different, then the number
that has a 1 in this bit is larger.
December 12th, 2003, 06:08 AM
hi TheRogue
If it has a check for zero,
could you decrement both variables until one of them is zero ?
Fortunatly, i have that instruction with me.
I can able to decrement both variable until one of them is ZERO.
now please clarify..
December 12th, 2003, 06:12 AM
decrement both variables
if neither variable has reached zero, goto start:
if one variable has reached zero it is smaller
if both variables have reached zero they are the same
ps. what is the microprocessor ?
December 12th, 2003, 06:13 AM
Do you have a carry flag in a FLAGS register. You can subtract the two numbers and see if a borrow was needed. | {"url":"http://forums.codeguru.com/printthread.php?t=275783&pp=15&page=1","timestamp":"2014-04-21T10:35:13Z","content_type":null,"content_length":"15204","record_id":"<urn:uuid:32e3abbb-5ed4-4e2b-b816-5f7217b161aa>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00394-ip-10-147-4-33.ec2.internal.warc.gz"} |
Uniform semantic treatment of default and autoepistemic logics
Results 1 - 10 of 31
- In Proceedings of ICLP-01, LNCS 2237 , 2001
"... is relatively straightforward. Another advantage of the ultimate approximation is that in cases where TP is monotone the ultimate well-founded model will be 2-valued and will coincide with the
least fixpoint of TP . This is not the case for the standard well-founded semantics. For example in the sta ..."
Cited by 44 (7 self)
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is relatively straightforward. Another advantage of the ultimate approximation is that in cases where TP is monotone the ultimate well-founded model will be 2-valued and will coincide with the least
fixpoint of TP . This is not the case for the standard well-founded semantics. For example in the standard well-founded model of the program: # p. p. p is undefined while the associated TP operator
is monotone and p is true in the ultimate well-founded model. One disadvantage of using the ultimate semantics is that it has a higher computational cost even for programs without aggregates. The
complexity goes one level higher in the polynomial hierarchy to # 2 for the well-founded model and to 2 for a stable model which is also complete for this class [2]. Fortunately, by adding aggregates
the complexity does not increase further. To give an example of a logic program with aggregates we consider the problem of computing the length of the shortest path between two nodes in a direc
- ACM transactions on computational logic , 2007
"... Well-known principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over well-founded sets and iterated
induction. In this work, we define a logic formalizing induction over well-founded sets and monotone and iterated i ..."
Cited by 28 (16 self)
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Well-known principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over well-founded sets and iterated induction. In
this work, we define a logic formalizing induction over well-founded sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the
principle of inflationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for Non-Monotone Inductive Definitions (ID-logic). The
semantics of the logic is strongly influenced by the well-founded semantics of logic programming. This paper discusses the formalisation of different forms of (non-)monotone induction by the
well-founded semantics and illustrates the use of the logic for formalizing mathematical and common-sense knowledge. To model different types of induction found in mathematics, we define several
subclasses of definitions, and show that they are correctly formalized by the well-founded semantics. We also present translations into classical first or second order logic. We develop modularity
and totality results and demonstrate their use to analyze and simplify complex definitions. We illustrate the use of the logic for temporal reasoning. The logic formally extends Logic Programming,
Abductive Logic Programming and Datalog, and thus formalizes the view on these formalisms as logics of (generalized) inductive definitions. Categories and Subject Descriptors:... [...]:... 1.
, 2000
"... In this paper we develop an algebraic framework for studying semantics of nonmonotonic logics. Our approach is formulated in the language of lattices, bilattices, operators and fixpoints. The
goal is to describe fixpoints of an operator O defined on a lattice. The key intuition is that of an approxi ..."
Cited by 18 (8 self)
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In this paper we develop an algebraic framework for studying semantics of nonmonotonic logics. Our approach is formulated in the language of lattices, bilattices, operators and fixpoints. The goal is
to describe fixpoints of an operator O defined on a lattice. The key intuition is that of an approximation, a pair (x, y) of lattice elements which can be viewed as an approximation to each lattice
element z such that x z y. The key notion is that of an approximating operator, a monotone operator on the bilattice of approximations whose fixpoints approximate the fixpoints of the operator O. The
main contribution of the paper is an algebraic construction which assigns a certain operator, called the stable operator, to every approximating operator on a bilattice of approximations. This
construction leads to an abstract version of the well-founded semantics. In the paper we show that our theory offers a unified framework for semantic studies of logic programming, default logic and
autoepistemic logic.
- Proceedings of the 23rd International Conference on Logic Programming (ICLP 2007), LNCS, Springer, 2007 (this , 2005
"... Abstract. We provide new perspectives on the semantics of logic programs with constraints. To this end we introduce several notions of computation and propose to use the results of computations
as answer sets of programs with constraints. We discuss the rationale behind different classes of computat ..."
Cited by 16 (2 self)
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Abstract. We provide new perspectives on the semantics of logic programs with constraints. To this end we introduce several notions of computation and propose to use the results of computations as
answer sets of programs with constraints. We discuss the rationale behind different classes of computations and study the relationships among them and among the corresponding concepts of answer sets.
The proposed semantics generalize the answer set semantics for programs with monotone, convex and/or arbitrary constraints described in the literature. 1
, 2008
"... Managing uncertainty and/or vagueness is starting to play an important role in Semantic Web representation languages. Our aim is to overview basic concepts on representing uncertain and vague
knowledge in current Semantic Web ontology and rule languages (and their combination). ..."
Cited by 16 (5 self)
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Managing uncertainty and/or vagueness is starting to play an important role in Semantic Web representation languages. Our aim is to overview basic concepts on representing uncertain and vague
knowledge in current Semantic Web ontology and rule languages (and their combination).
, 2005
"... Due to the usual incompleteness of information representation, any approach to assign a semantics to logic programs has to rely on a default assumption on the missing information. The stable
model semantics, that has become the dominating approach to give semantics to logic programs, relies on the C ..."
Cited by 8 (1 self)
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Due to the usual incompleteness of information representation, any approach to assign a semantics to logic programs has to rely on a default assumption on the missing information. The stable model
semantics, that has become the dominating approach to give semantics to logic programs, relies on the Closed World Assumption (CWA), which asserts that by default the truth of an atom is false. There
is a second well-known assumption, called Open World Assumption (OWA), which asserts that the truth of the atoms is supposed to be unknown by default. However, the CWA, the OWA and the combination of
them are extremal, though important, assumptions over a large variety of possible assumptions on the truth of the atoms, whenever the truth is taken from an arbitrary truth space. The topic of this
paper is to allow any assignment (i.e. interpretation), over a truth space, to be a default assumption. Our main result is that our extension is conservative in the sense that under the “everywhere
false ” default assumption (CWA) the usual stable model semantics is captured. Due to the generality and the purely algebraic nature of our approach, it abstracts from the particular formalism of
choice and the results may be applied in other contexts as well.
- Ceur-WS , 2003
"... The paper is an epistemological analysis of logic programming and shows an epistemological ambiguity. Many different logic programming formalisms and semantics have been proposed. Hence, logic
programming can be seen as a family of formal logics, each induced by a pair of a syntax and a semantics ..."
Cited by 8 (3 self)
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The paper is an epistemological analysis of logic programming and shows an epistemological ambiguity. Many different logic programming formalisms and semantics have been proposed. Hence, logic
programming can be seen as a family of formal logics, each induced by a pair of a syntax and a semantics, and each having a different declarative reading. However, we may expect that (a) if a program
belongs to different logics of this family and has the same formal semantics in these logics, then the declarative meaning attributed to this program in the different logics is equivalent, and (b)
that one and the same logic in this family has not been associated with distinct declarative readings.
- Principles of Knowledge Representation and Reasoning, Proceedings of the Tenth International Conference (KR2006 , 2006
"... We show that the concepts of strong and uniform equivalence of logic programs can be generalized to an abstract algebraic setting of operators on complete lattices. Our results imply
characterizations of strong and uniform equivalence for several nonmonotonic logics including logic programming with ..."
Cited by 8 (4 self)
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We show that the concepts of strong and uniform equivalence of logic programs can be generalized to an abstract algebraic setting of operators on complete lattices. Our results imply
characterizations of strong and uniform equivalence for several nonmonotonic logics including logic programming with aggregates, default logic and a version of autoepistemic logic. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=737228","timestamp":"2014-04-20T13:05:20Z","content_type":null,"content_length":"37359","record_id":"<urn:uuid:958c975a-bad7-4910-9575-414ff9460507>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00244-ip-10-147-4-33.ec2.internal.warc.gz"} |
Geometry and Euclid
Read pages 94-95 in Geometry.Theorem 3
Betweenness of Rays
You will have to study the proof on page 95 carefully. Ask for help if you need it.
Theorem 4
Angle Bisector Theorem
Look at
bisecting an angleTry this vocabulary quizTry measuring anglesExercises:
(these seem difficult; you may need help)
Set I
Set II#19-22
In this group, you are asked to match the statements with their theorems or definitions.
Hint: Look back at pages 94-95#23-26
-- skip these ones.
#27 -30
This is algebra review.
ASN = 5x
NSD = 3x + 28
and you know because SN is bisecting the angle that both these two angles are equal,
ASN = NSD
Knowing that, can you do the rest?
• Look at the facts about the figure.
• Hint: AJD + DJE = AJE (because of the Betweenness of Rays)
• Substitute the terms for the angle names, as you did in the last group of problems, and then you have an algebra problem that you can solve.
• When you see problems like this, ALWAYS look for ways to do this.
#35 - 36
This was a long lesson.... tomorrow will be an optional day. | {"url":"http://euclideanprep.blogspot.com/2010/08/betweenness-of-rays.html","timestamp":"2014-04-21T12:10:08Z","content_type":null,"content_length":"43703","record_id":"<urn:uuid:5131898f-63d1-4f0e-ac4f-e369e723cbc3>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00233-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hypothesis testing problem
March 21st 2010, 11:51 AM #1
Mar 2010
Hypothesis testing problem
At one office, the average amount of time that workers spend using computers is 21.6 hours with a standard deviation of 2.8 hours. One year later, hypothesis test was done to determine whether
the average amount of time spent using computers decreased. For sample of size 70 and significance level of 0.05, find the minimum value of sample mean that makes this test not to reject the null
The answer is 21.05(#1). How do I solve this problem.
null hypothesis is mu = 21.6
alternative hypothesis is mu < 21.6
Therefore it's a one sided test?
I get 21.0494777
BUT there is a HUGE flaw in this.
You really need to assume that the pop st deviation is 2.8
and not use LAST year's sample st deviation
The calculation is............
${\bar X-\mu\over \sigma/\sqrt{n}}=-1.645$
${\bar X-21.6\over 2.8/\sqrt{70}}=-1.645$
null hypothesis is mu = 21.6
alternative hypothesis is mu < 21.6
Therefore it's a one sided test?
YES to all of this
so, you put all the alpha=.05 on the left side as I did and get $(-\infty,-1.645)$ as the rejection region.
March 21st 2010, 08:28 PM #2
March 21st 2010, 08:31 PM #3 | {"url":"http://mathhelpforum.com/statistics/134880-hypothesis-testing-problem.html","timestamp":"2014-04-17T07:07:49Z","content_type":null,"content_length":"38953","record_id":"<urn:uuid:35b557b1-0b47-4241-af98-77ec01297ae5>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00584-ip-10-147-4-33.ec2.internal.warc.gz"} |
Field Precision software tips
Numerical solution of transcendental equations
In technical work, often we must deal with equations that do not have an algebraic solution. In this circumstance, we must turn to numerical methods to find approximate values at a set of points.
Powerful programs using sophisticated methods are available, but they are often an overkill. In this article I’ll describe some garage mathematics to get quick answers. Usually, it takes only a few
minutes if you are familiar with a script language like Perl or Python.
I will use a recent project as an illustration. The figure shows the Pierce geometry for a planar electron injector with gap width D. (The system extends an infinite distance out of the page.) The
theory to determine the electrode shapes is discussed in Sect. 7.1 of my book Charged Particle Beams. The equation for the anode surface outside the beam volume is given in terms of the local
coordinates (z, x) by
(r/D)^1.3333 cos(4θ/3) = 1,
where r^2 = z^2 + x^2, θ = atan(x/z). I wanted to find values of z at a set of x values. The coordinates could then be used in the Mesh program to define the anode. The surface at large values of x
has little effect on fields inside the beam. I picked a practical range 0.0 < x < 2.0.
As in all numerical work, the first task is to reduce the equations to a non-dimensional form so that the calculations have the greatest generality. Here, the task is simple. If we use the variables
R = r/D, X = x/D and Z = z/D, then the equations to solve are:
R^1.3333 cos(4θ/3) = 1, [1]
R^2 = X^2 + Y^2, [2]
θ = atan(X/Z) = asin(X/R). [3]
The point at the beam edge has coordinates X = 0.0, Z = 1.0 (R = 1.0, θ = 0.0).
Because we can’t solve the problem directly, we must use an iterative method that walks toward the solution in small steps. There are usually several choices to create an iterative loop, some of
which may be unstable. I choose to express the equation in this form
R‘ = 1.0/[cos(4θ/3)]^0.75, [4]
Z‘ = √(R’^2 – X’^2). [5]
The iteration proceeds as follows:
1. Start from the known point, X0 = 0.0, R0 = 1.0.
2. Using the next value X1, approximate the angle as θ1 = asin(X1/R0).
3. Solve Eqs. 4 and 5 to find R1.
4. Recalculate θ1 = asin(X1/R1) and determine a refined value of R1.
5. Repeat the procedure until the change in R1 between cycles is small. Find Z1 from Eq. 2.
Then move to the next point X2 using the value R1 for the initial estimate. Because the entire calculation takes less than a second, it is not necessary to worry about convergence rate or efficiency.
Here is a complete Python program:
import math
# Scaling
d = 1.0
x0 = 0.0
NCycle = 30
XMin = 0.0
XMax = 2.0
NStep = 20
Dx = (XMax-XMin)/NStep
RStart = 1.0
# Loop over X values
for n in range(NStep+1):
x = XMin + n*Dx
R = RStart
# Iteration loop
for m in range(NCycle):
theta = math.asin(x/R)
Rnew = 1.0/math.pow(math.cos(1.33333*theta),0.75)
# Averaging needed for stability
R = 0.25*Rnew + 0.75*R
z = math.sqrt(R*R-x*x)
xdisplay = x0 + d*x
zdisplay = d*z
print '%6i %9.6f %9.6f' % (n,xdisplay,zdisplay)
# Start search at the location of the previous search
RStart = R
Most of the time there is some iterative form of the initial equation that gives convergence. This case was more subtle. Initial values were correct, but the calculation became unstable as X
approached 1.0. I got valid results over the full range by increasing NStep and adding an averaging statement to slow convergence:
R = 0.25*Rnew + 0.75*R
It was not necessary to open a file for numerical output. I simply copied and pasted the output of the print statement from the Python interpreter window. The output is shown below. With a few simple
changes, the program can be modified to write the coordinates in a form that can be integrated directly into a Mesh script.
Results for d = 1.0, x0 = 0.0
0 0.000000 1.000000
1 0.100000 1.001662
2 0.200000 1.006604
3 0.300000 1.014700
4 0.400000 1.025755
5 0.500000 1.039534
6 0.600000 1.055782 | {"url":"http://fieldp.com/myblog/2010/numerical-solution-of-transcendental-equations/","timestamp":"2014-04-20T21:04:02Z","content_type":null,"content_length":"66276","record_id":"<urn:uuid:c0076cee-f1bb-4302-bc97-d505988fef99>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00602-ip-10-147-4-33.ec2.internal.warc.gz"} |
Free Downloadable Physics Software
* see Matlab Clones for an explanation of the fact that Matlab is great, however it is not only "Expensive" but "costs a lot". Many of the programs listed here are able to do MatLab-like
Linux/Unix/FreeBSD etc
*** Although this is Windows software, I have had no problems with it running in Linux using the free Windows compatability layer WINE
Free demo version of multilevel model software. Includes a circuit level for the modeling of Switched Mode Power Supplies, a component level for the modeling of electrical machines / loads and a
system level for the modeling of control algorithms.
A high-level language (very similar to Matlab), primarily intended for numerical computations. A free tool thats great for physicists.
I haven't tried this one, but it sounds similar to Octave and Scilab. The website says "Euler is not a MatLab clone, but similar to this program."
A high-level language (very similar to Matlab&Octave), primarily intended for numerical computations. Includes a MAPLE interface.
Dynamic Analyzer
Designed to allow students to model physical systems and compare the models with experimental data.
Instructional Software by RW Tarara
A Saint Mary's professor makes several instructional packages available for free.
A high-level language (very similar to Matlab&Octave), primarily intended for numerical computations. Includes a MAPLE interface.
Arbor Scientific
Software you can download and try-before-you-buy. Titles include: crocodile physics, interactive physics and oscillations & waves.
Free demo version of multilevel model software. Includes a circuit level for the modeling of Switched Mode Power Supplies, a component level for the modeling of electrical machines / loads and a
system level for the modeling of control algorithms.
Enables students to construct and analyze a wide variety of thermodynamic cycles.
I haven't tried this one, but it sounds similar to Octave and Scilab. The website says "Euler is not a MatLab clone, but similar to this program."
Dynamic Analyzer
Designed to allow students to model physical systems and compare the models with experimental data.
Instructional Software by RW Tarara
A Saint Mary's professor makes several instructional packages available for free.
A high-level language (very similar to Matlab), primarily intended for numerical computations. A free tool thats great for physicists.
Physics Collisions Simulator
Shows balls that bounce around the screen and perform oblique collisions. Can be used to demonstrate ideal gas laws.
PowerCalc - Bicycle Physics Software
Estimate a riders power and calorific requirements, taking into account wind resistance, rolling resistance, transmission losses etc. Also, GearCalc
A raytracing program written for teachers of geometrical optics and optical designers who want a quick tool for checking out their ideas.
Designed to be used by high school students. It includes, among others, calculations of wheel's diameters and number of teeth.
A high-level language (very similar to Matlab&Octave), primarily intended for numerical computations. Includes a MAPLE interface.
Allows students to investigate the superposition of two transverse traveling waves.
U of Arizona Math Software
Various software for math education. | {"url":"http://www.dctech.com/physics/software.php","timestamp":"2014-04-16T04:12:29Z","content_type":null,"content_length":"8292","record_id":"<urn:uuid:3ba0c0dd-98d3-446e-889f-52bdbccf16d3>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00128-ip-10-147-4-33.ec2.internal.warc.gz"} |
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the first resource for mathematics
Hasselblatt, B. (ed.) et al., Handbook of dynamical systems. Volume 1A. Amsterdam: North-Holland (ISBN 0-444-82669-6/hbk). 1091-1127 (2002).
Variational methods have been used very successfully to find periodic, homoclinic, or heteroclinic orbits of Hamiltonian systems. The author, a most influential contributor to this area for three
decades, surveys some of the representative problems, methods and results. He treats first-order Hamiltonian systems (HS) $J\stackrel{˙}{z}+{H}_{z}\left(t,z\right)=0$ as well as second-order systems
(HS2) $\stackrel{¨}{q}+{V}_{q}\left(t,q\right)=0$, both autonomous or nonautonomous. In the nonautonomous case it is required that $H,V$ depend periodically on $t$.
The paper consists of two parts: Part 1 is concerned with periodic solutions, Part 2 with homoclinic and heteroclinic orbits. After formulating without details a technical framework for periodic
solutions, the following topics are discussed: superquadratic autonomous Hamiltonian systems, fixed energy results, brake orbits, time dependent superquadratic fixed period problems, perturbations
from symmetry, subquadratic Hamiltonian systems, asymptotically quadratic Hamiltonians, singular potentials. Part 2 starts with the variational formulation for homoclinics to 0, and contains some
results for homoclinics, basic heteroclinic results, multibump solutions in the time dependent case, and multibump solutions in the autonomous case.
The author states a number of selected theorems precisely and gives sometimes ideas of proofs or of essential ingredients of the proofs. Other results are discussed informally. For all results
mentioned, references to the literature are given.
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
34C25 Periodic solutions of ODE
34C37 Homoclinic and heteroclinic solutions of ODE
58E05 Abstract critical point theory
37-02 Research exposition (Dynamical systems and ergodic theory) | {"url":"http://zbmath.org/?q=an:1048.37055","timestamp":"2014-04-20T08:57:40Z","content_type":null,"content_length":"22780","record_id":"<urn:uuid:4727886a-c56f-4dfe-9d84-a0726f5098e5>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00395-ip-10-147-4-33.ec2.internal.warc.gz"} |
MATHEMATICA BOHEMICA, Vol. 122, No. 3, pp. 249-255 (1997)
On $r$-extendability of the hypercube $Q_n$
Nirmala B. Limaye, Dinesh G. Sarvate
Nirmala B. Limaye, Department of Mathematics, University of Mumbai, India; Dinesh G. Sarvate, Department of Mathematics, University of Charleston, S. C., U.S.A
Abstract: A graph having a perfect matching is called $r$-extendable if every matching of size $r$ can be extended to a perfect matching. It is proved that in the hypercube $Q_n$, a matching $S$ with
$ |S|\leq n$ can be extended to a perfect matching if and only if it does not saturate the neighbourhood of any unsaturated vertex. In particular, $Q_n$ is $r$-extendable for every $r$ with $1\leq r\
leq n-1.$
Keywords: 1-factor, $r$-extendability, hypercube
Classification (MSC2000): 05C70
Full text of the article:
[Previous Article] [Next Article] [Contents of this Number] [Journals Homepage] © 1999--2000 ELibM for the EMIS Electronic Edition | {"url":"http://www.kurims.kyoto-u.ac.jp/EMIS/journals/MB/122.3/4.html","timestamp":"2014-04-18T16:14:11Z","content_type":null,"content_length":"1978","record_id":"<urn:uuid:4a10ea12-0bf1-4991-bb12-5e94b6ebf24b>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00268-ip-10-147-4-33.ec2.internal.warc.gz"} |
Essen1's NY-style pizza project
What I am talking about is to just use a poolish leavened with commercial yeast (IDY), not a combination of IDY and a natural starter/preferment. I think I should be able to come up with a dough
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the numbers. | {"url":"http://www.pizzamaking.com/forum/index.php/topic,8093.msg83503.html","timestamp":"2014-04-19T05:48:38Z","content_type":null,"content_length":"105175","record_id":"<urn:uuid:7d4540d7-a633-4e07-8896-ac692fce7e1f>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00520-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Vonmises Stress Vs Maximum Principle Stress
Hi all,
with regards to the max. principal stress vs von mises stress, does a material yield or fail if the von mises is greater than the max. principal stress or visa versa?
secondly, how is the von mises related to compressive strength (or minimun principal stress)?
your help will be greatly appreciated
The beauty of Von Mises stress is that in the real world "everything" fails by shear. That's why it has emerged as the favorite failure theory. Having said that, the world of material failure is
highly stochastic - subject to statistical variation. So as good as the theory is, you still need significant factors of safety if you don't want your project to come crashing down.
You find Von Mises stress from the principle stresses by using a big ol gnarly equation or three. It is always a smaller value than maximum principle stress (by definition) BUT it is aligned in the
direction that has to support the maximum shear load. This can be very helpful in design.
hth (and i hope that I'm not too rusty on this subject) | {"url":"http://www.physicsforums.com/showthread.php?t=206854","timestamp":"2014-04-16T16:12:00Z","content_type":null,"content_length":"39397","record_id":"<urn:uuid:7a9613f1-7fa7-4361-85c1-8b9ab8fe9b14>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00275-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Reaction Quotient
The reaction quotient (Q) measures the relative amount of products and reactants present during a reaction at a particular point in time. The reaction quotient aids in figuring out which direction a
reaction is likely to proceed, given either the pressures or the concentrations of the reactants and the products. The Q value can be compared to the Equilibrium Constant, K, to determine the
direction of the reaction that is taking place.
K vs. Q
The main difference between K and \(Q\) is that K describes a reaction that is presently in equilibrium, while Q describes a reaction that is not at equilibrium. To determine \(Q\), you must know the
concentrations of the reactants and products and this general equation:
\(Q_c = \dfrac{concentration \; of \; Products}{concentration \; of \; Reactants}\)
The first step is to write the \(Q\) equation based off of a general template. We are given this equation:
\(aA + bB \Longleftrightarrow cC + dD\)
We write the \(Q\) equation by multiplying the concentrations of the products and divide it by the concentrations of the reactants. If any component in the reaction has a coefficient, shown here with
lower case letters, we raise the concentration to the power of the coefficient. The completed \(Q\) equation is
\(Q_c = \dfrac{[C]^c[D]^d}{[A]^a[B]^b}\)
*Note: This equation only uses components in the gaseous or aqueous states. All pure liquids and solids have an activity of one and can be omitted. Equilibrium constants really contain a ratio of
concentrations (actual concentration divided by the reference concentration that defines the standard state). Since the standard state for concentrations is usually chosen to be 1 mol/L, it is not
written out in practical applications. Hence, the ratio of them does not contain units.
Based on how \(Q\) compares to K, we can tell which way the reaction will shift and which side of the reaction is favored.
• If Q>K, the reaction favors the reactants. This means that in the \(Q\) equation, the numerator, or the concentration or pressure of the products, is greater than the denominator, which is the
concentration or pressure of the reactants. Since reactions always try to reach equilibrium (Le Châtelier's Principle), the reaction will produce more reactants from the excess products,
therefore causing the system to shift to the LEFT. This allows the system to reach equilibrium.
• If Q<K, the reaction favors the products. This would mean that the denominator is greater that the numerator, giving us a small \(Q\) value. This translates to the concentration or the pressure
of the reactants is greater than the products. Since the reaction will want to reach equilibrium, the system will shift to the RIGHT to make more products.
• If Q=K, then the reaction is already at equilibrium. This is because the products are equal to the reactants and when placed in the \(Q\) equation, the answer is one. No side is favored and no
shift occurs.
• Q<K: Out of equilibrium: Reaction favors reactants and shifts Right.
• Q>K: Out of equilibrium: Reaction favors Products and shifts Left.
• Q=K: In equilibrium: No side favored and no shift.
Another important concept that is used in the calculation of the reaction quotient is called an activity. For example, write the \(Q\) equation for this acid/base reaction:
\(CH_3CH_2CO_2H{(aq)} + H_2O{(l)} \leftrightharpoons H_3O^+{(aq)} + CH_3CH_2CO_2^-{(aq)}\)
The Q equation is written as the concentrations of the products divided by the concentrations of the products, but only including components in the gaseous or aqueous states. We leave out pure liquid
or solid states. The Q equation for this example would be
\(Q_c = \dfrac{[H_3O^+{(aq)}][CH_3CH_2CO_2^-{(aq)}]}{[CH_3CH_2CO_2H{(aq)}]}\)
Example 1
What is the Q value for this equation? Which direction will the reaction shift?
Given: \(CO(g) + H_2O(g) \rightleftharpoons CO_2(g) + H_2(g)\)
\(K_c\) = 1.0
[CO[2](g)]= 2.0 M
[H[2](g)]= 2.0 M
[CO(g)]= 1.0 M
[H[2]O(g)]= 1.0 M
Step 1: Write the Q formula
\(Q_c = \dfrac{[CO_2][H_2]}{[CO][H_2O]}\)
Step 2: Plug in given concentration values.
\(Q_c = \dfrac{(2.0)(2.0)}{(1.0)(1.0)}\)
Q= 4.0
Step 3: Compare Q to K.
Since 4.0 is greater than 1.0, Q > K. This means that the reaction shifts left.
Answer: Q= 4.0 and shifts left.
Example 2
Find the value of \(Q\) and determine which side of the reaction is favored.
Given K=0.5
\(HCl(g) + NaOH(aq) \rightleftharpoons NaCl(aq) + H_2O(l)\)
[HCl]= 3.2
[NaOH]= 4.3
Step 1: Write \(Q\) formula. Since we know that the activity of a liquid is 1, we can omit the water component in the equation.
\(Q_c = \dfrac{[NaCl{(aq)}]}{[HCl{(g)}][NaOH{(aq)}]}\)
Step 2: Plug in given concentrations into the \(Q\) formula.
\(Q_c = \dfrac{[6]}{[3.2][4.3]}\)
Step 3: Calculate using the given concentrations
Q = 0.436
Step 4: Compare \(Q\) to K. Now that \(Q\) is found to be 0.436, we can compare it to the given K value of 0.5, so \(Q\) is less than K.
Since Q < K, the reaction is not at equilibrium and will proceed to the products side to reach dynamic equilibrium once again.
Answer: Q= 0.436 and the reaction favors the products.
Example 3
Given the equation, \(N_2(g) + 3H_2(aq) \rightleftharpoons 2NH_3(g)\) find Q and determine which direction the reaction will shift in order to reach equilibrium.
Given: \(N_2(g) + 3H_2(aq) \rightleftharpoons 2NH_3(g)\)
[N[2]]= 0.04M
[H[2]]= 0.09M
K= 0.040
Step 1: Write the Q formula
\(Q_c = \dfrac{[NH_3{(g)}]^2}{[N_2{(g)}][H_2{(g)}]^3}\)
Step 2: Plug in Values. Since the concentrations for N[2] and H[2] were given, we can plug those right in to the equation. However, no concentration value was given for NH[3] so we assume that there
is no concentration, which is denoted with a zero.
\(Q_c = \dfrac{(0)^2}{(.04)(.09)^3}\)
Step 3: Solve for Q.
Step 4: Compare Q to K. Since K=0.04 and Q is zero, K is greater than Q, meaning that the reaction will shift right in order to recreate equilibrium.
Answer: Q=0, shifts right. | {"url":"http://chemwiki.ucdavis.edu/Physical_Chemistry/Equilibria/Chemical_Equilibria/The_Reaction_Quotient","timestamp":"2014-04-17T04:16:06Z","content_type":null,"content_length":"52632","record_id":"<urn:uuid:89387c51-b9f6-4093-b3ae-35df55625be9>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00565-ip-10-147-4-33.ec2.internal.warc.gz"} |
Specification Hours
December 15th 2008, 09:46 PM #1
MHF Contributor
Jul 2008
Specification Hours
Random sample of 120 customers’s spent an average of 9.2 hours on their professional job with a sample standard deviation of 5.1 hours. Calculate the specification hours with confidence of 95%.
I have not heard of specification hours. It looks like you are supposed to calculate the confidence interval with 95% confidence level.
The Central Limit Theorem (which the general rule of thumb says is ok to invoke for $N\geq 30$) tells us that $\frac{\bar{X}-\mu}{S \sqrt{N}}$ is approximately normally distributed with mean 0
and std dev 1.
So then the confidence interval is given by $\bar{X}\pm z_{\alpha/2}S/\sqrt{N}$ where $z_{\alpha/2}$ is defined by $\mathrm{P}(-z_{\alpha/2}\leq Z \leq z_{\alpha/2})=1-\alpha$ (Z is from the
standard normal distribution). I'm assuming that you have already seen where these formulas came from.
So from a standard normal distribution table $z_{\alpha/2}=1.96$. So just plug away.
I have not heard of specification hours. It looks like you are supposed to calculate the confidence interval with 95% confidence level.
The Central Limit Theorem (which the general rule of thumb says is ok to invoke for $N\geq 30$) tells us that $\frac{\bar{X}-\mu}{S \sqrt{N}}$ is approximately normally distributed with mean 0
and std dev 1.
So then the confidence interval is given by $\bar{X}\pm z_{\alpha/2}S/\sqrt{N}$ where $z_{\alpha/2}$ is defined by $\mathrm{P}(-z_{\alpha/2}\leq Z \leq z_{\alpha/2})=1-\alpha$ (Z is from the
standard normal distribution). I'm assuming that you have already seen where these formulas came from.
So from a standard normal distribution table $z_{\alpha/2}=1.96$. So just plug away.
I thank you for your time and effort.
December 17th 2008, 03:02 PM #2
Jul 2008
December 21st 2008, 11:53 AM #3
MHF Contributor
Jul 2008 | {"url":"http://mathhelpforum.com/statistics/65200-specification-hours.html","timestamp":"2014-04-17T11:39:22Z","content_type":null,"content_length":"37131","record_id":"<urn:uuid:e03ca218-6388-4dd4-bd42-5287c678d39e>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00421-ip-10-147-4-33.ec2.internal.warc.gz"} |
Placement Exams
Placement Exams
Placement Exams in Mathematics and Chemistry are required for some students and help determine what level of mathematics or chemistry incoming students are prepared to take. Students should review
the information in each exam section below to help determine whether they need to take a Placement Exam. Placement exams are offered throughout the summer and during Orientation Week in the fall and
spring semesters.
Please note that each placement exam may only be taken once.
To view the Placement Exam schedule, or to register, click here.
Important Notes
• Calculators are not permitted.
• Students will receive their Placement Exam scores immediately following the completion of the exams, and are asked to wait for results. Students, however, will not be provided with a corrected
When to Take a Placement Exam
Students are encouraged to sit for a Placement Exam prior the Postbac Planning Session and meeting with their advisor, as the results of the Placement Exams will be important for discussing a course
Students who are still not sure whether they need to take a Placement Exam after reviewing the information below should consult with their advisor after attending a Postbac Planning Session.
Testing Accommodations for Students with Disabilities
In some cases, students with disabilities are entitled to accommodations related to administration of examinations, including placement exams. Such accommodations must be specified as well as
authorized by the Columbia University Office of Disability Services, based on an assessment of the disability and appropriate documentation. This authorization process can often take a month, once
all required documentation is received.
Students with learning disabilities or other disabilities that may warrant an accommodation should contact the Office of Disability Services immediately upon receipt of the GS Acceptance Packet.
Exams Administered by the Office of Disability Services
Students who take a Placement Exam through the Office of Disability Services will receive their scores within 24-48 hours via email.
Mathematics Exams
The Physics and Chemistry Departments at Columbia expect that students enrolling in General Chemistry or Physics have a working knowledge of Calculus, so students cannot enroll in these courses until
they have completed, or are prepared to take as a co-requisite, Calculus I. The Math Placement Exam consists of two parts, and helps determine if a student is ready for Calculus or if Pre-Calculus
course work is required.
Students are required to take the Mathematics Placement Exam if they:
• have not taken Calculus I during college.
• received a B- or lower in Calculus I during college.
• completed Advanced Placement Calculus in high school, but took no further Calculus coursework in college.
Students are not required to take the Mathematics Placement Exam if they:
• took Calculus I in the last five years during college and received a B or better.
• took specific mathematics courses that will transfer to Columbia. Students will need to discuss transfer credits with their advisor during the first advising appointment after the Postbac
Planning Session.
Math Part I
Passing Math Part I will place you into Pre-Calculus. You may take Part II only if you pass Part I. Students who pass the Mathematics Exam Part I must take Part II if they would like to enroll in
Calculus I.
Timing, Composition, and Scoring
• 45 minutes
• 20 multiple-choice problems
• 16/20 passing score
Topics and Preparation
The following topic are covered on Math Part I. Students should review a high school textbook covering these subjects to review for Math Part I. Calculators are not permitted.
• Order of Operations
• Manipulating Fractions
• Distributive Law
• Square Root
• Linear Equations in One Variable
• Factoring and Multiplying Algebraic Expressions
• Coordinates and Graphs (two variables)
• Inequalities
• Absolute Value
• Algebraic Fractions
• Properties of Straight Lines
• Functions
• Quadratic Equations
Math Part II
Passing Math Part II will place you into Calculus I. You may take Part II only if you pass Part I. We do not offer placement exams to determine placement in courses higher than Calculus I. Please
consult with your academic advisor for additional information on this matter.
Timing, Composition, and Scoring
• 40 minutes
• 12 multiple-choice problems
• 9/12 passing score
Topics and Preparation
The following topics are covered on Math Part II. Students should review a high school textbook covering these subject to review for Math Part II. Calculators are not permitted.
• Functions
• Trigonometric Identity
• Logarithms
• Graphing Functions
• Inequalities
• Distance
Exam Results
• Students who do not pass the Mathematics Exam Part I must take a course in Basic Mathematics [MATH S0065], which is offered only during the summer session.
• Students who pass the Mathematics Exam Part I, but not Part II will be required to take College Algebra and Analytic Geometry [MATH W1003], which is offered during the fall, spring and summer
• Students who pass both Part I and Part II of the Mathematics Placement Exam can enroll in Calculus I [MATH V1101].
Chemistry Exam
The Chemistry Placement Exam will test basic knowledge of chemistry.
Students are required to take the Chemistry Placement Exam if:
• the last time they took Chemistry was in high school*
• they completed Advanced Placement Chemistry in high school, but took no further Chemistry coursework in college.*
*Students are not required to take the Chemistry Placement Exam if they:
• plan to begin their Chemistry coursework by enrolling in Preparation for College Chemistry [CHEM W0001] or [CHEM S0001]; this course is offered only in the spring and summer terms.
• completed General Chemistry I with a C or better in college.
Timing, Composition, and Scoring
• 60 minutes
• 44 multiple-choice questions.
• 23/44 passing score.
• A periodic table is provided, but calculators are not permitted.
Topics and Preparation
The following topics are covered on the Chemistry Placement Exam. Students should review a high school chemistry book to prepare for the exam.
• Acids and bases
• Atomic and nuclear structure
• Balancing chemical equations
• Chemical bonding and its relation to molecular structure
• Chemical reactions
• Chemistry nomenclature
• Equilibria
• Gas laws
• Kinetics
• Periodic properties of elements
• Phase changes
• Stoichometry
• Thermodynamics
Exam Results
• Students who pass the Chemistry Placement Exam with a 23 or above may enroll in General Chemistry I [CHEM C1403/CHEM W1403].
• Students who receive a 20-22 on the Chemistry Placement Exam are encouraged to take Preparation for College Chemistry in the fall [CHEM W0001] or summer [CHEM S0001], as they may have difficulty
with General Chemistry I [CHEM C1403/CHEM W1403]. Students should speak with their advisor to determine a plan during their initial advising appointment.
• Students who receive a 19 or below on the Chemistry Placement Exam will be required to take Preparation for College Chemistry in the fall [CHEM W0001] or summer [CHEM S0001].
Advising Appointments and Course Registration
During the first term of enrollment, advisors complete course registration for students after their initial advising session. Students will discuss their Placement Exam results and arrive at a
decision regarding the appropriate course registration.
Advisors will register students for the appropriate courses after their appointment; students will not register themselves for their first semester. Note: Faculty members do not make determinations
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New Math LyricsSome of you who have small children may have perhaps been put in the embarrassing position of being unable to do your child's arithmetic homework because of the current revolution in
Hing known as the new math. so as a public service here tonight I thought I would offer a brief lesson in the new math. tonight we're going to cover subtraction. this is the first room I've work
R a while that didn't have a blackboard so we will have to make due with more primitive visual aids, as they say in the "ad biz." consider the following subtraction problem, which I wi
T up here: 342 - 173.
Now remember how we used to do that. three from two is nine; carry the one, and if you're under 35 or went to a private school you say seven from three is six, but if you're over 35 and went to
Lic school you say eight from four is six; carry the one so we have 169, but in the new approach, as you know, the important thing is to understand what you're doing rather than to get the right
Er. here's how they do it now.
You can't take three from two,
Two is less than three,
So you look at the four in the tens place.
Now that's really four tens,
So you make it three tens,
Regroup, and you change a ten to ten ones,
And you add them to the two and get twelve,
And you take away three, that's nine.
Is that clear?
Now instead of four in the tens place
You've got three,
'cause you added one,
That is to say, ten, to the two,
But you can't take seven from three,
So you look in the hundreds place.
From the three you then use one
To make ten ones...
(and you know why four plus minus one
Plus ten is fourteen minus one?
'cause addition is commutative, right.)
And so you have thirteen tens,
And you take away seven,
And that leaves five...
Well, six actually.
But the idea is the important thing.
Now go back to the hundreds place,
And you're left with two.
And you take away one from two,
And that leaves...?
Everybody get one?
Not bad for the first day!
Hooray for new math,
It won't do you a bit of good to review math.
It's so simple,
So very simple,
That only a child can do it!
Now that actually is not the answer that I had in mind, because the book that I got this problem out of wants you to do it in base eight. but don't panic. base eight is just like base ten really
You're missing two fingers. shall we have a go at it? hang on.
You can't take three from two,
Two is less than three,
So you look at the four in the eights place.
Now that's really four eights,
So you make it three eights,
Regroup, and you change an eight to eight ones,
And you add them to the two,
And you get one-two base eight,
Which is ten base ten,
And you take away three, that's seven.
Now instead of four in the eights place
You've got three,
'cause you added one,
That is to say, eight, to the two,
But you can't take seven from three,
So you look at the sixty-fours.
"sixty-four? how did sixty-four get into it? " I hear you cry.
Well, sixty-four is eight squared, don't you see?
(well, you ask a silly question, and you get a silly answer.)
From the three you then use one
To make eight ones,
And you add those ones to the three,
And you get one-three base eight,
Or, in other words,
In base ten you have eleven,
And you take away seven,
And seven from eleven is four.
Now go back to the sixty-fours,
And you're left with two,
And you take away one from two,
And that leaves...?
Now, let's not always see the same hands.
One, that's right!
Whoever got one can stay after the show and clean the erasers.
Hooray for new math,
It won't do you a bit of good to review math.
It's so simple,
So very simple,
That only a child can do it!
Come back tomorrow night. we're gonna do fractions.
Now I've often thought I'd like to write a mathematics text book because I have a title that I know will sell a million copies. I'll call it tropic of calculus.
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