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Pure Math, Pure Joy
1422555 story
Posted by
from the msri-loves-company dept.
e271828 writes
"The New York Times is carrying a nice little piece entitled Pure Math, Pure Joy about the beauty and applicability of pure math as carried out at the Mathematical Sciences Research Institute. There
is an accompanying slideshow of pictures of mathematicians in action; I particularly loved the picture titled Waging Mental Battle with a Proof."
This discussion has been archived. No new comments can be posted.
Pure Math, Pure Joy
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• Karma Whoring, Reg sucks (Score:1, Informative)
by jonman_d (465049) <nemilarNO@SPAMoptonline.net> on Sunday June 29, 2003 @02:44PM (#6325918) Homepage Journal
Pure Math, Pure Joy
By DENNIS OVERBYE
A mathematician, the Hungarian lover of numbers Paul Erdos once said, is a device for converting coffee into theorems. Here, then, are a few glimpses into the Truth Factory. The Mathematical
Sciences Research Institute, sustained mostly by the National Science Foundation, sits on a hill above the University of California at Berkeley, where it attracts people from around the world for
stints of up to a year to lose themselves in subjects like algebraic geometry or special holonomy.
Consider it an embassy of another world, a Platonic realm of clarity and beauty, of forms and relations, where the answers to questions not yet asked already exist.
Higher mathematics -- as opposed to what we do every April 15 -- has been relevant ever since Archimedes leaped out of his bath shouting "Eureka!" more than 2,000 years ago. Nobody knows when
some abstruse bit of math will float off a blackboard at a place like this and become -- often decades later -- a key tool in cryptography, biology, physics or economics (as in "A Beautiful
Take string theory, a mathematically labyrinthine effort to construct a so-called theory of everything out of the notion that the fundamental elements of nature are tiny strings flopping and
wriggling in an 11-dimensional space-time. It has been called a piece of 21st-century physics that fell by accident into the 20th century.
In their quest to negotiate this labyrinth, string theorists have made a hot topic of something called Riemann surfaces, invented by the German mathematician Georg Friedrich Bernhard Riemann 150
years ago, but they have also helped blaze new fields of mathematics.
"Since our theories are so far ahead of experimental capabilities, we are forced to use mathematics as our eyes," Dr. Brian Greene, a Columbia University string theorist, said recently. "That's
why we follow it where it takes us even if we can't see where we're going."
So in some ways the men and women seen here scrutinizing marks on their blackboards collectively represent a kind of particle accelerator of the mind.
But the "unreasonable effectiveness" of mathematics in explaining the world, as the physicist Eugene Wigner once put it, is a minor motivation at best for those immersed in the field. Most
mathematicians say they are in it for the math itself, for the delirious quest for patterns, the thrill of the detective chase and the lure of beautiful answers.
"Math is sense," said Dr. Robert Osserman, a Stanford professor and deputy director of the institute, quoting from the play "Copenhagen." "That's what sense is."
• You can trust the NYT (Score:3, Informative)
by CausticWindow (632215) on Sunday June 29, 2003 @03:28PM (#6326122)
I work in the maths department of a University, and yes.. it's very much like this. We sit around all day in small groups, staring at blackboards, "battling with proofs". Just like in that
wonderful movie with the violent australian, "A Beautiful Mind".
• misery loves company (Score:3, Informative)
by chloroquine (642737) on Sunday June 29, 2003 @04:29PM (#6326405) Journal
So, I just wanted to poke my head in here and note that MSRI (where the pictures are taken) is pronounced "misery" by the maths community.
My (insert close relative here) does minimal surfaces and hangs out with some of these guys. They look far too neatly dressed in the pictures. Anyway, for a good time, you might want to take a
look at some of the galleries of images that these crazy minimal surfaces guys do. I remember about ten years ago, one of my (insert close relative)'s colleagues sold a few images to the Grateful
Dead for their concerts.
http://www.msri.org/publications/sgp/jim/images/ [msri.org]
http://www.gang.umass.edu/ [umass.edu]
There is another site out at Minnesota but I'm too lazy to look for it today.
• Re:Is this really true? (Score:2, Informative)
by elizalovesmike (626844) on Sunday June 29, 2003 @04:34PM (#6326429)
"the beauty of this is that it is absolutely useless to anybody"
You're screwin' up the causal relationships again.
Pure math isn't a thing of beauty because discoveries yielded by it may have no *immediate* practicable value; nor is it a thing of beauty because it may be sourced in something other than a
desire to solve an immediate problem.
It's a thing of beauty because it has produced fascinating finds with respect to the relationships between various prime numbers and relatively prime numbers (Euler's Totient function). Modular
exponentiation is fascinating--how this works with primes (i.e. 3^1 mod 7 = 3; 3^2 mod 7 = 2; 3^3 mod 7 = 6; 3^4 mod 7 = 4; 3^5 mod 7 = 5; 3^6 mod 7 = 1; 3^7 mod 7 = 3 and it all REPEATs) -- so
is fast exponentiation, exponential inverses, modular inverses, Fermat's little theory etc.
That some of these finds combine to yield one-way (trapdoor) functions that can take advantage of the inability (for now!) to factor large numbers and provide a secure pub key system is a bonus
of monumental importance. And one that was only just recently (past 30 years or so) realized.
You can never know if a thing will be useful or not without understanding the essence of that thing; and there again "useful" is clearly a time-limited function... As one cannot perfectly predict
future needs and future landscapes, one cannot perfectly determine at any one point of time whether current work in number theory will be with or without practical value. Though what's so wrong
with discovery for discovery's sake! Isn't that part of the reason we are here?
• Some tests are public (Score:2, Informative)
by paugq (443696) <pgquiles@elpaTWAINuer.org minus author> on Sunday June 29, 2003 @05:50PM (#6326774) Homepage
You certainly don't know what you are talking about. Some tests are public and some even free.
For instance, here [mensa.es] (Mensa Spain) [mensa.es] you have a test publicly available.
And there are some books also publicly available sold as Mensa preparatory test books.
And that's not all, they sent me home a test (which I never filled), with solutions.
So, who is the liar?
• Funny... (Score:3, Informative)
by biostatman (105993) on Sunday June 29, 2003 @08:14PM (#6327490)
The title of the article is "Pure Math, Pure Joy" and it's about MSRI. While it is a phenomenal place, it is no picnic for young mathematicians for sure and is often referred to as "misery", as
in "yeah, I spent a year in misery (MSRI)".
• Re:Funny... (Score:2, Informative)
by D. J. Bernstein (313967) on Monday June 30, 2003 @02:41AM (#6328923)
Speaking as a mathematician who was around MSRI 1991-1995, 2000, and 2002-2003: We say ``misery'' because that's the easiest way to pronounce MSRI, not to express any negative sentiments towards
the place. When Bill Thurston took over as director in the early 1990s, he tried to get everyone to switch to a French-style ``emissary,'' but that word just isn't as easy to say as ``misery.''
Related Links Top of the: day, week, month. | {"url":"http://science.slashdot.org/story/03/06/29/183258/pure-math-pure-joy/informative-comments","timestamp":"2014-04-20T13:52:58Z","content_type":null,"content_length":"90886","record_id":"<urn:uuid:a32f34dc-ec65-4ab4-874f-26abd236a161>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00342-ip-10-147-4-33.ec2.internal.warc.gz"} |
Java - filling an overlapping polygon
up vote 1 down vote favorite
I'm trying to draw a 5 point star in AWT. Each point in the 2d grid is 72 degrees apart - so I thought I could draw the polygon using only 5 points by ordering the points 144 degrees apart, so the
polygon gets fed the points in order 1,3,5,2,4
Unfortunately, this involves a lot of intersecting lines, and the end result is that there are 5 triangles with my desired colour, circling a pentagon that has not been coloured.
Looking through, it has something to do with an even-odd rule, that intersecting points will not be filled.
I need my star to be drawn dynamically, and using the specific shape described (for scaling and such). If I manually plot the points where it intersects, I get some human error in my star's shape.
Is there any way to just turn this feature off, or failing that, is there a way to get the polygon to return an array of x[] and y[] where lines intersect so I can just draw another one inside it?
java awt polygon
add comment
2 Answers
active oldest votes
Draw it with ten points, 36 degrees apart, at two alternating radii.
up vote 0
down vote
I mentioned that in my original post - If I manually plot the points where it intersects, I get some human error in my star's shape. The second set of points seem to have a radius of
approximately 1/3 the radius of the peaks, but I don't know exactly how much it is or I'd have done this. – Brian Oct 16 '10 at 17:58
add comment
Establish the 10-point Polygon in cartesian coordinates, as suggested by relet and as shown in this example. Note how the coordinate system is centered on the origin for convenience in
rotating, scaling and translating. Because Polygon implements the Shape interface, the createTransformedShape() method of AffineTransform may be applied. A more advanced shape library may
be found here.
up vote 0 Is there a way to get the polygon to return an array of x[] and y[] where lines intersect?
down vote
Although typically unnecessary, you can examine the component coordinates using a Shape's PathIterator. I found it instructive to examine the coordinates before and after invoking
add comment
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How to tell gear pressure angles 20* vs 14 1/2*
02-16-2008, 11:26 PM #1
Join Date
Dec 2004
Thermopolis, Wyo
Bought a coil winder but need some different change gears to wind the size wire I want.
Question- how do I determine the pressure angle of the existing gears?? The catalogs warn that the pressure angles are not compatable.
Thanks, John
There are some brass geartooth gauges which are available for a small fee. These come as a bunch of different diametral pitches AND pressure angles. I even had one that was plastic at one time.
Reid Tool Supply has one (steel) at $88.00 ! ! ! You ought to be able to find a cheaper version somewhere!.
Jack C
A gauge sounds like the way to go! Does anyone have a spare they would be willing to part with?? I only need one for 32 pitch. Willing to pay of course.
TIA, John
Take the gear to your local bearing supplier they should have a set of guages to check the pressure angle.
I copied this tip from an earlier thread on the same subject. Sorry but I can't credit the original poster, I didn't copy his name!
as long as you know the 'DP' of the gear the following will give you a good guide to the 'PA'. take a measurement over any number of teeth with whatever equipment you have,multiply the cosine of
the 'PA' by 3.1416 and the divide by the 'DP' add this to your measurement, this should be the measurement over one more tooth if its not the same, change the cosine of the 'PA' and try again.
example 10dp 14.5pa 30 teeth, the meas over 3 teeth .776,cos.986147 multiplied by 3.1416 and divided by 10 = .304 add .776=1.080 over 4 teeth.hope this helps
20 degree pressure angle gears are much more "pyramid" shaped than 14.5's. They are very easily recognised once you take a look at the 2 types.
Thanks Mudflap for reposting that handy tip
Like Mr. Wilson says - 20° is more "pointy" and 14 1/2° is more square ended - same in the root.
INTERESTING- Jess
as long as you know the 'DP' of the gear the following will give you a good guide to the 'PA'. take a measurement over any number of teeth with whatever equipment you have,multiply the cosine of
the 'PA' by 3.1416 and the divide by the 'DP' add this to your measurement, this should be the measurement over one more tooth if its not the same, change the cosine of the 'PA' and try again.
example 10dp 14.5pa 30 teeth, the meas over 3 teeth .776,cos.986147 multiplied by 3.1416 and divided by 10 = .304 add .776=1.080 over 4 teeth.hope this helps
What does 'take a measurement' mean? Over pins, across the chords, at what depth?
What does 'take a measurement' mean? Over pins, across the chords, at what depth?
I took it to be a span over teeth measurement. Just use a dial caliper.
Like Mr. Wilson says - 20° is more "pointy" and 14 1/2° is more square ended - same in the root.
Here is a link to a web page with a diagram:
Get a worm of modelling clay on the bench and roll a gear along it, cut through the impression (along the length of the worm) and look at the tooth angle of the rack you have made in the clay, it
is easy to measure the angle, send the $88 to me!
02-17-2008, 11:57 AM #2
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Tools for large graph mining: structure and diffusion
Tools for large graph mining: structure and diffusion.
WWW 2008 tutorial, Beijing, China.
Jure Leskovec and Christos Faloutsos, Carnegie Mellon University
How do graphs look like? How do they evolve over time? How can we find patterns, anomalies and regularities in them? How to find influential nodes in the network? We will present both theoretical
results and algorithms as well as case studies on several real applications. Our emphasis is on the intuition behind each method, and on guidelines for the practitioner.
The tutorial has four parts: (a) Statistical properties and models and graph generators of static and evolving netoworks. (b) Diffusion and cascading behavior in networks, where a virus or
information spreads through the network. We present empirical observations, models and algorithms to find influential blogers or nodes to target for viral marketing. (c) Tools for the analysis of
static and dynamic graphs, like the Singular Value Decomposition, tensor decomposition for community detection, detecting anomalous nodes, and analyzing time evolving networks. (d) Case studies of
communication patterns of MSN Instant Messenger, how to find fraudsters on eBay, how to predict quality of web search results from the properties of webgraph, internet traffic analysis, and virus
propagation results.
Tutorial outline
• Part 1: Properties, models and tools to mine the structure of large networks [1.5 hours]
□ Statistical properties of static and evolving networks.
☆ Power law degree distributions found in static networks
☆ Small world phenomena and six degrees of separation
☆ Communities and clusters in networks
☆ Densification of time evolving networks
☆ Shrinking diameters of growing networks
☆ Evolution and link predictions in social networks. Where will the next edge appear in the evolution of the network?
□ Models and generators. We motivate each model by the properties of networks it generates, and explain the intuition behind it.
☆ Erdos-Renyi random graph model, giant components, properties and phase transitions
☆ Preferential attachment and variants (copying model)
☆ Forest Fire model for time evolving networks
☆ Kronecker graph generators and how to fit them to large real graphs so that one can generate realistic synthetic networks
• Part 2: Diffusion and influence propagation in web and social networks [1.5 hours]
□ Mathematical or virus propagation. We focus on SIS and SIR virus propagation models and on epidemic thresholds (how quickly does the virus need to spread to conquer the network?).
□ Models of diffusion in social networks: we review the independent cascade model and the threshold model. And show some very recent measurements on large product recommendation dataset that
help us validate the models.
□ Empirical studies of diffusion of information in blogs and cascading behavior in on-line viral marketing. We explain how does the network structure influence the propagation of information
and what are common topological patterns of information propagation. We also look into recommendation propagation from a large on-line retailer.
□ Algorithms for selecting influential nodes in networks (who are influential bloggers? who is good a summarizer?). Here we use submodularity to design theoretically sound algorithms to detect
information outbreaks in networks (what are the most informative blogs) and water distribution networks (where shall we position sensors to detect disease outbreaks quickly). E.g., see http:/
• Part 3: Tools for static and dynamic graphs [1.5 hours]
Focus is on powerful tools for analyzing large matrices and tensors. A matrix describes a graph, while tensor can describe a graph with nodes and edges of various types, time evolving graphs,
time series, etc. We will cover:
□ Singular Value Decomposition (SVD), HITS and Pagerank
□ Other matrix decompositions (CUR decomposition)
□ Tensors and higher-order SVD
□ Applications of random walk and spectral methods to measure proximity between nodes in networks, and to find best explanatory subgraph connecting a set of nodes.
• Part 4: Case studies [1.5 hours]
□ Communication patterns of MSN Messenger. The application of above mentioned tools and algorithms to a large network of communication on MSN Instant Messenger (30 billion conversations, 240
million people).
□ Detecting fraud on eBay. How to find fraudulent people on eBay. We present a belief propagation method that is able to find fraudulent people in large networks.
□ Monitoring social and communication networks over time -- intrusion and outlier detection. An application of tensor decomposition techniques to monitor multiple time series over time and
detect outliers and abnormal events
□ Web projections. Exploiting the structure of web graph to predict the quality of search results, user intention to reformulate queries and to find spam search results.
□ Connection subgraphs and CenterPiece subgraphs.
Who should attend
The target audience is data management, data mining and machine learning researchers and professionals who work on static or time-evolving graphs and want to know about tools and models when dealing
with large network datasets. There will be special emphasis on web, blogs and on-line social networks related topics.
About the instructors
Jure Leskovec is a PhD candidate in Machine Learning at Carnegie Mellon University graduating in summer 2008. He is also a Microsoft Research Graduate Fellow. He received the ACM KDD 2005 and ACM KDD
2007 best paper awards, he also won the ACM KDD cup competition in 2003 and the Battle of the Sensor Networks 2007 competitions. Jure holds three patents. His research interests include data mining
and machine learning on large real-world graphs and networks. He is especially interested in evolution, friendship formation, diffusion and cascading behavior in large social networks and the web.
Christos Faloutsos is a Professor at Carnegie Mellon University. He has received the Presidential Young Investigator Award by the National Science Foundation (1989), the ICDM 'research contributions'
award in 2006, and several ``best paper'' and teaching awards. He has delivered over 20 tutorials in database and data mining conferences, he has published over 160 refereed articles, one monograph,
and holds five patents. His research interests include data mining for graphs and streams, fractals, indexing methods for spatial and multimedia bases, and data base performance. | {"url":"http://cs.stanford.edu/people/jure/talks/www08tutorial/","timestamp":"2014-04-18T23:20:14Z","content_type":null,"content_length":"9307","record_id":"<urn:uuid:84f7da55-9dfc-4a18-b921-21971e13d839>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00612-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fitting The Main Effects Model
Next: Main Effects-- Internal Up: Dose-Response Analyses (DRA) Previous: Table with Marginal
The general Poisson regression model
was used to describe the joint effects of each of the explanatory variables of interest on cause-specific mortality. Maximum likelihood estimates of the parameters and LRT statistics were obtained
using widely available software [4,11]. The baseline rate 15,16] a parametric model was used to describe the baseline rates, i.e.
where 1], Chapter 4)
in which the baseline rates are assumed to be proportional to the known external standard rates
• Last Modified 3Jul97 FromeEL@ornl.gov(touches: 181313 ) | {"url":"http://www.csm.ornl.gov/~frome/ORMS/app/node12.html","timestamp":"2014-04-20T08:14:01Z","content_type":null,"content_length":"3143","record_id":"<urn:uuid:4ae82b27-2489-4a20-9d52-a21159bc49bc>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00447-ip-10-147-4-33.ec2.internal.warc.gz"} |
Emmaus Algebra 1 Tutor
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24 Subjects: including algebra 1, reading, English, GED
...I will do all I can to help students get up to speed on their math courses. I have a lot of extra materials to use if necessary. I am patient and kind and really do care about all of the
students I work with.
9 Subjects: including algebra 1, geometry, algebra 2, SAT math
...During this time I was able to cultivate my skills with working with children and learned how to make learning light and fun, but effective. In my last 11.5 years of professional employment I
have worked as a child care worker and provided advise on conflict resolution. As a bilingual social worker, I provided parenting education to young single parents.
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Models and calculations for the two-way ANOVA
The balanced 2-way factorial layout Factor A has 1, 2, ..., a levels. Factor B has 1, 2, ..., b levels. There are ab treatment combinations (or cells) in a complete factorial layout. Assume that each
treatment cell has r independent obsevations (known as replications). When each cell has the same number of replications, the design is a balanced factorial. In this case, the abrdata points {y[ijk]}
can be shown pictorially as follows:
Factor B  
1 2 ... b
1 y[111], y[112], ..., y[11r] y[121], y[122], ..., y[12r] ... y[1b1], y[1b2], ..., y[1br]
2 y[211], y[212], ..., y[21r] y[221], y[222], ..., y[22r] ... y[2b1], y[2b2], ..., y[2br]
Factor . ... .... ...
A .
a y[a11], y[a12], ..., y[a1r] y[a21], y[a22], ..., y[a2r] ... y[ab1], y[ab2], ..., y[abr]
How to obtain sums of squares for the balanced factorial layout Next, we will calculate the sums of squares needed for the ANOVA table.
• Let A[i] be the sum of all observations of level i of factor A, i = 1, ... ,a. The A[i] are the row sums.
• Let B[j] be the sum of all observations of level j of factor B, j = 1, ...,b. The B[j] are the column sums.
• Let (AB)[ij] be the sum of all observations of level i of A and level j of B. These are cell sums.
• Let r be the number of replicates in the experiment; that is: the number of times each factorial treatment combination appears in the experiment.
Then the total number of observations for each level of factor A is rb and the total number of observations for each level of factor B is raand the total number of observations for each interaction
is r.
Finally, the total number of observations n in the experiment is abr.
With the help of these expressions we arrive (omitting derivations) at
These expressions are used to calculate the ANOVA table entries for the (fixed effects) 2-way ANOVA. | {"url":"http://www.itl.nist.gov/div898/handbook/prc/section4/prc438.htm","timestamp":"2014-04-20T00:54:47Z","content_type":null,"content_length":"15598","record_id":"<urn:uuid:dadb0ec2-f645-4870-94e7-b1fa6520dc95>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00049-ip-10-147-4-33.ec2.internal.warc.gz"} |
nite time Turing machines
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- Int. J. Theoretical Phys , 2002
"... We investigate the Church–Kalmár–Kreisel–Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical
general relativity theory. We argue that (i) there are several distinguished Church–Turing-type Theses (not only ..."
Cited by 66 (8 self)
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We investigate the Church–Kalmár–Kreisel–Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general
relativity theory. We argue that (i) there are several distinguished Church–Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we
choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary,
hence certain forms of the Church– Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.) We also look at various
“obstacles ” to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the “design ” of our future
computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.
, 2002
"... In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the
key concepts in the field, tying together the disparate ideas and presenting them in a structure which al ..."
Cited by 31 (5 self)
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In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key
concepts in the field, tying together the disparate ideas and presenting them in a structure which allows comparisons of the many approaches and results. To this I add several new results and draw
out some interesting consequences of hypercomputation for several different disciplines. I begin with a succinct introduction to the classical theory of computation and its place amongst some of the
negative results of the 20 th Century. I then explain how the Church-Turing Thesis is commonly misunderstood and present new theses which better describe the possible limits on computability.
Following this, I introduce ten different hypermachines (including three of my own) and discuss in some depth the manners in which they attain their power and the physical plausibility of each
method. I then compare the powers of the different models using a device from recursion theory. Finally, I examine the implications of hypercomputation to mathematics, physics, computer science and
philosophy. Perhaps the most important of these implications is that the negative mathematical results of Gödel, Turing and Chaitin are each dependent upon the nature of physics. This both weakens
these results and provides strong links between mathematics and physics. I conclude that hypercomputation is of serious academic interest within many disciplines, opening new possibilities that were
previously ignored because of long held misconceptions about the limits of computation.
- Minds and Machines , 2002
"... Abstract. Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether
or not the decimal representation of π contains n consecutive 7s, for any n; solve the Turing-machine halti ..."
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Abstract. Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not
the decimal representation of π contains n consecutive 7s, for any n; solve the Turing-machine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically
impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary to a recent paper by
Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle’s Chinese room argument.
, 2003
"... A version of the Church-Turing Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical,
more than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing ..."
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A version of the Church-Turing Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more
than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, non-well-founded, analog, quantum, and retrocausal
computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem
would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard Church-Turing Thesis.
- Bulletin of the London Mathematical Society
"... We show that the halting times of infinite time Turing Machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This
gives a clarification of the nature of "supertasks" or infinite time computations. The proof further yields ..."
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We show that the halting times of infinite time Turing Machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This gives a
clarification of the nature of "supertasks" or infinite time computations. The proof further yields that the class of sets coded by outputs of halting computations coincides with a level of Godel's
constructible hierarchy: namely that of L where is the supremum of halting times. A number of other open questions are thereby answered. 1 Introduction: Infinite Time Turing Machines Hamkins and
Lewis in [4] give an account of the construction of these machines (first developed by Kidder and Hamkins in 1989) and develop the basic theory of this notion of computability. The reader should
refer to this paper for a clear and full account of their basic properties, from which all the results and definitions of this introduction are taken. But to summarise: an infinite time Turing
machine has...
- Applied Mathematics and Computation , 2006
"... This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. ..."
- Journal of Philosophical Logic , 2003
"... Abstract. We show that the set of ultimately true sentences in Hartry Field’s Revenge-immune solution to the semantic paradoxes is recursively isomorphic to the set of stably true sentences
obtained in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this ..."
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Abstract. We show that the set of ultimately true sentences in Hartry Field’s Revenge-immune solution to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained
in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second order number theory is needed to establish the
semantic values of sentences over the ground model of the standard natural numbers: ¢¡-Comprehension Axiom scheme) is insufficient. £-¤¦¥¨ § (second order number theory with a ©��
, 1984
"... A new model of parallel computation - a so called Parallel Turing Machine (PTM) - is proposed. It is shown that the PTM does not belong to the two machine classes suggested recently by van Emde
Boas, i.e., the PTM belongs neither to the first machine class consisting of the machines which are polyno ..."
Cited by 7 (0 self)
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A new model of parallel computation - a so called Parallel Turing Machine (PTM) - is proposed. It is shown that the PTM does not belong to the two machine classes suggested recently by van Emde Boas,
i.e., the PTM belongs neither to the first machine class consisting of the machines which are polynomial-time and linear-space equivalent to the sequential Turing Machine, nor to the second machine
class which consists of the machines which satisfy the parallel computation thesis. Further the notion of a pipelined PTM is introduced and the "period" is defied as a complexity measure suitable for
evaluating the efficiency of pipelined computations. It is shown that to within...
"... We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely !, and construct continuum many incomparable such degrees which are minimla
over 0. We show that we can apply an argument going back to that of H. Friedman to prove that the set 1-d ..."
Cited by 6 (5 self)
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We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely !, and construct continuum many incomparable such degrees which are minimla over 0.
We show that we can apply an argument going back to that of H. Friedman to prove that the set 1-degrees of certain \Sigma 1 2 -correct KP-models of the form L oe (oe ! ! L 1 ) have minimal upper
bounds. 1 Introduction Obtaining minimality results in degree theory has a long history: the methods go back to those of Spector when he constructed minimal Turing degrees, and to Gandy-Sacks, [2],
for minimal hyperdegrees. The perfect set construction is the common thread to these proofs. A further feature, which is shared, either directly or indirectly, by such arguments, is the use of a
selection principle in order to typically, directly shrink a perfect set T ` ! ! to a T 0 so that a particular function is either continuous one-to-one, or constant on the branches of T 0 . For ex... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=154819","timestamp":"2014-04-18T21:09:23Z","content_type":null,"content_length":"36455","record_id":"<urn:uuid:381d5d16-b431-4ba1-89fe-8252ef513517>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00047-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: October 2008 [00181]
[Date Index] [Thread Index] [Author Index]
Re: $Assumptions
• To: mathgroup at smc.vnet.net
• Subject: [mg92634] Re: $Assumptions
• From: Peter Breitfeld <phbrf at t-online.de>
• Date: Thu, 9 Oct 2008 06:35:00 -0400 (EDT)
• References: <gci2ia$ta$1@smc.vnet.net>
Ruth Lazkoz Sáez schrieb:
> Dear all,
> I seem to be unable to change some default options globally.
> At the top of my notebook I have
> $Assumptions = {MaxRecursion -> 30, AccuracyGoal -> 10};
> Then we I evaluate NIntegrate[Exp[-x^2], {x, 20., 40.}]
> I get 4.78196*10^-176
> But if I evaluate NIntegrate[Exp[-x^2], {x, 20., 40.}, MaxRecursion -> 30,
> AccuracyGoal -> 10]
> I get quite a different value, 1.36703*10^-177
> I had the same problem trying to change the options for fonts in plots,
> and then I had to add the options I wanted
> in every plot, I mean evaluation something like $TextStyle ={whatever}
> before doing the plots did not work.
> Hope you can help me. Thanks,
> Ruth Lazkoz
If you evaluate
NIntegrate[Exp[-x^2], {x, 20, 40}, Assumptions -> True]
you will get the error message:
NIntegrate::optx: Unknown option Assumptions in \
This means, that NIntegrate doesn't have an Assumptions option.
Setting $Assumtions to any value works only for functions with an
Assumtions optin which must default to $Assumptions.
The function Assuming works similiar to construct like this:
so using Assuming won't work with NItegrate either.
The following function, which I found in this group, lists all
functions in the System-context, which have an Assumptions option:
hatOption[option_] :=
ToExpression /@
MemberQ[ToExpression[#, StandardForm,
Options[Unevaluated[#]] &][[All, 1]], option] &]];
hA = hatOption[Assumptions]
The only function of these, which has not Assumptions->$Assumptions
as default option is PowerExpand. It's Assumptions Option defaults
to Automatic. So PowerExpand too will not reacht on setting
$Assumptions or using Assuming.
Gruss Peter
Peter Breitfeld, Bad Saulgau, Germany -- http://www.pBreitfeld.de | {"url":"http://forums.wolfram.com/mathgroup/archive/2008/Oct/msg00181.html","timestamp":"2014-04-17T13:12:49Z","content_type":null,"content_length":"27184","record_id":"<urn:uuid:80819c44-781f-439b-90b2-f0f54b03a202>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00544-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Homework Help
Posted by Jay on Wednesday, July 11, 2012 at 3:30pm.
A 100-g block hangs from a spring with k = 5.3 N/m. At t = 0 s, the block is 20.0 cm below the equilibrium position and moving upward with a speed of 194 cm/s. What is the block's speed when the
displacement from equilibrium is 31.5 cm?
• Physics - Elena, Wednesday, July 11, 2012 at 3:52pm
ω=sqrt(k/m)=sqrt(4.6/0.1) = 6.76rad/s.
Divide the first equation by the second and obtain
x/v= A•sinωt/ A•ω•cosωt= ω•tanωt,
tanωt = x•ω/v=20•6.76/194=0.7
ωt =arctanωt =0.61 rad,
sinωt = 0.571
A=x/sinωt = 20/0.571=35 cm.
For the second case:
sinωt1 =x1/A=31.5/35 =0.9.
cos ωt1=sqrt(1-sin²ωt1)=0.44.
v1= =A•ω•cosωt1=
=35•6.76•0.44=104 cm/s.
• Physics - Jay, Wednesday, July 11, 2012 at 4:04pm
K is 5.3 N/m. Why are you using 4.6 N/m?
• Physics - Elena, Wednesday, July 11, 2012 at 4:08pm
This is the problem similar to that I've solved for another post. Substitute your data and calculate yourself
• Physics - Elena, Wednesday, July 11, 2012 at 4:13pm
Check my calculations.
ω=sqrt(k/m)=sqrt(5.3/0.1) = 7.28rad/s.
Divide the first equation by the second and obtain
x/v= A•sinωt/ A•ω•cosωt= ω•tanωt,
tanωt = x•ω/v=20•7.28/194=0.75
ωt =arctanωt =0.64 rad,
sinωt = 0.6
A=x/sinωt = 20/0.571=33.3 cm.
For the second case:
sinωt1 =x1/A=31.5/33.3 =0.95.
cos ωt1=sqrt(1-sin²ωt1)=0.33.
v1= =A•ω•cosωt1=
=33.3•7.28•0.33=80 cm/s.
• Physics - Jay, Wednesday, July 11, 2012 at 4:22pm
I got a similar answer of 79.0 cm/s using the conservation of energy equation.
U(i) + K(i) = U(f) + U(f)
Thank you for the help. And sorry for asking this question. I didn't see the similar question when I searched mine in the search bar.
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Large cardinal axioms and the perfect set property
up vote 2 down vote favorite
It is known that if there is a measurable cardinal then every $\Pi_1^1$ set has the perfect set property (i.e it is either countable or contains a copy of $2^{\omega}$). Also if we have $\Pi_1^
1$-determinacy (or in other words $0^{\sharp}$) then we get that $\Sigma_2^1$ has the perfect set property. Note the result that $\Sigma_1^1$ has the perfect set property is a $ZFC$ result.
Is there a reason as to why we need stronger infinity axioms to prove the perfect set property for $\Pi$ classes in comparison to what we need to prove the perfect set property for $\Sigma$ classes?
This is weird because we have $\Pi_1^1 \subseteq \Sigma_2^1$. Do we actually need less than a measurable to prove the PSP for $\Pi_1^1$?
descriptive-set-theory set-theory lo.logic
Someone please change occurences of "prefect" to "perfect". Thank you in advance. Gerhard "Ask Me About System Design" Paseman, 2011.10.22 – Gerhard Paseman Oct 23 '11 at 3:24
Note that $2^\omega$ is not the only perfect set. Lightface $\Pi^1_1$-determinacy is equivalent to the existence of $0^\sharp$; you need all sharps for boldface $\boldsymbol{\Pi}^1_1$-determinacy.
Why does $\Pi^1_1 \subseteq \Sigma^1_2$ make this weird? – François G. Dorais♦ Oct 23 '11 at 3:25
@Gerard:changed. – Carlo Von Schnitzel Oct 23 '11 at 3:44
add comment
1 Answer
active oldest votes
Solovay showed that the following are equivalent:
1. $\boldsymbol{\Sigma}^1_2$ sets have the perfect set property
2. $\boldsymbol{\Pi}^1_1$ sets have the perfect set property
up vote 6 down 3. $\aleph_1^{L[a]} < \aleph_1$ for every real $a$
vote accepted
You only need an inaccessible to force (3).
Francois, maybe this should be asked as a full-fledged question, but have such "nice" characterizations been obtained further up in the projective hierarachy? My knowledge of
descriptive set theory tends to peter out around the $\mathbf{\Sigma}^1_3$ level... – Todd Eisworth Oct 23 '11 at 3:43
@Todd: That's what I was asking about. What happens on the projective hierarchy and why do we need more to prove the PSP for $\Pi$ classes in comparison to proving it for $\Sigma$
classes? – Carlo Von Schnitzel Oct 23 '11 at 3:47
Todd, I don't think so but I must admit I suffer from the same type of nearsightedness you do. There is a great paper by Brendle and Löwe that has a bunch of these
characterizations; I don't think they do large subscripts, but that's where I'd check first. – François G. Dorais♦ Oct 23 '11 at 3:58
@alephomega: The only way I know to prove PSP for large subscripts is using determinacy and the unfolding trick. Unfolding immediately gives the next $\Sigma$ pointclass up the
ladder. I have no idea how you could avoid the extra step and just have PSP for $\Pi$ pointclass and not the next $\Sigma$ pointclass. – François G. Dorais♦ Oct 23 '11 at 4:02
add comment
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Math in the Mall
Making a
Math in the Mall Display
Malgorzata Dubiel & Katherine Heinrich
Department of Mathematics and Statistics
Simon Fraser University
Burnaby, British Columbia, V5A 1S6
Fax: (604) 291-4947
Introduction: How did it start?
Our work in popularizing mathematics through displays with hands-on activities began with the teaching of our course Mathematics for Elementary School Teachers (MATH 190). Our goal was and is to
change the attitudes of student-teachers to mathematics. Our strategy is to prepare (and later let students prepare) various projects which would allow them to see a glimpse of what we see in
mathematics: the beauty of it; the excitement and amazement upon seeing something unexpected; the mathematics around us; and the people of mathematics.
The projects proved very succesful and we started to think about reaching younger children. Our first opportunity was the Homecoming event organized for the 25th anniversary of Simon Fraser
University in September 1990. We put up a display which we called Is this Math?, using some of our Math 190 projects; puzzles, games, geometrical models, videotape with a week's worth of Square One
TV (with permission of PBS), and a display of our choice of books. The display was even more succesful than we expected. When colleagues from computing science complained that our display was
diverting traffic from their exhibits, we knew we were on the right track. The president of SFU visited the display and was sufficiently impressed to promise funding a shopping mall version.
For our first foray outside the university we chose Lougheed Mall - a large mall in Burnaby, close to SFU and to several elementary schools. While preparing the event, we were contacted by organizers
of the Science and Technology Week 91 and, at their request, agreed to repeat the display three weeks later in another Burnaby mall, Metrotown. Since then, our mall appearances have been restricted
to Science and Technology Weeks and similar events, as this makes the organization much easier. We also take some of the activities to schools or occasionally have groups of children visit the
Suggested activities for a Math in the Mall display
Listed below are some activities we found succesful:
1. Kaleidocycles and hexaflexagons
We discovered them through Martin Gardner and Doris Schattschneider books. You can copy our designs, which are ready to cut, glue and decorate, or you can construct your own. Decorating is an
important part of the process as it allows the child to discover the symmetries of the object, and is the main part of the magic. The instructions are given on a separate page, but here is some
additional advice:
□ Construction paper is not a suitable material: it is too soft and falls apart quickly - too much effort wasted. Manilla tag is a reasonably inexpensive alternative. Ours is called cover stock
. Glue: for hexaflexagons, any glue stick or school glue will do. For kaleidocycles, we foud only rubber cement acceptable, since this model requires some elasticity to turn well. Smaller
children need to be supervised or helped when working with this glue.
□ Hexaflexagons are the easier of the two, and quicker to make. However, before gluing you have to make sure that you have three pairs of triangles, separated by folds, on each side - otherwise
it will not work. Do not start colouring before constructing your hexaflexagon: cut, fold and glue first, then decorate. If a design on a hexagonal face is a picture, or something which makes
a distinction between the centre and the outside of the face, the result will be more surprising and magical.
□ Kaleidocycles: Here, you can colour first and then cut and glue. The form creates a three-dimensional object, which has four faces. Like with hexaflexagons, some designs will make a more
interesting model than others, but you need to see first which triangles will show on each face - you need to make one model to understand it. When you cut the form, score all the lines and
fold according to the instructions, unfold and try to observe what is happening.
□ Necessary tools:
☆ scissors (have a pair of left-handed ones)
☆ knitting needles and rulers - for scoring the edges
☆ glue: rubber cement for kaleidocycles, to allow for flexibility, and glue sticks hexaflexagons
☆ coloured markers for decorating
2. Platonic solids
Forms to make them, tools as above, and lots of models.
3. Drinking straw models
Flexible plastic drinking straws are used to make geometrical models - the icosahedron being the most popular choice. You need several bags of straws (Safeway sells great straws in four solid
neon colours under its own brand, if you can find them), and masking tape - scotch tape does not work well on some types of straws, apart from being much more expensive. You cut through the
shorter end of a straw, squash it to make it narrower and insert it into the longer half of another. When you join three straws together, you make a triangle, four - a square. The polygons made
in this way can be assembled using masking tape to make three - dimensional models. This activity is very popular with small children, but everybody loves it, including our helpers, who always
manage to build a strange object during slower moments.
4. Pentagonal stars
A Math 190 student from Singapore taught us how to make these. The basic idea is that if you tie a knot on a rectangular strip of paper and flatten it, you get a regular pentagon. When you fold
the strip of paper around itself several times and then push the edges in to pop it out you get a pentagonal star. We use gift wrapping paper cut into long strips of various widths. You have to
make sure the strip is very even, and discover by trial and error how many times to fold so your star pops out well - this depends from the type of paper used, and the width of the strip. The
folds need to be tight. Notice also, that if you fold the paper tightly and carefully, it knows where to go. Common mistake is to force the paper to fold along a wrong side.
5. Mobius bands
6. Puzzles and games
You can find them in the Science Word gift store, some toy stores, Moyers' Teacher Store and Kids are Worth it!, Science and Nature and many others, but it may take time to build your collection.
There are different ones for every age. Mathematicians love them: they teach strategies, problem solving, geometric intuitions and much more! Erick Wong, who last year, at 17, got a degree in
math and computing science at SFU, told a Vancouver Sun reporter who intervieved him afterwards that his love of mathematics started with puzzles his mother was giving him in elementary school.
The card game described on another page teaches an important concept in mathematics: pattern recognition. It is really addictive! You can also buy a similar one, called SET, in Moyers' Teacher
Store and Kids are Worth it!, and possibly other places.
7. Posters
Have lots of poster boards and posters on geometry, Escher, math careers, your department, math humour, activities one does not usually associate with mathematics (eg. knitting, design,
architecture) etc. There are many good posters available. Start collecting now.
8. Books and mathematical toys
A display of your favourites (be careful - you may lose them). Or prepare a list of books, resources and places to get them, to give to teachers and parents. See the list of some of our
favourites at the end.
9. Computers with interesting software
Examples are tetris, fractals, Lifelab, Geometer's Sketchpad etc.
10. Videos
Many interesting math videos are available from various sources, including videos about platonic solids and others from Key Curriculum Press (toll-free number 1-800-338-7638). You can also get
permission from PBS to show recordings of Square 1 TV or other programs - they had an interesting Nova program about Fractals. Malls often will let you VCR for the day.
Help with the display
We recruit colleagues and graduate and undergraduate students. Students from Math 190 often like to help, partially because they have to demonstrate volunteer work for the admission to professional
programs in education. Talk to the mathematics specialists in the Faculty of Education. You will not have difficulties in finding help when repeating your display: people who help once usually come
back - they have fun participating.
Various work-study programs for undergraduate students have been used by us to pay student assistant for work done in preparing displays and presenting them to the public. We also get Science and
Technology Week T-shirts to give to volunteers.
Approach anybody you can think of: your department/university/college; professional organizations (CMS has a special fund for this activity during Science and Technology Week - see references 10 and
12), provincial (state) ministries of education, advanced education, science and technology; the mall (go to Promotions Director or similarperson), or other facility in which the display is to be
held. In Lougheed Mall we got a 40% discount in a stationery store (Willsons). If there is a science store in your mall, approach them for cooperation.
In Canada, Science and Technology Week provides easier access to help with organizing, financing and advertising, as well as attractive T-shirts for volunteers and as prizes.
Another possibility is Science Fairs: we've been invited to a few of them recently.
Local community papers and school boards are the most effective instruments, but try everything available. Most community papers are very enthusiastic, as are some local radio stations. Your college/
university Media office will help with advice and press releases.
Do not neglect word of mouth!
Math in the Malls across Canada
Kathy Heinrich described our displays in the article Science Awareness Week Project, in the May - June 1992 issue of CMS Notes. As the then chair of the CMS Education Committee, she invited
mathematics departments from other universities across Canada to organize such activities duringScience and Technology Week, suggesting that some funding might be available from CMS.
The first response was from the Department of Mathematics and Statistics at Memorial University. In October 1992, as a part of Science and Technology Week, Ed Williams and Bruce Shawyer organized an
event in the Village Shopping Mall in St. John's. The display included several fun activities aimed primarily at children aged 10 to 15: How to get to second base, a Mall Math Trail, a Math Challenge
with a prize for the best solution, and many others. The exhibit was extremely well received and became an annual event. In 1994 it was expanded to a Math and Science in the Mall multi-display, with
twelve different group participating, and attracted between four and five thousand people.
The Department of Mathematics, Statistics and Computing Science at Dalhousie organized a Math in the Mall display in October 1993, in the Halifax Shopping Centre. It included computer and video
displays, and exhibits borrowed from a local science museum: a giant Tower of Hanoi, and other games and puzzles. There was no display in 1994, but there are plans for 1995. For information contact
R.P. Gupta or Richard Nowakowski.
Simon Fraser's Mathematics and Statistics Department visits at least two malls yearly. In 1994 we were joined by the Physics Department with a very popular holography display, and by the School of
Engineering. The activity is also spreading to smaller communities. Salmon Arm had a mall display in 1994. Prince George is interested - there it will possibly be a cooperative effort between Prince
George Children's Festival, the Fort George Museum, and the University of Northern BC.
Final suggestions
Don't try to be too ambitious at first. Select 2 - 3 activities and have lots of models. Increase the activities with the experience and/or the number of helpers. Make sure you test all the
activities yourself first!
Be visible - have a big sign, one or two big models, and posters - make your own, especially about geometry, design, mathematical people, jokes etc. Also, make sure you choose the location well. Be
in the centre of the mall or near a major entrance! Some malls, though, have restrictions on the height of signs and poster boards.
• Have several long tables and chairs, so there is a lot of working space.
• Have lots of handouts for parents and teachers.
• Discuss the local mall requirements and rules - for example, they usually require that you carry sufficient insurance. Your university usually provides for such events. If you want to bring
computers or VCR monitors, make sure there is a power outlet nearby; bring your own power bar, ask the mall for an extension cord or bring your own if required.
• If you are ambitious and have enough help, consider organizing a Mathematics Trail through the Mall. (see references 12, 13) and the list of our favourite trails.
Suggested readings
The list below includes both articles about math displays and trails, and a selection of our favourite books:
1. B. Bolt, Mathematical Cavalcade, Cambridge University Press.
2. B. Bolt, Mathematical Funfair, Cambridge University Press.
3. B. Bolt, More Mathematical Activities, Cambridge University Press.
4. B. Bolt, The Amazing Mathematics Amusement Arcade, Cambridge University Press.
5. M. Gardner, Mathematical Carnival, MAA.
6. M. Gardner, Mathematical Magic Show, MAA.
7. M. Gardner, Mathematical Circus, MAA.
8. M. Gardner, The Scientific American Book of Mathematical Puzzles and Diversions, Simon and Schuster, 1959.
9. M. Gardner, The Second Scientific American Book of Mathematical Puzzles and Diversions, Simon and Schuster, 1961.
10. M. Gardner, Perplexing Puzzles and Tantalizing Teasers, MAA. (The easiest book for younger children.)
11. M. Gardner, Riddles of the Sphinx, MAA. (For older children, grade 7 and up.)
12. D. Schattschneider, W. Walker, M.C.Escher Kaleidocycles, Pomegranate Artbooks, 1987. (A book with several very ineresting ideas and wonderful models of kaleidocycles and solids to make, decorated
with Escher pictures - definitely worth buying! We buy a new one every two years.)
13. M. Burns, The I Hate Mathematics! Book, Little, Brown and Company.
14. M. Burns, Math for Smartypants, Little, Brown and Company.
15. K. Heinrich, Science Awareness Week Project CMS Notes, Vol.24 #4, May-June 1992.
16. E. Muller, Math Trails, CMS Notes, Vol.25 #2, March 1993.
17. E.R. Williams, Math in the Mall, CMS Notes, Vol.25 #1, January-February 1993.
Some interesting mathematics trails:
1. A Mathematics Trail around the City of Melbourne, by Dudley Blane and Doug Clarke, The Mathematics Education Centre, Monash University, Melbourne.
2. Niagara Falls Math Trail, by Eric Muller, Department of Mathematics, Brock University, St. Catherines, Ontario.
3. The Village Mall Mathematics Trail, by Ed Williams, Department of Mathematics and Statistics, Memorial University of Newfoundland, St. Johns, Newfoundland.
4. Loon Lake Mathematics Trail, by Malgorzata Dubiel and Katherine Heinrich, Department of Mathematics and Statistics, Simon Fraser University. | {"url":"http://cms.math.ca/Education/MallMath/","timestamp":"2014-04-18T00:28:11Z","content_type":null,"content_length":"36986","record_id":"<urn:uuid:0415a8e2-e6b8-4889-9fb5-a6c41c0a113d>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00555-ip-10-147-4-33.ec2.internal.warc.gz"} |
User Oblomov
bio website math.unice.fr/~cazanave
location Nice, France
visits member for 4 years
seen Apr 18 at 9:18
stats profile views 413
Oct Bass' stable range of $\mathbf Z[X]$
24 comment The answer is given in a paper of Grunewald, Mennicke and Vaserstein (On the groups $SL_2(\mathbf Z[x])$ and $SL_2(k[x,y])$). Israel J. Math. 86 (1994), no. 1-3, 157–193). One example of
unimodular row that is not reducible is the following $(21+ 4x, 12, x^2 + 20)$.
24 accepted Bass' stable range of $\mathbf Z[X]$
8 awarded Constituent
8 awarded Caucus
Jul An analogue of the Bass-Quillen conjecture with power or Laurent series
23 comment Indeed, sorry for my previous comment.
Jul An analogue of the Bass-Quillen conjecture with power or Laurent series
23 comment You may want to have a look at Lam's book titled "Serre's problem on projective modules", Section V.4 and V.5.
17 answered (Preferably rare) Audio/Video recordings of famous mathematicians?
Jun Bass' stable range of $\mathbf Z[X]$
17 revised added 23 characters in body
Jun Bass' stable range of $\mathbf Z[X]$
17 comment Yes, you are right, I'll edit my question.
Jun Bass' stable range of $\mathbf Z[X]$
7 comment Sorry, I was not able to find how to type matrices correctly
7 answered Bass' stable range of $\mathbf Z[X]$
Jun Bass' stable range of $\mathbf Z[X]$
6 comment Is $\mathbf Z[x_1, \dots, x_n]$ really of stable range $n+1$ (and not $n+2$)?
6 awarded Nice Question
Jun Bass' stable range of $\mathbf Z[X]$
6 comment I found Example 12.1.14 in Chen's book (it's on page 373). It states that a Dedekind domain is of stable range $2$. The typo you found seems to be rather a mistake.
Jun Bass' stable range of $\mathbf Z[X]$
6 comment Thanks for the reference (although I don't understand the argument too). I'll try to write to the author.
Jun Bass' stable range of $\mathbf Z[X]$
5 revised added 86 characters in body
Jun Bass' stable range of $\mathbf Z[X]$
5 comment It's the ideal generated by these elements. Sorry, I thought this was transparent.
5 asked Bass' stable range of $\mathbf Z[X]$
Apr Grothendieck 's question - any update?
29 comment For another solution to the (refined) question , you can have a look at: François Charles, Conjugate varieties with distinct real cohomology algebras J. Reine Angew. Math. 630 (2009),
pp. 125--139.
Apr awarded Teacher | {"url":"http://mathoverflow.net/users/5239/oblomov?tab=activity","timestamp":"2014-04-21T15:55:35Z","content_type":null,"content_length":"44971","record_id":"<urn:uuid:b73551e9-7fd3-4839-823a-7d63b9bd552c>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00376-ip-10-147-4-33.ec2.internal.warc.gz"} |
convergent series
convergent series
When the terms of the series live in $\mathbb{R}^{n}$, an equivalent condition for absolute convergence of the series is that all possible series obtained by rearrangements of the terms are also
convergent. (This is not true in arbitrary metric spaces.)
Let $\sum a_{n}$ be an absolutely convergent series, and $\sum b_{n}$ be a conditionally convergent series. Then any rearrangement of $\sum a_{n}$ is convergent to the same sum. It is a result due to
Riemann that $\sum b_{n}$ can be rearranged to converge to any sum, or not converge at all.
absolute convergence, conditional convergence, absolutely convergent, conditionally convergent, converges absolutely, convergent, divergent, divergent series
Series, HarmonicNumber, ConvergesUniformly, SumOfSeriesDependsOnOrder, UncoditionalConvergence, WeierstrassMTest, DeterminingSeriesConvergence
Mathematics Subject Classification
no label found
no label found
Added: 2002-02-20 - 22:22 | {"url":"http://planetmath.org/convergentseries","timestamp":"2014-04-18T15:40:28Z","content_type":null,"content_length":"41977","record_id":"<urn:uuid:a93c1ed1-d696-4bc8-9c83-4b31b207a029>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00268-ip-10-147-4-33.ec2.internal.warc.gz"} |
Selection of a shell model for a cylindrical shell immersed in
ASA 127th Meeting M.I.T. 1994 June 6-10
5aSA5. Selection of a shell model for a cylindrical shell immersed in fluid.
Martin G. Manley
Aerosp. and Mech. Eng. Dept., Boston Univ., 110 Cummington St., Boston, MA 02215
There are many thin shell models available to the investigator studying the vibrations of thin-walled, circular, cylindrical shells. A rational method for selecting a model for any given problem is
needed. An asymptotically exact dispersion relation for waves on fluid-loaded, elastic, hollow cylinders derived from the exact equations of elasticity previously [M. Manley, J. Acoust. Soc. Am. 93,
2335(A) (1993)] is applied to determine dominant terms for dispersion of axial waves. This is compared with dispersion relations obtained from shell models. The simplest shell model retaining the
dominant terms in the dispersion relation is recommended. Some examples are presented. [The author acknowledges the advice of A. D. Pierce.] | {"url":"http://www.auditory.org/asamtgs/asa94mit/5aSA/5aSA5.html","timestamp":"2014-04-16T20:12:58Z","content_type":null,"content_length":"1393","record_id":"<urn:uuid:ab9d93fb-45bb-4f24-811c-f9d84121f05a>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00485-ip-10-147-4-33.ec2.internal.warc.gz"} |
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16 Subjects: including algebra 2, calculus, geometry, algebra 1 | {"url":"http://www.purplemath.com/Nahant_algebra_2_tutors.php","timestamp":"2014-04-19T12:12:49Z","content_type":null,"content_length":"23958","record_id":"<urn:uuid:48f805e8-e678-441b-abde-f885e8c29dbb>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00168-ip-10-147-4-33.ec2.internal.warc.gz"} |
, is popularly used for a light ebullition, or a brisk intestine motion, produced in a liquor by the first action of heat, with any remarkable separation of its parts.
EFFICIENT Cause, is that which produces an effect. See Cause and Effect.
, in Arithmetic, are the numbers given for an operation of multiplication, and are otherwise called the factors. Hence the term coefficients in Algebra, which are the numbers prefixed to, or that
multiply the letters or algebraic quantities. | {"url":"http://words.fromoldbooks.org/Hutton-Mathematical-and-Philosophical-Dictionary/e/effervescence.html","timestamp":"2014-04-19T17:07:47Z","content_type":null,"content_length":"5554","record_id":"<urn:uuid:1b1c3478-71c5-4764-b4c3-170a4cc444af>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00032-ip-10-147-4-33.ec2.internal.warc.gz"} |
Method and Device for Implementing Automatic Frequency Control
Patent application title: Method and Device for Implementing Automatic Frequency Control
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The present invention discloses a method and apparatus for implementing automatic frequency control. The method comprises: calculating a correlation value of common pilot symbols in a reception
signal of each path and a frequency offset value, calculating a combined frequency offset value according to the frequency offset value of each path, and determining, according to the combined
frequency offset value, a frequency offset adjustment value for a voltage controlled oscillator, a frequency offset adjustment value for overall reception signals and a frequency offset adjustment
value for the reception signal of each path. Compared with prior art, with the method and apparatus of the present invention, a new method of small-scope frequency offset estimation is selected,
which makes frequency offset estimation more accurate; by using CORDIC, hardware resources are saved and calculation accuracy is improved; the stability of system is enhanced by the frequency offset
adjustment and control policy.
A method for implementing automatic frequency control, comprising the following steps of: calculating a correlation value of common pilot symbols in a reception signal of each path and a frequency
offset value, calculating and obtaining a combined frequency offset value according to the frequency offset value of the reception signal of each path, and determining, according to the combined
frequency offset value, a frequency offset adjustment value for a voltage controlled oscillator, a frequency offset adjustment value for overall reception signals and a frequency offset adjustment
value for the reception signal of each path.
The method according to claim 1, wherein, the step of calculating a correlation value of common pilot symbols in a reception signal of each path and a frequency offset value comprises: performing a
correlation calculation on a plurality of common pilot symbols received in a timeslot for the reception signal of each path to obtain the correlation value, wherein a symbol number L spaced between
two symbols on which the correlation calculation is performed is an integer greater than or equal to 2; and, calculating the frequency offset value corresponding to the correlation value according to
the correlation value and the symbol number L.
The method according to claim 1, wherein, the step of calculating and obtaining a combined frequency offset value according to the frequency offset value of the reception signal of each path
comprises: judging whether the frequency offset value of the reception signal of each path is valid according to a signal-to-interference ratio threshold of the reception signal of each path,
weighted combining the valid frequency offset values to obtain the combined frequency offset value, wherein a weighted parameter of the frequency offset value of the reception signal of each path is
determined by a proportion of a common pilot symbol signal-to-interference ratio corresponding to the frequency offset value to a sum of common pilot symbol signal-to-interference ratios of various
The method according to claim 3, further comprising: filtering the combined frequency offset value with a low-pass filter, and adjusting a value of a filtering coefficient used by the low-pass filter
according to an adjusting stage.
The method according to claim 1, wherein, the frequency offset adjustment value for the voltage controlled oscillator is determined by adjusting and controlling the combined frequency offset value in
the following way: Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur
- Δ f _ pre > κ Δ f _ pre ; ##EQU00014## wherein, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment.
The method according to claim 5, wherein, the frequency offset adjustment value for the overall reception signals is determined as Δf
- Δf
; the frequency offset adjustment value for the reception signal of each path is determined as Δf
, wherein, Δf
is the frequency offset value of the reception signal of path i.
The method according to claim 6, further comprising: performing a frequency offset compensation according to a phase angle corresponding to the determined frequency offset adjustment value and a
coordinate rotation digital computer algorithm.
An apparatus for implementing automatic frequency control, comprising a correlation calculation module and a frequency offset calculation module that are connected with each other, wherein the
correlation calculation module is configured to calculate a correlation value of common pilot symbols in a reception signal of each path; and the frequency offset calculation module is configured to
calculate a frequency offset value of a reception signal of each path; wherein the apparatus further comprises a frequency offset adjustment and control module that is connected with the frequency
offset calculation module; the frequency offset adjustment and control module is configured to calculate and obtain a combined frequency offset value according to the frequency offset value of the
reception signal of each path, and determine, according to the combined frequency offset value, a frequency offset adjustment value for a voltage controlled oscillator, a frequency offset adjustment
value of overall reception signals and a frequency offset adjustment value for the reception signal of each path.
The apparatus according to claim 8, wherein, the correlation calculation module is configured to calculate the correlation value of common pilot symbols in the reception signal of each path in the
following way: performing a correlation calculation on a plurality of common pilot symbols received in a timeslot for the reception signal of each path to obtain the correlation value, wherein a
symbol number L spaced between two symbols on which the correlation calculation is performed is an integer greater than or equal to 2; the frequency offset calculation module is configured to
calculate the frequency offset value of the reception signal of each path in the following way: calculating the frequency offset value corresponding to the correlation value according to the
correlation value and the symbol number L calculated by the correlation calculation module.
The apparatus according to claim 8, wherein, the frequency offset adjustment and control module is configured to calculate and obtain the combined frequency offset value in the following way: judging
whether the frequency offset value of the reception signal of each path is valid according to a signal-to-interference ratio threshold of the reception signal of each path, weighted combining the
valid frequency offset values to obtain the combined frequency offset value, wherein a weighted parameter of the frequency offset value of the reception signal of each path is determined by a
proportion of a common pilot symbol signal-to-interference ratio corresponding to the frequency offset value to a sum of common pilot symbol signal-to-interference ratios of various paths.
The apparatus according to claim 10, wherein, the frequency offset adjustment and control module is further configured to filter the combined frequency offset value with a low-pass filter, and adjust
a value of a filtering coefficient used by the low-pass filter according to an adjusting stage.
The apparatus according to claim 8, wherein, the frequency offset adjustment and control module is configured to determine the frequency offset adjustment value for the voltage controlled oscillator
by adjusting and controlling the combined frequency offset value in the following way: Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO
pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur - Δ f _ pre > κ Δ f _ pre ; ##EQU00015## wherein, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment.
The apparatus according to claim 12, wherein, the frequency offset adjustment and control module is configured to: determine the frequency offset adjustment value for the overall reception signals as
- Δf
; and determine the frequency offset adjustment value for the reception signal of each path as Δf
- Δf
, wherein, Δf
is the frequency offset value of the reception signal of path i.
The apparatus according to claim 13, further comprising a frequency offset compensation module, which is configured to perform a frequency offset compensation according to a phase angle corresponding
to the frequency offset adjustment value determined by the frequency offset adjustment and control module and a coordinate rotation digital computer algorithm.
The method according to claim 2, wherein, the frequency offset adjustment value for the voltage controlled oscillator is determined by adjusting and controlling the combined frequency offset value in
the following way: Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur
- Δ f _ pre > κ Δ f _ pre ; ##EQU00016## wherein, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment.
The method according to claim 15, wherein, the frequency offset adjustment value for the overall reception signals is determined as Δf
- Δf
; the frequency offset adjustment value for the reception signal of each path is determined as Δf
- Δf
, wherein, Δf
is the frequency offset value of the reception signal of path i.
The method according to claim 16, further comprising: performing a frequency offset compensation according to a phase angle corresponding to the determined frequency offset adjustment value and a
coordinate rotation digital computer algorithm.
The method according to claim 3, wherein, the frequency offset adjustment value for the voltage controlled oscillator is determined by adjusting and controlling the combined frequency offset value in
the following way: Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur
- Δ f _ pre > κ Δ f _ pre ; ##EQU00017## wherein, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment.
The method according to claim 18, wherein, the frequency offset adjustment value for the overall reception signals is determined as Δf
- Δf
; the frequency offset adjustment value for the reception signal of each path is determined as Δf
- Δf
, wherein, Δf
is the frequency offset value of the reception signal of path i.
The method according to claim 19, further comprising: performing a frequency offset compensation according to a phase angle corresponding to the determined frequency offset adjustment value and a
coordinate rotation digital computer algorithm.
The method according to claim 4, wherein, the frequency offset adjustment value for the voltage controlled oscillator is determined by adjusting and controlling the combined frequency offset value in
the following way: Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur
- Δ f _ pre > κ Δ f _ pre ; ##EQU00018## wherein, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment.
The method according to claim 21, wherein, the frequency offset adjustment value for the overall reception signals is determined as Δf
- Δf
; the frequency offset adjustment value for the reception signal of each path is determined as Δf
- Δf
, wherein, Δf
is the frequency offset value of the reception signal of path i.
The method according to claim 22, further comprising: performing a frequency offset compensation according to a phase angle corresponding to the determined frequency offset adjustment value and a
coordinate rotation digital computer algorithm.
The apparatus according to claim 9, wherein, the frequency offset adjustment and control module is configured to determine the frequency offset adjustment value for the voltage controlled oscillator
by adjusting and controlling the combined frequency offset value in the following way: Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO
pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur - Δ f _ pre > κ Δ f _ pre ; ##EQU00019## wherein, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment.
The apparatus according to claim 24, wherein, the frequency offset adjustment and control module is configured to: determine the frequency offset adjustment value for the overall reception signals as
- Δf
; and determine the frequency offset adjustment value for the reception signal of each path as Δf
- Δf
, wherein, Δf
is the frequency offset value of the reception signal of path i.
The apparatus according to claim 25, further comprising a frequency offset compensation module, which is configured to perform a frequency offset compensation according to a phase angle corresponding
to the frequency offset adjustment value determined by the frequency offset adjustment and control module and a coordinate rotation digital computer algorithm.
The apparatus according to claim 10, wherein, the frequency offset adjustment and control module is configured to determine the frequency offset adjustment value for the voltage controlled oscillator
by adjusting and controlling the combined frequency offset value in the following way: Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO
pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur - Δ f _ pre > κ Δ f _ pre ; ##EQU00020## wherein, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment.
The apparatus according to claim 27, wherein, the frequency offset adjustment and control module is configured to: determine the frequency offset adjustment value for the overall reception signals as
- Δf
; and determine the frequency offset adjustment value for the reception signal of each path as Δf
- Δf
, wherein, Δf
is the frequency offset value of the reception signal of path i.
The apparatus according to claim 28, further comprising a frequency offset compensation module, which is configured to perform a frequency offset compensation according to a phase angle corresponding
to the frequency offset adjustment value determined by the frequency offset adjustment and control module and a coordinate rotation digital computer algorithm.
The apparatus according to claim 11, wherein, the frequency offset adjustment and control module is configured to determine the frequency offset adjustment value for the voltage controlled oscillator
by adjusting and controlling the combined frequency offset value in the following way: Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO
pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur - Δ f _ pre > κ Δ f _ pre ; ##EQU00021## wherein, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment.
The apparatus according to claim 30, wherein, the frequency offset adjustment and control module is configured to: determine the frequency offset adjustment value for the overall reception signals as
- Δf
; and determine the frequency offset adjustment value for the reception signal of each path as Δf
- 66 f
, wherein, Δf
is the frequency offset value of the reception signal of path i.
The apparatus according to claim 31, further comprising a frequency offset compensation module, which is configured to perform a frequency offset compensation according to a phase angle corresponding
to the frequency offset adjustment value determined by the frequency offset adjustment and control module and a coordinate rotation digital computer algorithm.
TECHNICAL FILED [0001]
The present invention relates to the field of wireless communication, and in particular, to a method and an apparatus for implementing automatic frequency control.
BACKGROUND OF THE RELATED ART [0002]
In a Wideband Code Division Multiple Access (WCDMA) system, there is generally carrier frequency difference, which is also called as frequency offset, between a base station and a terminal due to the
influence of the accuracy and stability of a local crystal oscillator, and Doppler effect during the process of movement of the mobile terminal. Such frequency offset will have huge influence on the
demodulation performance of the terminal. Automatic Frequency Control (AFC) is widely used in the system as an efficient method for correcting and controlling frequency offset. However, the existing
method is very simple for Automatic Frequency Control in the WCDMA system, and lacks comprehensive consideration for the system; meanwhile, the frequency offset compensation algorithm adopts a
conventional table lookup method or triangular transformation method, which has a big consumption of hardware.
SUMMARY OF THE INVENTION [0003]
The present invention provides a method and an apparatus for implementing automatic frequency control to improve the accuracy, efficiency and stability of frequency offset control.
In order to solve the above problem, the present invention provides a method for implementing automatic frequency control, comprising the following steps of: calculating a correlation value of common
pilot symbols in a reception signal of each path and a frequency offset value, calculating and obtaining a combined frequency offset value according to the frequency offset value of the reception
signal of each path, and determining, according to the combined frequency offset value, a frequency offset adjustment value for a voltage controlled oscillator, a frequency offset adjustment value
for overall reception signals and a frequency offset adjustment value for the reception signal of each path.
The step of calculating a correlation value of common pilot symbols in a reception signal of each path and a frequency offset value may comprise: performing a correlation calculation on a plurality
of common pilot symbols received in a timeslot for the reception signal of each path to obtain the correlation value, wherein a symbol number L spaced between two symbols on which the correlation
calculation is performed is an integer greater than or equal to 2; and, calculating the frequency offset value corresponding to the correlation value according to the correlation value and the symbol
number L.
The step of calculating and obtaining a combined frequency offset value according to the frequency offset value of the reception signal of each path may comprise: judging whether the frequency offset
value of the reception signal of each path is valid according to a signal-to-interference ratio threshold of the reception signal of each path, weighted combining the valid frequency offset values to
obtain the combined frequency offset value, wherein a weighted parameter of the frequency offset value of the reception signal of each path is determined by a proportion of a common pilot symbol
signal-to-interference ratio corresponding to the frequency offset value to a sum of common pilot symbol signal-to-interference ratios of various paths. This method may further comprise: filtering
the combined frequency offset value with a low-pass filter, and adjusting a value of a filtering coefficient used by the low-pass filter according to an adjusting stage.
The frequency offset adjustment value for the voltage controlled oscillator may be determined by adjusting and controlling the combined frequency offset value in the following way:
Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur - Δ f _ pre > κ Δ f
_ pre ; ##EQU00001##
, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment.
The frequency offset adjustment value for the overall reception signals may be determined as Δf
- Δf
, the frequency offset adjustment value for the reception signal of each path may be determined as Δf
- Δf
, Δf
is the frequency offset value of the reception signal of path i. This method may further comprise: performing a frequency offset compensation according to a phase angle corresponding to the
determined frequency offset adjustment value and a Coordinate Rotation Digital Computer algorithm.
The present invention further provides an apparatus for implementing automatic frequency control, comprising a correlation calculation module and a frequency offset calculation module that are
connected with each other, wherein the correlation calculation module is configured to calculate a correlation value of common pilot symbols in a reception signal of each path; and the frequency
offset calculation module is configured to calculate a frequency offset value of a reception signal of each path; wherein the apparatus further comprises a frequency offset adjustment and control
module that is connected with the frequency, offset calculation module; the frequency offset adjustment and control module is configured to calculate and obtain a combined frequency offset value
according to the frequency offset value of the reception signal of each path, and determine, according to the combined frequency offset value, a frequency offset adjustment value for a voltage
controlled oscillator, a frequency offset adjustment value of overall reception signals and a frequency offset adjustment value for the reception signal of each path.
The correlation calculation module is configured to calculate the correlation value of common pilot symbols in the reception signal of each path in the following way: performing a correlation
calculation on a plurality of common pilot symbols received in a timeslot for the reception signal of each path to obtain the correlation value, wherein a symbol number L spaced between two symbols
on which the correlation calculation is performed is an integer greater than or equal to 2; the frequency offset calculation module is configured to calculate the frequency offset value of the
reception signal of each path in the following way: calculating the frequency offset value corresponding to the correlation value according to the correlation value and the symbol number L calculated
by the correlation calculation module.
The frequency offset adjustment and control module is configured to calculate and obtain the combined frequency offset value in the following way: judging whether the frequency offset value of the
reception signal of each path is valid according to a signal-to-interference ratio threshold of the reception signal of each path, weighted combining the valid frequency offset values to obtain the
combined frequency offset value, wherein a weighted parameter of the frequency offset value of the reception signal of each path is determined by a proportion of a common pilot symbol
signal-to-interference ratio corresponding to the frequency offset value to a sum of common pilot symbol signal-to-interference ratios of various paths. The frequency offset adjustment and control
module is further configured to filter the combined frequency offset value with a low-pass filter, and adjust a value of a filtering coefficient used by the low-pass filter according to an adjusting
The frequency offset adjustment and control module is configured to determine the frequency offset adjustment value for the voltage controlled oscillator by adjusting and controlling the combined
frequency offset value in the following way:
Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur - Δ f _ pre > κ Δ f
_ pre ; ##EQU00002##
, F,α,β,κ represents an adjusting and controlling parameter, Δf
is a combined frequency offset value in a process of the current frequency offset adjustment, Δf
is a combined frequency offset value in a process of the previous frequency offset adjustment, Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the current frequency offset adjustment, and Δf
is a frequency offset adjustment value for the voltage controlled oscillator in the process of the previous frequency offset adjustment. The frequency offset adjustment and control module is
configured to: determine the frequency offset adjustment value for the overall reception signals as Δf
- Δf
; and determine the frequency offset adjustment value for the reception signal of each path as Δf
- Δf
, wherein, Δf
is the frequency offset value of the reception signal of path i.
The apparatus may further comprise a frequency offset compensation module, which is configured to perform a frequency offset compensation according to a phase angle corresponding to the frequency
offset adjustment value determined by the frequency offset adjustment and control module and a coordinate rotation digital computer algorithm.
Compared with prior art, with the method and apparatus of the present invention, a new method of small-scope frequency offset estimation is selected, which makes frequency offset estimation more
accurate; by using Coordinate Rotation Digital Computer (CORDIC) algorithm, hardware resources are saved and calculation accuracy is improved; frequency offset adjustment and control policy enables
radio frequency adjustment to be more gentle and the stability of the system to be enhanced.
BRIEF DESCRIPTION OF DRAWINGS [0015]
FIG. 1 is a structure diagram of an apparatus for implementing automatic frequency control;
FIG. 2A is a structure diagram one of a correlation calculation of frequency offset estimation;
FIG. 2B is a structure diagram two of a correlation calculation of frequency offset estimation;
FIG. 2C is a structure diagram three of a correlation calculation of frequency offset estimation;
FIG. 3 is an overall scheme one of AFC in a Rake receiver;
FIG. 4 is an overall scheme two of AFC in a Rake receiver;
FIG. 5 is a pipeline structure diagram for implementing a CORDIC algorithm; and
FIG. 6 is a flow chart of a method for implementing automatic frequency control.
PREFERRED EMBODIMENTS OF THE INVENTION [0023]
As shown in FIG. 1, the apparatus for implementing automatic frequency control includes a correlation calculation module, a frequency offset calculation module, a frequency offset adjustment and
control module, and a frequency offset compensation module that are connected in turn.
Functions of each module will be described respectively.
(1) a correlation calculation module is used to extract common pilot symbols in the reception signal of each path, calculate the correlation value of the common pilot symbols in the reception signal
of each path; specifically, for the signal of each path, perform a correlation calculation on a plurality of common pilot symbols received in a timeslot to obtain the correlation value. The
calculating method will be described as follows:
Supposing that the CPICH symbol received by the correlation calculation module is S(t), the correlation calculation may be performed on part of the pilot symbols in a common pilot channel (CPICH)
received in one timeslot in order to eliminate the impact of the space-time transmit diversity (STTD) transmitting pattern, thereby achieving the purpose of eliminating the impact of the channel
through the correlation between symbols, for example, the correlation calculation is only performed on the first symbol to the eighth symbol.
The correlation calculation mode between the symbols relates to the range and accuracy of a frequency offset estimation result, including a large-range frequency offset estimation and a small-range
frequency offset estimation. The large-range frequency offset estimation has a low precision and is applicable to a frequency offset capture state; while the small-range frequency offset estimation
has a high precision and is applicable to a frequency offset tracking state. The frequency offset capture state is generally implemented inside an initial cell searching module. In a Rake receiver,
it mainly regards to the frequency offset tracking adjustment after undergoing initial frequency offset capture and compensation.
With regard to an estimation method relating to a larger range of frequency offset, the calculation of the correlation value C is as shown in FIG. 2A and the following formula:
= 1 4 i = 1 4 S 2 i - 1 * S 2 i ##EQU00003##
With regard to the estimation relating to a smaller range of frequency offset, the symbol number L spaced between the two symbols on which the correlation calculation is performed is an integer
greater than or equal to 2; a mode one taking the value of L as 4 is as shown in FIG. 2B and the following formula:
= 1 4 i = 1 4 S i * S i + 4 ##EQU00004##
A mode two taking the value of L as 4 is as shown in FIG. 2C and the following formula:
= ( S 1 + S 2 2 ) * ( S 5 + S 6 2 ) + ( S 3 + S 4 2 ) * ( S 7 + S 8 2 ) ##EQU00005##
In addition, considering the reason that the influence of noise is too big when performing frequency offset estimation for correlation of the CPICH symbols of a single timeslot, the correlation
values of several timeslots may also be added and averaged to weaken the influence.
(2) The frequency offset calculation module is used to calculate the frequency offset value of the reception signal of each path;
the frequency offset calculation module is further used to calculate the frequency offset value corresponding to the correlation value according to the correlation value output by the correlation
calculation module and the value of L used in the correlation calculation.
The specific calculating method is as shown in the following formula:
Δ ω = 1 TL arctan ( Im ( C ) Re ( C ) ) ##EQU00006##
, L is the symbol number spaced between the two symbols on which the correlation calculation is performed; T is the duration time of a unit symbol, which is 1/15000. Furthermore, the obtained
frequency offset is:
Δ f = Δ ω 2 π = 7500 π L arctan ( Im ( C ) Re ( C ) ) ##EQU00007##
It can be seen from the above formula that the calculating range of the frequency offset is determined by L. For example: when L is 1, the measured frequency offset limit is ±7.5 KHz with a large
range of frequency offset estimation but a limited estimation accuracy, and thus is applicable to the initial frequency offset capture and can be applied in the initial cell searching module. When L
is 4, the measured frequency offset limit is ±1.75 KHz with a high estimation accuracy and can meet the requirement for estimation range of Doppler frequency offset caused by high-speed movement, and
thus is applicable to the frequency offset tracking and adjusting state. The present invention relates to the precise AFC adjustment in the Rake after undergoing the initial frequency offset
adjustment. The AFC control with different accuracies and ranges can be achieved in the WCDMA system through calculation by selecting a suitable correlation module.
(3) The frequency offset adjustment and control module is used to calculate and obtain a combined frequency offset value according to the frequency offset value of the reception signal of each path,
and determine, according to the combined frequency offset value, a frequency offset adjustment value for a voltage controlled oscillator, a frequency offset adjustment value of overall reception
signals (i.e., the overall received I, Q data) and a frequency offset adjustment value for the reception signal of each path.
The frequency offset adjustment and control module judges whether the frequency offset value of the signal of each path is valid according to a signal-to-interference ratio (SIR) threshold of the
signal of each path to eliminate the invalid frequency offset values. Herein, the SIR threshold may be estimated according to the received CPICH symbols as shown in FIG. 3, or may be directly
provided by the multi-path searching module in the system as shown In FIG. 4.
The frequency offset adjustment and control module weighted combines the valid frequency offset values to obtain the combined frequency offset value according to a formula as shown below:
Δ f _ = i = 0 N - 1 w i Δ f i ; ##EQU00008##
, w
+ . . . +w
-1=1, Δf
refers to the i
valid frequency offset value. The weighted parameter of the frequency offset value of the signal of each path is determined by the proportion of the corresponding common pilot symbol SIR to a sum of
common pilot symbol SIRs of various paths. If SIR
denotes the SIR of finger i, then,
w i
= SIR i i = 0 M - 1 SIR i and SIR i > λ ##EQU00009##
, λ is a control threshold (M is the number of the valid paths that exceed the threshold).
Next, the adjustment and control module uses a low-pass filter, which may be a first order infinite impulse response (IIR) low-pass filter, to filter the combined frequency offset value, and adjusts
its filtering coefficient according to different adjusting stages. For example, a larger filtering coefficient, e.g., 1/2, is initially selected, and after several adjusting cycles, a smaller
filtering coefficient, e.g., 1/4, is selected; the purpose is to enable the filtering effect to rapidly follow the current frequency offset value at the beginning of adjustment, and have a good
filtering action after following.
Next, adjustment on the frequency offset value by the frequency offset adjustment and control module is divided into three parts:
1) Frequency Offset Compensation for the Voltage Controlled Oscillator (VCO)
The combined frequency offset value is adjusted and controlled in the following way:
Δ f _ VCO cur = { 0 Δ f _ cur ≦ F α Δ f _ cur Δ f _ cur > F , and Δ f _ cur - Δ f _ pre ≦ κ Δ f _ pre Δ f _ VCO pre + β ( Δ f _ cur - Δ f _ VCO pre ) Δ f _ cur > F , and Δ f _ cur - Δ f _ pre > κ Δ f
_ pre ; ##EQU00010##
F,α,β,κ are adjusting and controlling parameters, F is a frequency value set by the system, α and β are real numbers greater than 0 and smaller than 1, and κ is a real number greater than 0.
Generally, the value of F is 50 Hz, α is 1/2, β is 1/4, and κ is 2. Δf
is the combined frequency offset value in the process of current frequency offset adjustment, Δf
is the combined frequency offset value in the process of previous frequency offset adjustment, Δf
is the frequency offset adjustment value for the voltage controlled oscillator in the process of current frequency offset adjustment, and Δf
is the frequency offset adjustment value for the voltage controlled oscillator in the process of previous frequency offset adjustment.
2) Frequency Offset Compensation for Overall Reception Signals
It is acquired from the combined frequency offset vale and the VCO frequency offset adjustment value that the frequency offset compensation value for the overall reception signals is Δf
- Δf
, which is compensated by using a CORDIC algorithm in the frequency offset compensation module.
3) Frequency Offset Compensation for the Reception Signal of each Path
The reception signal of each path is compensated using the CORDIC algorithm, i.e., the internal frequency offset of finger (finger refers to the finger in the RAKE receiver, and each finger in the
RAKE receiver is responsible for receiving and tracking each multi-path signal) is compensated, and the frequency offset compensation value is the difference between the frequency offset value of
each path and the combined frequency offset adjustment value, i.e., Δf
- Δf
, wherein, Δf
is the frequency offset value of path i, i is an integer greater than or equal to 0 and smaller than or equal to N-1, and N is the number of fingers, thereby achieving rapid and fine compensation for
residual frequency offset inside the finger.
(4) The frequency offset compensation module is used to perform frequency offset compensation according to a phase angle corresponding to the frequency offset adjustment value and a Coordinate
Rotation Digital Computer algorithm.
If the input reception signal is indicated as x+jy, then performing the frequency offset compensation obtains:
-.sub.rot=x cos φ-y sin φ+j(y cos φ+x sin φ)
, φ is the phase rotation amount of frequency offset compensation of reception signal samples, and its value is calculated through 2πfΔt, wherein Δt is the time interval between the signal samples; f
is the calculated frequency offset compensation value. It can be seen from above that, the value for overall input signals is an aggregate-value of frequency offset adjustment Δf
- Δf
after AFC is initiated, and the value for the reception signal of each path is an aggregate-value of frequency offset adjustment Δf
- Δf
after AFC is initiated. With regard to the processing of the above formula, the conventional method adopts a table lookup algorithm or triangular transformation method, which only has limited
precision in view of expense of resources. The present invention is implemented by using the CORDIC (Coordinate Rotation Digital Computer) algorithm, which is featured in simple implementation, low
expense of resources and high accuracy. The specific algorithm is as follows:
[ x i + 1 y i + 1 ] = { i = 0 b - 1 1 1 + 2 - 2 i } [ 1 - σ i 2 - i σ i 2 - i 1 ] [ x i y i ] ##EQU00011## z i + 1 = z i - σ i arctan ( 2 - i ) ##EQU00011.2##
The number of times of iteration is generally determined by the calculation accuracy, and can be quantified to a definite value.
In the above algorithm, x
=x, y
=y, z
σ i = sign [ Φ - i Φ i ] , ##EQU00012##
i is the iteration sequence
, and the value output by the final iteration is precisely the value obtained by the reception signal data through frequency offset compensation. For the amplitude factor
K b
= i = 0 b - 1 1 1 + 2 - 2 i , ##EQU00013##
it is a constant when the number of times of iteration b is sufficient
, and it can be also converted into constant shift and addition operation. In the process of implementing the CORDIC, it can be totally implemented by using a pipeline structure of the shift and
addition operation, which is highly efficient and can be easily implemented. The pipeline unit structure for implementing the CORDIC is as shown in FIG. 5.
As shown in FIG. 6, the method for implementing automatic frequency control comprises: calculating a correlation value of common pilot symbols in a reception signal of each path and a frequency
offset value, calculating and obtaining a combined frequency offset value according to the frequency offset value of the reception signal of each path, and determining, according to the combined
frequency offset value, a frequency offset adjustment value for a voltage controlled oscillator, a frequency offset adjustment value of overall reception signals (i.e., overall received I, Q data)
and a frequency offset adjustment value for the reception signal of each path.
The implementation order of the above method is the same with the flow of the correlation calculation module, the frequency offset calculation module, the frequency offset adjustment and control
module and the frequency offset compensation module in the above description on the apparatus performing the corresponding functions in sequence, the specific implementing method corresponds to the
functions of the modules, and thus will not be described here.
The above embodiments are only preferred embodiments of the present invention, and they are not intended to limit the present invention. For a person having ordinary skills in the art, the present
invention may have various modifications and variations. Any modification, equivalent, improvement and so on made within the sprit and principle of the present invention shall be within the
protection scope of the present invention.
INDUSTRIAL APPLICABILITY [0055]
Compared with the prior art, the present invention has the following beneficial effects: frequency offset estimation is more accurate; hardware resources are saved and calculation accuracy is
improved; and the stability of the system is enhanced.
Patent applications by Liqiang Yi, Shenzhen CN
Patent applications by ZTE CORPORATION
Patent applications in class Correlative or matched filter
Patent applications in all subclasses Correlative or matched filter
User Contributions:
Comment about this patent or add new information about this topic: | {"url":"http://www.faqs.org/patents/app/20120288040","timestamp":"2014-04-25T00:33:39Z","content_type":null,"content_length":"80634","record_id":"<urn:uuid:698598e9-e96c-4ed1-8b07-11a81ecd4944>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00309-ip-10-147-4-33.ec2.internal.warc.gz"} |
Volume of a sphere
Definition: The number of cubic units that will exactly fill a sphere.
Try this Drag the orange dot to adjust the radius of the sphere and note how the volume changes.
The volume enclosed by a sphere is given by the formula
This formula was discovered over two thousand years ago by the Greek philosopher Archimedes. He also realized that the volume of a sphere is exactly two thirds the volume of its circumscribed
cylinder, which is the smallest cylinder that can contain the sphere.
If you know the volume
By rearranging the above formula you can find the radius:
Note Most calculators don't have a cube root button. Instead, use the calculator's "raise to a power" button and raise the inner part to the power one third.
Interesting fact
For a given surface area, the sphere is the one solid that has the greatest volume. This why it appears in nature so much, such as water drops, bubbles and planets.
Things to try
□ In the figure above, click "hide details".
□ Drag the orange dot to resize the sphere.
□ Calculate the volume of the sphere
□ Click "show details" to check your answer.
□ In the figure above, click "reset" then uncheck "show radius"
□ Drag the orange dot to resize the sphere.
□ Calculate the radius of the sphere from the volume
□ Click "show radius" to check your answer.
Related topics
(C) 2009 Copyright Math Open Reference. All rights reserved | {"url":"http://www.mathopenref.com/spherevolume.html","timestamp":"2014-04-19T09:26:03Z","content_type":null,"content_length":"11981","record_id":"<urn:uuid:3ceb29cd-8fa0-44de-a20d-3d50d54fe50b>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00551-ip-10-147-4-33.ec2.internal.warc.gz"} |
Second Derivatives and Beyond
Most of the functions we've been dealing with in this unit are differentiable, or at least have only a few problem spots where the derivative doesn't exist. We know that if a function is
differentiable it must be continuous, but a continuous function doesn't need to be differentiable.
Here's a strange fact: it's possible to build a continuous function that isn't differentiable anywhere. That is, the whole function consists of corners. There's a little dot you can slide that starts
at n = 0. As n increases the number of corners on the function also increases, and the limit of these functions as n approaches ∞ consists entirely of corners.
Here's an even stranger fact: most continuous functions aren't differentiable anywhere. This is similar to the fact that most real numbers are irrational - after all, we can count the rational
numbers but we can't count the irrational ones.
All this means those few functions that are infinitely differentiable are special. | {"url":"http://www.shmoop.com/second-derivatives/abstract-things-summary.html","timestamp":"2014-04-16T04:49:43Z","content_type":null,"content_length":"26864","record_id":"<urn:uuid:dd978998-d925-493b-b9ea-64131197da2d>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00232-ip-10-147-4-33.ec2.internal.warc.gz"} |
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The Zeeman Effect in Finance: Libor Spectroscopy and Basis Risk Management
Marco, Bianchetti (2011): The Zeeman Effect in Finance: Libor Spectroscopy and Basis Risk Management.
Download (850Kb) | Preview
Once upon a time there was a classical financial world in which all the Libors were equal. Standard textbooks taught that simple relations held, such that, for example, a 6 months Libor Deposit was
replicable with a 3 months Libor Deposits plus a 3x6 months Forward Rate Agreement (FRA), and that Libor was a good proxy of the risk free rate required as basic building block of no-arbitrage
pricing theory. Nowadays, in the modern financial world after the credit crunch, some Libors are more equal than others, depending on their rate tenor, and classical formulas are history. Banks are
not anymore “too big to fail”, Libors are fixed by panels of risky banks, and they are risky rates themselves. These simple empirical facts carry very important consequences in derivative’s trading
and risk management, such as, for example, basis risk, collateralization and regulatory pressure in favour of Central Counterparties. Something that should be carefully considered by anyone managing
even a single plain vanilla Swap. In this qualitative note we review the problem trying to shed some light on this modern animal farm, recurring to an analogy with quantum physics, the Zeeman effect.
Item Type: MPRA Paper
Original The Zeeman Effect in Finance: Libor Spectroscopy and Basis Risk Management
Language: English
Keywords: crisis; liquidity; credit; counterparty; risk; fixed income; Libor; Euribor; Eonia; yield curve; forward curve; discount curve; single curve; multiple curve; collateral; CSA-discounting;
liquidity; funding; no arbitrage; pricing; interest rate derivatives; Deposit; FRA; Swap; OIS; Basis Swap; Zeeman; Lorentz; quantum mechanics; atomic physics
E - Macroeconomics and Monetary Economics > E4 - Money and Interest Rates > E43 - Interest Rates: Determination, Term Structure, and Effects
Subjects: G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing; Trading volume; Bond Interest Rates
G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing; Futures Pricing
Item ID: 42247
Depositing Marco Bianchetti
Date 28. Oct 2012 03:50
Last 15. Feb 2013 08:16
[1] See Zeeman’s biography at the Nobel Prize website http://nobelprize.org and Wikipedia at http://en.wikipedia.org/wiki/Pieter_Zeeman.
[2] See P. Zeeman, "The Effect of Magnetisation on the Nature of Light Emitted by a Substance", Nature 55, p. 347, 11 February 1897; R. P. Feynman, R. Leighton, M. Sands, “The Feynman
Lectures on Physics”, vol. 3, ch. 12-4; Wikipedia at http://en.wikipedia.org/wiki/Zeeman_effect.
[3] D. Wood, “Libor fix”, Risk Magazine, 1 Jul. 2011.
[4] C. Snider, T. Youle, “Does the Libor reflect banks’ borrowing costs ?”, 2 Apr. 2010, SSRN working paper, http://ssrn.com/abstract=1569603
[5] M. Morini, ”Solving the Puzzle in the Interest Rate Market”, Oct. 2009, SSRN working paper, http://ssrn.com/abstract=1506046.
[6] B. Tuckman and P. Porfirio, “Interest Rate Parity, Money Market Basis Swap, and Cross-Currency Basis Swap”, Lehman Brothers Fixed Income Liquid Markets Research – LMR Quarterly ,
2004, Q2.
[7] ISDA, “ISDA Margin survey 2011”, 14 Apr. 2011, http://www.isda.org.
[8] V. Piterbarg, “Funding beyond discounting: collateral agreements and derivatives pricing“, Risk, Feb. 2010.
[9] M. Morini, A. Prampolini, ”Risky Funding with counterparty and liquidity charges”, Risk, Mar. 2011.
[10] C. Burgard, M. Kjaer, “In the Balance”, 14 Mar. 2011, SSRN working paper, http://ssrn.com/abstract=1785262
[11] C. Fries, “Discounting Revisited – Valuations under Funding Costs, Counterparty Risk and Collateralization”, 15 May 2010, SSRN working paper, http://ssrn.com/abstract=1609587.
[12] A. Castagna, “”Funding, Liquidity, Credit and Counterparty Risk: Links and Implications”, DefaultRisk.com working paper, http://www.defaultrisk.com/pp_liqty_53.htm.
[13] M. Fujii and A. Takahashi, “Choice of Collateral Currency”, Risk, Jan. 2011.
[14] F. Ametrano, M. Bianchetti, “Bootstrapping the Illiquidity: Multiple Yield Curves Construction For Market Coherent Forward Rates Estimation”, in “Modeling Interest Rates: Latest
Advances for Derivatives Pricing”, edited by F. Mercurio, Risk Books, 2009.
[15] H. Lipman, F. Mercurio, “The New Swap Math”, Bloomberg Markets, Feb. 2010.
[16] M. Bianchetti, “Two Curves, One Price”, Risk, August 2010.
[17] M. Bianchetti, M. Carlicchi “Interest Rates after the Credit Crunch: Multiple Curve Vanilla Derivatives and SABR”, SSRN working paper, http://ssrn.com/abstract=1783070.
[18] N. Sawyer, “ISDA working group to draw up new, standardised CSA”, Risk, 15 Feb. 2011.
[19] D. Brigo, A. Capponi, A. Pallavicini, V. Papatheodorou, “Collateral Margining in Arbitrage-Free Counterparty Valuation Adjustment including Re-Hypotecation and Netting”, SSRN
working paper, http://ssrn.com/abstract=1744101.
[20] F. Mercurio, “Modern LIBOR Market Models: Using Different Curves for Projecting Rates and for Discounting”, International Journal of Theoretical and Applied Finance, Vol. 13, No. 1,
2010, pp. 113-137.
[21] D. Brigo and F. Mercurio, “Interest Rate Models: Theory and Practice”, 2nd edition, 2006, Springer.
[22] Leif B.G. Andersen and Vladimir V. Piterbarg, “Interest Rate Modeling”, 1st edition, 2010, Atlantic Financial Press.
URI: http://mpra.ub.uni-muenchen.de/id/eprint/42247 | {"url":"http://mpra.ub.uni-muenchen.de/42247/","timestamp":"2014-04-17T19:06:07Z","content_type":null,"content_length":"26342","record_id":"<urn:uuid:aecface1-1c31-4c12-93d9-8bd550d55261>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00002-ip-10-147-4-33.ec2.internal.warc.gz"} |
ing your
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Hedging your bets
01 June 2006
In the second of the series on trading strategies, Jim Hanly and John Cotter* ask, how effective are hedging strategies?
The terms risk management and hedging are often used together. Hedging is a component of risk management as it is one of the methods available to control risk, and there are many hedging tools and
strategies that may be used to try and engineer an efficient hedging outcome. Hedging1 whereby the underlying spot asset is hedged by the corresponding futures contract is an important tool in the
management of risk.
This article examines the hedging effectiveness of different hedging strategies in risk management. It details and compares a number of risk measures that can be used to measure the effectiveness of
different hedging strategies. A key issue is the extent to which hedging reduces the volatility of investors positions. Since hedging can be interpreted as a form of insurance, it is worth
considering how effective that insurance is in terms of risk reduction, and it is this issue we focus our attention on.
Risk measures and hedging performance metrics
While risk management tends to mean the implementation of strategies designed to reduce risk, an important consideration is the measure that is used to define risk. We examine three commonly applied
measures of risk in terms of their attributes, and their
suitability as risk measures in the context of evaluating hedging strategies from the perspective of risk reduction.
Variance and the standard deviation
The use of standard deviation2 is widespread in terms of measuring risk. Moreover, it is extensively used by hedgers following Ederington (1979), who suggests that the main purpose of hedging is to
minimise the variance associated with the hedge, referred to as the Minimum Variance Hedge Ratio (MVHR). However, this measure has a number of shortcomings.
Firstly, it is not an intuitive measure, as it deals with manipulating deviations by first squaring them and then taking the square root of the outcomes. More importantly, the standard deviation
cannot distinguish between positive and negative returns and therefore it does not provide an accurate measure of risk for asymmetric distributions (non-normal)3. Given asymmetry, hedging
effectiveness metrics that cannot distinguish between tail probabilities may be inaccurate in terms of risk measurement. Also, since it cannot measure the left and right tails of the distribution, it
fails as a risk measure to differentiate in terms of hedging effectiveness between short and long hedgers4. Where asset return distributions are asymmetric as is generally the case, the standard
deviation will over or underestimate tail risk. It is therefore not an adequate measure of risk for hedgers except in the event that the return distribution is symmetric (normal). Even in this case
the semi-variance is more appropriate5.
Value at Risk (VaR)
Recently VaR has become a very popular market risk measure. We can use it as an objective function in hedging when the user tries to minimise the VaR associated with their hedge. VaR is essentially a
quantile of a loss distribution6 and is widely used in financial risk management and the evaluation of the effectiveness of hedge strategies7. In particular, unlike the variance, it is estimated
separately for upside and downside risk (see figure 1 for downside VaR). VaR has become popular despite a number of limitations as a risk measure. Its most serious theoretical shortcoming is that it
is not a coherent measure of risk, as it is not sub-additive. A risk measure, r(.) is sub-additive if r(X+Y)<r(X) + r(Y) implying that aggregating risk does not increase the risk of the portfolio
over the sum of the risks of the constituent sub-portfolios. The fact that VaR is not sub-additive leads to strange 'negative' diversification effects in the context of portfolio theory. Also, in
practice, two portfolios may have the same VaR but exhibit very different potential losses. In other words, the VaR cannot tell us what the likely losses will be in the event that the VaR is exceeded
(see figure 1). It is of limited use therefore as a valid measure of risk to use in the evaluation of hedge strategies in situations where hedging is concerned with protecting against extreme losses
such as those associated with tail events in excess of the VaR. These shortcomings can be addressed by the use of the Conditional Value at Risk (CVaR) measure that has the property of sub-additivity.
CVaR is the expected loss conditional that we have exceeded the VaR, which is essentially a weighted average of the losses that exceed the VaR. CVaR is preferable to the VaR because it estimates not
only the probability of a loss, but also the magnitude of a possible loss. Furthermore, CVaR exhibits the sub-additive property and is thus coherent. Risk measures such as CVaR are increasingly being
incorporated into risk management systems as an additional tool that may be viewed in conjunction with a VaR statement, to give not only the probability of a tail loss, but also some indication of
the potential losses that may arise from a tail event (see figure 1). We can use CVaR as an alternative objective function to VaR where the hedger tries to minimise the CVaR associated with the
hedging strategy.
Hedging effectiveness
Having examined these three different risk measures that a hedging strategy may attempt to reduce, we now consider the effectiveness of various hedging strategies in terms of their ability to reduce
these risk measures, and to examine whether there are differences between different hedging strategies in terms of hedging effectiveness. We illustrate our results for the Nymex8 crude oil futures
contract that is much in demand at present given uncertain energy prices.
Hedging strategies
We examine hedging effectiveness for three separate strategies. First, investors may of course choose not to hedge their exposures, that we call a No hedge strategy. Second, in the event that hedging
is considered, the simplest hedge strategy using futures is a Naïve hedge. This strategy uses a hedge ratio9 of 1:1 where each unit of the crude oil contract is hedged with equivalent units in an
opposite position in a futures contract. Third, model based hedges such as moving window Ordinary Least Squares or GARCH models have become popular as a means of running a dynamic hedging strategy.
These strategies involve continually monitoring the hedge over time and changing the hedge to reflect changes that may occur in the relationship between the spot asset being hedged and the underlying
hedging instrument.
We now turn to some examples to investigate whether hedging is as effective at reducing VaR and CVaR as it is in reducing a more general risk measure such as the standard deviation. We examine the
three risk measures outlined to determine both the possible losses and the hedging effectiveness based on a No hedge, a Naïve hedge and a dynamic daily hedging strategy10 (model hedge) as applied to
crude oil exposures. The hedging instrument used is the corresponding futures contract and hedging effectiveness is measured not just by the standard deviation, but also by VaR and CVaR. We also
examined11 hedging effectiveness for both symmetric and asymmetric distributions to demonstrate how different distributional characteristics that are found in real world hedging situations would
affect the various risk measures. Figure 2 presents estimated daily losses as measured by the Standard Deviation, VaR and CVaR metrics respectively.
A number of interesting points arise. Firstly, we can see that both the VaR and CVaR risk measures are useful in that they can distinguish performance for short hedgers from those of long hedgers.
Using these risk measures we can see that there are significant differences in terms of the potential losses associated with the separate tails of the distribution. For example, the one-day CVaR
figures (symmetric distribution) using a Naïve Hedge are $37,330 for short hedgers as compared with $29,780 for long hedgers. Secondly, we can see that potential tail losses are quite different for
symmetric as compared with asymmetric distributions. For example, while the estimated losses are similar for opposite tails of the symmetric distribution, there are large differences between left and
right tails in the case of the asymmetric distribution. This means that the estimated losses of short and long hedgers differ significantly, and differences would become more pronounced for the
(more) skewed distributions. This implies that hedgers who fail to use tail specific hedging performance metrics may chose inefficient hedging strategies that result in them being mishedged vis a vis
their hedging objectives.
Figure 3 demonstrates the benefits of hedging as compared with leaving crude oil exposures unhedged. The percentage reductions in the relevant risk measure are calculated using the figures presented
in figure 2. For example, a Naïve Hedge strategy reduces the VaR for a short hedger under the symmetric distribution by 42% (i.e. from a one-day VaR of $51,150 for No Hedge to $29,660 for a Naïve
Hedge). The model based hedge is even better with a 46% reduction in the one-day VaR. Also from figure 3, we can see that both the Naïve and Model based hedges outperform a No Hedge position in all
cases bar one. This demonstrates the value of hedging as a method of reducing risk across each of the different risk metrics employed.
A second point relates to the hedging performance for symmetric as compared with the asymmetric distributions. We can see that hedging performance is significantly better for the symmetric
distribution with reductions in standard deviation of around 60% and VaR and CVaR reductions of the order of 40-50%. For the asymmetric distribution however, the story changes with significantly
worse hedging effectiveness observed across each risk measure. Using the case of short hedgers for example, the Model Based Hedge will only reduce the VaR by 21% and the CVaR by 10%. This may
indicate that hedging may not be as effective during periods of high volatility associated with asymmetric return distributions. Thus hedgers may face the risk that their hedges may not be as
effective during periods when they most require them. Again demonstrated are the different hedging outcomes for short as compared with long hedgers. In this example, the long hedgers benefit more
than short hedgers as measured by larger reductions in both VaR and CVaR. This demonstrates the ability of the VaR and CVaR metrics to model tail events and to differentiate between the tails of the
distribution whereas the standard deviation is limited in this respect.
This article has put forward some justifications for the use of a number of risk measures as part of an overall risk management strategy. We have highlighted that traditional measures based on
standard deviation are not capable of measuring risk in the same way as tail specific ones, as they cannot distinguish between left and right tail probabilities as required by short and long hedgers.
While the VaR is tail specific, we have also noted some potential shortcomings of the measure and put forward an alternative measure - the CVaR which addresses some weaknesses of VaR. The message for
investors is that a number of risk measures should be considered when designing a hedging strategy, but more importantly, they need to decide which risk measure they are seeking to minimise, as
hedging effectiveness may vary, depending on the risk measure used. Also, hedges may not be as
effective at reducing risk in volatile markets that are skewed. In hedging terms, this means that investors may face the risk that their hedges will not fulfill their function of risk reduction
during stressful markets conditions when they are most needed. q
Cotter, J & Hanly, J (2006a). Re-examining Hedging Performance. Journal of Futures Markets.
Cotter, J & Hanly, J (2006b). Hedging Effectiveness under Conditions of Asymmetry. Working Paper. UCD.
Ederington, L (1979). 'The Hedging Performance of the New Futures Markets'. Journal of Finance.
1 Hedging can be static or dynamic by definition. Static hedging refers to a hedging strategy that does not change to accommodate changes in the underlying relationship between the asset being hedged
and the hedging instrument. Dynamic hedging accounts for the changing relationship between the spot asset and the hedging instrument by changing the ratio of futures contracts to spot exposure based
on a time-varying relationship.
2 The standard deviation is the square root of the variance. Minimising the variance is equivalent to minimising the standard deviation.
3 The variance or standard deviation makes no distinction between positive and negative deviations from the mean and it does not measure separate loss probabilities for non-symmetric distributions.
For example, assume the mean return is 0% and there are four returns [1, -2, 6, -12]. The standard deviation is 13.8 and does not say anything about the greater potential for downside risk (negative
deviations from the mean) than upside risk (positive deviations from the mean).
4 Short hedgers are investors who hold the spot asset and seek to go short the futures contract in order to reduce risk whereas long hedgers are short the spot asset and long the futures contract.
5 The semi-variance is half the variance for a symmetric distribution.
6 A quantile is the quantity associated with a particular cumulative probability. This is the loss level of a portfolio over a certain period that will not be exceeded with a specified probability.
VaR has two parameters, the time horizon (N) and the confidence level (x). Generally VaR is the (100-x)th percentile of the portfolio over N days.
7 See Cotter and Hanly (2006a).
8 We use the Nymex West Texas light sweet crude contract to examine energy prices. This is the most liquid and highly traded contract by volume and is used as an oil price benchmark.
9 The hedge ratio refers to the number of futures contracts used to hedge a given spot exposure. It is usually chosen to minimise an appropriate risk measure, for example the variance (standard
deviation), the VaR or the CVaR.
10 We used both a Rolling window Ordinary Least Square (OLS) and two GARCH models, the DVECH and the Asymmetric DVECH models to calculate optimal hedging strategies. The results reported are from the
best performing model.
11 Results shown are out-of-sample based on forecasted hedging strategies to reflect real world hedging situations. See Cotter and Hanly (2006b). | {"url":"http://www.fow.com/Article/1385896/Issue/26557/Hedging-your-bets.html","timestamp":"2014-04-18T10:33:38Z","content_type":null,"content_length":"84173","record_id":"<urn:uuid:f715a251-ac4d-4f67-9456-3c24468bb0d5>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00554-ip-10-147-4-33.ec2.internal.warc.gz"} |
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• Four pumpkins to a page, with numbers 1-12. Includes minus, add, and equal signs. Room to draw pumpkin faces.
• Fill in the squares below by adding the two numbers that meet in the square. Use your finger, pencil, or ruler to help you track the grid.
• [member-created with abctools] 27 pages of worksheets introducing multiplication to 12.
• Introduction to the concept, with several pages for practice. These are in color so you can make a transparency, but they will look fine in black and white for the students.
• Fill in the squares below by adding the two numbers that meet in the square. Use your finger, pencil, or ruler to help you track the grid.
• Fill in the appropriate operators (addition or subtraction).
• Make 8 copies of the kites and bows. Write an "answer" to a math problem on each kite. Write various combinations of math problems (that match up with each answer) on the bows.
• Two addends, one digit, columns. Numbers 0, 1, and 2. Four pages plus answers.
Worksheet practicing rounding numbers to the nearest 100 and estimating the sums. Common Core: 4.NBT.A.3
Double-six dominoes, but with weather words instead of dots. Great for early literacy/language instruction.
Our math "machines" make addition drills fun. Cut out the two shapes and practice simple addition.
"Last year Lucas collected 159 items trick-or-treating. This year, he collected a total of 249 items. How many items did he collect altogether in both years?" Five math word problems with a
Halloween theme.
Addition Activity:; Add the sums (up to 18) and use the key to color the picture of two runners.
Clearly explains how to add partial sums, followed by a page of practice sums.
This set of worksheets introduces the fundamentals of addition with concrete examples, story problems, and colorful pictures.
Students fill in missing sums for illustrated basic facts number sentences
Students understand the concept of addition by adding numbers, with sums up to twenty. Printable PDF packet with answers. Coordinated Smart Notebook interactive available.
"Use the number from the table above to complete the problems below."
Match the math problems (with sums to 15) to the answers to win the game.
Match the math problems (with sums to 15) to the answers to win the game.
"Help the knight find a path to the castle. Draw a path through each stone with an answer of 8."
Help the knight find a path to the dragon. Draw a path through each stone with an answer of 3.
"Here is a table of what Jenny planted in the new flower garden. Read the table and answer the questions."
"The duckling lived in a farm with 45 animals. There were 10 pigs, 12 goats, 5 dogs, 2 horses and the rest were sheep." Choose the equation that best expresses the word problem.
"Carol and Linda are making candy apples for a party. They made sixty-two. Today they made eight more. How many candy apples have they made in all?" Five math word problems with a Halloween
"Mike and his brother Pat are selling pumpkins they grew. If Mike sells nine and Pat sells seven, how many do they sell in all?" Five Thanksgiving-themed addition word problems.
"Ed and his sister are helping cook Thanksgiving dinner. They decide to have a potato-peeling contest. Ed peels twenty-two. His sister peels six more than that. How many did she peel?" Five
Thanksgiving-themed addition word problems.
"Myron analyzed the caloric content of his Thanksgiving dinner for his science class. His serving of stuffing had 619 calories. His serving of pumpkin pie had 780 calories. His serving of turkey
had 282 calories. How many calories were there in his stuffing and turkey combined?"
Solve the simple addition problems. Use the answers to color the picture.
Solve the simple addition problems. Use the answers to color the picture.
Add the sums (up to 8) and use the key to color the picture of a sunflower turned to the sun.
Addition and subtraction practice and assessment. Includes; practice worksheets, in and out boxes, and matching game.
Students write addition problems (with sums up to 18). When the answer to the problem is called, they cover the square. Comes with a page of suggested equations.
25 cards, each with equations with sums up to 18. The calling card has the sums.
Roll the die and then record the coin shown on the face. Add the coin values. Three cubes for a range of games: pennies and nickels; pennies, nickels, and dimes; pennies, nickels, dimes, and
Roll a die with coin pictures on the faces (available on abcteach) or draw coins out of a bag. Write the value of each coin in the row, then add up the row.
Clearly explains how to add partial sums, followed by a page of practice.
A crayon shaped booklet of addition and subtraction word problems about crayons.
Students fill in missing sums for illustrated basic fact number sentences.
Students fill in missing sums for illustrated basic fact number sentences.
Students fill in missing sums for illustrated basic facts number sentences.
Students fill in missing sums for illustrated basic facts number sentences.
Students fill in missing sums for illustrated basic facts number sentences.
Students fill in missing sums for illustrated basic facts number sentences.
Students fill in missing sums for illustrated basic facts number sentences.
Students fill in missing sums for illustrated basic facts number sentences.
Students fill in missing sums for illustrated basic facts number sentences.
Students fill in missing sums for illustrated basic facts number sentences.
Students fill in missing sums for illustrated basic facts number sentences.
Complete the In and Out boxes to solve for numbers within 20. CC: Math: 1.OA.A.1, K.CC.B.5
Solve ten story problems. Add and subtract numbers in the thousands and estimate the answer. Includes answer key.
Solve six each difficult addition and subtraction problems, answer sheet included.
Solve six each medium difficulty addition and subtraction problems,answer sheet included.
Solve four subraction problems using base 10 blocks. Includes answer key.
Set of 9 insect-themed posters illustrating the addition of ten ones and further ones to find sums 11-19. Correlates to common core math standards.
Common Core Math: K.NBT.1
Commutative property addition worksheet. Includes ten frames, addition, and word problems using the common core standards in first grade math. Common Core Math: 1.OA.B.3, 2.OA.C.4
Students use the rule to fill in the blank boxes..
Common Core 3.OA.9
Fill in the appropriate operators, addition or subtraction symbols.
Fill in the appropriate operator symbols, addition or subtraction.
Seven pages of practice worksheets, multiple choice answers. This is formatted for testing practice.
Solve simple addition problems using In and Out boxes. Numbers are within 10.
This penguin theme unit is a great way to practice counting and adding to ten. This 16 page unit includes, tracing numbers 1-10, counting in order, finding patterns, ten frame activity, in and
out boxes and much more! CC: Math: K.CC.B.4
This penguin theme unit is a great way to practice counting and adding to 20. This 21 page unit includes; tracing numbers, cut and paste, finding patterns, ten frame activity, in and out boxes
and much more! CC: Math: K.CC.B.4 | {"url":"http://www.abcteach.com/directory/subjects-math-addition-650-2-0","timestamp":"2014-04-21T07:31:18Z","content_type":null,"content_length":"405942","record_id":"<urn:uuid:acba1e0b-613e-4bda-bc9b-69a8c94454a7>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00026-ip-10-147-4-33.ec2.internal.warc.gz"} |
Browse by 1. Referees
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Yener, Serkan (2012): Nonparametric estimation of the jump component in financial time series. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
Yener, Tina (2011): Risk management beyond correlation. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
Racheva-Iotova, Borjana (2010): An Integrated System for Market Risk, Credit Risk and Portfolio Optimization Based on Heavy-Tailed Medols and Downside Risk Measures. Dissertation, LMU München:
Faculty of Mathematics, Computer Science and Statistics
Wohlrabe, Klaus (2009): Forecasting with mixed-frequency time series models. Dissertation, LMU München: Faculty of Economics
Reese, Christof (2007): Grenzen der Quantifizierung operationeller Risiken. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
Pigorsch, Christian (2007): Estimation of Continuous–Time Financial Models Using High–Frequency Data. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
Kalcheva, Katerina (2006): Essays on the Empirics of Transition. Dissertation, LMU München: Faculty of Economics
Grzybowski, Lukasz (2005): Essays on Economics of Network Industries: Mobile Telephony. Dissertation, LMU München: Faculty of Economics
Cerquera, Daniel (2005): Dynamic R&D Incentives with Network Externalities. Dissertation, LMU München: Faculty of Economics | {"url":"http://edoc.ub.uni-muenchen.de/view/gutachter/Mittnik=3AStefan=3A=3A.html","timestamp":"2014-04-18T20:44:19Z","content_type":null,"content_length":"13348","record_id":"<urn:uuid:68248646-84df-4dd7-837b-5dc7c71f906a>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00094-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hilbert Space
November 19th 2009, 03:24 AM #1
Aug 2009
Hilbert Space
Consider the Hilbert Space $H = L^2 (-1,1)$ of Lebesgue integrable real valued function on $(-1,1)$ with inner product $(f,g) = \int_{(-1,1)} fg dm$.
Prove that the set of all even function in $H$ is a closed subspace of $H$.
Well, you know that all you have to prove is that a non-empty subset of vector space is closed under addition and scalar multiplication to prove it is a subspace, don't you? So you want to prove
that the sum of two even functions is an even function and that the product of a number and an even function is an even function.
Well, proving it's a subspace is quite easy. For the fact that it's closed you'll need the following:
1) If $(f_n) \subset L^1 (\Omega)$ is a sequence of integrable functions such that $\lim_{n\rightarrow \infty } \int_{\Omega } \vert f_n \vert =0$ then there exist a subsequence $(f_{n_k} )$ such
that $f_{n_k} \rightarrow 0$ a.e. on $\Omega$
Take $n_k<n_{k+1}$ such that $\int_{\Omega } \vert f_{n_k} < \frac{1}{2^k}$ then $\int_{\Omega } \sum_{k=1}^{\infty } \vert f_{n_k} \vert \leq \sum_{k=1}^{\infty } \int_{\Omega } \vert f_{n_k} \
leq 1$ and so $f_{n_k} \rightarrow 0$ a.e. on $\Omega$.
2)If $(f_n) \subset L^p(\Omega )$ is a sequence of integrable functions such that $f_n\rightarrow f$ in $L^p(\Omega )$ then there exists a subsequence $(f_{n_k})$ such that $f_{n_k}(x) \
rightarrow f(x)$ a.e. on $\Omega$.
Since $\lim_{n\rightarrow \infty } \int_{\Omega } \vert f_n -f \vert ^p =0$ by (1) we have that there exists a subsequence $f_{n_k}$ such that $\vert f_{n_k}(x)-f(x) \vert ^p \rightarrow 0$ a.e.
on $\Omega$ and so the result follows. Notice that $p\in [1,\infty )$ but in fact the case $p=\infty$ is easier since you don't even need to pick a subsequence (you see why?).
Now let $(f_n) \subset L^2(-1,1)$ be even functions such that $f_n\rightarrow f$ in $L^2(-1,1)$ then there exists a subsequence $f_{n_k}$ such that $f_{n_k}(x) \rightarrow f(x)$ a.e. on $(-1,1)$
and so let $Z:=\{ x\in (-1,1) : f_{n_k}(x) rightarrow f(x) \}$ then it's obvious $\mu (Z)=0$ and if $x\in (-1,1) \setminus Z$ then $f(x) = \lim_{k\rightarrow \infty } f_{n_k}(x)= \lim_{k\
rightarrow \infty } f_{n_k}(-x)=f(-x)$. Now just define $g(x)=f(x)$ if $x\in (-1,1)\setminus Z$ and $g(x)=0$ otherwise then $g=f$ in $L^2(-1,1)$ and $g$ is even.
November 19th 2009, 04:28 AM #2
MHF Contributor
Apr 2005
November 19th 2009, 09:42 AM #3
Super Member
Apr 2009 | {"url":"http://mathhelpforum.com/differential-geometry/115554-hilbert-space.html","timestamp":"2014-04-18T12:20:50Z","content_type":null,"content_length":"46022","record_id":"<urn:uuid:94a4a0e6-8f0a-4ef1-a1e2-692b3895599e>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00030-ip-10-147-4-33.ec2.internal.warc.gz"} |
[SOLVED] Prove log x < e^x
September 3rd 2009, 08:01 PM #1
Senior Member
Feb 2008
[SOLVED] Prove log x < e^x
Hey guys. I seem to have forgotten how to work with log functions. Can someone please remind me how to show that
$\log x < e^x$
Either a generalized solution for all $x\in\mathbb{R}$ or else a more specific solution for $x>k$ for some constant $k$ will work fine for my purposes.
The statement $\log x < e^x$ is satisfied if and only if $e^{\log x} < e^{e^x}$, so $x < e^{e^x}$.
This is true because $y<e^y$ so $x < e^x$ and $e^x < e^{e^x}$ by setting $y=e^x$. But then it means $x<e^{e^x}$.
September 3rd 2009, 08:04 PM #2
Global Moderator
Nov 2005
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floating point division
Author floating point division
Ranch Hand
Joined: Mar 01, 2004
Posts: 100 Why is the output infinity
and not -infinity
Ranch Hand
To get -Infinity you have to use -0.0 as the denominator.
Joined: Aug 03, 2002
Posts: 7729 My interpretation of your example is as follows:
-0 is done with integer arithmetic, so -0 -> -(0) -> 0 and that gets converted to 0.0 before the division is done. Therefore (+)Infinity is the result.
Ask a Meaningful Question and HowToAskQuestionsOnJavaRanch
Getting someone to think and try something out is much more useful than just telling them the answer.
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Mplus Discussion >> Why slope variance changes in growth models
Why slope variance changes in growth ...
George Howe posted on Wednesday, October 03, 2012 - 7:52 am
Hi folks,
(Congrats on getting 7 up and out, by the way!).
I'm modeling growth on two separate variables simultaneously. Three time points, using TSCORE for varying times of measurement. Runs fine.
I had run a separate grwoth model using only one of the variables, and I noticed that the estimated variance for the slope of that variable was quite different when it was run alone, compared to when
it was run in the parallel growth model. THe parallel model allows the slopes and intercepts to correlate, but nothing is regressed on anything, so I can't figure out why the variance estimate would
I did notice that the N is slightly different (421 vs 423), so I also ran a simple growth model for the first variable but included the three indicators of the second growth variable only as measured
variables, allowing them to correlate with each other (this made the Ns equivalent to those in the parallel growth model). This gave the same slope variance estimate as in the simple growth model, so
it doesn't look like it could be different N's causing the different variance estimates.
This does lead to substantive differences (in the parallel growth model the slope variance is over twice as large, and robustly significant, while in the single growth model it is smaller and
Any thoughts as to why this might be happening?
Bengt O. Muthen posted on Wednesday, October 03, 2012 - 10:19 am
It sounds like it may have to do with the restrictions imposed by the parallel process model, namely that correlations between the outcomes of the two processes have to be channeled through the
growth factors. You can try to relax that by for instance correlating the concurrent residuals of the outcomes of the two processes. And see if the growth factor estimates are sensitive to those
residual correlations being allowed or not.
George Howe posted on Wednesday, October 03, 2012 - 10:52 am
You're right, although the results are more complex. In the parallel growth model, if I leave the correlations among slopes and intercepts but also include correlated residuals with concurrent
measures of the two variables, the slope variance decreases some but is still greater than that in the single variable growth model. However if I simply force the correlations to zero between slope
and intercept across variables (but keep within-variable slope-intercept correlations free), the variance estimate is almost identical to that in the single variable growth model.
So it does seem to be due to the cross-process correlations.
However, I'm still not clear why this is the case, or what the implication is. These findings would make sense if I was somehow independently accounting for residual variance in the first variable
process, allowing for a more sensitive test of the slope variance by removing error, but that's not the case with this model.
The correlated parallel process model indicates that there is systematic slope variance worth exploring, which I like, but if I hadn't included the other process this wouldn't be the case. Bottom
line: am I justified in testing covariate associations with that slope, keeping the parallel process in the model but only because the slope variance remains significant, or is that tantamount to
cherry-picking the model that serves my goals?
Bengt O. Muthen posted on Wednesday, October 03, 2012 - 11:24 am
You may want to check how things look if you include a list of key time-invariant covariates. That sometimes changes the variance assessment for growth factors. Perhaps results are more similar then,
comparing single- and parallel-processes.
If you have key time-varying covariates, that may also change the picture.
I assume that your parallel-process model fits the data well.
Harald Gerber posted on Sunday, December 02, 2012 - 9:34 am
I have a variable of main interest and I want to correlate its growth factors with the growth factors of three covariates (5 measurement points in total). To do this, I have modeled three separate
bivariate growth models. I would like to compare the growth factor correlations (esp. the "between processes"-correlations) of these models but what makes this difficult is the issue of correlated
concurrent residuals, which is more or less prevalent in each of the three bivariate models. My strategy was to estimate the concurrent residual correlations in each bivariate model first and to fix
those correlations to zero which proved to be insignificant (p > .05). In one bivariate model, for instance, there were no significant concurrent residual correlations (thus, all 5 correlations were
fixed to zero) and in another bivariate model there were three significant correlations (thus, I only fixed 2 concurrent residual correlations to zero). The result is, that there are higher
"between-processes"-correlations in the first model as compared to second (and I'm afraid that this is mostly due to fixed concurrent correlations between residuals in the first model). Is my
approach nevertheless appropriate?
Harald Gerber posted on Thursday, December 06, 2012 - 5:35 am
May be my question was a bit too difficult to answer if one does not have the data, sorry.
Easier: Would you estimate "all" correlations between contemperanous residuals "by default" in bivariate growth models (even if some of them are insignificant?) or would you only estimate
statistically significant correlations between concurrent residuals (ps < .05). My observation is that it makes quite a big difference between both procedures (with respect to the strength and
significance of growth factor correlations between the processes) although both approaches appear acceptable at a first glance.
Bengt O. Muthen posted on Thursday, December 06, 2012 - 8:35 am
I would take an "a priori" approach and free all concurrent residual covariances - and let them stay in the model even if some are non-significant.
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A New Model Of Quantum Dots: Rethinking The Electronics
"One of the interesting things about quantum dots is that their band gaps are much larger than the same material in bulk. At the same time their overall dielectric constants are much smaller," says
Lin-Wang Wang of Berkeley Lab's Computational Research Division. "Therefore it was natural to assume that the size of the band gap in a quantum dot is what determines its overall dielectric
Recently French researchers led by Christophe Delerue of the Institut Supérieur d'Electronique du Nord raised doubts about this assumed relationship, however, basing their argument on approximate
calculations. To test the questions posed by the French group, Wang and postdoctoral fellow Xavier Cartoixà performed, for the first time, ab initio ("from first principles") microscopic studies of
the dielectric function in quantum dots. To do so they used PEtot, a quantum-mechanical electronic-structure program developed by Wang, on the Seaborg supercomputer at the Department of Energy's
National Energy Research Scientific Computing Center (NERSC), based at Berkeley Lab.
Wang and Cartoixá's results, published in the June 17, 2005 issue of Physical Review Letters, led them to devise a simple mathematical model, the first that nanoscience researchers can use for quick,
consistent calculations of the dielectric function in nanocrystals.
Tunable band gaps and a rainbow of colors
"One useful feature of quantum dots is that the colors of light they absorb and emit can be tuned simply by varying their size," says Wang. "This is because dots of the same material but different
sizes have different band gaps, which absorb and emit different frequencies."
The band gap of a semiconductor like silicon or gallium arsenide is the energy required to lift an electron from its valence band, filled with electrons, to its conduction band, which is empty. For
example, an incoming photon whose energy matches or exceeds the band gap can boost an electron into the conduction band, leaving behind a "hole" of opposite charge. This is the principle that
underlies photovoltaic cells, which generate electrical current when stimulated by light.
Conversely, when an electron falls from the conduction band back down to the valence band, eliminating a hole, the lost energy is emitted as light whose color corresponds to the band gap -- this is
the principle behind light-emitting diodes, LEDs.
Each semiconductor has a characteristic band gap, but when the diameter of a piece of the material is shorter than the quantum-mechanical wave function of its electrons, the "squeezed" electron wave
function makes the band gap wider. For an electron to jump from the valence band to the conduction band now requires more energy.
"In a classical picture this would be like the electron, which is free to meander through the bulk material, suddenly being forced to speed up in a confined space," Lin-Wang Wang says -- analogous to
a circus motorcycle rider moving faster inside a steel cage.
The smaller the quantum dot, the wider the band gap. The band gap of gallium arsenide in bulk, for example, is 1.52 electron volts (eV), while a quantum dot consisting of 933 atoms of gallium and
arsenic has a band gap of 2.8 eV, and a dot half as big, with 465 atoms, has a band gap of 3.2 eV -- about twice that of the bulk material. Changing the band gap, and thus the color of light a
quantum dot absorbs or emits, requires only adding or subtracting atoms from the quantum dot.
Enter the dielectric constant
The electron-hole pair formed when an incoming photon boosts an electron out of the valence band into the conduction band is called an exciton. An exciton's energy (which corresponds to the color of
the quantum dot) is not identical with the band gap; instead it depends on a number of other factors.
Most important is the dielectric function inside the quantum dot, which mediates how strongly the exciton's negatively charged electron and positively charged hole attract each other. Calculating the
dielectric function is thus essential to understanding how excitons behave in a quantum dot (including its exact color) and how its electronic states can be manipulated -- for example by adding
dopant atoms that seed the semiconductor with extra electrons or holes.
In 1994 Wang, then at DOE's National Renewable Energy Laboratory, and his colleague Alex Zunger found a consistent relationship between a quantum dot's band gap and its overall dielectric constant, a
relationship suggestive of the observed scaling between a dot's size and its band gap. A quantum dot's electric constant is the average of the dielectric function inside the dot. Advances in
computing now make it possible to calculate the dielectric function on the microscopic scale -- virtually atom by atom.
In the recent study, Wang and Cartoixà calculated what would happen if a single-electron "perturbation" -- caused by a dopant atom, for example -- were introduced into the center of a 933-atom
quantum dot of gallium arsenide. To replicate a realistic quantum dot, they "passivated" the atoms on its surface with fractionally charged hydrogen-like atoms, mimicking reactions between the dot
and its surroundings.
Using the Seaborg supercomputer at NERSC, the researchers were able to determine the electron charge density of the perturbation throughout the dot, using an ab initio calculation technique called
local density approximation. In the presence of a weak electric field the results were virtually identical to similar measurements of the bulk material -- at least until the responses were measured
near the surface of the dot.
They repeated the calculations for a 465-atom gallium-arsenide quantum dot, and also for a 465-atom quantum dot made of silicon. In the smaller dots, measurements near the center of the dot were
still similar to the bulk measurements -- but varied significantly where the perturbation vanishes, near the surface.
A simple model
Measured microscopically, the dielectric function inside a quantum dot is the same as it is in the bulk material; measurements near a perturbation in the center of the dot show no significant
difference, but in a small dot the differences are large near the boundary. Averaging makes it appear that the dielectric constant mimics size-dependent changes in the band gaps. But in fact there is
no direct relationship.
"Using many hours of supercomputer time, we calculated all the electronic states in these quantum dots when they were perturbed by a single electron in the middle," says Wang. "We found they were the
same as in the bulk." The electronic response of a quantum dot thus depends on where it is measured, and on the dot's size.
"If the response of the dot had been different from the bulk, it would have been hard to model," Wang says. "Instead we were able to devise a simple model for calculating the dielectric function on
the microscopic scale that gives virtually the same results as ab initio calculations with a supercomputer. This should be very useful in future calculations."
"Microscopic response effects in semiconductor quantum dots," by Xavier Cartoixà and Lin-Wang Wang, appears in the June 17, 2005, issue of Physical Review Letters (volume 94, number 23, article
236804) and is available online as of June 15 at http://prl.aps.org/.
Berkeley Lab is a U.S. Department of Energy national laboratory located in Berkeley, California. It conducts unclassified scientific research and is managed by the University of California. Visit our
website at http://www.lbl.gov.
Story Source:
The above story is based on materials provided by DOE/Lawrence Berkeley National Laboratory. Note: Materials may be edited for content and length.
Cite This Page:
DOE/Lawrence Berkeley National Laboratory. "A New Model Of Quantum Dots: Rethinking The Electronics." ScienceDaily. ScienceDaily, 17 June 2005. <www.sciencedaily.com/releases/2005/06/
DOE/Lawrence Berkeley National Laboratory. (2005, June 17). A New Model Of Quantum Dots: Rethinking The Electronics. ScienceDaily. Retrieved April 16, 2014 from www.sciencedaily.com/releases/2005/06/
DOE/Lawrence Berkeley National Laboratory. "A New Model Of Quantum Dots: Rethinking The Electronics." ScienceDaily. www.sciencedaily.com/releases/2005/06/050616060710.htm (accessed April 16, 2014). | {"url":"http://www.sciencedaily.com/releases/2005/06/050616060710.htm","timestamp":"2014-04-17T01:47:50Z","content_type":null,"content_length":"91401","record_id":"<urn:uuid:f611827f-d90b-445c-b8a9-df2bebbb084d>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00008-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rules for Counting Significant Figures
Continuing on from my
previous post on Significant Figures
, here I will go over some of the general rules. The rules that help us identify a sig fig are not difficult, and there are not many... but they are very important to understand. As always,
understanding the basics is absolutely essential to being able to move forward.
As I mentioned
you DO NOT count a zero as a sig fig if it starts the number
0.1 has 1 sig fig
0.0045 has 2 sig figs
All non-zero digits ARE significant.
123 has 3 sig figs
0.007654 has 4 sig figs
A zero that is found between other non-zero digits DOES count as significant.
101 has 3 sig figs
0.3056 has 4 sig figs
Numbers with a zero at the end MAY count it as significant. The distinction is whether the number contains a decimal place. If it does, then a zero at the end DOES count as significant. If there is
no decimal place, then a zero at the end does NOT count as significant.
100 has 1 sig fig
100.0 has 4 sig figs
100. has 3 sig figs
50.30 has 4 sig figs
2.03000 has 6 sig figs
0.0098700 has 5 sig figs
9,885,000 has 4 sig figs
As I mentioned, the number of significant figures is often a result of the degree of precision in a measurement. However, sometimes the value must be taken in context to find the correct number of
sig figs. For example, look at the number 100 above. I said it has 1 sig fig. However, if we are talking about a count of something... say, the number of houses on a street, then that value is
precise to 3 digits, and it would be fair to say that, in this case, it would have 3 sig figs. Sometimes, this can be noted with a decimal point at the end, without any extra zeros (100.).
And those are all the rules for counting Significant Figures. They're not difficult once you practice for a while and recognize where the various rules apply.
Now that we have figured out just what counts as a sig fig, the
next step
is being able to do math with them... add them, multiply them, etc... and to do that, of course, we have an entirely different set of rules. This is the part that ALWAYS confuses people, so I will do
my best to lay it out for you so that it makes sense.
2 comments:
1. Thanks for the easy to understand explanation, really helped me to understand significant figuresjer :)
2. Thank you soooo much | {"url":"http://sk19math.blogspot.com/2010/09/rules-for-counting-significant-figures.html","timestamp":"2014-04-21T07:10:40Z","content_type":null,"content_length":"114139","record_id":"<urn:uuid:179e6f86-2bfd-4db3-868d-93f54891fac2>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00533-ip-10-147-4-33.ec2.internal.warc.gz"} |
All NYC A Sleepaway Camp Where Math Is the Main Sport [Archive] - The School Boards
07-28-2011, 08:34 AM
From the N.Y. Times (http://www.nytimes.com/2011/07/28/nyregion/a-sleepaway-camp-for-low-income-ny-math-whizzes.html) by Rachel Cromidas:
As camps go, the Summer Program in Mathematical Problem Solving might sound like a recipe for misery: six hours of head-scratching math instruction each day and nights in a college dorm far from
But Mattie Williams, 13, who attends Middle School 343 in the Bronx, was happy to attend, giving up summer barbecues with her parents and afternoons in the park with her Chihuahua, Pepsi. She and 16
other adolescents are spending three weeks at Bard College here in a free, new camp for low-income students gifted in mathematics.
All are entering eighth grade at New York City public middle schools where at least 75 percent of the student body is eligible for free lunches. And all love math. At this camp, asking “What kind of
math do you like, algebra or geometry?” is considered an appropriate icebreaker, and invoking the newly learned term “the multiplication principle” elicits whoops and high-fives.
In a Bard classroom one afternoon, it seemed for a moment that Arturo Portnoy had stumped everyone. Dr. Portnoy, a math professor visiting from the University of Puerto Rico, posed this question:
“The length of a rectangle is increased by 10 percent and the width is decreased by 10 percent. What percentage of the old area is the new area?”
The 17 campers whispered and scribbled. One crumpled his paper into a ball. Mattie Williams may have looked as if she was doodling as she drew dozens of tiny rectangles in her notebook, but she was
hard at work on the problem, which was taken from the American Mathematics Competitions, a contest series known for its difficulty.
In less than 10 minutes, she had the answer — 99 percent — and was ready for the next question.
For some schoolchildren, mathematics is a competitive sport, and summer is the time for training — poring over test-prep books, taking practice exams and attending selective math camps. But for
students who cannot afford such programs, or have not been exposed to many advanced math concepts, the avenues to new skills are limited.
Daniel Zaharopol, the director of the camp at Bard (http://www.artofproblemsolving.org/spmps/program.html), is trying to change that. He has brought four math educators to the Bard campus to teach
the middle school students concepts as varied as number theory and cryptography. Among the instructors is Dr. Portnoy, a director of the Puerto Rico Mathematical Olympiads.
read more>> (http://www.nytimes.com/2011/07/28/nyregion/a-sleepaway-camp-for-low-income-ny-math-whizzes.html) | {"url":"http://www.theschoolboards.com/archive/index.php/t-1632.html","timestamp":"2014-04-25T05:42:43Z","content_type":null,"content_length":"6330","record_id":"<urn:uuid:598e9f6f-1dd9-4a1b-9293-e678cd5869ea>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00634-ip-10-147-4-33.ec2.internal.warc.gz"} |
Copyright © University of Cambridge. All rights reserved.
'Ante Up' printed from http://nrich.maths.org/
You can read more about the solution to this problem on the Plus website
Mark Yao from the British School of Manila correctly reasoned his way through this problem using tree diagrams and conditional probability; his results were similar to that obtained using the
systematic enumeration shown below. A systematic enumeration of the 64 possible continuations really helps with this problem. A key aspect is that of all possible continuations, each is equally
First part
Note that Turing CANNOT lose if a Head emerges next (This would be the same if the game continued indefinitely) and will win 31 out of 32 times if a Head emerges first; a draw will occur if six Heads
emerge. Moreover, Turing will win on at least two occasions if Tails emerges first. So, Turing is more likely to win than me. The following image will help you to see this:
Mark computed the probabilities as P(HTT wins) = 21/64; P(HHT wins) = 39/64.
Second part
We need to beat THT to a win, so it makes sense to see when THT would occur. Using the systematic enumeration from the previous question we can see that THT emerges first in the following places
It is quite easy to see that HHT will win over THT from this list; exact probabilities could be computed by counting.
Third part
Suppose that I choose TTT. Then, consider the first time TT has emerged. Neither Turing nor I will have won prior to this point, and the game will certainly be over after the next square. We will
each win with 50% probability.
Suppose I do not choose TTT. If Turing chooses TTH then he will win for certain if the first two squares are TT. There is nothing I can do about this! Suppose therefore that a H emerges before TT.
Turing will need TT to start his sequence; so if my sequence is (H)(TT) then my sequence will end at the same time Turing's starts. Thus, I am guaranteed a win if a H emerges before TT. Since the
chance of TT on the first two turns is exactly 1 in 4, I can guarantee a win by choosing HTT 3/4 of the time.
Lewis from The Greneway School sent in these thoughts on the infinite parts of the problem, along with a version that can be played with cards
A HTT and HHT is in favour of the second player with odds of 2 to 1. Although, this is one of the four strongest first player choices, the odds still go in the 2nd player's favour. The weakest hand
for player 1 to choose would be HHH or TTT. Player 2 needs to put the opposite letter in the first position (THH or HTT) to have 7 to 1 odds of winning. A variation of this game can be played with
playing cards. Interestingly, if Player 1 chooses BBB and Player 2 chooses RBB (using the "other letter" rule mentioned above), there are odds of 99.49% of a win for Player 2, just 0.11% for Player 1
and 0.40% for a draw. The best odds player 1 can get on the cards variation is 11.61% or 1.99 to 1 in favour of Player 2. Player 2's process is to move the first 2 of Player 1's selection to the very
right and add the opposite letter of the 2nd of the 2 moved to the left. The main weakness of choosing HHH is if tails is tossed once, Player 1 cannot win. This explains why the odds are stacked in
favour of Player 2 | {"url":"http://nrich.maths.org/7295/solution?nomenu=1","timestamp":"2014-04-19T07:01:14Z","content_type":null,"content_length":"6799","record_id":"<urn:uuid:1cecd6e5-10ef-4961-9aaf-51b868491a44>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00029-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Table of Contents
Real Numbers And Algebraic Expressions
Tips for Success in Mathematics
Algebraic Expressions and Sets of Numbers
Operations on Real Numbers
Properties of Real Numbers
Equations, Inequalities, And Problem Solving
Linear Equations in One Variable
An Introduction to Problem Solving
Formulas and Problem Solving
Linear Inequalities and Problem Solving
Compound Inequalities
Absolute Value Equations
Absolute Value
Graphs and Functions
Graphing Equations
Introduction to Functions
Graphing Linear Functions
The Slope of a Line
Equations of Lines
Graphing Linear
Systems of Equations
Solving Systems of Linear Equations in Two Variables
Solving Systems of Linear Equations in Three Variables
Systems of Linear Equations and Problem Solving
Solving Systems of Equations by Matrices
Solving Systems of Equations by Determinants
Exponents, Polynomials, and Polynomial Functions
Exponents and Scientific Notation
More Work with Exponents and Scientific Notation
Polynomials and Polynomial Functions
Multiplying Polynomials
The Greatest Common Factor and Factoring by Grouping
Factoring Trinomials
Factoring by Special Products
Solving Equations by Factoring and Problem Solving
Rational Expressions
Rational Functions and Multiplying and Dividing Rational Expressions
Adding and Subtracting Rational Expressions
Simplifying Complex Fractions
Dividing Polynomials
Synthetic Division and the Remainder Theorem
Solving Equations Containing Rational Expressions
Rational Equations and Problem Solving
Variation and Problem Solving
Rational Exponents, Radicals, and Complex Numbers
Radicals and Radical Functions
Rational Exponents
Simplifying Radical Expressions
Adding, Subtracting, and Multiplying Radical Expressions
Rationalizing Denominators and Numerators of Radical Expressions
Radical equations and Problem Solving
Complex Numbers
Quadratic Equations and Functions
Solving Quadratic Equations by Completing the Square
Solving Quadratic Equations by the Quadratic Formula
Solving equations by Using Quadratic Methods
Nonlinear Inequalities in One Variable
Quadratic Functions and Their Graphs
Further Graphing of Quadratic Functions
Exponential and Logarithmic Functions
The Algebra of Functions
Composite Functions
Inverse Functions
Exponential Functions
Logarithmic Functions
Properties of Logarithms
Common Logarithms, Natural Logarithms, and Change of Base
Exponential and Logarithmic Equations and Applications
Conic Sections
The Parabola and the Circle
The Ellipse and the Hyperbola
Solving Nonlinear Systems of Equations
Nonlinear Inequalities and Systems of Inequalities
Sequences, Series, and the Binomial Theorem
Sequences. Arithmetic and Geometric Sequences
Series. Partial Sums of Arithmetic and Geometric Sequences
The Binomial Theorem
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A Simple Trig Challenge
This is a fun and simple trigonometry challenge.
No calculators allowed!
Which of the following is the largest?
(a) $\frac{\sqrt{3}}{2}$ (b) $sin (\frac{7\pi}{12})$
(c) $sin (\frac{7\pi}{16})$ (d) $cos (\frac{\pi}{10})$
There are a couple of nice ways to solve this without using a calculator, and plenty of good ways to extend this question.
The real challenge is to come up with your own version of this problem!
3 Comments
1. I got C.
2. The question is how did you get it?
□ The way I thought about it was sin pi/2 is 1. sin pi/3 is root 3 over 2. 7 pi over six is closer to one than pi over 3 so its larger than root 3 over 2. From there I just tried to find which
is closest to 1. (D)pi over ten inserted into cos is pi over ten away which is larger then pi over 12 (B), and pi over sixteen is the smallest (c) so it seems to be the answer to me. | {"url":"http://mrhonner.com/archives/7717","timestamp":"2014-04-16T21:52:14Z","content_type":null,"content_length":"47771","record_id":"<urn:uuid:bf71d582-06f6-4425-8760-39d8b08d1683>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00282-ip-10-147-4-33.ec2.internal.warc.gz"} |
rbeezer / fcla - bc60749
Edits to save 6 pages in print
<p>This example illustrates the proof of <acroref type="theorem" acro="EMHE" />, so will employ the same notation as the proof <mdash /> look there for full explanations. It is <em>not</em> meant to be an example of a reasonable computational approach to finding eigenvalues and eigenvectors. OK, warnings in place, here we go.</p>
<p>It is important to notice that the choice of $\vect{x}$ could be <em>anything</em>, so long as it is <em>not</em> the zero vector. We have not chosen $\vect{x}$ totally at random, but so as to make our illustration of the theorem as general as possible. You could replicate this example with your own choice and the computations are guaranteed to be reasonable, provided you have a computational tool that will factor a fifth degree polynomial for you.</p>
<p>The $n\times n$ square matrix $A$ is <define>upper triangular</define> if $\matrixentry{A}{ij} =0$ whenever $i>j$.</p>
<p>The $n\times n$ square matrix $A$ is <define>lower triangular</define> if $\matrixentry{A}{ij} =0$ whenever
<p>Obviously, properties of a lower triangular matrices will have analogues for upper triangular matrices. Rather than stating two very similar theorems, we will say that matrices are <q>triangular of the same type</q> as a convenient shorthand to cover both possibilities and then give a proof for just one type.</p>
-<p>For Exercises C21<ndash />C28, find the solution set of the given system of linear equations. Identify the values of $n$ and $r$, and compare your answers to the results of the theorems of this section.</p>
+<p>For Exercises C21<ndash />C28, find the solution set of the system of linear equations. Give the values of $n$ and $r$, and interpret your answers in light of the theorems of this section.</p>
-If you cannot conclude anything about an entry, then say so. (See <acroref type="exercise" acro="TSS.M46" />[DEL: for inspiration:DEL].)
+If you cannot conclude anything about an entry, then say so. (See <acroref type="exercise" acro="TSS.M46" />.)
-and apply $\vectrepname{B}$ to each vector in the linear combination. This gives us a new computation, now in the vector space $\complex{6}$,
-which we can compute with operations in $\complex{6}$ (<acroref type="definition" acro="CVA" />, <acroref type="definition" acro="CVSM" />), to arrive at
+and apply $\vectrepname{B}$ to each vector in the linear combination. This gives us a new computation, now in the vector space $\complex{6}$, which we can compute with operations in $\complex{6}$ (<acroref type="definition" acro="CVA" />, <acroref type="definition" acro="CVSM" />),
<p>We are after the result of a computation in $M_{32}$, so we now can apply $\ltinverse{\vectrepname{B}}$ to obtain a $3\times 2$ matrix,
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Combination Calculation
March 30th 2009, 11:11 AM #1
Mar 2009
Combination Calculation
OK...glad you are all here to help.
8 of item a
16 of item b
4 of item c
How many combinations can I make? It takes all 3 together to make 1.
Please post the question exactly as it is written.
What you have posted is meaningless.
Always give the exact wording.
It is a waste of time to guess at meaning only to be wrong.
there is no question...
the item contains 1 of each of a, b, and c.
there are 8 versions of a
16 versions of b
and 4 versions of c
How many items are possible?
Thanks for your polite and prompt response.
Ok...starting to get the feel of the dialogue here.
Keep in mind I am a marketing guy...calculating margins is about as heavy as I get into math.
If there are
8 different versions of a
16 different versions of b
and 4 different versions of c
how many unique items can be put together?
There are 8 different types of a's; 16 different types of b's; and 4 different types of c's.
So to have one of each a, b, & c, we can choose the a in 8 ways, the b in 16 ways, and the c in 4 ways. That is 512 ways to make up a package of one a, one b, and one c. But each package is
different because the types are themselves different.
March 30th 2009, 11:28 AM #2
March 30th 2009, 11:34 AM #3
Mar 2009
March 30th 2009, 11:39 AM #4
March 30th 2009, 11:43 AM #5
Mar 2009
March 30th 2009, 11:55 AM #6 | {"url":"http://mathhelpforum.com/statistics/81494-combination-calculation.html","timestamp":"2014-04-19T10:11:11Z","content_type":null,"content_length":"46459","record_id":"<urn:uuid:248bf3e0-696a-4fab-928d-c487fa5a0104>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00428-ip-10-147-4-33.ec2.internal.warc.gz"} |
Power series
From Encyclopedia of Mathematics
Power series in one complex variable
A series (representing a function) of the form
such that if
Figure: p074240a
Within analytic function at least inside
or its sum is an entire transcendental function, which is regular in the entire place essential singular point at infinity.
Conversely, the very concept of analyticity of a function
which is the Taylor series for
Consequently, the uniqueness property of a power series is important: If the sum
Thus, every power series is the Taylor series for its own sum.
Let there be another power series apart from (1):
with the same centre
The laws of commutativity, associativity and distributivity hold, whereby subtraction is the inverse operation of addition. Thus, the set of power series with positive radii of convergence and a
fixed centre is a ring over the field
where the coefficients
For the sake of simplicity, let
The coefficient
(for a more general inversion problem see Bürmann–Lagrange series).
If the power series (1) converges at a point
i.e. the sum of the series, Angular boundary value). This theorem, dating back to 1827, can be seen as the first major result in the research into the boundary properties of power series. Inversion
of Abel's second theorem without extra restrictions on the coefficients of the power series is impossible. However, if one assumes, for example, that Tauberian theorems.
For other results relating to the boundary properties of power series and particularly to the location of singular points of power series, see Hadamard theorem; Analytic continuation; Boundary
properties of analytic functions; Fatou theorem (see also –).
Power series in several complex variables
where Reinhardt domain of
while the domain of convergence is
Figure: p074240b
The uniqueness property of power series is preserved in the sense that if
Operations with multiple power series are carried out, broadly speaking, according to the same rules as when
Power series in real variables
where abbreviated notations are used, as in , and
to the analytic function
[1] A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian)
[2] A.I. Markushevich, "Theory of analytic functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[3] E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979)
[4] L. Bieberbach, "Analytische Fortsetzung" , Springer (1955)
[5] E. Landau, D. Gaier, "Darstellung und Begrundung einiger neuerer Ergebnisse der Funktionentheorie" , Springer, reprint (1986)
[6] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[7] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) (In Russian)
[8] S. Bochner, W.T. Martin, "Several complex variables" , Princeton Univ. Press (1948)
[9] A.I. Yanushauskas, "Double series" , Novosibirsk (1980) (In Russian)
The approach to analytic functions via power series is the so-called Weierstrass approach. For a somewhat more abstract setting ([a3], Chapts. 2–3 (cf. also Formal power series).
For [a1]–[a2].
[a1] W. Rudin, "Function theory in polydiscs" , Benjamin (1969)
[a2] R. Narasimhan, "Several complex variables" , Univ. Chicago Press (1971)
[a3] K. Diederich, R. Remmert, "Funktionentheorie" , I , Springer (1972)
How to Cite This Entry:
Power series. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Power_series&oldid=15309
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
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Upgraded to Pandemic
At least in my own head. No, I
can't shut it off
! Not sure I would want to even if I could.
I am getting gas the other day and this shot is screaming at me:
So today, I put the slide up and the kids immediately start talking about slope. I like that they thought slope, but slope isn't really going to do much when it comes to fences. So I asked them:
"What would you want to know if you had to build the fence?"
They caught on pretty quickly that one would want to know the length of each board.
"Alright, then tell me the length of each board."
"But, Mr. Cox, we don't have enough information."
"What do you want to know?"
"We need to know how long the shortest board is."
"Alright, now tell me how long each board is."
"We can't. We need more information."
So I had them discuss with their groups what information they had to have in order to figure out how long each board was. Once they had an exhaustive list, they were to write it on their group's
easel. We quickly came up with the following:
• The length of the next board.
• The length of one more board.
• The width of each board.
So do we need the next board, or will any board do? We eventually settled on any other board.
It didn't take long before students had listed all the board lengths. Many used the rate of change and then added the increase to each board to find the length of the next. But it did't take much
prodding for them to realize that having an equation would be nice. We came up with y = 2.5x + 36 fairly quickly. The interesting discussion came about when I asked what x represented.
"X is the number of boards."
"Okay, so go up to the board and point to board #1."
Which board do you think they pointed to? (You guessed it, the one labeled 36". )
"So, plug 1 into your equation and check it out. Tell me if your equation works."
You would have thought that I asked them to stand in the corner of a round room. But once the "but this equation haa-aas to work" wore off. They realized that it wasn't the equation's fault. It was
how they defined x. Board 0 is important because we aren't actually counting boards, we are counting the number of increases.
: I think that the lesson went really well, but it was very telling how many students wanted to impress with their knowledge of the vocabulary as opposed to just looking at the problem and asking the
obvious questions. They were trying to be mathematicians rather than someone who just needs to build a fence. Next year I want to do a better job of introducing concepts a bit more organically as
opposed to "here are the rules, here are some examples, let's get to it." Students are much more engaged when the information is given a little at a time. It keeps them from answer chasing and allows
them to think a little. It may take a bit longer to deliver the lesson, but the benefit of having kids think about the math is priceless.
What else could I have done with this image?
3 comments:
Good work. When I first saw this picture, I was afraid it was going to be forced (I saw "slope" as I skimmed and thought you had forced the issue). Upon reading, I think you really weren't very
helpful... bravo. So, apart from WCYDWT, you asked "how would I build this fence" and the kids did the rest of the work.
I guess the goal now is to make it into a lesson. You've got the workings of fulfilling standards on slope, two-point form of a line, or even linear regression. So, the question would be, what
would best extend to one of those areas, without being forced, contrived, or even just uninteresting?
Oddly enough, the fence looks like it could be parallel to the roof in the background... how tall is that house?
David Cox said...
I didn't notice the roofline, but it wouldn't be difficult to find out how tall it is. I drive by it every day. What would you do with that though?
Nick said...
What if instead of telling them everything, you asked them to find all possible measurements with the catch that you will only give them a single measurement. For example, if you provide the
length of an object in the picture, they should be able to use proportions, calculate angles, and find the length of pretty much everything else. In a geometry class, they should be able to use
trig ratios to find the length of the succeeding boards using proportional measurements. It might be more engaging if they calculated a bunch of measurements, and you ended class with a slide of
all the measurements you took at the site, and a comparison of the calculated values and the real values. Just a thought. | {"url":"http://coxmath.blogspot.com/2009/05/upgraded-to-pandemic.html","timestamp":"2014-04-21T13:03:14Z","content_type":null,"content_length":"84540","record_id":"<urn:uuid:bb11723b-ecd8-4b31-9bee-6252338d3bd3>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00524-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Tools Discussion: Research Area, upper middle schoolers that haven't yet mastered multiplication facts
Discussion: Research Area
Topic: upper middle schoolers that haven't yet mastered multiplication facts
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Subject: RE: upper middle schoolers that haven't yet mastered multiplication facts
Author: GED Guy
Date: Dec 17 2004
On Dec 6 2004, Bethy wrote:
> I'm about at wits end. I teach in a very small rural school (one
> school district) and we have a number of students who, at 7th and
> 8th grade, still don't know 75% or more of the multiplication facts.
> I'm not a proponent of rote memorization, but these students need
> some means of quickly retrieving/calculating this information. What
> does the research indicate? Does anyone know of an effective method
> for equipping these students with a procedure so that they can move
> ahead?
This is long. Sorry.
I work as an administrator in Adult Education - GED Prep, Alternative Education
for Youth and ESL. I always hate to say I have the solution but I do think that
have the keys to the soloution for most students. In adult education few of our
students know their math facts. Most do have the capacity although not the
desire to learn them.
I observed the solution years ago when my wife was asked to tutor a child one
summer. I saw a 5th grader who knew few facts at the beginning of the summer go
on to blaze through all flash cards in lightning speed by the end of the summer.
When struggling with 16-19 year old students working on their GED I remembered
what my wife did and recreated major parts of it.
One issue to memorization is intellegence. That is out of your control.
Another issue is "time on task". That is in your control. Increase time on
task through motivation. But motivation is hard. My wife used candy. We use a
variety of motivators in Adult Education.
The real genius to my wife's method was coming up with a way for students to see
incrmental improvement. That was done by measuring time. It is objective and
very small increments show very small progress. She worked with one student but
I am working with a class so I "systemitized" her method.
The basic technique is this. Have students measure how long it takes them to
complete an operation. This is the baseline. Have them work to memorize the
facts. Remeasure. The new, improved time shows their progress. They see this
and they are motivated. Of course it is the "art" of teaching that leads to
success. The mechanics and the motivators we use are individualized to our
My wife used a specific approach with flashcards which worked very well and
which we use in our class.
Work on 1 operation (e.g. +) at a time. We actually divide each operation into
4 parts. Thus a student masters 1/4 of addition at a time.
In pairs (one student flashes cards, one responds) go through 1/4 of the deck
and measure the time. We have digital egg timers. Don't spend more than 5
seconds on unknown facts - give them the answer and go on.
Go through the same cards again, this time sorting cards into 2 piles - known
and unknown. Work on the unknown pile for 3-5 minutes WITHOUT a timer
(reduces stress).
Recombine piles, go through whole pile once or twice then again using a timer.
"Wow! I know them better. Took me 70 seconds before. This time I did them in
45 seconds!" I think that my wife gave out candy for each day an operation was
improved. We give Hershey Kisses.
Specifically this is what I did:
Divide each operation deck into 4 parts identifing each 1/4 with a different
colored dot. This allows students to repeat the exact same set of facts and
thus always have comparable times. We used black, blue, green and red permenant
marker. Every time someone does the "black addition" they get the exact same
facts. Bill did black addition in 45 seconds, Jane did black addition in 42
seconds. Jane knows her facts better.
Create tracking charts for studnets to record their times and see their
progress. Standard paper. Rows for each operation and color, columns for date
of time. We have minimum standars which are:
30 seconds for 1/4 deck on + and -
40 seconds for 1/4 deck on X and /
We had a student that learned that when they hold the deck it is much faster so
for final timings, observed by a teacher, students hold the deck and flip it
themselves. One student got 1/4 X blue dot (it happens to be easiest for some
reason) in 21 seconds. You try it. Wow.
Obviously you can create charts on the wall or for students to keep. Give
rewards and awards for getting an initial timing on an operation, for various
incriments (say 100 seconds, 75 seconds, 50 seconds and 30 seconds). Heck, if it
is a big problem, let a student be exempt from one quiz for each operation they
reach the minimum. They could get out of 4 quizzes. Perhaps they get to
replace that grade with an A. Or 2 A's for the second operation. 3 A's for the
3rd. 4 A's for the 4th. Whatever works.
Key components:
Visible progress to the student
Make the learning part risk free
small steps, genuine praise for small success
carefull with student psyche
Personally I had difficulty with my math tables. People memorize but then
forget. I have told my students that doing math without knowing the facts is
like driving a car with 5 lbs of pressure in the tires. You will get there but
it will take you a lot longer and will do damage (for people the damage is to
the psychie) along the way.
Good luck. Perhaps they won't then enter my class in a few years!
Reply to this message Quote this message when replying?
yes no
Post a new topic to the Research Area Discussion discussion
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quadratic equations
July 31st 2007, 01:52 PM
quadratic equations
Hi, I am trying to teach myself algebra from a book. It was going fairly smooth and I understand most of what I have read and practiced. However, I came to a section on quadratic equations
whereupon I was presented with a basic form x^2 = 3. It was simple enough to understand that the answer must be either x = 1.73 or -1.73 without any experience with quadratic equations.
Now, the very next paragraph I was presented with this (x - 1)^2 - 4 = 0 and the simple solution that x = 3 or -1. Then my problem begins as the author goes on to say that the equation (x - 1)^2
- 4 can be "simplified" to x^2 - 2x - 3. How is this possible? where does he get 2x? I have checked the numbers and they are ofcourse correct but how has he made this assumption without knowing
the value of x? Is there some sort of mathematical trick I have missed out on here?
My level of Math is far from impressive so any advice would be helpful please. I would normally just read or re-read to try to understand but one minute the author is making sense then it's asif
he just skips a page and assumes you understand :)
July 31st 2007, 02:07 PM
Hi, I am trying to teach myself algebra from a book. It was going fairly smooth and I understand most of what I have read and practiced. However, I came to a section on quadratic equations
whereupon I was presented with a basic form x^2 = 3. It was simple enough to understand that the answer must be either x = 1.73 or -1.73 without any experience with quadratic equations.
Now, the very next paragraph I was presented with this (x - 1)^2 - 4 = 0 and the simple solution that x = 3 or -1. Then my problem begins as the author goes on to say that the equation (x - 1)^2
- 4 can be "simplified" to x^2 - 2x - 3. How is this possible? where does he get 2x? I have checked the numbers and they are ofcourse correct but how has he made this assumption without knowing
the value of x? Is there some sort of mathematical trick I have missed out on here?
My level of Math is far from impressive so any advice would be helpful please. I would normally just read or re-read to try to understand but one minute the author is making sense then it's asif
he just skips a page and assumes you understand :)
Now expand this.
July 31st 2007, 02:16 PM
Now, the very next paragraph I was presented with this (x - 1)^2 - 4 = 0 and the simple solution that x = 3 or -1. Then my problem begins as the author goes on to say that the equation (x - 1)^2
- 4 can be "simplified" to x^2 - 2x - 3. How is this possible? where does he get 2x? I have checked the numbers and they are ofcourse correct but how has he made this assumption without knowing
the value of x? Is there some sort of mathematical trick I have missed out on here?
Do you know the FOIL method?
As TPH said:
$(x - 1)^2 = (x - 1)(x - 1) = x \cdot x + x \cdot (-1) + (-1) \cdot x + (-1) \cdot (-1) = x^2 - x -x + 1 = x^2 -2x + 1$
July 31st 2007, 02:46 PM
We can use the difference of perfect squares as follows
<br /> \begin{aligned}<br /> \left( {x - 1} \right)^2 - 4 &= 0\\<br /> \left[ {\left( {x - 1} \right) + 2} \right]\left[ {\left( {x - 1} \right) - 2} \right] &= 0\\<br /> \left( {x + 1} \right)\
left( {x - 3} \right) &= 0\\<br /> x^2 - 2x - 3 &= 0<br /> \end{aligned}<br />
$\emph{As desired}~\blacksquare$
P.S.: actually, the last step isn't necessary to get the roots of the equation. | {"url":"http://mathhelpforum.com/algebra/17372-quadratic-equations-print.html","timestamp":"2014-04-20T11:11:59Z","content_type":null,"content_length":"9743","record_id":"<urn:uuid:abd8ab88-6448-4888-b059-4debaa8b719d>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00056-ip-10-147-4-33.ec2.internal.warc.gz"} |
Prove a transitive closure
October 30th 2012, 04:48 AM
Prove a transitive closure
Hello everyone. I have a proof that I am having trouble beginning. Should try to solve this proof directly or by induction?
Let R be a binary relation on a set A, and let R^T be the transitive closure of R . Prove that for all x and y in A, (x,y) ∈ R^T if and only if there is a sequence of elements of A say x[1, ]x
[2],…x[n], such that x = x[1], x[1]Rx[2], x[2]Rx[3], …, x[n-1]Rx[n] = y
October 30th 2012, 07:12 AM
Re: Prove a transitive closure
The property that you need to prove is often given as a definition of transitive closure. Since you need to prove this, I guess you define R^T as the smallest transitive relation containing R. Is
this so? | {"url":"http://mathhelpforum.com/discrete-math/206376-prove-transitive-closure-print.html","timestamp":"2014-04-20T06:00:22Z","content_type":null,"content_length":"4988","record_id":"<urn:uuid:2318c933-b489-429b-8b3b-9f2da67fa993>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00458-ip-10-147-4-33.ec2.internal.warc.gz"} |
Concavity/Convexity of a function
August 6th 2010, 01:42 PM
Concavity/Convexity of a function
I need help determining whether the following function is concave or convex:
f(x,y) = x^a * y^b ; a>0, b>0
The way to solve this is to get the determinant of the Hessian. I have narrowed it down the the result below.
[ (a-1)a(x^(a-2))y^b * (b-1)b(y^(b-2))x^a ] - [ a(x^(a-1)) * a(x^(a-1)) ]^2
I guess it is now a matter of determining whether the determinant below is >0, <0, >=0, or<=0. It is convex if the determinant is >= 0 and concave if it is <= 0. I know that when x or y = 0, the
determinant is 0. I am having trouble though finding out what the result would be if I look at other combinations of of x and y values without plugging in different values and solving. If that is
the only solution then I could do that, but I have a feeling there is perhaps a quicker way which I would appreciate if someone could shed some light on. Also, I hope my assumptions above are
Thank you. | {"url":"http://mathhelpforum.com/calculus/152942-concavity-convexity-function-print.html","timestamp":"2014-04-17T10:08:22Z","content_type":null,"content_length":"3985","record_id":"<urn:uuid:1169d112-53ec-488e-b88b-cd026b4e3710>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00578-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Advaita-l] Elementary Particles and Nature of Reality
Shrinivas Gadkari sgadkari2001 at yahoo.com
Wed Nov 28 06:55:50 CST 2007
In quantum physics we have something that is
fundamental and then there is a mathematical
framework that is built using this fundamental
Based on my understanding elementary particles are
almost like some constants within the mathematical
framework, while Wave-particle duality is the
fundamental principle. (In my personal study
I have not tried to understand any elementary
particle other than electron and photon).
Here is a brief summary of this concept for
benefit of list members (this is formulation of
the Schrodinger's equation)
1. In general there are waves that constitute
everything in this universe.
2. Wave has two basic parameters: frequnecy(F),
and wavelength (L).
3. F = E/h (Planck's hypothesis)
E is some representative of energy of the
wave - for light this is energy of a single
photon, for matter waves I think we should be
able to take relativistic total energy or
something similar here.
h = Planck's constant
4. L = h/p (de-broglie's hypothesis)
p = momentum - of the particle that we
will construct with this wave(s)
5. Particles are constructed using superposition
of waves to create wave-packets that are
roughly localized in space.
Note: complex exponentials of the form
exp(j * theta) = cos(theta) + j * sin (theta)
enter into quantum physics in a fundamental
manner. This is a SURPRISE - in classical
physics and engineering these are introduced
purposely to simplify things.
6. Bigger Surprise: Wave packet can never
we completely localized - i.e. made to
vanish everywhere except in a small region
of space where we feel the particle is located.
The more we want it to be localized, the
more are the wavelengths we need to add to
the wavepacket. And since wavelengths are
related to momentum, the more uncertain the
momentum becomes. This is the uncertainty
7. Biggest surprise however is trying to understand
how the mathematical process described in point
6 manifests physically in the process of
obseravtion, and the role of the observer -
collapse of a wave function.
To understand nature of reality one can focus
on understanding this wave-particle duality
rather than the elementary particles.
I believe one can get some insights into the
nature of reality from wave-particle duality.
This if supplemented with yoga sutras, Vedanta
and sankhya the understanding can get a sound
Needless to say, understanding physics is a not
a necessary condition to understand the nature of
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Cleary, Sean - Department of Mathematics, City College, City University of New York
• Syllabus for Math 203 Dr. Cleary, Fall 2010
• Some Math 360 sample questions for review Incidence, Euclid's and Hilbert's axioms will be provided if needed.
• Mathematica License Agreement For Students of City College of NewYork
• Exam 2 solutions Q1: parameterized surface, use ru rv method
• Information Sheet for Math 392 Course: Math 392, Linear Algebra and Vector Analysis for Engineers, Fall 2010
• Bounding restricted rotation distance Sean Cleary 1
• Restricted Rotation Distance between Binary Sean Cleary 1
• Information Sheet for Math 360 Course: Math 360, Introduction to Modern Geometry, Spring 2011
• Three points making a triangle in the upper half-plane: In[32]:= p = 80, 1<
• Some Math 360 sample questions for review Incidence, Euclid's and Hilbert's axioms will be provided if needed.
• Information Sheet for Math 203 Course: Math 203 LL, Calculus 3, Fall Semester 2010
• Math 392 Halloween Question, Dr. Cleary The Great Pumpkin just got back to the North Pole after delivering presents to all the
• Dr. Cleary's Helpful Guide to Symmetry and Integrals Often if we have a region with some symmetry and a function with some symmetry we can
• Dr. Cleary's Guide to Symmetry and Mutiple Integrals Often if we have a region with some symmetry and a function with some symmetry we can
• Information Sheet for Math A6100 / Math 461 Course: Math A6100/ Math 46100 , Differential Geometry, Spring 2011
• The City College of New York Department of Mathematics MATH 39200 Course Syllabus Fall 2010 | {"url":"http://www.osti.gov/eprints/topicpages/documents/starturl/11/072.html","timestamp":"2014-04-19T02:28:24Z","content_type":null,"content_length":"9104","record_id":"<urn:uuid:f9f04437-9b53-4942-9b65-856bb4aceb6b>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00562-ip-10-147-4-33.ec2.internal.warc.gz"} |
Induction, recursion, replacement and the ordinals
Paul Taylor
My chief contribution to this subject is the notion of "well founded coalgebra", which is described briefly in the "extended abstract" below and also Sections 2.5, 6.3, 6.7 and 9.5 of my book,
Practical Foundations of Mathematics. The detailed treatment is in the full paper that is the second item below.
In particular, this proves
• the equivalence between my notion of well-foundedness (induction) and the recursion scheme due to Osius, for endofunctors of Set that preserve inverse images,
• that, if the functor has an initial algebra, a coalgebra for it is well founded iff it has a coalgebra homomorphism to the initial algebra, and
• that extensional well founded coalgebras behave in many ways like sets (of the set-theoretic kind), in particular their coproducts "overlap" in a similar way, and are idempotent like joins in a
Parametric recursion is covered in the Exercises for Chapter VI of the book.
The earlier JSL paper on Intutionistic Sets and Ordinals treats them in the normal mathematical way, as carriers together with structure (a binary relation called ε), in any elementary topos.
Mostowski's theorem, in the form of the extensional quotient of any well founded relation, is proved without replacement.
The classical notion of ordinal splits into several versions when considered intuitionistically. These are characterised in a quasi-set-theoretic fashion in my JSL paper, categorically in Section 6.7
of the book and algebraically by Joyal and Moerdijk in their book Algebraic Set Theory.
Section 9.5 of the book concludes with a sketch of how extensional well founded coalgebras may be used to characterise ordinal iteration of functors in an elementary way, and thereby formulate a
version of the axiom-scheme of replacement.
PDF (156 kb) Presented at Logical Foundations of Mathematics, Computer Science and Physics - Kurt Gödel's Legacy (Gödel '96), Brno, 26 August 1996. This extended abstract was circulated at the
DVI (30 kb) meeting, as the paper did not appear in the printed proceedings. This work had also been presented at Category Theory 1995 in Cambridge on 8 August 1995. Here are the slides that I
PostScript used at these two conferences and other seminars. NB: This file is a 21Mb scanned image of a merged collection of overhead projector slides from several lectures.
(compressed) (54
A5 PS booklet
(compressed) (48
What are these?
[12 Feb 2009]
PDF (349 kb) Written in 1995-6, circulated in Brno and elsewhere and available on my web page 1996-2003 and from 2006. The recursive construction of a function f:A→Θ consists, paradigmatically, of
DVI (206 kb) finding a functor T and maps α:A→ T A and θ:TΘ→Θ such that f=α;T f;θ. The role of the functor T is to marshall the recursive sub-arguments, and apply the function f to them in parallel.
PostScript This equation is called partial correctness of the recursive program, because we have also to show that it terminates, i.e. that the recursion (coded by α) is well founded. This may be
(compressed) done by finding another map g:A→ N, called a loop variant, where N is some standard well founded srtucture such as the natural numbers or ordinals. In set theory the functor T is the
(242 kb) covariant powerset; in the study of the free algebra for a free theory Ω (such as in proof theory) it is the polynomial Σ[r ∈ Ω]−^ar(r), and it is often something very crude. We identify
A5 PS the properties of the category of sets needed to prove the general recursion theorem, that these data suffice to define f uniquely. For any pullback-preserving functor T, a structure
booklet similar to the von Neumann hierarchy is developed which analyses the free T-algebra if it exists, or deputises for it otherwise. There is considerable latitude in the choice of ambient
(compressed) category, the functor T and the class of predicates admissible in the induction scheme. Free algebras, set theory, the familiar ordinals and novel forms of them which have arisen in
(225 kb) theoretical computer science are treated in a uniform fashion.
What are
[12 Feb
PDF (411 kb) Journal of Symbolic Logic, 61 (1996) 705-744. Presented at Category Theory and Computer Science 5, Amsterdam, September 1993. Transitive extensional well founded relations provide an
DVI (175 kb) intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is poorly behaved, leading to problems of
PostScript functoriality. We show how to make the successor monotone by introducing plumpness, which strengthens transitivity. This clarifes the traditional development of successors and unions,
(compressed) making it intuitionistic; even the (classical) proof of trichotomy is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals grow very
(153 kb) rapidly. Directedness must be defined hereditarily. It is orthogonal to the other four conditions, and the lower powerdomain construction is shown to be the universal way of imposing it.
A5 PS We treat ordinals as order-types, and develop a corresponding set theory similar to Osius' transitive set objects. This presents Mostowski's theorem as a reflection of categories, and
booklet set-theoretic union is a corollary of the adjoint functor theorem. Mostowski's theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of
(compressed) replacement, but this seems to be unavoidable for the plump rank. The comparison between sets and toposes is developed as far as the identification of replacement with completeness and
(141 kb) there are some suggestions for further work in this area. Each notion of set or ordinal defines a free algebra for one of the theories discussed by Joyal and Moerdijk, namely joins of a
What are family of arities together with an operation s satisfying conditions such as x ≤ s x, monotonicity or s(x∨y) ≤ s x∨s y. Finally we discuss the fixed point theorem for a monotone
these? endofunction s of a poset with least element and directed joins. This may be proved under each of a variety of additional hypotheses. We explain why it is unlikely that any notion of
[12 Feb ordinal obeying the induction scheme for arbitrary predicates will prove the pure result.
This paper is Presented at Logic in Computer Science 6, Amsterdam, July 1991. We present an elementary axiomatisation of synthetic domain theory and show that it is sufficient to deduce the fixed
not available to point property and solve domain equations. Models of these axioms based on partial equivalence relations have received much attention, but there are also very simple sheaf models
download. based on classical domain theory. In any case the aim of this paper is to show that an important theorem can be derived from an abstract axiomatisation, rather than from a particular
model. Also, by providing a common framework in which both PER and classical models can be expressed, this work builds a bridge between the two.
This is www.PaulTaylor.EU/ordinals/index.php and it was derived from ordinals/index.tex which was last modified on 13 February 2009. | {"url":"http://www.paultaylor.eu/ordinals/index.php","timestamp":"2014-04-19T22:43:49Z","content_type":null,"content_length":"12945","record_id":"<urn:uuid:3084e83e-e4f2-47f6-b5db-95c25b46e55b>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00395-ip-10-147-4-33.ec2.internal.warc.gz"} |
Find the sum of two numbers that are from two different arrays.
The sum is a specific number. Is there any way to solve this problem in linear-time algorithm?
Do you know the location of the two elements? Would it be as simple as int sum = arrayOne[1] + arrayTwo[1]; or something of the sort?
Thank you for replying. we do not know the location of two elements.
I need to add that there are n integers in each array, and each integer is between 0 and n^5.
I'm sorry but I don't think I understand exactly what you are trying to do.... If you have two arrays: A[5] = [1,2,3,4,4] B[5] = [4,5,6,3,2] what are you trying to do? Ignoring the actual coding
Sorry, i should explain it. Find an element a[ i ] and b[ j ] from two arrays a and b where the sum of the two elements equals another number, like z. Can a linear-time algorithm solve this problem?
Without sorting the arrays first? Are the arrays unordered? Think of it this way: If you have two arrays, one with 5 elements, and one with 10 elements. In the worst case, if you need to find both
numbers before summing them (though, if you don't know what they are, how are you going to search for them?), then in the worst case, you'll have to search 5 numbers in the first array, and 10 in the
second. Thats linear (length of array A + length of array B). But I think your problem is different. I think from reading your comments, that you know the sum, but not the two operands. You need to
sum different values in the two arrays until you arrive at the right sum. Am I correct? If this is the case, then the number of combinations is a permutation. If you take every element in array A and
try adding them to every element in array B, in the best case you will have 2 reads, 1 add, and 1 comparison. Thats a scaler so it'd essentially be O(1). But in the worst case, you would have to
compare every element in A to every element in B. Thats A*B operations. For every extra number in A or B, you will have to recheck every single number in the other array.
It can be done but you need a Map for that: stick all the values of array a in a Map as the key value; the index of the number is the associated value in the Map. This completes step 1 of the
algorithm. Step 2 walks over array b and tries to find a value z-b[ j ] where z is the total you're looking for. If found b [ j ] + (z - b[ j ]) is the sum and the value z-b[ j ] in the map tells you
where you can find that number in array a. To be more exact, the Map should be a Map<Integer, List<Integer>> kind regards, Jos | {"url":"http://www.java-forums.org/new-java/38487-find-sum-two-numbers-two-different-arrays-print.html","timestamp":"2014-04-17T19:35:48Z","content_type":null,"content_length":"9586","record_id":"<urn:uuid:1313b97d-fa87-467c-833b-07383e8b014b>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00291-ip-10-147-4-33.ec2.internal.warc.gz"} |
Curve Resolution and Evolving Factor Analysis
From Eigenvector Documentation Wiki
als - Alternating Least Squares computational engine.
comparelcms_simengine - Calculational Engine for comparelcms.
comparelcms_sim_interactive - Interactive interface for COMPARELCMS.
coda_dw_interactive - Interactive version of CODA_DW.
coda_dw - Calculates values for the Durbin_Watson criterion of columns of data set.
corrspec - Resolves correlation spectroscopy maps.
dispmat - Calculates the dispersion matrix of two spectral data sets.
evolvfa - Evolving factor analysis (forward and reverse).
ewfa - Evolving window factor analysis.
mcr - Multivariate curve resolution with constraints.
purity - Self-modeling mixture analysis method based on purity of variables or spectra.
purityengine - calculates purity values of columns of data set.
wtfa - Window target factor analysis.
(Sub topic of Categorical_Index) | {"url":"http://wiki.eigenvector.com/index.php?title=Curve_Resolution_and_Evolving_Factor_Analysis","timestamp":"2014-04-18T13:05:39Z","content_type":null,"content_length":"13841","record_id":"<urn:uuid:c56d82f2-eb9d-4727-ac27-1bb6fc77aa94>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00280-ip-10-147-4-33.ec2.internal.warc.gz"} |
Learning to Multiply Matrices
A matrix is a rectangular array of number. A matrix's dimension is in the form rows x columns. Both addition and subtraction of matrices worked in a way you might expect. Simply line up values in
the same position and find their sum or difference. Matrix multiplication isn't so simple, here is how to learn to multiply matrices in 12 straight forward steps. Once you understand how to
multiply the matrix, try a few questions on this worksheet.
November 5, 2012 at 6:37 pm
(1) Carla says:
I dont know how to do this homework I need help
November 11, 2012 at 10:49 am
(2) Michael Paul Goldenberg says:
Any reason that anyone would want to know how to multiply matrices other than because it comes up in math class?
November 11, 2012 at 12:13 pm
(3) Teko says:
That’s the only reason I do math, it comes up in math class. Why else would I need tutorials? I take math because I have to, otherwise I wouldn’t.
November 25, 2012 at 2:37 pm
(4) Benedict says:
Ms. Russell, it seems you got a right on hour hands! | {"url":"http://math.about.com/b/2012/11/05/learning-to-multiply-matrices.htm","timestamp":"2014-04-20T08:22:37Z","content_type":null,"content_length":"39854","record_id":"<urn:uuid:f95fac08-daac-497c-97d1-680354fd9cb4>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00097-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Why is arithmetic difficult?
Posted by: Alexandre Borovik | May 14, 2008
Why is arithmetic difficult?
My colleague EHK told me today about a difficulty she experienced in her first encounter with arithmetic, aged 6. She could easily solve “put a number in the box” problems of the type
$7 + \square = 12$,
buy counting how many 1′s she had to add to 7 in order to get 12
but struggled with
$\square +6 =11$,
because she did not know where to start. Worse, she felt that she could not communicate her difficulty to adults.
A brief look at Peano axioms for formal arithmetic provides some insight in EHK’s difficulties. I quote Wikipedia, with slight changes:
The Peano axioms define the properties of natural numbers, usually represented as a set N or $\mathbb{N}.$ [I skip axioms for equality relation -- AB.]
The [...] axioms define the properties of the natural numbers. The constant 1 is assumed to be a natural number, and the naturals are assumed to be closed under a “successor” function S.
1. 1 is a natural number.
2. For every natural number n, S(n) is a natural number.
Axioms 1 and 2 define a unary representation of the natural numbers: the number 2 is S(1), and, in general, any natural number n is S^n-1(1). The next two axioms define the properties of this
1. For every natural number n other than 1, S(n) ≠ 1. That is, there is no natural number whose successor is 1.
2. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.
The final axiom, sometimes called the axiom of induction, is a method of reasoning about all natural numbers; it is the only second order axiom.
1. If K is a set such that:
☆ 1 is in K, and
☆ for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.
Thus, Peano arithmetic is a formalisation of that very counting by one that little EHK did, and addition is defined in a precisely the same way as EHK learned to do it: by a recursion
$m + 1 =S(m); \quad m+S(n) = S(m+n).$
Commutativity of addition is a non-trivial (although still accessible to a beginner) theorem. Try to prove it — you will be forced to feel some sympathy to poor little EHK. If it is a trivial task
for you, write a recursive rule for multiplication and prove, from Peano axioms, commutativity of multiplication. Then write a recursion for exponentiation and try to explain, why this time
commutativity fails, even if the recursive scheme appears to be the same.
In an ideal world (I emphasise, in an IDEAL world), primary school teachers should be taught Peano arithmetic — of course not because they have to teach it to their pupils, but because they have to
appreciate intellectual challenges that their pupils have to overcome.
It is a strange concept of British model of teachers training that teachers need to know only the stuff that they pass to pupils. For a successful teaching, at least in mathematics, a teacher has to
know much, much, more.
I have to emphasise: I do not propose to introduce Peano arithmetic into teacher training courses. I talk about an IDEAL world.
That might not be an IDEAL way of making student thinking numbers like that. I believe students in their early days should learn math though intuition.
But since 4th grade calculus is possible… that won’t be such a bad idea…
By: Mgccl on May 14, 2008
at 11:27 pm
Mgccl: I repeat again, formal arithmetic should be taught to TEACHERS, not pupils — because at least some of the pupils are likely to reinvent it is intuitive level. I emphasise , Peano arithmetic is
just simple principles of counting.
By: Alexandre Borovik on May 15, 2008
at 4:05 am
I slightly changed Axioms, bringing them closer to original form proposed by Peano: in most modern books natural number start with 0, not with 1. Children start their arithmetic with 1.
By: Alexandre Borovik on May 15, 2008
at 4:13 am
The poor girl hasn’t figured out yet that $\square + 6 = 6 + \square$. I’m sure a careful teacher would have figured out what the difficulty had been and would have explaiend. As for the Peano axioms
for teachers, I’m not sure that it is such a great idea, it’s a bit dry. Maybe they need more practical discussions about the difficulties that the students have with arithmetics and how to help
To Mgccl: 4th grade calculus? What are they going to do with it? Isn’t it better to develop some problem solving skills?
By: misha on May 15, 2008
at 7:27 am
I don’t think it is correct to say that in the “British model of teachers training that teachers need to know only the stuff that they pass to pupils”. To be a primary school teacher you need to have
a higher level pass in GCSE Maths (a much higher level than taught in primary school). To be a secondary school teacher you need to have a maths (or heavily mathematical) degree, which is again a
much higher level than taught in secondary school.
The situation you describe does crop up at university level though: I recall when I was an undergraduate going to my tutor for help on an algebra problem. He replied: “look, I’ve not done any algebra
since I was a undergrad! you know much more about it than me”. It was even worse for a course on knot theory, most of which he was (understandably for someone who was an undergraduate in the 70s)
completely unaware of.
By: matthew on May 15, 2008
at 7:54 am
Misha: yes, EHK confirms that she did not understand precisely commutativity of addition, because addition done via counting by ones did not appear to be commutative. Of course, the teacher had to
help. The whole issue is HOW the teacher helps a child in this situation: in case of EHK the best solution would be to give her sufficient number of examples like 2+3, 3+2, etc. However, in most
cases, the teacher uses a simplest form of proof: by intimidation. EHK told me that that was the last time she was trying to think for herself, from that unfortunate episode on she just blindly
followed what teachers told her.
At one time of my life I was involved in running maths competitions for children and the selection procedure for FMSh, Physics and Matheatics School at Novosibirsk University (a preparatory boarding
school). There was a whole genre of interview problems, designed and selected to pick particular traits of mathematical thinking in the interviewees. There was an interesting subgenre of problems on
“hidden counting”.
For example: a rectangle 19 by 99 is divided, by lines parallel to its sides, into $19 \times 99$ equal squares. How many of these squares does a diagonal of the rectangle intersect?
To solve this problem, it could be useful remember that “to count” means, in the initial meaning of this world, “to count by ones”.
Elementary mathematics contains a number of hidden structures; usually they sit in ambush exactly in dangerous points where lower leve concepts and techniques are integrated into a higher level
context, like counting by one into addition. Children normally jump over such stfethes of white water as salmons over a waterfal, by process of reification (you can find more on that in my book). For
a teacher, it is not enough to understand psychological difficulties experienced by a child in reification, the teacher should also be able to understand mathematical difficulties involved and hidden
dangers awaiting his/her pupils.
By: Alexandre Borovik on May 15, 2008
at 10:56 am
This is fascinating! Just one thing… I think it was probably the last time I tried to think for myself at primary school. I had a wonderful teacher from age 11 who gave us problems to investigate
rather than exercises from text books. She certainly encouraged us to think for ourselves and (something which is absolutely not emphasized enough in school maths) write down our arguments.
By: EHK on May 15, 2008
at 1:35 pm
Sasha: Giving more examples, as you suggested, would certainly help, but I’m not sure it’s the best way to explain. With your suggestion, here we go again, theorem-proof-example or fact-example
approach again, formalizing too early, not giving enough explanations of where the formalism came from, not making enough with something outside of the formalism. After all, what does it mean to
explain? Doesn’t it mean to point out a connection to something that the student already knows? Whay does it mean to understand something? Doesn’t it mean to make a connection to something else that
we are already familiar with? What does it mean to master a topic? Doesn’t it mean to make enough connections it with the other topics, how to use the developed tools to solve problems, to freely go
from one formulation to the other?
I suspect that EHK had not been exposed enough to the word problems that would connect the formal addition to something familiar to a child, would give her some informal ways of thinking about
addition, rather than in terms of counting to 7 and then continue counting 5 more times to get to 7+5. Such a sipmle problem as “there are 2 baskets of apples, 5 apples in one and 7 apples in the
other, how many apples are in two baskets together?” would make clear that addition is commutative. Thinking about addition in terms of adding lengths (like putting one stool on top of the other, how
high the top of the stool on top will be?) would also help.
My diagnosis: too many topics and factoids taught in isolation, too much mechaniccal drilling, too few informal explanations, too few connections with other topics and subjects, too few word
problems. After all, the bulk of our brain is made of connections, that’s what makes us smart. Teaching should be easy on facts and heavy on connections between the facts, on explanations of these
facts, on letting students think about what they learn and letting them find their own explanations and connections, on figuring things out. The most interesting thing about something new is usually
not the fact itself, but some innovative way to get to it, some novel way to figure it out. Unfortunately it’s not what is discussed in most of the articles and textbooks, especially in mathematics,
the true springs of discovery and understanding are obscured by dry formalism and heavy terminology.
And now we are wondering why education, the way it is, doesn’t work so well. It’s like cutting out a big part of a frog’s brain and then wondering why it doesn’t jump. Calculus, by the way, is a
prime example of this phenomenon.
To EHK: I see I guessed it right.
By: misha on May 16, 2008
at 3:35 am
In my previous comment: “…not making enough with something outside of the formalism” should be “…not making enough connections with something outside of the formalism.” sorry for sloppy editing
By: misha on May 16, 2008
at 3:41 am
I agree with Misha. My chiild started to count at 6.5 (in Europe we start much later that in Russia for exalmple). The first rule I explained him was X+Y=Y+X. It’s a basis of arithmetics.
Now, 8 months later, he counts billons, multiplies, knows decimals, fractions, his IQ is 143 and my youngest is 150. We live in Europe and constate that education is much poorere that in Russia. I
have to follow Russian normal program in math, because the local one is too easy for him. In Russia they have a great math program by Peterson. Arythmetics is teached by using groups theory. Child
understand how to groups objects by diferent characteristics: by color or by shape or size – same group, and N° of objects in each group would differ, but som is the same, because they are still same
objects. My son does it easily, because it’s anyway 5+3 and he could count 365.48+154.65, etc, but logic points inclusion, repetition, creative presentation and challenge makes this program fun for
him and he asks for “math work”. The prof of Gifted kids school liked Russian materials I’ve shown her. The copybooks are in Russian and, normally, the concept should be teached in a class before
done at home, but I just give him several pages with saying nothing. he could not read Russian and just started to read Dutch, but he understand everything himself.
If you need to make child progress very quickly in math, to understand concepts and have more fun – use Peterson’s books!
By: Inna on January 9, 2009
at 8:15 pm
Sorry – I was in a hurry and made many mistakes. Just to add that due to my experience, simply following Peterson’s copybooks would solve problem with simple repetition, because there is a repetition
in a book, but in different forms and that’s why is not boring. And also 2+3 is explained on a level of apples, but it could be 5 apples – some of them are green, other yellow, then some big, other –
small, some with leaves, some – without. And then you create from the whole set 2 groups of big-smal, which would represent 2+3 for ex, or group of red-green (1+4), etc.
Then you know WHY do you need to count 1+4 and 2+3, etc. It’s better than apples. You could think it’s more complicated, but there are different levels and in the beginning children compare groups of
shapes, then count them and then have same example using numbers and same example using expressions (X+Y=Z, then Y=Z-X). So, form the beginning they have the whole picture. I’d say Russian level for
5-6 y.o. kids should be equalent to 7-8 y.o. european program taking into the account that parents in Russian work with their kids at home on school subjects and that Russian kids should read 25
words per minute when they go to first grade, count and write NICELY. I do not think kids living in Europe are stupid – they simply need to have a proper program, motivation, challenge and some
effort from parents.
By: Inna on January 9, 2009
at 8:30 pm
Inna — thanks for your comments. Best luck to your children!
By: Alexandre Borovik on January 10, 2009
at 2:06 pm
[...] Alexandre Borovik “Why is arithmetic difficult?” [...]
By: re: Alexandre Borovik Why is arithmetic difficult? | New Math Done Right on May 9, 2012
at 12:58 am
Posted in Uncategorized | {"url":"https://micromath.wordpress.com/2008/05/14/why-is-arithmetic-difficult/","timestamp":"2014-04-17T13:14:45Z","content_type":null,"content_length":"78751","record_id":"<urn:uuid:1e99c3cc-6b34-4b01-a711-320b0ec04496>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00052-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Help
April 13th 2009, 11:44 PM #1
Junior Member
Mar 2009
We have polynomials f(x),g(x) (degree>1), their GCD has degree>1.
I have to prove that there exist polynomials u(x),v(x) such that:
f(x).u(x)=g(x).v(x) and deg(u)<deg(g),deg(v)<deg(f). $a^-1(x)$
Am I right?
let d(x) be their GCD, then we have:
so $f(x).a^-1(x)=g(x).b^-1(x)$ where and $b^-1(x)$ are polynomials u(x),v(x)
is the proof of existence ok?
I need help with showing the inequality about degrees. Can anybody help me please?
Thank you for your help.
where did $a^{-1}(x)$ come from?? $\deg(v) < \deg(f).a^{-1}(x)$ has no meaning because the RHS of the inequality is not a number!!
let d(x) be their GCD, then we have:
so $f(x).a^-1(x)=g(x).b^-1(x)$ where and $b^-1(x)$ are polynomials u(x),v(x)
the inverse of a polynomial is not necessarily a polynomial. so $a^{-1}(x)$ and $b^{-1}(x)$ might not even exist in your polynomial ring!
the problem, as you posted, doesn't make any sense and you certainly won't get help if your problem has itself a problem!
Thank you, there was a mistake.
So the problem is:
We have polynomials f(x),g(x) (degree>1), their GCD has degree>1.
I have to prove that there exist polynomials u(x),v(x) such that:
f(x).u(x)=g(x).v(x) and deg(u)<deg(g),deg(v)<deg(f)
f(x),g(x) are from F[x], where F is a field
Thank you for your help.
Thank you, there was a mistake.
So the problem is:
We have polynomials f(x),g(x) (degree>1), their GCD has degree>1.
I have to prove that there exist polynomials u(x),v(x) such that:
f(x).u(x)=g(x).v(x) and deg(u)<deg(g),deg(v)<deg(f)
f(x),g(x) are from F[x], where F is a field
Thank you for your help.
that's good now! so suppose $d(x)=\gcd(f(x),g(x)).$ we have $f(x)=d(x)f_1(x), \ g(x)=d(x)g_1(x).$ now let $u(x)=xg_1(x)$ and $v(x)=xf_1(x).$ see that
$f(x)u(x)=g(x)v(x).$ we also have: $\deg u(x)= 1 + \deg g_1(x) < \deg d(x) + \deg g_1(x) = \deg (d(x)g_1(x))=\deg g(x).$ similarly $\deg v(x) < \deg f(x).$
April 14th 2009, 01:21 AM #2
MHF Contributor
May 2008
April 14th 2009, 01:27 AM #3
Junior Member
Mar 2009
April 14th 2009, 01:57 AM #4
MHF Contributor
May 2008 | {"url":"http://mathhelpforum.com/advanced-algebra/83645-gcd.html","timestamp":"2014-04-19T14:50:00Z","content_type":null,"content_length":"39713","record_id":"<urn:uuid:81731580-ad29-413f-a798-74096c5dc5f7>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00511-ip-10-147-4-33.ec2.internal.warc.gz"} |
Scattered neutron tomography based on a neutron transport problem
Abstract (Summary)
Tomography refers to the cross-sectional imaging of an object from either transmission or reflection data collected by illuminating the object from many different directions. Classical tomography
fails to reconstruct the optical properties of thick scattering objects because it does not adequately account for the scattering component of the neutron beam intensity exiting the sample. We
proposed a new method of computed tomography which employs an inverse problem analysis of both the transmitted and scattered images generated from a beam passing through an optically thick object.
This inverse problem makes use of a computationally efficient, two-dimensional forward problem based on neutron transport theory that effectively calculates the detector readings around the edges of
an object. The forward problem solution uses a Step-Characteristic (SC) code with known uncollided source per cell, zero boundary flux condition and Sn discretization for the angular dependence. The
calculation of the uncollided sources is performed by using an accurate discretization scheme given properties and position of the incoming beam and beam collimator. The detector predictions are
obtained considering both the collided and uncollided components of the incoming radiation. The inverse problem is referred as an optimization problem. The function to be minimized, called an
objective function, is calculated as the normalized-squared error between predicted and measured data. The predicted data are calculated by assuming a uniform distribution for the optical properties
of the object. The objective function depends directly on the optical properties of the object; therefore, by minimizing it, the correct property distribution can be found. The minimization of this
multidimensional function is performed with the Polack Ribiere conjugate-gradient technique that makes use of the gradient of the function with respect to the cross sections of the internal cells of
the domain. The forward and inverse models have been successfully tested against numerical results obtained with MCNP (Monte Carlo Neutral Particles) showing excellent agreements. The reconstructions
of several objects were successful. In the case of a single intrusion, TNTs (Tomography Neutron Transport using Scattering) was always able to detect the intrusion. In the case of the double body
object, TNTs was able to reconstruct partially the optical distribution. The most important defect, in terms of gradient, was correctly located and reconstructed. Difficulties were discovered in the
location and reconstruction of the second defect. Nevertheless, the results are exceptional considering they were obtained by lightening the object from only one side. The use of multiple beams
around the object will significantly improve the capability of TNTs since it increases the number of constraints for the minimization problem.
Bibliographical Information:
Advisor:Adams, Marvin; Charlton, William; Beskok, Ali; Nelson, Paul
School:Texas A&M University
School Location:USA - Texas
Source Type:Master's Thesis
Keywords:scattered tomography transport theory inverse problem adjoint calculation conjugate gradient
Date of Publication:08/01/2004 | {"url":"http://www.openthesis.org/documents/Scattered-neutron-tomography-based-transport-220779.html","timestamp":"2014-04-20T00:40:53Z","content_type":null,"content_length":"10868","record_id":"<urn:uuid:6ff7524a-e37e-4425-b6f9-45d1e6f2e9f2>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00025-ip-10-147-4-33.ec2.internal.warc.gz"} |
Developer: RhinoScript Summary: Discusses the logic, or lack of, in VBScript.
If blnResult = True Then Print "True!" Else Print "False!"
Yes, there is a big difference. If blnResult is True or False, then both statements do what you would expect – the same thing. But, the first statement is asking “Is blnResult equal to True?” whereas
the second question is asking “Is blnResult not equal to False?”
In a strictly Boolean world, those are equivalent statements. But the VBScript type system is richer than just Booleans.
For example, what if blnResult is the string “True”? The string “True” is not equal to the Boolean True, so the first statement is false. But the string is also not equal to False, so the second
statement is true, and therefore the statements have different semantics.
The same goes for numbers. When converted to a number, True converts to -1 (for reasons which will become clear in a moment) and False converts to 0. Therefore, if blnResult is 1, again the first
statement is false because 1 <> -1, and the second statement is true because 1 <> 0.
What's going on is that VBScript is not logical. VBScript is bitwise. All the so-called logical operators work on numbers, not on Boolean values. Not, And, Or, XOr, Eqv and Imp all convert their
arguments to four-byte integers, do the logical operation on each pair of bits in the integers, and return the result. If True is -1 and False is 0 then everything works out, because -1 has all its
bits turned on and 0 has all its bits turned off. But if other numbers get in there, all bets are off.
This can lead to some strange situations if you're not careful. In VBScript, it is certainly possible for
to be false, if blnResult is 1 and blnAnswer is 2, for example.
Conditional statements should always take Booleans. Or, in other words, use Booleans as Booleans, use nothing else as Booleans.
Suppose you've got a method that returns a number and you want to do something if it doesn't return zero. Don't do this, even though it does exactly what you want:
it's clearer to call it out and make the conditional take a Boolean:
Conversely, if a value is a Boolean and you know that, there's no need to compare it. When I see
If blnResult can only contain True or False, then you can just say
Use the same practice with logical operators. Do not mix-n-match - either every argument should explicitly be a number, and you're doing bitwise comparisons, or every argument is a Boolean. Mixing
the two makes the code harder to read and more bug-prone. | {"url":"http://wiki.mcneel.com/developer/scriptsamples/vbslogic","timestamp":"2014-04-16T04:25:32Z","content_type":null,"content_length":"16394","record_id":"<urn:uuid:1c5ee6c5-e752-429d-9c8b-43b2498e0981>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00008-ip-10-147-4-33.ec2.internal.warc.gz"} |
Solving for a Reactant Using a Chemical Equations
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that the setcfw_x Rn : Ax = b , x 0is a convex se
Wisconsin Milwaukee - CS - 525
CS 525 - Fall 2011 - Homework 9For extra creditassigned 11/30/11, due 12/8/111. Do Exercise 7-1-32. Do Exercise 7-2-23. (a) Write down the KKT conditions for the problemminimize x2 + x2 +
x2321subject to x1 + x2 + x3 1x1 x2 x3 1x1 + x 2 x3 1x
Wisconsin Milwaukee - CS - 525
525 Computing Project 1, Fall 2011Its a small world, after allIn the 1960s, Stanley Milgram conducted an experiment to determine thediameter of the social network of the united states. Beginning with
a fewpeople in Kansas and Nebraska, participants we
Wisconsin Milwaukee - CS - 525
525 Computing Project 2, Spring 2012The Disputed Federalist PapersLinear and quadratic programming can be used to solve problems in manyapplications. In this project, we will apply quadratic
programming to amachine learning formulation to determine th
Wisconsin Milwaukee - CS - 525
525 Computing Project 2, Fall 2011DMTF with Compressed Sensing:The Past Meets the FutureWay back in the day when we all had land lines, the phone companygured out who you were calling using a clever
scheme called dual-tone multifrequency signaling (DT
USC - ECON - 421
AB1 Purchase or Rent Analysis23 InputsLength of ownership/renting (years)4Personal interest rate (percent)56Renting7Deposit (returned at departure)8Monthly payments (first year)9Monthly rent change
each year10Renters insurance (monthly)
USC - ECON - 421
1005s0.750.770.790.810.830.850.870.890.910.930.95Zu51.2754.5157.8561.3164.8868.5772.3776.2980.3384.4988.77100.0090.0080.00ZuKu70.0060.0050.0040.0030.000.70.750.8slope paramet er, s0.850.90.951
USC - ECON - 421
sK0.92$27Home12345678910111213141516MaterialCumulative CumulativeCost per
USC - ECON - 421
Credit Card Balance:Credit Card APR:$17,00012%Plan 1: Pay interest due at end of each month and principal at the end of fourth month.1 $17,000$170 $17,170$0$1702 $17,000$170 $17,170$0$1703
$17,000$170 $17,170$0$1704 $17,000$170 $17,170$
USC - ECON - 421
Note: This spreadsheet assumes monthly compounding.Loan Amount# of
USC - ECON - 421
USC - ECON - 421
Given:Inputs:Loan AmountInterest RateRepayment PeriodsAnnual Expenses(per apartment)Number of unitsAnnual Rental FeeOccupancy RateIntermediate Calculations:Annual Loan PaymentResults:Annual Profit
USC - ECON - 421
Investment AmountInterest Rate$4,0005%Class A:012345678910Class B:Fee5%0.61%0.61%0.61%0.61%0.61%0.61%0.61%0.61%0.61%0.61%Account Valueat
USC - ECON - 421
ABC$28,000 $55,000 $40,000$15,000 $13,000 $22,000$23,000 $28,000 $32,000$6,000$8,000 $10,00010 years 10 years 10 yearsInvestment costAnnual expensesAnnual revenuesMarket valueUseful lifeMARR20%A -
DNC-AB-A0 ($28,000) ($12,000) ($27,000)
USC - ECON - 421
Table Entries are Before-tax Rate of Returns on taxable bonds.After-Tax Rateof Return onMunicipal Bonds4%5%6%Federal Income Tax Rate15%28%35%4.71%5.56%6.15%5.88%6.94%7.69%7.06%8.33%9.23%
USC - ECON - 421
Natural Gas-Fired PlantInvestmentCapacity FactorMax CapacityEfficiencyAnnual o&MCost of gasCO2 taxCO2 emittedConversionCoal-Fired Plant$1.12 billion80%800 MW40%$0.01 / kWhr$8.00 per million
Btu$15 / MT CO255 MT CO2 / billion Btu1 kWhr =
USC - ECON - 421
InflationRateErosion of Money's Purchasing Power(years)1015252%-18%-26%-39%3%-26%-36%-52%4%-32%-44%-62%
USC - ECON - 421
Starting Salary =Annual Salary Increase =Savings Interest Rate =Average Inflation Rate =Desired amount in 2027 (R$) =$60,0008.00%7.50%3.75%$500,000Desired Amount in 2027 (A$) =% Salary
USC - ECON - 421
MARR15%Defender:Percent Change in Annual Expenses =EOY k0123MV4,0003,0002,5002,000Expenses CR
USC - ECON - 421
Single Factor Change:Extra Cost$ / galMiles/yrFuel Economy (mpg):Gas engine% ImprovementDiesel engineMARRAW(fuel savings)Net AW$1,200$3.0020,0002533%33.2514%$595.49$245.95MARRFuel Economy of Gas
Engine (mpg)2425262712% $287.41 $2
USC - ECON - 421
MARR20%Useful LifeInstallationExpense ($/in.)150Operating Hoursper YearAnnual Tax andInsurance Rate5%Cost of HeatLoss ($/Btu)InsulationThickness (in.)345678Change in Costof Heat | {"url":"http://www.coursehero.com/file/6726296/Solving-for-a-Reactant-Using-a-Chemical-Equations/","timestamp":"2014-04-18T03:46:27Z","content_type":null,"content_length":"48604","record_id":"<urn:uuid:ba71b6e4-71c7-4c92-aebc-f9751b2c5a08>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00067-ip-10-147-4-33.ec2.internal.warc.gz"} |
Derivatives of trig functions
November 12th 2010, 05:28 AM
Derivatives of trig functions
Having a ball teaching myself calculus for DSP...
I'm working with derivatives of trig functions. I'm trying to solve the following limit:
$\lim_{x\rightarrow 0} \frac{x \csc 2x}{\cos 5x}$
My first thought was to just set x to 0, since the equation would still remain valid. This is obviously wrong, when checking the answer in the back of the book, which gives this limit as 1/2.
This question immediately follows a proof regarding the limit of sin(x)/x as x approaches zero always being 1, so I thought it might be helpful to look for ways to reduce this to something along
those lines, but I'm just not seeing it. Can someone point me in the right direction?
Many thanks,
November 12th 2010, 05:34 AM
You should realize that:
$\displaystyle x\cdot\csc(2x)=\dfrac{2x}{2\cdot \sin(2x)}$
that limit is $\frac{1}{2}$
November 12th 2010, 06:14 AM
Well, I had gotten far enough to realise that
$x\csc{2x} = x\dfrac{1}{\sin 2x}$
which is essentially what you've posted. What I don't see, however, is how you take the limit of this as x approaches zero to be $\frac{1}{2}$. I don't see, unless my following logic here is
correct (but it seems fishy to me):
$\lim_{x\rightarrow 0}\dfrac{2x}{2\sin (2x)} = \lim_{x\rightarrow 0}\dfrac{1}{2}\cdot \dfrac{2x}{\sin (2x)} = \dfrac{1}{2}\cdot\lim_{x\rightarrow 0}\dfrac{2x}{\sin (2x)}$
Furthermore, since
$\lim_{x\rightarrow 0}\dfrac{\sin x}{x} = 1$
it (should?) follow that also,
$\lim_{x\rightarrow 0}\dfrac{x}{\sin x} = 1$
$\dfrac{1}{2}\cdot\lim_{x\rightarrow 0}\dfrac{2x}{\sin (2x)} = \dfrac{1}{2}\cdot {1} = \dfrac{1}{2}$
Am I on the right track?
Thanks again,
November 14th 2010, 06:48 AM
Knowing I'm doing the right thing is just as helpful as knowing I've done something stupid... ;) Anyone?
November 14th 2010, 08:00 AM
It actually doesn't remain valid. The reason is that $\csc 2x$ goes toward $\pm\infty$ as $x\to0$. So you're left with a term $x$ that goes to 0 and a term $\csc 2x$ that goes toward $\infty$
multiplied together... in other words, this is the $0 \cdot \infty$ indeterminate form.
That's why direct substitution does not work.
Yes, you did it correctly. | {"url":"http://mathhelpforum.com/calculus/162967-derivatives-trig-functions-print.html","timestamp":"2014-04-19T23:53:14Z","content_type":null,"content_length":"10350","record_id":"<urn:uuid:52b074ba-4509-490e-b100-6ae8dad15e1a>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00646-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: March 2003 [00541]
[Date Index] [Thread Index] [Author Index]
RE: Ellipse Drawing
• To: mathgroup at smc.vnet.net
• Subject: [mg40201] RE: [mg40170] Ellipse Drawing
• From: "David Park" <djmp at earthlink.net>
• Date: Tue, 25 Mar 2003 14:49:56 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com
If you want to try out the ConicSections package from my web site below you
will find it easy to plot your ellipses.
Here is an example.
The following generates all the basic information about a standard ellipse
with semi-major axis 2 and ellipticity e = 7/8. The information is returned
as a set of rules. It represents the ellipse in "standard position".
information = StandardConic[{2, 7/8}]
{conictype -> "Ellipse",
conicequation -> x^2/4 + (16*y^2)/15 == 1,
coniccurve -> {2*Cos[t], (1/4)*Sqrt[15]*Sin[t]},
coniccurvedomain -> {-Pi, Pi},
coniccenter -> {0, 0},
conicfocus -> {{7/4, 0}, {-(7/4), 0}},
conicdirectrix -> {x == -(16/7), x == 16/7},
conicvertex -> {{2, 0}, {-2, 0}}}
The following command will transform the ellipse to a nonstandard position.
TransofrmEllipseRules[P,T,R][information] will transform the rules generated
by StandardConic for an ellipse in standard position to one in actual
position as rotated, translated and reflected by P, T and R.
Say that we want to rotate it by 60 degrees. We generate new information
rules using a rotation matrix for 60 degrees, a zero translation vector, and
an identity matrix for no reflection of x and y axes.
ang = 60*Degree;
newInformation = TransformEllipseRules[
{{Cos[ang], -Sin[ang]}, {Sin[ang], Cos[ang]}},
{0, 0}, IdentityMatrix[2]][information]
{conictype -> "Ellipse",
conicequation ->
(1/240)*(207*x^2 - 98*Sqrt[3]*x*y + 109*y^2) == 1,
coniccurve -> {Cos[t] - (3/8)*Sqrt[5]*Sin[t],
Sqrt[3]*Cos[t] + (1/8)*Sqrt[15]*Sin[t]},
coniccurvedomain -> {-Pi, Pi},
coniccenter -> {0, 0},
conicfocus -> {{7/8, (7*Sqrt[3])/8},
{-(7/8), -((7*Sqrt[3])/8)}},
conicdirectrix ->
{(1/2)*(x + Sqrt[3]*y) == -(16/7),
(1/2)*(x + Sqrt[3]*y) == 16/7},
conicvertex -> {{1, Sqrt[3]}, {-1, -Sqrt[3]}}}
We can then extract the parametrization curve by...
ellipse[t_] = coniccurve /. newInformation
{Cos[t] - (3/8)*Sqrt[5]*Sin[t], Sqrt[3]*Cos[t] +
Now we can plot the curve.
ParametricPlot[Evaluate[ellipse[t]], {t, -Pi, Pi},
AspectRatio -> Automatic,
Frame -> True,
AxesStyle -> GrayLevel[0.7]];
Which is the same as
ParametricPlot[{Cos[t] - (3/8)*Sqrt[5]*Sin[t], Sqrt[3]*Cos[t] +
(1/8)*Sqrt[15]*Sin[t]}, {t, -Pi, Pi},
AspectRatio -> Automatic, Frame -> True,
AxesStyle -> GrayLevel[0.7]];
if you want to try it out.
David Park
djmp at earthlink.net
From: caroline nyhan [mailto:caznyhan at yahoo.com]
To: mathgroup at smc.vnet.net
I have a question concerning using mathematica to draw
the cross-sectional pattern of the polarisation
I want to know how I would draw an ellipse, in the XY
plane, with propagation in the positive z-direction,
by specifying its
- azimuth (angle that the major axis of the
cross-sectional ellipse makes with the horizontal
x-axis (positive when counterclockwise from x-axis))
- ellipticity (measure of the fatness of the ellipse,
(ratio of the lengh of the semi-minor axis to that of
the semi-major axis))
Do you Yahoo!?
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st: RE: accuracy and preserving uniqueness of id
[Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index]
st: RE: accuracy and preserving uniqueness of id
From "Nick Cox" <n.j.cox@durham.ac.uk>
To <statalist@hsphsun2.harvard.edu>
Subject st: RE: accuracy and preserving uniqueness of id
Date Wed, 26 Feb 2003 09:41:46 -0000
Radu Ban wrote
> i'm using -infix- to read in a large dataset into stata.
> each line of the
> dataset begins with an 18 character, numeric, company identification
> block. each company occupies several lines, that all start
> with the same
> identification code. to make things clearer here's my sample code:
> infix id 1-18 reccat 19-20 var1 21-25 var2 26-30 ... if reccat=11
> infix id 1-18 reccat 19-20 var3 21-23 var4 24-27 ... if reccat=12
> after i ran this i took a look at my resulting dataset and
> to my surprise,
> the id displayed by Stata looked very different from the id
> i originally
> had in my flat text file.
> for example:
> in text, id = 200101380110999991
> in stata, id= 200101375269404672
> or
> in text, id = 200101380206999991(different from above)
> in stata, id= 200101375269404672(same as above)
> what's bothering me is that ids that are different in text
> become the same
> in stata. is there a way to preserve the accuracy and hence
> uniqueness of the ids in this situation?
and Devra Golbe, Phil Ryan and Erik Sorensen all firmly
advised the use of a string variable for this purpose.
I concur.
Here are some extracts from a paper "On numbers
and strings" in Stata Journal 2(3):314--329 (2002).
... unique identifiers will often conveniently be held in
string variables. There is little point in defining a
value label if that value label occurs once only. It is
also less likely that you would want to use such a
variable as defining one axis of a graph.
Less obviously, identifiers which consist entirely of numeric
codes are often better held as string variables. U.S. Social
Security Numbers (SSNs) are one of the most frequently
discussed examples on Statalist. .... When stored without
hyphens, these SSNs can be read into Stata as numeric variables,
but small problems often arise later. More generally, to hold
multi-digit identifiers without numeric precision problems
(that is, holding every digit exactly) may require the use of a
-long- variable. To display such a variable (as with -list-) may
require changing format to avoid most digits being lost
whenever identifiers are presented in scientific notation.
(See [R] format.) For example, a -float- numeric variable set equal
to 123456789 will by default be -list-ed as 1.23e+08, shorthand for
1.23 * 10^8. These are small and soluble problems, but they often
cause puzzlement to Stata users. Holding such identifiers as
strings, even though every character is numeric, solves those
problems, with no apparent downside.
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2003-02/msg00776.html","timestamp":"2014-04-18T05:50:32Z","content_type":null,"content_length":"7589","record_id":"<urn:uuid:2935803b-8ac0-43ce-9b10-afc904e39bdb>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00131-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Help
April 16th 2010, 08:45 AM #1
Junior Member
Oct 2008
Maple Help
In Maple I need to produce a plot of chance of winning at least w on the y-axis and won the x-axis. With w ranging from $0 to $30.
The equation for determining the odds is given by...
int(1/((2*Pi)^(1/2)*11.13)*exp(-(x+7.87)^2/(2*11.13^2)),x= w .. 300)
I have been playing with forever now and can't figure it out. Any help is greatly appreciated. And for the record I cannot stand maple...
Actually I got it now. Thanks though for this great forum.
April 16th 2010, 08:47 AM #2
Junior Member
Oct 2008 | {"url":"http://mathhelpforum.com/math-software/139512-maple-help.html","timestamp":"2014-04-17T14:21:55Z","content_type":null,"content_length":"31063","record_id":"<urn:uuid:d4643619-74cd-482c-8e1e-7a99e31d7460>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00338-ip-10-147-4-33.ec2.internal.warc.gz"} |
Numerical Simulation of Baroclinic Jovian Vortices
This thesis consists of two papers on the dynamics of Jovian planet atmospheres. The first paper discusses the uses of a normal-mode expansion in the vertical for modeling the dynamics of Jupiter's
atmosphere. The second paper uses a non-linear numerical model based on the normal-mode expansion of the first paper to study the dynamics of baroclinic vortices. The abstracts for the two papers are
reproduced below.
Paper 1:
We propose a non-linear, quasi-geostrophic, baroclinic model of Jovian atmospheric dynamics, in which vertical variations of velocity are represented by a truncated sum over a complete set of
orthogonal functions obtained by a separation of variables of the linearized quasi-geostrophic potential vorticity equation. A set of equations for the time variation of the mode amplitudes in the
non-linear case is then derived. We show that for a planet with a neutrally stable, fluid interior instead of a solid lower boundary, the barotropic mode represents motions in the interior, and is
not affected by the baroclinic modes. One consequence of this is that a normal mode model with one baroclinic mode is dynamically equivalent to a one layer model with solid lower topography. We also
show that for motions in Jupiter's cloudy lower troposphere, the stratosphere behaves nearly as a rigid lid, so that the normal-mode model is applicable to Jupiter. We test the accuracy of the
normal-mode model for Jupiter using two simple problems: forced, vertically propagating Rossby waves, using two and three baroclinic modes, and baroclinic instability, using two baroclinic modes. We
find that the normal-mode model provides qualitatively correct results, even with only a very limited number of vertical degrees of freedom.
Paper 2:
We examine the evolution of baroclinic vortices in a time dependent, nonlinear numerical model of a Jovian atmosphere. The model uses a normal-mode expansion in the vertical, using the barotropic and
first two baroclinic modes (Achterberg and Ingersoll 1989). Our results for the stability of baroclinic vortices on an ƒ-plane in the absence of a mean zonal flow are consistent with previous results
in the literature, although the presence of the deep fluid interior on the Jovian planets appears to shift the stability boundaries to smaller length scales. The presence of a mean zonal shear flow
acts to stabilize vortices against instability, significantly modifies the finite amplitude form of baroclinic instabilities, and combined with internal barotropic instability (Gent and McWilliams
1986) produces periodic oscillations in the latitude and longitude of the vortex as observed at the level of the cloud tops. This instability may explain some, but not all, observations of
longitudinal oscillations of vortices on the outer planets. Oscillations in aspect ratio and orientation of stable elliptical vortices in a zonal shear flow are observed in this baroclinic model, as
in simpler two-dimensional models (Kida 1981). The meridional propagation and decay of vortices on a β-plane is inhibited by the presence of a mean zonal flow. The direction of propagation of a
vortex relative to the mean zonal flow depends upon the sign of the meridional potential vorticity gradient; combined with observations of vortex drift rates, this may provide a constraint on model
assumption for the flow in the deep interior of Jupiter.
Item Type: Thesis (Dissertation (Ph.D.))
Subject Keywords: Geology
Degree Grantor: California Institute of Technology
Division: Geological and Planetary Sciences
Major Option: Geology
Thesis Availability: Restricted to Caltech community only
Research Advisor(s): • Ingersoll, Andrew P. (advisor)
• Yung, Yuk L. (co-advisor)
Thesis Committee: • Unknown, Unknown
Defense Date: 3 October 1991
Record Number: CaltechTHESIS:09262011-111706613
Persistent URL: http://resolver.caltech.edu/CaltechTHESIS:09262011-111706613
Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code: 6688
Collection: CaltechTHESIS
Deposited By: John Wade
Deposited On: 26 Sep 2011 18:42
Last Modified: 26 Dec 2012 04:38
Thesis Files
PDF - Final Version
Restricted to Caltech community only
See Usage Policy.
Repository Staff Only: item control page | {"url":"http://thesis.library.caltech.edu/6688/","timestamp":"2014-04-17T00:48:27Z","content_type":null,"content_length":"27588","record_id":"<urn:uuid:2c237248-70a8-4ca7-b43a-0a82e217337e>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00456-ip-10-147-4-33.ec2.internal.warc.gz"} |
SRINIVASA RAMANUJAN - The Greatest Mathematical Genius of the Twentieth Century
SRINIVASA RAMANUJAN – The Greatest Mathematical Genius of the Twentieth Century
by Siddharth
I recently discovered that 2012 has been declared as the National Mathematics year by the government, respecting the contributions made by the mathematics wizard S Ramanujan. A rare feat by Indian
government but too late none the less but Ramanujan still is no more than a lost hero of Indian history at least for many Indians who are busy rewarding their bollywood stars and phony leaders with
the slogan India shining with most of its luster muffled up in the shadow of its grand history.
This is one of the reasons for me to write this post but not the only one. I have been intrigued by his personality from a long time and also had an idea about a film based on his life but I just
found that recently Pressman Film Corp acquired rights of biography The Man Who Knew Infinity: A Life of a Genius Ramanujan by Robert Kanigel to be directed by Matthew Brown. (Looks like I will have
to delete the idea from my biopic films folder). Anyways, I will share more about my film ideas later in another post but I’m glad that a film is being made on the life of this genius who grew up in
poverty and isolation in madras with a career like a supernova briefly lighting up the heavens with his mathematical brilliance.
Srinivasa Ramanujan one of India’s great mathematical genius made contributions to the analytical theory of number and worked on elliptic functions, continued fractions and infinite series was born
on 22^nd December 1887 in Erode Tamil Nadu and died at a young age of 33 of tuberculosis on 26^th April 1920 in Kumbakonam. The story of Ramanujan is a story of human triumph and an example of what
genius can accomplish against the odds says Robert Kanigel, Ramanujan’s biographer.
In one of my favorite movie “Goodwill Hunting” the professors at MIT are shocked to find that the street tough played by Matt Damon is actually a mathematical genius who can simply write down the
answers to seemingly intractable mathematical problems and realizing this street kid has learned advanced mathematics on his own, one of them blurts out that he is the next Ramanujan. In fact
goodwill hunting is loosely based on the life of Srinivasa Ramanujan.
Despite his miserable financial situation and ill health, he managed to secure a scholarship in madras university and later in Cambridge university where after his graduation was elected a fellow of
the Cambridge philosophical society. He has been proposed by impressive list of mathematicians namely Hardy, Macmohan, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood, Nocholson, Young, Whitaker,
Forsyth, and Whitehead
Mathematicians are still trying to decipher the lost notebooks of Ramanujan found after his death. Looking back at Ramanujan’s work, we see that it can be generalized to eight dimensions, which is
directly applicable to String Theory. Physicists add two more dimensions in order to construct a physical theory. When we add two more dimensions to Ramanujan’s functions, the magic numbers of
mathematics become 10 and 26 precisely the magic numbers of String Theory. So, in some sense, Ramanujan was doing String Theory before World War I.
I’m quite excited about this upcoming film based on his life from Madras to Cambridge. If not through the book but at-least with film’s popularity many Indians and people around the world will get
acquainted with the rags to the intellectual riches story of genius incarnate.
4 Comments to “SRINIVASA RAMANUJAN – The Greatest Mathematical Genius of the Twentieth Century”
1. great post! He was a tragic figure in Mathematics…
□ Thanks for dropping by and your comments Meenakshi:) He was a genius nevertheless!!
2. The article is good. But how can 1887 to 1920 make him 37 years..As per all other reliable sources he passed away at the age of 32 +.
□ Thanks for dropping by Sukanya and also for pointing out the error. You’re right, he was in his 33rd year when he died. | {"url":"http://anignorantatom.wordpress.com/2012/10/10/srinivasa-ramanujan-the-greatest-mathematical-genius-of-the-twentieth-century/","timestamp":"2014-04-19T17:02:13Z","content_type":null,"content_length":"62455","record_id":"<urn:uuid:fc6c8de4-1fa3-423b-a255-a61272c03442>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00066-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Category:Order statistic algorithms
From LiteratePrograms
Order statistic problems include finding the minimum, maximum, median, and, most importantly, the nth largest (or smallest) element of a list. There are worst-case linear algorithms for finding each
of these, which is ideal, but in practice expected linear-time algorithms for finding the median and nth largest or smallest element outperform the known worst-case linear algorithms. Related
problems include finding the k largest (or smallest) elements of a list and sorting a list.
hijacker hijacker
This category has only the following subcategory. | {"url":"http://en.literateprograms.org/Category:Order_statistic_algorithms","timestamp":"2014-04-19T13:13:50Z","content_type":null,"content_length":"16756","record_id":"<urn:uuid:06e2fa74-8d2c-45a1-bd4f-89d2d32165ab>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00488-ip-10-147-4-33.ec2.internal.warc.gz"} |
Basic Algebra/Working with Numbers/Subtracting Rational Numbers
From Wikibooks, open books for an open world
Subtracting fractions when the denominators are equal is also easy
$\frac{3}{7} - \frac{2}{7} = \frac{1}{7}$
Notice that when the denominators are the same for either adding or subtracting fractions we only add or subtract the numerators. With this in mind we can now use the techniques learned in Lesson 4
to subtract more complex fractions by finding the factors of the following fractions
$\frac{121}{456} - \frac{61}{570} = \frac{11 \times 11}{2 \times 2 \times 2 \times 3 \times 19} - \frac{61}{2 \times 3 \times 5 \times 19}$
We can now see that multiplying 456 by 5 and 570 by 4 gives the smallest possible denominator
$\frac{121 \times 5}{5 \times 456} - \frac{4 \times 61}{4 \times 570} = \frac{605 - 244}{2280} = \frac{361}{2280}$
Now the question is, is this fraction irreducible? We know the factors in the denominator so checking if 361 is divisible by the numbers 2, 3, 5, 19 will show that 361/19 = 19 therefore
$\frac{361}{2280} = \frac{19}{120}$
Which is now an irreducible fraction.
Example Problems[edit]
$\frac{20(8-3)-4(10-3)}{10(2-6)} + 2(7+4)$
$=-\frac{9}{5}+\frac {5(22)}{5}$
$=-\frac{9}{5}+\frac {110}{5}$
Practice Games[edit]
put links here to games that reinforce these skills
Practice Problems[edit]
(Note: put answer in parentheses after each problem you write) | {"url":"https://en.wikibooks.org/wiki/Basic_Algebra/Working_with_Numbers/Subtracting_Rational_Numbers","timestamp":"2014-04-18T19:02:53Z","content_type":null,"content_length":"29546","record_id":"<urn:uuid:2fbc01a8-08ad-499d-b0ad-4de1073469c1>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00207-ip-10-147-4-33.ec2.internal.warc.gz"} |
Precalculus Help and Problems
Topics in precalculus will serve as a transition between algebra and calculus, containing material covered in advanced algebra and trigonometry courses. Precalculus consists of insights needed to
understand calculus.
Still need help after using our precalculus resources? Use our service to find a precalculus tutor.
A brief overview of sets, one of the fundamental principles of mathematics. The topic of sets introduces grouping of objects and number classifications.
An extension of exponents in terms of functions, as well as introducing the constant e. Topics include exponential growth and decay, solving with logs, compound and continuously compounded interest,
and the exponential function of e.
A more in depth look at logarithms and logarithmic functions, as well as how they relate to exponents. Topics include the standard and natural log, solving for x and the inverse properities of
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roots, and functions with no solution.
Sequences and Series always go hand in hand and they introduce the concept of Mathematical Patterns and how to deal with them. This section deals with the different types of Series and Sequences as
well as the methods of finding the next term in a sequence or the sum of a series.
The major types of series and sequences include:
Introduction to the factorial notation. The concept of combinations and permutations is introduced and explained.
Statement of the Binomial Theorem. Pascals' Triangle and its relation to Binomial Theorem. Expanding polynomials using the Binomial Theorem.
An overview of the use of parametric equations, including parametrizing functions and finding a function for a set of parametric equations. Topics include parametrizing lines, segments, circles and
ellipses, and peicewise functions.
An introduction to the polar coordinate system. Topics include graphing points, converting from rectangular to polar and polar to rectangular coordinates, converting degrees and radians, and polar
Introduction to the concept of matrices. Different types of matrices discussed. Matrix algebra including addition, subtraction and multiplication. Matrix inverses and determinants.
Subpages include:
Matrix Addition and Subtraction
Many times we come cross systems of equations and we're usually at odds with how to solve them. Most times we deal with two-variable or three-variable systems of equations but the methods explained
in this section can be used to solve any number variable system of equations. The trick is picking which one would work better and faster for you.
The methods explained include:
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Items tagged with geometry2d
Let the polygon P:=polygon([[0,2],[1,4],[2,3.5],[4,4],[5,1],[4,0.75],[3,0]])
be given.
How to find the rectangle of the minimal area which contains P? Of course, with Maple. Is it a rectangle having a side parallel to the longest diagonal of P?
The same problem in the general case: a procedure is required. | {"url":"http://www.mapleprimes.com/tags/geometry2d","timestamp":"2014-04-18T21:49:56Z","content_type":null,"content_length":"86688","record_id":"<urn:uuid:7fb97eff-9e39-4559-a301-7564bcdc2ddd>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00543-ip-10-147-4-33.ec2.internal.warc.gz"} |
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On Pre-Hilbert Noncommutative Jordan Algebras Satisfying
ISRN Algebra
VolumeΒ 2012Β (2012), Article IDΒ 328752, 11 pages
Research Article
On Pre-Hilbert Noncommutative Jordan Algebras Satisfying
DΓ©partement de MathΓ©matiques et Informatique, FacultΓ© des Sciences, B.P. 2121, TΓ©touan, Morocco
Received 17 April 2012; Accepted 30 May 2012
Academic Editors: A. Jaballah, A. Kiliçman, D. Sage, K. P. Shum, F. Uhlig, A. Vourdas, and H. You
Copyright Β© 2012 Mohamed Benslimane and Abdelhadi Moutassim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Let be a real or complex algebra. Assuming that a vector space is endowed with a pre-Hilbert norm satisfying for all . We prove that is finite dimensional in the following cases. (1) is a real weakly
alternative algebra without divisors of zero. (2) is a complex powers associative algebra. (3) is a complex flexible algebraic algebra. (4) is a complex Jordan algebra. In the first case is
isomorphic to or and is isomorphic to in the last three cases. These last cases permit us to show that if is a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite
dimensional and is isomorphic to . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .
1. Introduction
Let A be a real or complex algebra not necessarily associative or finite dimensional. Assuming that a vector space A is endowed with a pre-Hilbert norm satisfying for all . Zalar (1995, [1]) proved
that. (1)If is a real alternative algebra containing a unit element such that , then is finite dimensional and is isomorphic to , or . (2)If is a real associative algebra satisfying , then is finite
dimensional and is isomorphic to , or . (3)If is a complex normed algebra containing a unit element such that , then is finite dimensional and is isomorphic to . These results were extended,
respectively, to the following cases. (1)If is a real alternative algebra containing a nonzero central element such that , then is finite dimensional and is isomorphic to , or (2008, [2]). (2)If is a
real alternative algebra satisfying , then is finite dimensional and is isomorphic to , or (2008, [2]). (3)If is a complex normed algebra without divisors of zero and containing an invertible element
such that , then is finite dimensional and is isomorphic to (2010, [3]). In the present paper, we extend the above results to more general situation. Indeed, we prove that, if is a real or complex
pre-Hilbert algebra satisfying for all , then is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero and satisfying for all (Theorem 3.5). (2)
is a real weakly alternative algebra without divisors of zero and containing a nonzero central element such that for all (Theorem 3.7). (3) is a complex powers associative algebra satisfying for all
(Theorem 4.8). In the first two cases is isomorphic to or and is isomorphic to in the last two cases. This last allows us to show that if is a complex pre-Hilbert noncommutative Jordan algebra
(resp., flexible algebraic algebra or Jordan algebra) satisfying for all , then is finite dimensional and is isomorphic to (Theorems 4.9 and 4.10). Moreover, we give an example of an
infinite-dimensional real pre-Hilbert Jordan algebra (weakly alternative algebra) with divisors of zero and satisfying for all .
2. Notation and Preliminary Results
Throughout the paper, the word algebra refers to a nonnecessarily associative algebra over or .
Definitions 1. Let be an arbitrary algebra and is a field of characteristic not .(1)(i) is called alternative if it is satisfied the identities and (where means associator), for all (1966, [4]).(ii)
is called a powers associative if, for every in , the subalgebra generated by is associative.(iii) is called flexible if for all .(iv) is called a Jordan algebra if it is commutative and satisfied
the Jordan identity: (J) for all .(v) is called a noncommutative Jordan algebra if it is flexible and satisfied the Jordan identity (J).(vi) is called weakly alternative if it is a noncommutative
Jordan algebra and satisfied the identity (where means commutator). An alternative algebra or Jordan algebra is evidently weakly alternative.(vii) is called quadratic if it has an identity element
and satisfied the identity for all and .(2)(viii) We say that is algebraic if, for every in , the subalgebra of generated by is finite dimensional (1947, [5]).(ix) A symmetric bilinear form over is
called a trace form if for all .(x) is termed normed (resp., absolute valued) if it is endowed with a space norm such that (resp., ), for all .(xi) is called a pre-Hilbert algebra if it is endowed
with a space norm comes from an inner product .(xii) We mean by a nonzero central element in , a nonzero element which commute with all elements of the algebra .
The most natural examples of absolute valued algebras are (the algebra of Hamilton quaternion) and (the algebra of Cayley numbers), with norms equal to their usual absolute values (1991, [6]) and
(2004, [7]). The algebra (1949, [8]) was obtained by replacing the product of with the one defined by , where means the standard involution of .
We have the following very known results.
Lemma 2.1 (see [4]). Let be a powers associative algebra over and without divisors of zero. If is a nonzero idempotent in , then has an identity element .
Proposition 2.2 (see [9]). If is a set of commuting elements in a flexible algebra over , then the subalgebra generated by the is commutative.
Proposition 2.3 (see [10]). Let be a noncommutative Jordan algebra over , then is a powers associative algebra.
Lemma 2.4 (see [11]). Let be a quadratic algebra over . Then flexible if and only if is symmetric and the following equivalent statements hold. (1) is a trace form over .(2) is a trace over .(3) = 0
for every .
Theorem 2.5 (see [4]). The subalgebra generated by any two elements of an alternative algebra is associative.
We need the following results.
Theorem 2.6 (see [1]). Let be a real pre-Hilbert associative algebra satisfying for all . Then is finite dimensional and is isomorphic to , or .
Theorem 2.7 (see [2]). Let be a real pre-Hilbert commutative algebra without divisors of zero and satisfying for all . Suppose that containing a nonzero central element such that for all . Then is
isomorphic to , or .
Theorem 2.8 (see [1]). Let be a real pre-Hilbert alternative algebra with identity . Suppose that for all and . Then is isomorphic to , or .
3. Real Pre-Hilbert Weakly Alternative Algebras
In this subparagraph, we prove that, if is a real pre-Hilbert algebra satisfying for all . Then is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors
of zero. (2) is a real Jordan algebra without divisors of zero. In the first case is isomorphic to , or , and is isomorphic to or in the last case. Moreover, we give an example of an
infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .
Lemma 3.1 (see [12]). Let be a real pre-Hilbert algebra with identity such that for all and let then.(1). (2) for all .
Remark 3.2. (i) The product , for all , provides the structure of an anticommutative algebra.
(ii) If is flexible, then for all .
Proof. (i) Let , we have
(ii) As is a flexible algebra, then This implies that for all , and by Lemma 3.1, we have . Thus, .
Theorem 3.3. Let be a real pre-Hilbert weakly alternative algebra with identity and without divisors of zero. Suppose that for all and . Then is finite dimensional and is isomorphic to , or .
Proof. It is sufficient to prove that is an alternative algebra.
Let such that , according to Lemma 3.1 we have This implies that So As has nonzero divisors, then Therefore, . Now we take two arbitrary elements , and let . Or , then Let and two elements in , with
and , we have . Therefore , thus . So is a left alternative algebra. Now we show that is a right alternative algebra, if are two orthogonal elements. Then And (Remark 3.2), thus, Similarly, we prove
that for all , then is a right alternative algebra. Thus, is an alternative algebra, the result ensues then of Theorem 2.8.
Corollary 3.4. Let be a real pre-Hilbert Jordan algebra with identity and without divisors of zero. Suppose that for all and , then is finite dimensional and is isomorphic to or .
Theorem 3.5. Let be a real pre-Hilbert weakly alternative algebra without divisors of zero. Suppose that for all , then is finite dimensional and is isomorphic to , or .
Proof. is a powers associative algebra (Proposition 2.3) then the subalgebra of , generated by , is associative and verifying the conditions of Theorem 2.6. Therefore, is isomorphic to or , thus
there is a nonzero idempotent such that ; that is, is a unital algebra of unit (Lemma 2.1). So the result is a consequence of Theorem 3.3.
Corollary 3.6. Let be a real pre-Hilbert Jordan algebra without divisors of zero. Suppose that for all , then is finite dimensional and is isomorphic to or .
We give an extension of Theorem 3.3.
Theorem 3.7. Let be a real pre-Hilbert weakly alternative algebra without divisors of zero and satisfying for all . Suppose that containing a nonzero central element such that for all . Then is
finite dimensional and is isomorphic to , or .
Proof . Let , the subalgebra of generated by is commutative. Theorem 2.7 implies that , thus the result is a consequence of Theorem 3.5.
Corollary 3.8. Let be a real pre-Hilbert Jordan algebra without divisors of zero and satisfying for all . Suppose that contains a nonzero central element such that for all . Then is finite
dimensional and is isomorphic to or .
Remark 3.9. In the previous results the hypothesis without divisors of zero is necessary. The following example proves it.
Let be an infinite-dimensional real Hilbert space, we define the multiplication on the vector space by: And the scalar product by So is a commutative algebra satisfying and for all . Indeed, we put
and . We have Then , moreover, Then Thus,
Similarly, Thus,
From the two equalities (3.15) and (3.17), we conclude that ; that is, for all . This implies that is an infinite-dimensional real pre-Hilbert Jordan (weakly alternative) algebra with identity
satisfying and has a zero divisors. Indeed, let and be two orthogonal nonzero elements in , as defined multiplication of , we have . Hence, is an algebra with zero divisors.
4. Complex Pre-Hilbert Noncommutative Jordan Algebras Satisfying
We show that if is a noncommutative Jordan complex pre-Hilbert algebra satisfying for all , then is finite dimensional and is isomorphic to .
4.1. Complex Pre-Hilbert Alternative Algebras Satisfying
We need the following results.
Proposition 4.1 (see [3]). Let be a complex pre-Hilbert commutative associative algebra satisfying for all . Then is finite dimensional and is isomorphic to .
Theorem 4.2 (see [3]). Let be a complex pre-Hilbert algebra with identity . Suppose that for all . Then is finite dimensional and is isomorphic to .
Lemma 4.3 (see [3]). Let be a complex pre-Hilbert commutative algebra satisfying for all . Then has nonzero divisors.
Theorem 4.4 (see [3]). Let be a complex pre-Hilbert commutative algebraic algebra satisfying for all . Then is finite dimensional and is isomorphic to .
Lemma 4.5. Let be a complex pre-Hilbert alternative algebra satisfying for all . Then has nonzero divisors.
Proof. Let be a nonzero element in and let an element in such that . The subalgebra of generated by is associative (Theorem 2.5). We have then . Thus, is a commutative and associative, therefore, the
Proposition 4.1 complete the demonstration.
Theorem 4.6. Let be a complex pre-Hilbert alternative algebra satisfying for all , then is finite dimensional and is isomorphic to .
Proof. Let , the subalgebra of generated by is commutative and associative (Theorem 2.5). Proposition 4.1 proves that is isomorphic to , then there exists a nonzero idempotent . According to Theorem
4.2 it is sufficient to prove that is a unit element of . Let , we have and . As is without divisors of zero (Lemma 4.5), then . Thus, is finite dimensional and is isomorphic to .
4.2. Complexes Pre-Hilbert Powers Associative Algebras Satisfying
In this subparagraph we show that if (, ) is a complex pre-Hilbert powers associative algebra (resp., flexible algebraic algebra, noncommutative Jordan algebra, or weakly alternative algebra)
satisfying for all . Then is finite dimensional and is isomorphic to .
We have the following importing result.
Lemma 4.7. Let be a complex pre-Hilbert powers associative algebra satisfying for all . Then has nonzero divisors.
Proof. Let be a nonzero element in , the subalgebra of is associative. According to Theorem 4.6, is isomorphic to . Therefore, there exist a nonzero idempotent and such that . Suppose there is a
nonzero element , as is isomorphic to (Theorem 4.6), then there exist a nonzero idempotent and such that . We have , and This implies that , because . Thus, or This is absurd and hence, has nonzero
Theorem 4.8. Let be a complex pre-Hilbert powers associative algebra satisfying for all , then is finite dimensional and is isomorphic to .
Proof. According to Lemma 4.7, has a nonzero divisors. Let be a nonzero element in , then the subalgebra of is associative. Theorem 4.6 implies that is isomorphic to . Hence, containing a nonzero
idempotent, this gives that has a unit element (Lemma 2.1). The result is a consequence of Theorem 4.2.
Theorem 4.9. Let be a complex pre-Hilbert flexible algebraic algebra satisfying for all , then is finite dimensional and is isomorphic to .
Proof. Let be a nonzero element, according to Proposition 2.2 and Lemma 4.3, the subalgebra of is commutative, algebraic, and without divisors of zero. Thus is isomorphic to (Theorem 4.4). This
implies that is a powers associative algebra, then the result is a consequence of Theorem 4.8.
We state the main theorem.
Theorem 4.10. Let be a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to .
Proof. Proposition 2.3 implies that is a powers associative algebra, and hence, is isomorphic to (Theorem 4.8).
Corollary 4.11. Let be a complex pre-Hilbert weakly alternative algebra satisfying for all , then is finite dimensional and is isomorphic to .
Proof. is a noncommutative Jordan algebra. By Theorem 4.10, is finite dimensional and is isomorphic to .
Corollary 4.12. Let be a complex pre-Hilbert Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to .
Proof. is a weakly alternative algebra. By Corollary 4.11, is finite dimensional and is isomorphic to .
The authors are very grateful to professor A. M. Kaidi for his advice and help. This paper is dedicated to the memory of professor Khalid Bouhya.
1. B. Zalar, β On Hilbert spaces with unital multiplication,β Proceedings of the American Mathematical Society, vol. 123, no. 5, pp. 1497β 1501, 1995. View at Publisher Β· View at Google Scholar
Β· View at Zentralblatt MATH
2. A. Moutassim and A. Rochdi, β Sur les algèbres préhilbertiennes vérifiant $\parallel {x}^{2}{\parallel \le \parallel x\parallel }^{2}$,β Advances in Applied Clifford Algebras, vol. 18, no. 2,
pp. 269β 278, 2008. View at Publisher Β· View at Google Scholar
3. M. R. Hilali, A. Moutassim, and A. Rochdi, β $ℂ$-algebres normées préhilbertiennes vérifiant $\parallel {x}^{2}{\parallel =\parallel x\parallel }^{2}$,β Advances in Applied Clifford Algebras,
vol. 20, no. 1, pp. 33β 41, 2010. View at Publisher Β· View at Google Scholar
4. R. D. Schafer, An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, NY, USA, 1966.
5. A. A. Albert, β Absolute valued real algebras,β Annals of Mathematics. Second Series, vol. 48, pp. 495β 501, 1947. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
6. H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch et al., Numbers, vol. 123 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1991.
7. A. Rodriguez, β Absolute-valued algebras, and absolute-valuable Banach spaces,β in Advanced Courses of Mathematical Analysis I, pp. 99β 155, World Scientific, Hackensack, NJ, USA, 2004. View
at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
8. A. A. Albert, β A note of correction,β Bulletin of the American Mathematical Society, vol. 55, p. 1191, 1949. View at Publisher Β· View at Google Scholar
9. G. M. Benkart, D. J. Britten, and J. M. Osborn, β Real flexible division algebras,β Canadian Journal of Mathematics, vol. 34, no. 3, pp. 550β 588, 1982. View at Publisher Β· View at Google
Scholar Β· View at Zentralblatt MATH
10. H. Braun and M. Koecher, Jordan-Algebren, Springer, Berlin, Germany, 1966.
11. J. M. Osborn, β Quadratic division algebras,β Transactions of the American Mathematical Society, vol. 105, pp. 202β 221, 1962. View at Publisher Β· View at Google Scholar Β· View at
Zentralblatt MATH
12. A. Moutassim, β Quelques résultats sur les algèbres flexibles préhilbertiennes sans diviseurs de zéro vérifiant $\parallel {x}^{2}{\parallel =\parallel x\parallel }^{2}$,β Advances in Applied
Clifford Algebras, vol. 18, no. 2, pp. 255β 267, 2008. View at Publisher Β· View at Google Scholar | {"url":"http://www.hindawi.com/journals/isrn.algebra/2012/328752/","timestamp":"2014-04-16T22:47:50Z","content_type":null,"content_length":"418450","record_id":"<urn:uuid:c5831012-d315-487e-bda6-7d8ac8e3cfaf>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00552-ip-10-147-4-33.ec2.internal.warc.gz"} |
Post a reply
Its this function:
i wanted to say if x is 0 then the f(x) is 0 and if x is not 0 then the f(x) is the one above
Our professor of math told me that the taylor series arent same as this function, but he didnt explain me why and how. He told me a hint to proof that this function and its taylor series arent same
should be to make sme basic derivation npot for numbers but globally for n. which is pretty hard and im not sure this way i cna make the proof that the taylor series arent same as the function
By taylor series not being same as function i mean that usually if you derivate your taylor series its looking more and more like the function from derivation to derivation and in n infinite
derivation it should be same as the function, and in this function it doesnt work, its taylor series doesnt going to look like the function
Hope wrote it easy to understand :-) and hope i havent made any theoretical mistakes | {"url":"http://www.mathisfunforum.com/post.php?tid=18431&qid=240306","timestamp":"2014-04-18T00:25:04Z","content_type":null,"content_length":"25883","record_id":"<urn:uuid:9bca1dd2-61ad-4564-9089-d0fc34b13356>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00522-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Retract (algebraic topology)
September 7th 2009, 03:06 PM #1
Sep 2009
Retract (algebraic topology)
Hi everyone, excuse me about my bad english but it isn't my natural language.
I have tried so much but I canīt solve the next problem:
Let $K$ be a closed orientable surface. Let $\alpha \subset K$ a loop such that $K - \alpha$ is connected. Show that $\alpha$ is retract of $K$.
I would like to know how to solve this.I hope sombody could help me. Thanks anyway.
Choose two distinct points $x,y\in (K-a)^o$. The homogeneity lemma (cf Milnor's "Topology from the Differentiable Viewpoint") guarantees that there exists a
diffeomorphism of $K-a$ onto itself that maps one point to the other, and that this diffeomorphism is smoothly isotopic to the identity.
Last edited by Rebesques; December 29th 2010 at 03:04 PM.
cut off one of the two generator circles of a torus, the resulting space is a cylinder which is connected, while a torus does not retract to a circle since its fundamental group is Z x Z. Did I
understand your question correctly?
A "deformation retract" and a "retract" are not the same thing. Your argument works if the question asked about a "deformation retract".
In other words, a torus does not deformation retract to a circle, but it retracts to a circle.
December 29th 2010, 02:49 PM #2
December 29th 2010, 07:13 PM #3
Senior Member
Mar 2010
Beijing, China
December 29th 2010, 08:26 PM #4
May 2010 | {"url":"http://mathhelpforum.com/differential-geometry/101005-retract-algebraic-topology.html","timestamp":"2014-04-17T02:45:53Z","content_type":null,"content_length":"39175","record_id":"<urn:uuid:b642aa13-80d7-44ed-82a7-1b8fb69c7bc5>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00561-ip-10-147-4-33.ec2.internal.warc.gz"} |
Results 1 - 10 of 28
- Proceedings 22nd International Colloquium on Automata, Languages and Programming , 1995
"... We describe the first parallel algorithm with optimal speedup for constructing minimum-width tree decompositions of graphs of bounded treewidth. On n-vertex input graphs, the algorithm works in
O((logn)^2) time using O(n) operations on the EREW PRAM. We also give faster parallel algorithms with opti ..."
Cited by 33 (10 self)
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We describe the first parallel algorithm with optimal speedup for constructing minimum-width tree decompositions of graphs of bounded treewidth. On n-vertex input graphs, the algorithm works in O
((logn)^2) time using O(n) operations on the EREW PRAM. We also give faster parallel algorithms with optimal speedup for the problem of deciding whether the treewidth of an input graph is bounded by
a given constant and for a variety of problems on graphs of bounded treewidth, including all decision problems expressible in monadic second-order logic. On n-vertex input graphs, the algorithms use
O(n) operations together with O(log n log n) time on the EREW PRAM, or O(log n) time on the CRCW PRAM.
, 1995
"... ) Artur Czumaj 1 , Friedhelm Meyer auf der Heide 2 , and Volker Stemann 1 1 Heinz Nixdorf Institute, University of Paderborn, D-33095 Paderborn, Germany 2 Heinz Nixdorf Institute and Department
of Computer Science, University of Paderborn, D-33095 Paderborn, Germany Abstract. We conside ..."
Cited by 21 (4 self)
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) Artur Czumaj 1 , Friedhelm Meyer auf der Heide 2 , and Volker Stemann 1 1 Heinz Nixdorf Institute, University of Paderborn, D-33095 Paderborn, Germany 2 Heinz Nixdorf Institute and Department of
Computer Science, University of Paderborn, D-33095 Paderborn, Germany Abstract. We consider the problem of simulating a PRAM on a distributed memory machine (DMM). Our main result is a randomized
algorithm that simulates each step of an n-processor CRCW PRAM on an n-processor DMM with O(log log log n log n) delay, with high probability. This is an exponential improvement on all previously
known simulations. It can be extended to a simulation of an (n log log log n log n)- processor EREW PRAM on an n-processor DMM with optimal delay O(log log log n log n), with high probability.
Finally a lower bound of \Omega (log log log n=log log log log n) expected time is proved for a large class of randomized simulations that includes all known simulations. 1 Introduction Para...
, 1993
"... All algorithms below are optimal alphabet-independent parallel CRCW PRAM algorithms. In one dimension: Given a pattern string of length m for the string-matching problem, we design an algorithm
that computes a deterministic sample of a sufficiently long substring in constant time. This problem use ..."
Cited by 19 (10 self)
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All algorithms below are optimal alphabet-independent parallel CRCW PRAM algorithms. In one dimension: Given a pattern string of length m for the string-matching problem, we design an algorithm that
computes a deterministic sample of a sufficiently long substring in constant time. This problem used to be a bottleneck in the pattern preprocessing for one- and two-dimensional pattern matching. The
best previous time bound was O(log 2 m= log log m). We use this algorithm to obtain the following results. 1. Improving the preprocessing of the constant-time text search algorithm [12] from O(log 2
m= log log m) to O(log log m), which is now best possible. 2. A constant-time deterministic string-matching algorithm in the case that the text length n satisfies n = \Omega\Gamma m 1+ffl ) for a
constant ffl ? 0. 3. A simple probabilistic string-matching algorithm that has constant time with high probability for random input. 4. A constant expected time Las-Vegas algorithm for computing t...
- In Proc. Israel Symp. on Theory and Computing Systems (ISTCS'95 , 1995
"... We show that extremely accurate approximation to the prefix sums of a sequence of n integers can be computed deterministically in O(log log n) time using O(n= log log n) processors in the Common
CRCW PRAM model. This complements randomized approximation methods obtained recently by Goodrich, Matias ..."
Cited by 14 (0 self)
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We show that extremely accurate approximation to the prefix sums of a sequence of n integers can be computed deterministically in O(log log n) time using O(n= log log n) processors in the Common CRCW
PRAM model. This complements randomized approximation methods obtained recently by Goodrich, Matias and Vishkin and improves previous deterministic results obtained by Hagerup and Raman. Furthermore,
our results completely match a lower bound obtained recently by Chaudhuri. Our results have many applications. Using them we improve upon the best known time bounds for deterministic approximate
selection and for deterministic padded sorting. 1 Introduction The computation of prefix sums is one of the most basic tools in the design of fast parallel algorithms (see Blelloch [9] and J'aJ'a
[33]). Prefix-sums can be computed in O(logn) time and linear work in the EREW PRAM model (Ladner and Fischer [34]) and in O(log n= log log n) and linear work in the Common CRCW PRAM model (Cole and
- In Proc 23rd ICALP , 1996
"... The su#x tree of a string, the fundamental data structure in the area of combinatorial pattern matching, has many elegant applications. In this paper, we present a novel, simple sequential
algorithm for the construction of su#x trees. We are also able to parallelize our algorithm so that we settl ..."
Cited by 14 (3 self)
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The su#x tree of a string, the fundamental data structure in the area of combinatorial pattern matching, has many elegant applications. In this paper, we present a novel, simple sequential algorithm
for the construction of su#x trees. We are also able to parallelize our algorithm so that we settle the main open problem in the construction of su#x trees: we give a Las Vegas CRCW PRAM algorithm
that constructs the su#x tree of a binary string of length n in O(log n) time and O(n) work with high probability. In contrast, the previously known work-optimal algorithms, while deterministic, take
# (log n) time.
, 1996
"... This paper examines some computational aspects of different arrays enhanced with optical pipelined buses. The array processors with optical pipelined buses (APPB) are shown to be extremely
flexible, as demonstrated by their ability to efficiently simulate different variants of PRAMs and bounded degr ..."
Cited by 13 (2 self)
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This paper examines some computational aspects of different arrays enhanced with optical pipelined buses. The array processors with optical pipelined buses (APPB) are shown to be extremely flexible,
as demonstrated by their ability to efficiently simulate different variants of PRAMs and bounded degree networks. A model of computation is introduced, the array with reconfigurable optical buses
(AROB), which combines some of the advantages and characteristics of the classical reconfigurable networks (RN) and the APPB. A number of applications of the APPB and AROB are presented, and their
power is investigated. It is shown that beside AROB's capability of simulating classical reconfigurable networks, the enhanced communication mechanisms allow for an important system reduction when
compared with the classical RNs. Keywords: optical interconnections, pipelined optical buses, reconfigurable networks, bounded degree networks, PRAM models. 1 Introduction Interprocessor
communication networks...
, 1991
"... The first half of the paper is a general introduction which emphasizes the central role that the PRAM model of parallel computation plays in algorithmic studies for parallel computers. Some of
the collective knowledge-base on non-numerical parallel algorithms can be characterized in a structural way ..."
Cited by 11 (4 self)
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The first half of the paper is a general introduction which emphasizes the central role that the PRAM model of parallel computation plays in algorithmic studies for parallel computers. Some of the
collective knowledge-base on non-numerical parallel algorithms can be characterized in a structural way. Each structure relates a few problems and technique to one another from the basic to the more
involved. The second half of the paper provides a bird's-eye view of such structures for: (1) list, tree and graph parallel algorithms; (2) very fast deterministic parallel algorithms; and (3) very
fast randomized parallel algorithms. 1 Introduction Parallelism is a concern that is missing from "traditional" algorithmic design. Unfortunately, it turns out that most efficient serial algorithms
become rather inefficient parallel algorithms. The experience is that the design of parallel algorithms requires new paradigms and techniques, offering an exciting intellectual challenge. We note
that it had...
"... In this paper we study the problem of simulating shared memory on the Distributed Memory Machine (DMM). Our approach uses multiple copies of shared memory cells, distributed among the memory
modules of the DMM via universal hashing. Thus the main problem is to design strategies that resolve cont ..."
Cited by 9 (3 self)
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In this paper we study the problem of simulating shared memory on the Distributed Memory Machine (DMM). Our approach uses multiple copies of shared memory cells, distributed among the memory modules
of the DMM via universal hashing. Thus the main problem is to design strategies that resolve contention at the memory modules. Developing ideas from random graphs and very fast randomized algorithms,
we present new simulation techniques that enable us to improve the previously best results exponentially. Particularly, we show that an n-processor CRCW PRAM can be simulated by an n-processor DMM
with delay O(log log log n log n), with high probability. Next we show a general technique that can be used to turn these simulations to time-processor optimal ones, in the case of EREW PRAMs to be
simulated. We obtain a time-processor optimal simulation of an (n log log log n log n)-processor EREW PRAM on an n-processor DMM with O(log log log n log n) delay. When a CRCW PRAM with (n...
, 1992
"... Problems involving strings arise in many areas of computer science and have numerous practical applications. We consider several problems from a theoretical perspective and provide efficient
algorithms and lower bounds for these problems in sequential and parallel models of computation. In the sequ ..."
Cited by 8 (6 self)
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Problems involving strings arise in many areas of computer science and have numerous practical applications. We consider several problems from a theoretical perspective and provide efficient
algorithms and lower bounds for these problems in sequential and parallel models of computation. In the sequential setting, we present new algorithms for the string matching problem improving the
previous bounds on the number of comparisons performed by such algorithms. In parallel computation, we present tight algorithms and lower bounds for the string matching problem, for finding the
periods of a string, for detecting squares and for finding initial palindromes.
- in Proc. 7th ACM Symp. on Parallel Algorithms and Architectures, (ACM , 1995
"... In this paper we study the question: How useful is randomization in speeding up Exclusive Write PRAM computations? Our results give further evidence that randomization is of limited use in these
types of computations. First we examine a compaction problem on both the CREW and EREW PRAM models, and w ..."
Cited by 7 (0 self)
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In this paper we study the question: How useful is randomization in speeding up Exclusive Write PRAM computations? Our results give further evidence that randomization is of limited use in these
types of computations. First we examine a compaction problem on both the CREW and EREW PRAM models, and we present randomized lower bounds which match the best deterministic lower bounds known. (For
the CREW PRAM model, the lower bound is asymptotically optimal.) These are the first non-trivial randomized lower bounds known for the compaction problem on these models. We show that our lower
bounds also apply to the problem of approximate compaction. Next we examine the problem of computing boolean functions on the CREW PRAM model, and we present a randomized lower bound which improves
on the previous best randomized lower bound for many boolean functions, including the OR function. (The previous lower bounds for these functions were asymptotically optimal, but we improve the
constant multiplicat... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=197871","timestamp":"2014-04-21T02:14:16Z","content_type":null,"content_length":"39374","record_id":"<urn:uuid:aad8a0fc-fa3f-4819-a96a-c70409ca644a>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00210-ip-10-147-4-33.ec2.internal.warc.gz"} |
Maths General Knowledge Quiz
1. Subtract the second even number from the third square number.
2. Divide a thousand by a hundred.
3. What is the name for the longest side of a right angled triangle?
4. If a square is also a rectangle find 16 squared otherwise find the square root of 16.
5. How many grams are in one kilogram?
6. How many seconds are there in four and a half minutes?
7. How many dots are there on a dice?
8. What is the largest prime number less than 20?
9. How mant sides does a decagon have?
10. Multiply this year by last year without using a calculator | {"url":"http://www.transum.org/software/SW/Starter_of_the_day/students/General_Knowledge.asp","timestamp":"2014-04-20T09:22:16Z","content_type":null,"content_length":"33659","record_id":"<urn:uuid:01050a08-193f-4cc8-8f91-d8fe145395cd>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00229-ip-10-147-4-33.ec2.internal.warc.gz"} |
The University of Insubria
All publications are in English.
A. Mira and S. Petrone Bayesian hierarchical nonparametric inference for change point problems
1996, Bayesian Statistics 5 , pp. 693-703
R. Bellazzi, C. Larizza, A. Riva, A. Mira and S. Fiocchi Distribuited intelligent data analysis in diabetic patients management
1996 , Journal of the American Medical Informatics Association , pp. 194 - 198
L. Tierney and A. Mira Some adaptive Monte Carlo methods for Bayesian inference
Statistics in Medicine, 1999, 18, pp. 2507-2515
A. Mira and C.J. Geyer On non-reversible Markov chains
Fields Institute Communications Volume 26: Monte Carlo Methods
2000, pp. 93-108 Providence , RI : American Mathematical Society, N. Madras (Ed.)
A. Mira Distribution-free test for symmetry based on Bonferroni's measure 1999, Journal of Applied Statistics , V. 26, No. 8, pp. 959 - 971
A. Mira Ordering and improving the performance of Monte Carlo Markov chains Statistical Science, 2001, Vol. 16, n. 4, pp. 340-350
A. Mira and P.J. Green Invited discussion of `The art of data augmentation' by David A. van Dyk and Xiao-Li Meng Journal of Computational and Graphical Statistics, 2001, Vol 10:1, pp. 94 - 98
P. J. Green and A. Mira Delayed Rejection in Reversible Jump Metropolis-Hastings Biometrika , 2001 - vol 88, no. 3
A. Mira On Metropolis-Hastings algorithms with delayed rejection Metron, 2001, Vol. LIX, n. 3-4, pp. 231-241
A. Mira, J. Møller and G. O. Roberts Perfect Slice Sampler
Journal of the Royal Statistical Society Ser. B , volume 63 (2001), part 3, pp. 593 - 606
A. Mira e G. Nicholls, Bridge estimation of the probability density at a point, Statistica Sinica, Vol. 14, N. 2, pp. 603-612
A. Mira, Efficiency of finite state space Monte Carlo Markov chains, Statistics & Probability Letters, 2001, Vol. 54:4, pp. 405-411
A. Mira e L. Tierney, Efficiency and Convergence Properties of Slice Samplers. Scandinavian Journal of Statistics, 2002, Vol. 29:1, pp. 1-12
D. Bressanini, A. Morosi, S. Tarasco e A. Mira, Delayed Rejection Variational Monte Carlo, Journal of Chemical Physics, 2004, Vol. 121, n. 8, pp. 3446-3451
A. Mira e G. Roberts, Invited discussion of `Slice sampling' by R. Neal, Annals of Statistics, 2003, Vol. 31:3, pp. 705-767
A. Mira Efficiency increasing and stationarity preserving probability mass transfers for MCMC
Statistics and Probability Letters , 2001, Vol. 54:4, pp. 405-411
A. Mira and L. Tierney On the use of auxiliary variables in Markov chain Monte Carlo sampling
Scandinavian Journal of Statistics, 2002, Vol. 29:1, pp. 1-12
A. Mira and D. Sargent A new strategy for speeding Markov chain Monte Carlo algorithms
Statistical Methods & Applications, 2003, Vol. 1:12, pp. 49-60
A. Mira and G. Roberts Invited discussion of `Slice sampling' by R. Neal Annals of Statistics 2003, Vol. 31:3, pp. 705-767
A. Mira and G. Nicholls Bridge estimation of the probability density at a point Statistica Sinica 2004, Volume 14, Number 2, pp. 603-612
D. Bressanini, A. Morosi, S. Tarasco and A. Mira Delayed Rejection Variational Monte Carlo Journal of Chemical Physic 2004, Vol. 121, n. 8, pp. 3446-3451
A. Mira MCMC methods to estimate Bayesian parametric models Handbook of Statistics Vol 25. "Bayesian Statistics: Modelling and Computation", D.K. Dey and C.R. Rao Ed. 2005, pp. 419-439
F. Audrino, G. Barone-Adesi and A. Mira The stability of factor models of interest rates | {"url":"http://www.uninsubria.eu/research/econ/cv_Mira.htm","timestamp":"2014-04-19T12:09:33Z","content_type":null,"content_length":"23145","record_id":"<urn:uuid:bdc3bf69-4361-49c6-90e1-961f26c6d5f3>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00555-ip-10-147-4-33.ec2.internal.warc.gz"} |
Flipkart Interview Question Site Reliability Engineers
• Suppose there is a linked list ,how to find it is circular and also find the node where it becomes circular..like say
| |
so here 4 is the head where circular linklist starts.
Team: GGn
Country: United States
Interview Type: In-Person
Comment hidden because of low score. Click to expand.
1. Take two pointers, fast and slow. At each iteration, the fast pointer moves forward two nodes and the slow pointer moves forward one node. If the two pointers meet, that means there is a loop.
Break out of the loop.
2. Now start a counter to find out the length of the loop. Initialize the counter to 0. Don't move the slow pointer, only move the fast pointer forward. For every iteration, increment the counter by
1. When the slow and fast pointers meet, the counter's value will give you the length of the loop.
3. Now that you have the length of the loop (loopLength), initialize both slow and fast pointers to the head of the linked list. Start a for loop and move the fast pointer forward loopLength number
of times. At this point, the slow pointer points to the head and the fast pointer is loopLength nodes ahead of the slow pointer.
4. Now just move the slow and fast pointers forward till they meet. Where they meet is the beginning of the loop.
public Node getBeginingOfLoop() {
Node slowIterator = this.head; // slow pointer
Node fastIterator = this.head; // fast pointer
* slow pointer moves ahead one node
* fast pointer moves ahead two nodes.
* If the two pointers meet, break out of the loop
while(fastIterator.getNext().getNext() != null) {
slowIterator = slowIterator.getNext();
fastIterator = fastIterator.getNext().getNext();
if(slowIterator == fastIterator) {
* find out the length of the loop by keeping slow
* pointer stationary and moving the fast pointer
* forward till they meet
int loopLength = 0;
while(slowIterator != fastIterator) {
fastIterator = fastIterator.getNext();
slowIterator = this.head; // initialize pointers to head
fastIterator = this.head;
/* move the fast pointer loopLength nodes ahead */
for(int iii = 0; iii < loopLength; iii++) {
fastIterator = fastIterator.getNext();
*now the slow pointer is at the head and fast pointer is
* loopLength nodes ahead of slow pointer.
* Start moving pointers ahead till they meet
while(slowIterator != fastIterator) {
fastIterator = fastIterator.getNext();
slowIterator = slowIterator.getNext();
return fastIterator; // where they meet is the starting of the loop
Comment hidden because of low score. Click to expand.
geeksforgeeks.org/archives/12225 code for detect and remove loop.
Comment hidden because of low score. Click to expand.
the diagram is not coming fine here..take it as 4 is joined to 7 ,and 0 joined to 9.
Comment hidden because of low score. Click to expand.
circular_LL(node *root){
}while(p->data!=q->data && p!=null);
This code will show the meeting point and detects the circular linked list
Comment hidden because of low score. Click to expand.
Comment hidden because of low score. Click to expand.
Let each node contains a flag 'isVisited' which is set to 'False' initially..
while traversing the list modify the 'isVisited' variable to 'True' in each node....
when you find the node with 'isVisited' variable with 'True'.. then it is circularly linked list and this node is the meeting point.
Comment hidden because of low score. Click to expand.
1. take two pointers fast & slow.
2. Move slow once & fast twice until they do not meet or fast becomes NULL(No cycle)
3. Move slow to root & move slow & fast one step at a time until they do not meet. This is the start of the cycle. Here is pseudo code:
struct node* slow,*fast;
while(fast && fast->next && fast!=slow)
if(!fast || !(fast->next)) return NULL;
return fast;
Comment hidden because of low score. Click to expand.
did not understand your third point here.
Comment hidden because of low score. Click to expand.
Fix fast at the meeting point, move slow to the root.
Now move both slow & fast once until they do not meet.
Comment hidden because of low score. Click to expand.
@shondik: Even I did not understand your 3rd point here. Consider a case where cycle exists for the entire linked list i.e., 1->2->3->4->1 (the first '1') then the fast and slow pointer will never
meet and the last while loop will not come out.
Comment hidden because of low score. Click to expand.
download cracking coding interviews pdf . this questions is explained well in it.
Comment hidden because of low score. Click to expand.
use 2 stacks
pop out stacks when found the meeting point
O(n) space and time Complexity
Comment hidden because of low score. Click to expand.
Have two pointer which is pointed to head of the linked list initially.Then Increment the first pointer by 1, and increment the second pointer by 2 . If these 2 pointers then theres is a circle and
meeting point is value to be print.
Comment hidden because of low score. Click to expand.
but I don't think they will meet at 4 ..which we need. | {"url":"http://www.careercup.com/question?id=14245687","timestamp":"2014-04-16T08:26:33Z","content_type":null,"content_length":"56668","record_id":"<urn:uuid:c20b2215-22ac-4be7-9eb4-55c36d05352d>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00002-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rearrange product of two series
November 1st 2009, 10:28 AM #1
Nov 2008
Rearrange product of two series
I know that:
$<br /> \sum _{i=0}^{\infty } \frac{ X^i }{i!}e^{-X} \sum _{m=0}^{N-1+i} \frac{Y^m}{m!}e^{-Y}<br />$
$\sum _{m=0}^{N-1} \frac{Y^m}{m!}e^{-Y}+\sum _{m=N}^{\infty } \frac{Y^m}{m!}e^{-Y}\left(1- \sum _{i=0}^{m-N} \frac{ X^i }{i!}e^{-X}\right)$
should be equivalent, but how do you get from the first form to the second? Y and X are positive and real.
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FOM: Quine's NF
Matt Insall montez at rollanet.org
Wed Mar 8 16:26:17 EST 2000
Several years ago, I read several books on different set theories. Quine's
NF was mentioned, and I seem to recall that in at least one reference, it
was claimed that NF is inconsistent. I have been unable to find such a
reference lately, but in the text ``Foundations of Set Theory'' by
Bar-Hillel and Levy, the following facts are mentioned:
1. Quine's system presented in his ``Mathematical Logic'', called ``ML'',
was shown to be inconsistent by Rosser in 1942, because it implied the
Burali-Forti Paradox. (Fraenkel, Bar-Hillel and Levy, footnote on page
2. NF contradicts certain ``simple and obvious facts of classical set
theory'' (Fraenkel, Bar-Hillel and Levy, page 163, line 2):
a) In NF, some sets X are such that if Y = {{x}|x \in X}, then X and Y
are not equinumerous.
b) In NF, some sets X are such that X is equinumerous with the power
set of X.
Perhaps I misremembered the quote about ML as a quote about NF. In any
case, item 2 seems a good deal more serious to me (as it seems to have
appeared to Fraenkel, Bar-Hillel and Levy, cf. page 163, line 1). To me,
I expect is the case for most mathematicians, a) and b) are just
counterintuitive, especially since no sets of real numbers satisfy either
or b). It is true that in the NBG extension of ZF to include classes,
are *classes* X of *sets* such that the class P_S(X) of all *subsets* of X
is equinumerous with X, but all such classes are proper classes and cannot
be members of P_S(X) for that reason. In a technical (or formalistic)
sense, these objections may be ``mere linguistic problems'' (and my gut
reaction is to say they are just that), but that is quite an unsatisfying
answer, in my opinion. The fact that one must always be careful which kind
of set one is working with in NF in order to use such fundamental intuitive
notions as the equinumerosity of a set with its set of singletons would
automatically cause me to prefer ZF, or, even better, NBG. (Note that in
NBG, every class is equinumerous with the class of its singletons, and the
``linguistic distinction'' between classes and sets is actually not just a
technical advantage, but an intuitive one as well. Witness all the
following topics in Mathematics: the use of varieties of algebraic
structures in universal algebra, universal spaces in topology, category
theory, algebraic K-theory (I think), etc., etc., etc.)
Steve asked whether NF (or some suitably related system) can be interpreted
in ZF (or some extension, such as NBG), or vice versa. Apparently, this
asked also years ago, for the similar questions appear on page 166 of
``Foundations of Set Theory''. I guess this hasn't been answered yet? Who
might be working on this?
Matt Insall
Associate Professor
Mathematics and Statistics Department
University of Missouri - Rolla
insall at umr.edu
montez at rollanet.org
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/2000-March/003859.html","timestamp":"2014-04-19T17:07:31Z","content_type":null,"content_length":"5224","record_id":"<urn:uuid:5078f7ee-fc3c-4574-b0e8-8529c1695246>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00167-ip-10-147-4-33.ec2.internal.warc.gz"} |
How many dollars is 16 pounds?
You asked:
How many dollars is 16 pounds?
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Evi is our best selling mobile app that can answer questions about local knowledge, weather, books, music, films, people and places, recipe ideas, shopping and much more. Over the next few months we
will be adding all of Evi's power to this site.
Until then, to experience all of the power of Evi you can download Evi for free on iOS, Android and Kindle Fire. | {"url":"http://www.evi.com/q/how_many_dollars_is_16_pounds","timestamp":"2014-04-19T17:04:23Z","content_type":null,"content_length":"57705","record_id":"<urn:uuid:3815d1be-651d-4c9b-91a2-4eb08bcb9215>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00523-ip-10-147-4-33.ec2.internal.warc.gz"} |
This class implements the Hilbert Schmidtd Independence Criterion based independence test as described in [1].
Given samples
The HSIC is a kernel based independence criterion, which is based on the largest singular value of a Cross-Covariance Operator in a reproducing kernel Hilbert space (RKHS). Its population expression
is zero if and only if the two underlying distributions are independent.
This class can compute empirical biased estimates:
Note that computing the statistic returns m*MMD; same holds for the null distribution samples.
Along with the statistic comes a method to compute a p-value based on different methods. Bootstrapping, is also possible. If unsure which one to use, bootstrapping with 250 iterations always is
correct (but slow).
To choose, use set_null_approximation_method() and choose from
HSIC_GAMMA: for a very fast, but not consistent test based on moment matching of a Gamma distribution, as described in [1].
BOOTSTRAPPING: For permuting available samples to sample null-distribution. Bootstrapping is done on precomputed kernel matrices, since they have to be stored anyway when the statistic is computed.
A very basic method for kernel selection when using CGaussianKernel is to use the median distance of the underlying data. See examples how to do that. More advanced methods will follow in the near
future. However, the median heuristic works in quite some cases. See [1].
[1]: Gretton, A., Fukumizu, K., Teo, C., & Song, L. (2008). A kernel statistical test of independence. Advances in Neural Information Processing Systems, 1-8.
Definition at line 67 of file HSIC.h.
Public Member Functions
CHSIC ()
CHSIC (CKernel *kernel_p, CKernel *kernel_q, CFeatures *p_and_q, index_t q_start)
CHSIC (CKernel *kernel_p, CKernel *kernel_q, CFeatures *p, CFeatures *q)
virtual ~CHSIC ()
virtual float64_t compute_statistic ()
virtual float64_t compute_p_value (float64_t statistic)
virtual float64_t compute_threshold (float64_t alpha)
virtual const char * get_name () const
SGVector< float64_t > fit_null_gamma ()
virtual SGVector< float64_t > bootstrap_null ()
virtual void set_bootstrap_iterations (index_t bootstrap_iterations)
virtual void set_null_approximation_method (ENullApproximationMethod null_approximation_method)
virtual CSGObject * shallow_copy () const
virtual CSGObject * deep_copy () const
virtual bool is_generic (EPrimitiveType *generic) const
template<class T >
void set_generic ()
void unset_generic ()
virtual void print_serializable (const char *prefix="")
virtual bool save_serializable (CSerializableFile *file, const char *prefix="", int32_t param_version=VERSION_PARAMETER)
virtual bool load_serializable (CSerializableFile *file, const char *prefix="", int32_t param_version=VERSION_PARAMETER)
DynArray< TParameter * > * load_file_parameters (const SGParamInfo *param_info, int32_t file_version, CSerializableFile *file, const char *prefix="")
DynArray< TParameter * > * load_all_file_parameters (int32_t file_version, int32_t current_version, CSerializableFile *file, const char *prefix="")
void map_parameters (DynArray< TParameter * > *param_base, int32_t &base_version, DynArray< const SGParamInfo * > *target_param_infos)
void set_global_io (SGIO *io)
SGIO * get_global_io ()
void set_global_parallel (Parallel *parallel)
Parallel * get_global_parallel ()
void set_global_version (Version *version)
Version * get_global_version ()
SGStringList< char > get_modelsel_names ()
void print_modsel_params ()
char * get_modsel_param_descr (const char *param_name)
index_t get_modsel_param_index (const char *param_name)
void build_parameter_dictionary (CMap< TParameter *, CSGObject * > &dict) | {"url":"http://www.shogun-toolbox.org/doc/en/2.0.0/classshogun_1_1CHSIC.html","timestamp":"2014-04-18T08:50:12Z","content_type":null,"content_length":"107833","record_id":"<urn:uuid:c9f07584-d200-45bd-af67-ba0907d4f341>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00536-ip-10-147-4-33.ec2.internal.warc.gz"} |
How to test if a number is an integer with Javascript
Testing if a value is an integer in Javascript
I decided to post this article after doing a Google search for "Javascript is_int()" and "Javascript integer test" on Google.
I got plenty of results back, but each result that I looked at contained a function which don't always work as intended, either because the logic was flawed, or because it won't work with variables
which are typed as a string, such as form input values.
Let me explain...
Javascript is_int() function example 1
Some sites proposed a solution which involved doing a pattern matching each for a decimal point in the variable. Something along the lines of this:
function is_int(value){
for (i = 0 ; i < value.length ; i++) {
if ((value.charAt(i) < '0') || (value.charAt(i) > '9')) return false
return true;
Why this won't work
If I enter 1.00 or 2.00 as my variable, the function will return false, even though the number I entered is in fact an integer.
Javascript is_int() function example 2
Some solutions propose doing a parseInt() on the value, and returning false if this is NaN:
function is_int(value){
return !isNaN(parseInt(value * 1)
Why this won't work
Whilst this will work with standard floats and integers, it won't work with string values (for example, form submission data).
Finally: a Javascript is_int() function which WORKS:
If the value of a variable is an integer, then the numeric value of it's parseFloat() and parseInt() equivalents will be the same, so:
function is_int(value){
if((parseFloat(value) == parseInt(value)) && !isNaN(value)){
return true;
} else {
return false;
NOTE: the above function uses WEAK VARIABLE TYPING - if you need to check is_int() using strong variable typing, just use example 2 instead.
There you go. If you can find fault with me on this, let me know. | {"url":"http://www.inventpartners.com/javascript_is_int","timestamp":"2014-04-17T02:10:28Z","content_type":null,"content_length":"9653","record_id":"<urn:uuid:5452b101-8831-4a88-a86a-c855a027408f>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00005-ip-10-147-4-33.ec2.internal.warc.gz"} |
Comment on
Hey kiat,
I noticed that your algorithm skips the "1 1" row. I made some changes to fix that bug:
sub pascal {
my ($rows) = @_;
print "1\n";
for (my $outer = 1; $outer < $rows; $outer++) {
my $inner = $outer;
print "1";
for ($i = 1; $i < $inner; $i++) {
my $denominator = factorial($i)*(factorial($inner-$i));
my $pascalnum = factorial($inner)/$denominator if ($denominator
+!= 0);
print " $pascalnum" if ($outer > 1);
print " 1\n";
I also think that using the Binomial Theorem when printing out an entire Pascal's triangle is inefficient. I would rather add numbers from the previous row, as this is much, much faster for large
numbers of rows. So I wrote up some code to do this:
sub ton_pascal
my $rows = shift;
my $last_row = [ ];
my $this_row = [ 1 ];
last unless ($rows > 0);
print "1\n";
for (my $i = 1; $i < $rows; ++$i) {
$last_row = $this_row;
$this_row = [ 1 ];
for (my $j = 1; $j < $i; $j++) {
push(@$this_row, $last_row->[$j - 1] + $last_row->[$j]);
push(@$this_row, 1);
print join(' ', @$this_row) . "\n";
I'm using array references instead of arrays for extra speed (when we copy the results of $this_row into $last_row), but the references could be removed without affecting the algorithm.
Hope this was helpful... I had fun coding up ton_pascal, so thanks for the problem!
Be bloody, bold, and resolute; laugh to scorn
The power of man...
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Improved Criteria on Delay-Dependent Stability for Discrete-Time Neural Networks with Interval Time-Varying Delays
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 285931, 16 pages
Research Article
Improved Criteria on Delay-Dependent Stability for Discrete-Time Neural Networks with Interval Time-Varying Delays
^1School of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 361-763, Republic of Korea
^2Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Gyeongsan 712-749, Republic of Korea
^3School of Electronic Engineering, Daegu University, Gyeongsan 712-714, Republic of Korea
^4Department of Biomedical Engineering, School of Medicine, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 361-763, Republic of Korea
Received 16 August 2012; Accepted 5 October 2012
Academic Editor: Tingwen Huang
Copyright © 2012 O. M. Kwon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
The purpose of this paper is to investigate the delay-dependent stability analysis for discrete-time neural networks with interval time-varying delays. Based on Lyapunov method, improved
delay-dependent criteria for the stability of the networks are derived in terms of linear matrix inequalities (LMIs) by constructing a suitable Lyapunov-Krasovskii functional and utilizing
reciprocally convex approach. Also, a new activation condition which has not been considered in the literature is proposed and utilized for derivation of stability criteria. Two numerical examples
are given to illustrate the effectiveness of the proposed method.
1. Introduction
Neural networks have received increasing attention of researches from various fields of science and engineering such as moving image reconstructing, signal processing, pattern recognition, and
fixed-point computation. In the hardware implementation of systems, there exists naturally time delay due to the finite information processing speed and the finite switching speed of amplifiers. It
is well known that time delay often causes undesirable dynamic behaviors such as performance degradation, oscillation, or even instability of the systems. Since it is a prerequisite to ensure
stability of neural networks before its application to various fields such as information science and biological systems, the problem of stability of neural networks with time delay has been a
challenging issue [1–10]. Also, these days, most systems use digital computers (usually microprocessor or microcontrollers) with the necessary input/output hardware to implement the systems. The
fundamental character of the digital computer is that it takes computed answers at discrete steps. Therefore, discrete-time modeling with time delay plays an important role in many fields of science
and engineering applications. With this regard, various approaches to delay-dependent stability criteria for discrete-time neural networks with time delay have been investigated in the literature [11
In the field of delay-dependent stability analysis, one of the hot issues attracting the concern of the researchers is to increase the feasible region of stability criteria. The most utilized index
to check the conservatism of stability criteria is to get maximum delay bounds for guaranteeing the globally exponential stability of the concerned networks. Thus, many researchers put time and
efforts into some new approaches to enhance the feasible region of stability conditions. In this regard, Liu et al. [11] proposed a unified linear matrix inequality approach to establish sufficient
conditions for the discrete-time neural networks to be globally exponentially stable by employing a Lyapunov-Krasovskii functional. In [12, 13], the existence and stability of the periodic solution
for discrete-time recurrent neural network with time-varying delays were studied under more general description on activation functions by utilizing free-weighting matrix method. Based on the idea of
delay partitioning, a new stability criterion for discrete-time recurrent neural networks with time-varying delays was derived [14]. Recently, some novel delay-dependent sufficient conditions for
guaranteeing stability of discrete-time stochastic recurrent neural networks with time-varying delays were presented in [15] by introducing the midpoint of the time delay’s variational interval. Very
recently, via a new Lyapunov functional, a novel stability criterion for discrete-time recurrent neural networks with time-varying delays was proposed in [16] and its improvement on the feasible
region of stability criterion was shown through numerical examples. However, there are rooms for further improvement in delay-dependent stability criteria of discrete-time neural networks with
time-varying delays.
Motivated by the above discussions, the problem of new delay-dependent stability criteria for discrete-time neural networks with time-varying delays is considered in this paper. It should be noted
that the delay-dependent analysis has been paid more attention than delay-independent one because the sufficient conditions for delay-dependent analysis make use of the information on the size of
time delay [17, 18]. That is, the former is generally less conservative than the latter. By construction of a suitable Lyapunov-Krasovskii functional and utilization of reciprocally convex approach [
19], a new stability criterion is derived in Theorem 3.1. Based on the results of Theorem 3.1 and motivated by the work of [20], a further improved stability criterion will be introduced in Theorem
3.4 by applying zero equalities to the results of Theorem 3.1. Finally, two numerical examples are included to show the effectiveness of the proposed method.
Notation. is the -dimensional Euclidean space, and denotes the set of all real matrices. For symmetric matrices and , (resp., ) means that the matrix is positive definite (resp., nonnegative).
denotes a basis for the null space of . denotes the identity matrix with appropriate dimensions. refers to the Euclidean vector norm or the induced matrix norm. denotes the block diagonal matrix.
represents the elements below the main diagonal of a symmetric matrix.
2. Problem Statements
Consider the following discrete-time neural networks with interval time-varying delays: where denotes the number of neurons in a neural network, is the neuron state vector, denotes the neuron
activation function vector, means a constant external input vector, is the state feedback matrix, are the connection weight matrices, and is interval time-varying delays satisfying where and are
known positive integers.
In this paper, it is assumed that the activation functions satisfy the following assumption.
Assumption 2.1. The neurons activation functions, , are continuous and bounded, and for any , , where and are known constant scalars.
As usual, a vector is said to be an equilibrium point of system (2.1) if it satisfies . From [10], under Assumption 2.1, it is not difficult to ensure the existence of equilibrium point of the system
(2.1) by using Brouwer’s fixed-point theorem. In the sequel, we will establish a condition to ensure the equilibrium point of system (2.1) is globally exponentially stable. That is, there exist two
constants and such that . To confirm this, refer to [16]. For simplicity, in stability analysis of the network (2.1), the equilibrium point is shifted to the origin by utilizing the transformation ,
which leads the network (2.1) to the following form: where is the state vector of the transformed network, and is the transformed neuron activation function vector with and . From Assumption 2.1, it
should be noted that the activation functions satisfy the following condition [10]: which is equivalent to and if , then the following inequality holds:
Here, the aim of this paper is to investigate the delay-dependent stability analysis of the network (2.4) with interval time-varying delays. In order to do this, the following definition and lemmas
are needed.
Definition 2.2 (see [16]). The discrete-time neural network (2.4) is said to be globally exponentially stable if there exist two constants and such that
Lemma 2.3 ((Jensen inequality) [21]). For any constant matrix , integers and satisfying , and vector function , the following inequality holds:
Lemma 2.4 ((Finsler’s lemma) [22]). Let , , and such that . The following statements are equivalent: (i),,,(ii),(iii),.
3. Main Results
In this section, new stability criteria for the network (2.4) will be proposed. For the sake of simplicity on matrix representation, are defined as block entry matrices (e.g., ). The notations of
several matrices are defined as
Now, the first main result is given by the following theorem.
Theorem 3.1. For given positive integers and , diagonal matrices and , the network (2.4) is globally exponentially stable for , if there exist positive definite matrices , , , , , positive diagonal
matrices , any symmetric matrices , and any matrix satisfying the following LMIs: where , , and are defined in (3.1).
Proof. Define the forward difference of and as Let us consider the following Lyapunov-Krasovskii functional candidate as where The forward differences of and are calculated as By calculating the
forward differences of and , we get For any matrix , integers and satisfying , and a vector function where is the discrete time, the following equality holds: It should be noted that From the
equalities (3.12) and (3.13), by choosing as , and , the following three zero equations hold with any symmetric matrices ,, and : By adding three zero equalities into the results of , we have where
By Lemma 2.3, when , the sum term in (3.18) is bounded as where .
By reciprocally convex approach [19], if the inequality (3.3) holds, then the following inequality for any matrix satisfies which implies It should be pointed out that when or , we have or ,
respectively. Thus, the following inequality still holds: Then, has an upper bound as follows: Here, if the inequalities (3.4) hold, then is bounded as From (2.7), for any positive diagonal matrices
, the following inequality holds: Therefore, from (3.8)–(3.16) and by application of the -procedure [23], has a new upper bound as where and are defined in (3.1).
Also, the system (2.4) with the augmented vector can be rewritten as where is defined in (3.1).
Then, a delay-dependent stability condition for the system (2.4) is Finally, by utilizing Lemma 2.4, the condition (3.29) is equivalent to the following inequality From the inequality (3.30), if the
LMIs (3.2)-(3.4) hold. From (ii) and (iii) of Lemma 2.4, if the stability condition (3.29) holds, then for any free maxrix with appropriate dimension, the condition (3.29) is equivalent to Therefore,
from (3.31), there exists a sufficient small scalar such that By using the similar method of [11, 12], the system (2.4) is globally exponentially stable for any time-varying delay from Definition 2.2
. This completes our proof.
Remark 3.2. In Theorem 3.1, the stability condition is derived by utilizing a new augmented vector including . This state vector which may give more information on dynamic behavior of the system (2.4
) has not been utilized as an element of augmented vector in any other literature. Correspondingly, the state vector is also included in (3.26).
Remark 3.3. As mentioned in [10], the activation functions of transformed system (2.4) also satisfy the condition (2.6). In Theorem 3.4, by choosing in (2.6) as and , , more information on
cross-terms among the states , , , , and will be utilized, which may lead to less conservative stability criteria. In stability analysis for discrete-time neural networks with time-varying delays,
this consideration has not been proposed in any other literature. Through two numerical examples, it will be shown that the newly proposed activation condition may enhance the feasible region of
stability criterion by comparing maximum delay bounds with the results obtained by Theorem 3.1.
As mentioned in Remark 3.3, from (2.6), we add the following new inequality with any positive diagonal matrices to be chosen as where , , and . We will add this inequality (3.33) in Theorem 3.4. Now,
we have the following theorem.
Theorem 3.4. For given positive integers and , diagonal matrices and , the network (2.4) is globally exponentially stable for , if there exist positive definite matrices , , , , , positive diagonal
matrices , any symmetric matrices , and any matrix satisfying the following LMIs: where , , and are defined in (3.1) and is in (3.33).
Proof. With the same Lyapunov-Krasovskii functional candidate in (3.6), by using the similar method in (3.8)–(3.16), and considering inequality (3.36), the procedure of deriving the condition (3.34)–
(3.36) is straightforward from the proof of Theorem 3.1, so it is omitted.
4. Numerical Examples
In this section, we provide two numerical examples to illustrate the effectiveness of the proposed criteria in this paper.
Example 4.1. Consider the discrete-time neural networks (2.4) where The activation functions satisfy Assumption 2.1 with For various , the comparison of maximum delay bounds () obtained by Theorems
3.1 and 3.4 with those of [12, 16] is conducted in Table 1. From Table 1, it can be confirmed that the results of Theorem 3.1 give a larger delay bound than those of [12] and are equal to the results
of [16]. However, the results obtained by Theorem 3.4 are better than the results of [16] and Theorem 3.1, which supports the effectiveness of the proposed idea mentioned in Remark 3.3.
Example 4.2. Consider the discrete-time neural networks (2.4) having the following parameters: When , for different values of , maximum delay bounds obtained by [12–14, 16] and our Theorems are
listed in Table 2. From Table 2, it can be confirmed that all the results of Theorems 3.1 and 3.4 provide larger delay bounds than those of [12–14]. Also, our results are better than or equal to the
results of [16]. For the case of , another comparison of our results with those of [15, 16] is conducted in Table 3, which shows all the results obtained by Theorems 3.1 and 3.4 give larger delay
bounds than those of [15, 16].
5. Conclusions
In this paper, improved delay-dependent stability criteria were proposed for discrete-time neural networks with time-varying delays. In Theorem 3.1, by constructing the suitable Lyapunov-Krasovskii’s
functional and utilizing some recent results introduced in [19, 20], the sufficient condition for guaranteeing the global exponential stability of discrete-time neural network having interval
time-varying delays has been derived. Based on the results of Theorem 3.1, by constructing new inequalities of activation functions, the further improved stability criterion was presented in Theorem
3.4. Via two numerical examples, the improvement of the proposed stability criteria has been successfully verified.
This paper was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0000479), and by a
grant of the Korea Healthcare Technology R & D Project, Ministry of Health & Welfare, Republic of Korea (A100054).
1. O. Faydasicok and S. Arik, “Equilibrium and stability analysis of delayed neural networks under parameter uncertainties,” Applied Mathematics and Computation, vol. 218, no. 12, pp. 6716–6726,
2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
2. O. Faydasicok and S. Arik, “Further analysis of global robust stability of neural networks with multiple time delays,” Journal of the Franklin Institute, vol. 349, no. 3, pp. 813–825, 2012. View
at Publisher · View at Google Scholar
3. T. Ensari and S. Arik, “New results for robust stability of dynamical neural networks with discrete time delays,” Expert Systems with Applications, vol. 37, no. 8, pp. 5925–5930, 2010. View at
Publisher · View at Google Scholar · View at Scopus
4. C. Li, C. Li, and T. Huang, “Exponential stability of impulsive high-order Hopfield-type neural networks with delays and reaction-diffusion,” International Journal of Computer Mathematics, vol.
88, no. 15, pp. 3150–3162, 2011. View at Publisher · View at Google Scholar
5. C. J. Li, C. D. Li, T. Huang, and X. Liao, “Impulsive effects on stability of high-order BAM neural networks with time delays,” Neurocomputing, vol. 74, no. 10, pp. 1541–1550, 2011. View at
Publisher · View at Google Scholar · View at Scopus
6. D. J. Lu and C. J. Li, “Exponential stability of stochastic high-order BAM neural networks with time delays and impulsive effects,” Neural Computing and Applications. In press. View at Publisher
· View at Google Scholar · View at Scopus
7. C. D. Li, C. J. Li, and C. Liu, “Destabilizing effects of impulse in delayed bam neural networks,” Modern Physics Letters B, vol. 23, no. 29, pp. 3503–3513, 2009. View at Publisher · View at
Google Scholar · View at Scopus
8. C. D. Li, S. Wu, G. G. Feng, and X. Liao, “Stabilizing effects of impulses in discrete-time delayed neural networks,” IEEE Transactions on Neural Networks, vol. 22, no. 2, pp. 323–329, 2011. View
at Publisher · View at Google Scholar · View at Scopus
9. H. Wang and Q. Song, “Synchronization for an array of coupled stochastic discrete-time neural networks with mixed delays,” Neurocomputing, vol. 74, no. 10, pp. 1572–1584, 2011. View at Publisher
· View at Google Scholar · View at Scopus
10. Y. Liu, Z. Wang, and X. Liu, “Global exponential stability of generalized recurrent neural networks with discrete and distributed delays,” Neural Networks, vol. 19, no. 5, pp. 667–675, 2006. View
at Publisher · View at Google Scholar · View at Scopus
11. Y. Liu, Z. Wang, A. Serrano, and X. Liu, “Discrete-time recurrent neural networks with time-varying delays: exponential stability analysis,” Physics Letters A, vol. 362, no. 5-6, pp. 480–488,
2007. View at Publisher · View at Google Scholar · View at Scopus
12. Q. Song and Z. Wang, “A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays,” Physics Letters A, vol. 368, no. 1-2, pp. 134–145,
2007. View at Publisher · View at Google Scholar · View at Scopus
13. B. Zhang, S. Xu, and Y. Zou, “Improved delay-dependent exponential stability criteria for discrete-time recurrent neural networks with time-varying delays,” Neurocomputing, vol. 72, no. 1–3, pp.
321–330, 2008. View at Publisher · View at Google Scholar · View at Scopus
14. C. Song, H. Gao, and W. Xing Zheng, “A new approach to stability analysis of discrete-time recurrent neural networks with time-varying delay,” Neurocomputing, vol. 72, no. 10-12, pp. 2563–2568,
2009. View at Publisher · View at Google Scholar · View at Scopus
15. Y. Zhang, S. Xu, and Z. Zeng, “Novel robust stability criteria of discrete-time stochastic recurrent neural networks with time delay,” Neurocomputing, vol. 72, no. 13-15, pp. 3343–3351, 2009.
View at Publisher · View at Google Scholar · View at Scopus
16. Z. Wu, H. Su, J. Chu, and W. Zhou, “Improved delay-dependent stability condition of discrete recurrent neural networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 21,
no. 4, pp. 692–697, 2010. View at Publisher · View at Google Scholar · View at Scopus
17. S. Xu and J. Lam, “A survey of linear matrix inequality techniques in stability analysis of delay systems,” International Journal of Systems Science, vol. 39, no. 12, pp. 1095–1113, 2008. View at
Publisher · View at Google Scholar · View at Zentralblatt MATH
18. H. Y. Shao, “Improved delay-dependent stability criteria for systems with a delay varying in a range,” Automatica, vol. 44, no. 12, pp. 3215–3218, 2008. View at Publisher · View at Google Scholar
· View at Zentralblatt MATH
19. P. Park, J. W. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, no. 1, pp. 235–238, 2011. View at Publisher · View at Google
Scholar · View at Zentralblatt MATH
20. S. H. Kim, “Improved approach to robust ℋ[∞] stabilization of discrete-time T–S fuzzy systems with time-varying delays,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 5, pp. 1008–1015, 2010.
View at Publisher · View at Google Scholar · View at Scopus
21. X. L. Zhu and G. H. Yang, “Jensen inequality approach to stability analysis of discrete-time systems with time-varying delay,” in Proceedings of the American Control Conference (ACC '08), pp.
1644–1649, Seattle, Wash, USA, June 2008. View at Publisher · View at Google Scholar · View at Scopus
22. M. C. de Oliveira and R. E. Skelton, Stability Tests for Constrained Linear Systems, Springer, Berlin, Germany, 2001.
23. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994.
View at Publisher · View at Google Scholar | {"url":"http://www.hindawi.com/journals/aaa/2012/285931/","timestamp":"2014-04-18T01:52:48Z","content_type":null,"content_length":"671206","record_id":"<urn:uuid:0c015630-999e-4716-adc6-d35adc531c55>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00016-ip-10-147-4-33.ec2.internal.warc.gz"} |
Table 6: Correlation coefficient and Kolmogorov-Smirnov test statistic between empirical and theoretical CDF of the Mahalanobis distance around each of the targets and the condition number of local .
Name Preprocessing Correlation mean ± std KS statistic mean ± std Condition number of local mean ± std
CAM SB
KDR (5kcp)
OSLO1 SB
KDR (5kcp)
BJO SB
SQRT + SB
OSLO2 None
Name Preprocessing Correlation mean ± std KS statistic mean ± std Condition number of local mean ± std
CAM SB
KDR (5kcp)
OSLO1 SB
KDR (5kcp)
BJO SB
SQRT + SB
OSLO2 None | {"url":"http://www.hindawi.com/journals/jece/2012/162106/tab6/","timestamp":"2014-04-17T19:22:52Z","content_type":null,"content_length":"165819","record_id":"<urn:uuid:1ebba0a3-739c-46e0-9419-bf70bb0b44ea>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00186-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Does anyone have any nice ways of handling differentials like \[ \frac{d^2}{dx^2}\left(\frac{20}{1+x^2}\right)\]? And, the more extreme cases like \[\frac{d^4}{dx^4}\left(\frac{20}{1+x^2}\right)\].
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Have you considered using a negative exponent? \[\frac{d^2}{dx^2}(20)(1+x^2)^{-1}\]
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Yes, but the problem gets ugly, quickly, especially considering we are taking the second derivative. So, I'm guessing, there is none? All just straight up computation. Hmm.
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i usually use a "u-substitution" in derivatives...i suppose that makes me weird....
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Enlighten me...?
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@LolWolf \[\frac{ d ^{2} }{ dx ^{2} }(\frac{ 20 }{ 1+x ^{2} })\]implies that you find the second derivative of the function y, if\[y =\frac{ 20 }{ 1+x ^{2} }\]I assume you know your derivative
rules, correct?
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Yes, I'm not wondering so much how to find it, but, rather, how to do it neatly.
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(If someone has some ingenious way of managing it, et al)
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y'=-40x/(1+x^2)^2 now do it again the second time
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No, I am a math teacher and I will tell you that you must use the quotient rule to find the second derivative, but it would be wise to use the chain rule (power of a function rule) to find the
first derivative.
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use the quotient rule
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@mark_o. Yes, I know. I'm not talking about how to find it, finding it is simple, I'm referring to taking this twice or thrice without having an utter mess of calculations, or without it taking
more than some short time. I don't know if there even *is* such a way, I'm just guessing.
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@mark_o. please stop giving out answers. You're not helping!
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I already have the answer. That's not what I'm looking for. I'm wondering if there is some way to manage this in a quick and nice manner, rather than crunching numbers.
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(Again, there might *not* be, but I'm not the most imaginative person in Calculus--I deal with Number Theory--and this question came up)
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Those derivative rules are in place for a reason. You're looking for a short cut and I'm telling you that those derivative rules are the shortcut.
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im sorry i didnt want to give the answer on the second derivative., i thought he didnt knoe how to arrive at the first derivative.... continue plzz
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@mark_o. That's alright! He said he already got the answer so no harm done.
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Yes, I am not referring to the derivative rules. Of course, most people could come up with the proofs, and I am not looking for a shortcut, at least, not directly. The point that I ask is *not*
how to find them, nor to get a lesson on the shortcuts, and whether they should be used or not, I'm asking if there *is* a nice way of *handling* the operations, or if there is some much nicer
(not necessarily *easier*) way of dealing with problems that are similar.
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ok continue plzz...
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@mark_o. there is nothing to continue LOL because @LolWolf the answer to your question is a very nice NO.
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That's all I wanted to know. Thank you.
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well, there no way of getting around it,or to short cut them, just continue solving them wether you started on quotient rule or product rule.you will arrive on the same answer,......
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Yes, thank you.
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ok YW, have fun now
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Actually, there IS, indeed a much nicer way to do this derivative, if we split it into separate functions. Not to beat a dead horse, or anything, but please don't say that one 'cannot' do
something without actually knowing so, even if you're a teacher. If we simply assume the function of \(x^2=l\), then, by repeated application of the chain rule, we reduce the laborious task of
handling the repeated quotient rule to a simple product rule with a single multiplication per step, along with a much easier-to-compute chain rule. Enjoy.
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How to get a regression curve for an x-y value pair where a single x can have multiple y values in the set?
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I am trying to generate a regression curve for a set of x-y values where a single x value can have multiple y values. Although in the end, only one y value should be the correct one and rest all
should be categorized as outliers. Is least square linear regression the right way to do this? I was reading a paper which said I could apply linear regression here; but when I use an existing code
library and input my data, it gives an error. How do I get the line for this data?
st.statistics ca.analysis-and-odes
It is a normal situation that doesn't make a difference to the analysis. The software shouldn't complain, except perhaps if you give it degenerate case like all $x$ values equal. – Brendan McKay
Oct 18 '12 at 11:44
1 This question really belongs here... stats.stackexchange.com/?as=1 – Kjetil B Halvorsen Oct 18 '12 at 14:58
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Pop Quiz Friday - It's not what you think it is edition
In ‘crossword format’
The government just announced a reduction in VAT from 17.5% to 15%. How much money do you save per £100?
You’re at a magic show. The magician asks you to shuffle the cards (a normal 52 card deck) and hand them back to you. He asks a member of the audience to name a card. The audience member names
the Ace of Spades. The magician draws a single card out of the deck. What is the chance (odds?) of the card drawn being the card named?
EOD: Answer Update (after 9 comments)
The papers/press/media have all been reporting that with a 2.5% reduction in VAT you’ll only be saving £2.50 in every £100 by working out the complex equation of “What’s 2.5% of 100?”. Of course
this is clearly wrong.
In £100 payment inclusive 17.5% tax you’re paying £14.89 tax such that the item value would have been £85.11
A 15% tax on £85.11 would result in £12.77 tax.
£14.89 - £12.77 = £2.12 savings.
Of course as Nigel points out, there is actually a more complex answer here too (based on my question), and that even for a specific right answer, there can be multiple right answers.
Also everyone has been writing in to letters pages commenting how pointless the reduction is, as £2.50 is not a lot of money. Which proves my point that most people are idiots who don’t
understand anything. I wont explain here why a 2.5% decrease is significant, just to say image how people would have reacted had VAT gone UP by 2.5%.
The question really is about how we evaluate risk. And how we consider the factors involved. There is no real right answer here, the right answer is in how we explain the answer. Of course some
answers are more right than others.
Marc was the most correct here, or put another way, had a most correct answer.
A simple way of looking at it is like this
□ The random chance of pulling a specific card out of a deck is 1 in 52
□ However we know we are at magic show and we know a magician probably knows what we are doing so we could consider the odds are then really 1 in 1.
□ However looking at the factors again, we know a magician probably is going to not pull the card out the first time to build suspense, so we could say the odds are 0 in 1.
□ Or we can factor together different tricks (as Marc did) and calculate even more precise odds.
So those are the two answers (or two of the answers) to today’s Friday Pop Quiz questions. I’ll be honest I stole the questions from the BBC R4 More or Less podcast. If you want to listen to more
detail you can watch it on iPlayer or download here
I quite liked pop quiz Friday. More for the debate and interesting perspectives (including @Jack (2)’s). I might do this every week. If I can think of enough interesting questions.
9 Comments
1. Will
£2.13 I think?
To the cryptic: if the magician is aiming to pick it out, probably 1 - if they’re planning to get it wrong, 0. And if it’s just random, 1 in 52.
1. Charles Dickens
2. The Polish–Swedish War of 1600–1611
3. almostwitty.com
1. £2.12 I think?
2. No doubt this will trigger QI-style WOOGA WOOGA YOU ARE WRONG alerts, but I’m guessing 1 in 52?
4. Marc
1) If you are looking at an item that cost £100 after VAT then it will £2.127 cheaper with the new rate.
2) My guess would be somewhere between evens to 1 in 5. Let’s call it 2 in 7.
5. Adrian
@Marc (4), Can you explain your thinking for point 2?
(unless you want to wait till other people have answered a bit, and see what people are saying, which is cool too)
6. Marc
Correction: £2.128
Although if you were looking to spend your whole £100 then you can afford things that are £1.850 more expensive pre-VAT.
7. Marc
It’s a bit plucked out of the air on the probability calculation front but considering it’s a magician then I assume the trick is to reveal the card. A straight forward card trick would reveal the
card straight away which means it’s an even chance. But then I thought about one of my card tricks and there’s three iterations before the reveal. And I can think of a trick that goes through five
iterations before the selected card is revealed.
Assuming there are card tricks that present the reveal instantly and after 5 iterations. We will discount tricks with more iterations as too few to be statistically relevant as they approach
interminably long and tedious tricks.
So if the tricks with 1 to 5 iterations were evenly distributed there would be 1 in 5 chance that it would be your card.
But as I plucked my number out of the air I’m going to prove it. In fact I actually know that the distribution of card tricks looks like this:
Reveals | # of Tricks*
1 | 240
2 | 180
3 | 210
4 | 90
5 | 120
Total: 840
So for immediate reveals: 240/840 = 28.57% = 2/7.
* Officially recognised reveal card tricks - Magic Circle Census 2003
8. Nigel
1. There’s actually no simple answer to this.
The government is funding this VAT cut with extra borrowing, which will need to be paid off in the future by increased taxes. So how much you “save” (or, potentially “lose”) depends on the
combination of your future overall tax burden (in turn dependent on the shape of future tax policy to pay off the extra borrowing) and your current consumption (of products on which you pay VAT).
2. I don’t think there’s much to add to Marc’s explanation here.
9. Matt
Quick answer: Not much.
Cryptic answer: The chance is high, but the odds are low. | {"url":"http://sevitz.com/2008/12/pop_quiz_friday_-_its_not_what_you_think_it_is_edition","timestamp":"2014-04-19T01:48:55Z","content_type":null,"content_length":"24764","record_id":"<urn:uuid:9fd048db-fddc-4140-ad03-d036e95bfb48>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00611-ip-10-147-4-33.ec2.internal.warc.gz"} |
Blindfold Cubing
Blindfold cubing requires that the problem of solving the cube be split into a number of independent problems using moves that only affect a sub-portion of the cubes. A normal approach would be along
these lines:
orient corners
place corners (perhaps swapping 2 edges in the process)
orient edges
place edges
Making the solving of the corners and the solving of the edges 'almost' independent puts the problem of solving the cube blindfold within reach of the average human being.
Normally you solve the cube in the LRFBUD group. This means that you can make turns of any faces. This may result in flipping of edges. Solving the cube in the L2R2FBUD group means that turns of the
L and R face must be double turns, whilst all other faces may be single or double moves. This group is of great use in solving blindfold since edge pieces cannot be flipped by moves within this
Orientating the corners so they lie in the L2R2F2B2UD group makes it irrelevant in which order the corners are placed in the correct location.
Which edges will require flipping can be determined by seeing whether the pieces can be placed by moving them through the middle layer using F and B turns. Note that the group that the edges are to
be solved in means that the cube can be orientated in 3 different ways. Choosing your orientation of the cube affects which pieces are considered flipped, and which aren't.
The PnXsnX'sn'Pn' technique
The PnXsnX'sn'Pn' technique is a simple way of finding moves that swap/rotates edges/corners. What you do is put all the pieces that you are interested in finding an operator on in the same layer,
and call this sequence Pn. The inverse of this sequence is called Pn'. Now, by trial and error find a sequence of moves that affects some of the pieces in the layer, but leaves all other pieces in
that layer unmoved. This sequence is called X, and the inverse is called X'. Do a turn of the layer, called sn, which has inverse sn'. Doing X' will now unscramble the other 2 layers, and doing sn'
and then Pn' will return the cube to its previous state with just a few cubes altered.
It isn't necessary for there to be moves in all of the stages, and if necessary the moves used could be from a subgroup of all moves by, say, restricting moves of the R face to R2.
Finding corners orientating algorithms to put corners in L2R2F2B2UD group
The simplest method to find algorithms is to use the PnXsnX'sn'Pn' technique for finding moves.
X=LD2L'F'D2F with S1=U, S2=U2, S2=U' and Pn=nothing where n=1,2 leaves U unchanged except FLU rotated clockwise
These algorithms are well known already, and rotate clockwise the corner in the LFU position before X is applied, and rotate anticlockwise the corner in the LFU position before X' is applied. By
choosing Pn from (U, U2, U', none) and sn from (U, U2, U', none) appropriately, which corner from the top layer is rotated clockwise and which is rotated anticlockwise can be chosen. Further, by
choosing Pn from (R2U, F2U, R2U2, F2U2, R2U', F2U', R2, F2) and sn from (U, U2, U', none) [note that there are no other allowable moves for sn since it is a rotation of the layer chosen earlier] we
can choose any pair of corners on the cube (one from the D layer and one from the U layer) to be rotated with one going clockwise and the other anticlockwise.
Finding corners moving algorithms in L2R2F2B2UD group
Again, the simplest method to find algorithms is to use the PnXsnX'sn'Pn' technique for finding moves.
X=R2D' R2D2 R2D R2D' R2D2 R2D R2 with S1=U, S2=U2, S2=U' and Pn=nothing where n=1,2 leaves U unchanged except FRU and BRU swapped
Note that this example has X=X'. X was found by trying to split the corners from the edge piece and then recombining the corners the other way round.
Using this sequence you can determine the corner moving sequences possible. It may be possible to utilise a sequence that uses moves other than L2R2F2B2UD that takes less moves than using moves
L2R2F2B2UD; if you have shown that the result of the sequence is achievable using those moves then that is allowable. The whole point is to find allowable swaps; to speed the solving up and shorten
the solution length use shorter algorithms that briefly take the cube out of the group before returning the cube back to a state within the group.
Determining the parity of the corners
If an even number of swaps will place the corners in the correct place then the corners can be placed without swapping the edges. If an odd number of swaps is required then an odd number of pairs of
edges will also need swapping. A rotation of a slice will achieve a change of parity. Since L2R2F2B2UD is being used to solve the corners, in this method the parity can be changed by rotating U or D
a total of an odd number of turns. The other way to swap the parity of the corners is to do a swap of 2 corners and 2 edges.
How to determine the parity:
Number the corners positions 1,2,...,8 in any fashion that you find easy to remember and work out which position the corner that should be in that position actually is.
As an example: 12345678 83265417
Convert this to express in terms of cycles. In this example (187)(23)(46)(5). For each n cycle the parity of the cycle is n-1, so the parity of the position described is 2+1+1+0=4. So this example
has even parity, therefore no U or D turns are required. If the parity was odd then a single turn of U or D either way would change the parity to even.
Here is step-by-step how I would determine the cycles in disjoint notation for the example quoted above (write the numbers down until you get the hang of it, then move on to doing it in your head):
We have some corners we haven't dealt with yet so write down (
You have (
Find the lowest numbered corner not written down. Since we haven't written down any it will always be 1. Write that down
You have (1
Look at the piece in position 1 and see where it belongs. It belongs in position 8. Write that down
You have (18
Look in position 8 and see where the piece there belongs. It belongs in position 7. Write that down
You have (187
Look in position 7 and see where the piece there belongs. It belongs in position 1, but we have already written down that number so we write down ) instead
You now have (187)
There are some corners we haven't dealt with yet so we write down (
You have (187)(
Now find the lowest numbered corner we haven't written down, we have written down 1, so the next lowest is 2. Write that down
You have (187)(2
Look in position 2 and see where that piece belongs. It belongs in position 3. Write that down
You have (187)(23
Look in position 3 and see where that piece belongs. It belongs in position 2, but we have already written down that number so we write down ) instead
You have (187)(23)
There are some cornerw we haven't dealt with yet so we write down (
You have (187)(23)(
The lowest numbered corner we haven't written down yet is 4, so write that down
You have (187)(23)(4
Look in position 4 and see where that piece belongs. It belongs in position 6 so write that down
You have (187)(23)(46
Look in position 6 and see where that piece belongs. It belongs in position 4, but we have written down that number so we write down ) instead
You have (187)(23)(46)
We have corners that we haven't dealt with yet so write down (
You have (187)(23)(46)(
The lowest numbered corner we haven't written down is 5, so write that down
You have (187)(23)(46)(5
Look in position 5 and see where that piece there belongs. It belongs in position 5, but we have written down that number so we write down 5 instead
You have (187)(23)(46)(5)
You have written down 8 numbers so you are finished, and the cycles in disjoint notation are (187)(23)(46)(5)
Calculating the corner twist of a layer
To determine the twist of a layer, for each of the four corners in that layer determine how many clockwise rotations of that corner from untwisted would put it in that rotational orientation (which
will be 0, 1 or 2), and add up each twist. Discard any multiples of 3, and you are left with the twist of the layer.
As an example, on a solved cube, do RU2R2U'R2U'R2U2R. Considering the twist of the upper layer: the twist of the FLU corner is 2, for the BLU it is 1, for BRU it is 1, for FRU it is 2, giving a total
of 6=(3+3+0), so U twist is 0. The twist of the L layer in this example is FLD+BLD+BLU+FLU=0+0+1+2=3=(3+0)=0. | {"url":"http://homepage.ntlworld.com/angela.hayden/cube/blind2.html","timestamp":"2014-04-18T16:04:11Z","content_type":null,"content_length":"9408","record_id":"<urn:uuid:d6245588-26be-4ace-9a37-9694bf1fbcbd>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00247-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Special & General Relativity Questions and Answers
What are P-branes?
The folks that are studying super string theory for clues to the 'Theory of Everything' say that P-branes are 'p-dimensional subspaces in the 10- dimensional string manifold' which apparently serve
the mathematical function of 'anchoring' the endpoints of the strings which represent the fundamental particles. Evidently by mathematically manipulating these objects instead of the strings
themselves, you can investigate how the strings interact in ways that a direct approach doesn't let you. I personally have not the foggiest idea what they are talking about, and I view all of these
elements as computational tools, not necessarily a statement of what is actually going on in a physical way. Ultimately, they are studying the geometric properties of space-time, and this in turn
represents the gravitational field itself. These mathematical representations are simply the tools we are now using to motivate certain kinds of calculations we would most like to perform.
Return to the Special & General Relativity Questions and Answers page.
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Topic: Cantor's absurdity, once again, why not?
Replies: 77 Last Post: Mar 19, 2013 11:02 PM
Messages: [ Previous | Next ]
Virgil Re: Cantor's absurdity, once again, why not?
Posted: Mar 19, 2013 2:34 PM
Posts: 7,011
Registered: 1/6/11 In article
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 17 Mrz., 07:11, fom <fomJ...@nyms.net> wrote:
> > On 3/16/2013 10:55 AM, WM wrote:
> >
> > > On 16 Mrz., 16:01, fom <fomJ...@nyms.net> wrote:
> >
> > >> perhaps you could explain what you mean
> > >> by "given object" and how an immaterial
> > >> object can be given.
> >
> > > It cannot be given other than by naming it (except from clumsy
> > > approaches by means of sign language). How to name some numbers, and
> > > rules how to invent further names, that can be understood by others,
> > > who were taught the same rules, is taught in school, university and
> > > other sources.
> >
> > What then are some examples
> > of rules that invent these
> > further names?
> If 5 and 6 are given, mathematics defines how to produce 11.
Not until decimal notation, at least up to 11, is also given.
> >
> > The point of this question is
> > that you claim such rules but
> > ignore the work of others who
> > have steadfastly worked at clarifying
> > the nature of such rules as a
> > matter of scientific principle
> > (in the wider epistemological
> > sense).
> I do not ignore these rules, but in some instances I can show that
> they are contradictory.
WM often claims to be able to show things, but the only thong he shows
with any reliability is his incompetence as a mathematician.
> The isomorphism is from |R,+,* to |R,+,*. Only in one case the
> elements of |R are written as binary sequences and the other time as
> paths of the Binary Tree. Virgil is simply too stupid to understand
> that.everal flaws in WM's claim that the identity map on induces a linear map on 2^|N.
WM's flaws in making that claim work include, but are not necessarily
limited to:
(1) not all members of |R will have any such binary expansions, only
those between 0 and 1, so that not all sums of vectors will "add up" to
be vectors within his alleged linear space, and
(2) some reals (the positive binary rationals strictly between 0 and 1)
will have two distinct and unequal-as-vectors representations, requiring
that some real numbers not be equal to themselves as a vectors, and
(3) WM's method does not provide for the negatives of any of the vectors
that he can form.
On the basis of the above problems, and possibly others as well that I
have not yet even thought of, I challenge WM's claim to have represented
the set |R as the set of all binary sequences, much less to have imbued
that set of all binary sequences with the structure of a real vector
space or the showed the identity mapping to be a linear mapping on his
set of "vectors".
WM has several times claimed that the standard bijection from the set of
all binary sequences to the set of all paths of a Complete Infinite
Binary Tree is a linear mapping from the set of all binary sequences
regarded as a linear space over |R to the set of all paths of a CIBT.
While the obvious mapping is easily shown to be bijective, it fails to
be a linear mapping as WM describes it:
> The isomorphism is from |R,+,* to |R,+,*. Only in one case the
> elements of |R are written as binary sequences and the other time as
> paths of the Binary Tree. Virgil is simply too stupid to understand
> that.everal flaws in WM's claim that the identity map on induces a linear map on 2^|N.
WM's flaws in making that claim work include, but are not necessarily
limited to:
(1) not all members of |R will have any such binary expansions, only
those between 0 and 1, so that not all sums of vectors will "add up" to
be vectors within his alleged linear space of binaries, and
(2) some reals (the positive binary rationals strictly between 0 and 1)
will have two distinct and unequal-as-vectors representations, requiring
that some real numbers not be equal to themselves as a vectors, and so
that two such pairs of binary sequences can only map to a single real
thus also only to a single path, so that the mapping cannot be a
bijection, and
(3) WM's method does not provide for the negatives of any of the vectors
that he can form, so his "space" does not qualify as a linear space in
that way, either.
On the basis of the above problems, and possibly others as well that I
have not yet even thought of, I challenge WM's claim to have linearly
injected the set of binary sequences into the set |R or the image of the
set of binary sequnces in |R linearly ONTO the set of all paths.
Note that while WM's model doe not achieve what he claims for it, there
is another model, which a reasonably competent mathematician should be
able to find, which does make the mapping into an isomorphism of linear
spaces. I will produce this model when WM concedes his error, or at
least no longer claims it is not an error.
It is a shame that someone so obviously of limited ability at
mathematics as WM should feel himself so driven to try and correct his
> Regards, WM
Date Subject Author
3/14/13 Cantor's absurdity, once again, why not? David Petry
3/14/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? David Petry
3/14/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? David Petry
3/14/13 Re: Cantor's absurdity, once again, why not? fom
3/17/13 Re: Cantor's absurdity, once again, why not? Shmuel (Seymour J.) Metz
3/17/13 Re: Cantor's absurdity, once again, why not? ross.finlayson@gmail.com
3/18/13 Re: Cantor's absurdity, once again, why not? fom
3/18/13 Re: Cantor's absurdity, once again, why not? Shmuel (Seymour J.) Metz
3/19/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? Virgil
3/14/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/14/13 Re: Cantor's absurdity, once again, why not? Virgil
3/14/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/14/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/14/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? Virgil
3/14/13 Re: Cantor's absurdity, once again, why not? Virgil
3/14/13 Re: Cantor's absurdity, once again, why not? Virgil
3/14/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/14/13 Re: Cantor's absurdity, once again, why not? harold james
3/14/13 Re: Cantor's absurdity, once again, why not? Virgil
3/14/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/14/13 Re: Cantor's absurdity, once again, why not? Virgil
3/14/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? Jesse F. Hughes
3/14/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/14/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? David Petry
3/15/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/15/13 Re: Cantor's absurdity, once again, why not? Virgil
3/15/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/15/13 Re: Cantor's absurdity, once again, why not? Virgil
3/15/13 Re: Cantor's absurdity, once again, why not? fom
3/16/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/16/13 Re: Cantor's absurdity, once again, why not? FredJeffries@gmail.com
3/16/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/16/13 Re: Cantor's absurdity, once again, why not? Virgil
3/16/13 Re: Cantor's absurdity, once again, why not? fom
3/16/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/16/13 Re: Cantor's absurdity, once again, why not? Virgil
3/16/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/16/13 Re: Cantor's absurdity, once again, why not? Virgil
3/17/13 Re: Cantor's absurdity, once again, why not? fom
3/19/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/19/13 Re: Cantor's absurdity, once again, why not? fom
3/19/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/19/13 Re: Cantor's absurdity, once again, why not? Virgil
3/19/13 Re: Cantor's absurdity, once again, why not? fom
3/19/13 Re: Cantor's absurdity, once again, why not? fom
3/19/13 Re: Cantor's absurdity, once again, why not? fom
3/19/13 Re: Cantor's absurdity, once again, why not? fom
3/19/13 Re: Cantor's absurdity, once again, why not? fom
3/19/13 Re: Cantor's absurdity, once again, why not? fom
3/19/13 Re: Cantor's absurdity, once again, why not? Virgil
3/16/13 Re: WM's absurdity, once again, why not? Virgil
3/17/13 Re: Cantor's absurdity, once again, why not? fom
3/14/13 Re: Cantor's absurdity, once again, why not? Virgil
3/14/13 Re: Cantor's absurdity, once again, why not? Jesse F. Hughes
3/15/13 Re: Cantor's absurdity, once again, why not? mueckenh@rz.fh-augsburg.de
3/15/13 Re: Cantor's absurdity, once again, why not? Virgil
3/14/13 Re: Cantor's absurdity, once again, why not? David Petry
3/14/13 Re: Cantor's absurdity, once again, why not? Jesse F. Hughes
3/14/13 Re: Cantor's absurdity, once again, why not? David Petry
3/14/13 Re: Cantor's absurdity, once again, why not? Jesse F. Hughes
3/15/13 Re: Cantor's absurdity, once again, why not? David Petry
3/15/13 Re: Cantor's absurdity, once again, why not? Jesse F. Hughes
3/15/13 Re: Cantor's absurdity, once again, why not? David Petry
3/15/13 Re: Cantor's absurdity, once again, why not? Virgil
3/15/13 Re: Cantor's absurdity, once again, why not? fom
3/15/13 Re: Cantor's absurdity, once again, why not? fom
3/15/13 Re: Cantor's absurdity, once again, why not? fom
3/15/13 Re: Cantor's absurdity, once again, why not? Jesse F. Hughes
3/14/13 Re: Cantor's absurdity, once again, why not? ross.finlayson@gmail.com | {"url":"http://mathforum.org/kb/message.jspa?messageID=8681499","timestamp":"2014-04-21T03:16:08Z","content_type":null,"content_length":"114017","record_id":"<urn:uuid:eca70e8b-3a3b-4ed8-b77f-bb7c5cddf78b>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00219-ip-10-147-4-33.ec2.internal.warc.gz"} |
One Half-Angle Queston
May 7th 2006, 02:52 PM #1
One Half-Angle Queston
For the Half-Angle Problems:
how do you know if the answer is positve or negative...
(i circled it in RED)
For the Half-Angle Problems:
how do you know if the answer is positve or negative...
(i circled it in RED)
It depends in which quadrant is $\frac{a}{2}$.
NOT $a$ but $\frac{a}{2}$.
If $\frac{a}{2}$ is in the first or second quadrant then you use positive, otherwise use the negative.
Your question, is a classic mistake which I seen many people make, hopefully you will not make it.
for example, what if a=4/5. so it'll be (4/5)/2 and that'll turn out to be 2/5. how do i know if 2/5 is in the first or second quadrant?
for example, what if a=4/5. so it'll be (4/5)/2 and that'll turn out to be 2/5. how do i know if 2/5 is in the first or second quadrant?
I presume that is in radians?
The first and second quadrant span $0^o\leq x\leq 180^o$
Or in Radians, $0\leq x\leq \pi$
$0\leq .4\leq 3.14$ it is in the first and second quadrants.
In fact it is in first quadrant because it spans,
$0\leq x\leq \frac{\pi}{2}\approx 1.57$
May 7th 2006, 03:01 PM #2
Global Moderator
Nov 2005
New York City
May 7th 2006, 03:08 PM #3
May 7th 2006, 03:14 PM #4
Global Moderator
Nov 2005
New York City | {"url":"http://mathhelpforum.com/trigonometry/2865-one-half-angle-queston.html","timestamp":"2014-04-16T17:41:24Z","content_type":null,"content_length":"40994","record_id":"<urn:uuid:f4d9b6f7-1489-4ba4-b082-83af6dd497ae>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00416-ip-10-147-4-33.ec2.internal.warc.gz"} |
Aug 01
Math and Computer Science Club
The purpose of the Math/CS Club is to provide its members and, whenever possible, the Seton Hall University community with an education relating to mathematics and computer science. This is often
done with programs that promote these interests including a series of lectures where students learn about various fields of mathematics, current research, and raise a greater interest in the fields
of mathematics and computer science. The Math/CS Club is also devoted to making opportunities for those students at Seton Hall to represent Seton Hall University at various math and academic
conferences as teams and as individuals.
In 2010 the Math/CS Club constructed three teams of its members to compete in two different scholarship competitions (The NSCS Academic Bowl on March 25th and the Garden State Undergraduate
Mathematics Competition on April 10th) toward with goal. The Math/CS Club is also devoted to raising funds for various philanthropic causes through mathematics and computer science related
programming like the annual Pi Day events giving Seton Hall students access to the ability to help others.
1 comment
Job Opportunity for those interested in Computer Science! We’re right here on Campus, and could use Your skills this summer. iD Tech Camps is the World’s #1 summer technology program for ages
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We only accept online applications so, while you’re here browsing the internet, why not take the first step to a great summer? Apply here. | {"url":"http://blogs.shu.edu/clubs/2011/08/01/math-and-computer-science-club/comment-page-1/","timestamp":"2014-04-21T14:41:25Z","content_type":null,"content_length":"36519","record_id":"<urn:uuid:0e9afc31-47af-4dfc-ac39-4bb2d4a27416>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00214-ip-10-147-4-33.ec2.internal.warc.gz"} |
inequality proof
September 30th 2008, 09:25 PM #1
inequality proof
Using the triangle inequality show $a$ and $b$ are in $\mathbb{R}$ with $a eq b$, then $\exists$ open intervals $U$ centered at $a$ and $V$ centered at b, both with radius $\epsilon=\frac{1}{2}|
so far all I have is
$||a|-|b|| \leq |a-b|$
now I'm not sure if what I'm doing is right:
$-\epsilon<||a|-|b|| < \epsilon \rightarrow ||a|-|b|| < 2\epsilon \rightarrow ||a|-|b|| < |a-b|$ ?
You do realise that as this stands it is not the question that you think it is, don't you?
Do you want to show that:
For all $a,b \in \mathbb{R},\ a e b$, then the two disjoint open intervals $U$ centered at $a$ and $V$ centered at $b$, both with radius $\epsilon=\frac{1}{2}|a-b|$ are disjoint?
I completely overlooked that part of the question. But I did found a prove for this statement but am still lost as to how it was done, even after asking my prof.
This is what I found:
$U=(x-\epsilon, \ x+\epsilon), \ V = (y-\epsilon, \ y+\epsilon)$
$U \bigcap V = \emptyset$
suppose $z$ is in $U \bigcap V$
$|x-y| \leq |x-z|+|z-y| < \epsilon + \epsilon = 2\epsilon = |x-y|$
which is apparently a contradiction.
If anyone could clarify this, it would help me out a lot.
I completely overlooked that part of the question. But I did found a prove for this statement but am still lost as to how it was done, even after asking my prof.
This is what I found:
$U=(x-\epsilon, \ x+\epsilon), \ V = (y-\epsilon, \ y+\epsilon)$
$U \bigcap V = \emptyset$
suppose $z$ is in $U \bigcap V$
$|x-y| \leq |x-z|+|z-y| < \epsilon + \epsilon = 2\epsilon = |x-y|$
which is apparently a contradiction.
If anyone could clarify this, it would help me out a lot.
The bit that is missing is:
Now by the triangle inequality:
$|x-y|=|(x-z)-(y-z)| \le |x-z|+|y-z|$
That solves a part of the proof, but where is the contradiction?
It lies is the fact that $z$ is assumed to be in $U \cap V$ so lies in $U$ and in $V$ so $|x-z|<\epsilon,$ and $|y-z|<\epsilon$, so:
$|x-y|<2 \epsilon$
But $|x-y|=2 \epsilon$, so we have:
which is a contradiction.
Last edited by CaptainBlack; October 4th 2008 at 11:19 PM.
September 30th 2008, 11:09 PM #2
Grand Panjandrum
Nov 2005
October 3rd 2008, 11:03 PM #3
October 3rd 2008, 11:08 PM #4
Grand Panjandrum
Nov 2005
October 4th 2008, 03:22 PM #5 | {"url":"http://mathhelpforum.com/discrete-math/51464-inequality-proof.html","timestamp":"2014-04-24T10:05:33Z","content_type":null,"content_length":"54240","record_id":"<urn:uuid:c3160895-3161-40e2-8353-459e55179441>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00532-ip-10-147-4-33.ec2.internal.warc.gz"} |
Today's calculation of integral #15
Find $\int_{0}^{\pi }{x\cos ^{2}x\,dx}.$ It seems pretty trivial, but find a quick solution.
That's not a quick solution, you omitted details that you should explain at least.
Ok I owe it at least that much... $\cos(x)^2=\frac{1+\cos(2x)}{2}\therefore{x\cos(x)^ 2=x\bigg(\frac{1+\cos(2x)}{2}\bigg)}=\frac{x}{2}+\ frac{x\cos(2x)}{2}$ Now using integration by parts on the
second and standard techniques on the first we obtain the antiderivative O...and just because I know you love it $\frac{x}{2}+\frac{x\cos(2x)}{2}=\frac{x}{2}+\sum_{ n=0}^{\infty}\frac{(-1)^{n}2^{2n}x
^{2n+1}}{(2n)!}$ Integrating we get $\int{x\cos(x)^2}dx=\frac{x^2}{4}+\sum_{n=0}^{\inft y}\frac{(-1)^{n}2^{2n}x^{2n+2}}{(2n+2)(2n)!}+C$ That was obviously a stupid joke^...me attempting to be funny
Partial integration is involved. Find another solution which does not involve integration by parts.
Let: $<br /> u = \pi - x<br />$ Then: $<br /> I = \int_0^\pi {x \cdot \cos ^2 \left( x \right) \cdot dx} = \int_0^\pi {\left( {\pi - u} \right) \cdot \cos ^2 \left( u \right)du} <br /> <br />$
Summing the expressions above : $<br /> I = \left( {\tfrac{\pi }<br /> {2}} \right) \cdot \int_0^\pi {\cos ^2 \left( x \right) \cdot dx} <br /> <br />$ And: $<br /> \int_0^\pi {\cos ^2 \left( x \
right) \cdot dx} = \int_0^\pi {\tfrac{{1 + \cos \left( {2x} \right)}}<br /> {2} \cdot dx} = \tfrac{\pi }<br /> {2}<br />$ Thus: $<br /> I = {\tfrac{\pi^2 }<br /> {4}} <br /> <br />$
I got confused with the method shown. here's another: note that $\int_0^\pi \cos^2 x ~dx = \int_0^\pi \sin^2 x ~dx$ Thus, $\int_0^\pi x \cos^2 x ~dx = \frac 12 \left[ \int_0^\pi x \sin^2 x ~dx + \
int_0^\pi x \cos^2 x~dx \right]$ $= \frac 12 \left[ \int_0^\pi x ~dx\right]$ $= \frac 12 \cdot \frac {x^2}2 \bigg|_0^\pi$ $= \frac {\pi^2}4$
HiWhat conditions have to be respected so that one can say "if $\int_a^b f(x)\,\mathrm{d}x = \int_a^b g(x)\,\mathrm{d}x$ then $\int_a^b f(x)\cdot h(x)\,\mathrm{d}x=\int_a^b g(x)\cdot h(x)\,\mathrm{d}
x$" ? I ask this question because there are some cases in which it does not work. For example : $\int_0^{2\pi} \sin x\,\mathrm{d}x= \int_0^{2\pi}\cos x\,\mathrm{d}x=0$ but $\int_0^{2\pi} \sin^2x\,\
mathrm{d}x=\pi$ and $\int_0^{2\pi} \cos x \cdot \sin x\,\mathrm{d}x=\left[\frac{sin^2x}{2}\right]_0^{2\pi}=0eq \pi$
Here's my solution: Let $\lambda =\int_{0}^{\pi }{x\cos ^{2}x\,dx}.$ Substitute $x\to x-\frac{\pi }{2}\implies \lambda =\pi \int_{0}^{\pi /2}{\sin ^{2}x\,dx}.$ In general $\int_{0}^{\pi /2}{f(\sin x)
\,dx}=\int_{0}^{\pi /2}{f(\cos x)\,dx},$ hence $\lambda =\frac{\pi ^{2}}{4}.$ | {"url":"http://mathhelpforum.com/calculus/37807-today-s-calculation-integral-15-a.html","timestamp":"2014-04-16T16:57:26Z","content_type":null,"content_length":"76770","record_id":"<urn:uuid:3eacd3ec-6974-4fa2-b39d-05e7b0ce9f8c>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00132-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts from November 2012 on Chris Webb's BI Blog
Archive for November 2012
I’ve now arrived in Seattle for the PASS Summit 2012, and as always I’m looking forward to a busy week. There are a few things I wanted to highlight:
• There are always a few big announcements at PASS, and while I’ll be blogging every day you’ll hear the news first if you follow me on Twitter – I’m @Technitrain. For example I’ve just noticed a
session on “Using Power View with Multidimensional Models” on the schedule which means, I suspect, that the long-promised support for Power View over SSAS Multidimensional, via DAX queries, will
be announced this week!
• I’m doing a session on “What’s New in SSAS 2012?” (BIA-303) on Wednesday morning at 10:15am. Unfortunately, the printed schedule and summit guide booklet have the wrong session title and/or
abstract at the moment; the session details online are correct. This means that contrary to what you might see in some places I will not be doing a pure SSAS Multidimensional session; there will
be about 15-20 minutes of 2012 Multidimensional content (I’d struggle to fill more than that) and the rest will cover terminology, Tabular, and choosing between Tabular and Multidimensional.
• I’m doing a book signing with Marco and Alberto for our new book “SQL Server Analysis Services 2012: The BISM Tabular Model” on Thursday lunchtime from 12:00pm to 12:30pm at the Summit bookstore.
• I’m doing a second session on “The Best Microsoft BI Tools You’ve Never Heard Of!” (BI-202) on Thursday at 1:30pm. This will be a fun session showing how you can use tools like NodeXL and
Layerscape alongside PowerPivot for BI purposes; and yes, I know, if you’re a regular reader of my blog you will have heard of these tools… but hopefully you’ll find it enjoyable nonetheless.
If you see me wandering around at any point, feel free to say hello! I’m quite friendly and I won’t bite. I might be a bit jet-lagged though – I’m wide awake at 4:45am right now…
The combination of the filtering functionality built into PivotTables, and the ability to delete and reorder tuples in a set without needing to edit the set expression itself that the Excel named set
functionality gives you, means that you can usually implement the filters you need in PowerPivot without needing to resort to MDX. However there are some scenarios where knowing the MDX functions
that allow you to filter a set are useful and in this post I’ll show a few of them.
The Filter() function is the Swiss-Army penknife of filtering in MDX: it can do pretty much anything you want, but isn’t always the most elegant method. It’s quite simple in that it takes two
parameters, the set that is to be filtered and a boolean expression that is evaluated for every item in the set and which determines whether that item passes through the filter or not.
Here’s a simple example. Consider a simple PivotTable (using my example model, described here) with FullDateAlternateKey on rows and the Sum of SalesAmount measure on columns:
The set of members on the FullDateAlternateKey level of the FullDateAlternateKey hierarchy can be obtained by using the .Members() function as I showed earlier in this series:
This set can then be filtered by passing it to the Filter function, which itself returns a set, so it too can be used to create a named set. Let’s say we only wanted the set of dates where Sum of
SalesAmount was greater than £10000; we could get it using the following expression:
, ([Measures].[Sum of SalesAmount])>10000)
What I’m doing is passing the set of all dates into the first parameter of Filter() and then, in the second parameter, testing to see if the value of the tuple ([Measures].[Sum of SalesAmount]) is
greater than 10000 for each item in that set.
Here’s the result:
As I’ve mentioned before, it’s very important that you remember to check the ‘Recalculate set with every update’ button if you want the filter to be re-evaluated every time you change a slicer, which
you almost always want to do.
Where is the filter function actually useful though? Here’s the same PivotTable but with Color on columns and with only Red and Black showing:
It’s not possible to filter this PivotTable to show only the dates where sales for Black products are greater than Sales of Red products using native functionality, but it is using MDX. Here’s the
set expression:
, ([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Black])
([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Red])
Here I’m doing something similar to what I did in the first example, but now comparing two tuple values for each date: the tuple that returns the value of Sum of SalesAmount for Black products, which
([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Black])
and the tuple that returns the value of Sum of SalesAmount for Red products, which is
([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Red])
The NonEmpty() function also does filtering, but it’s much more specialised than the Filter() function – it filters items from a set that have empty values for one or more tuples. As with the Filter
() function its first parameter is the set to be filtered, but its second parameter is another set, each of whose items are evaluated for each item in the first set. If one item in the second set
evaluates to a non empty value for an item in the first set then that item passes through the filter.
That explanation is, I know, quite hard to digest so let’s look at an example. Here’s our PivotTable with no filter applied on rows and all Colors displayed on columns:
If you wanted to see only the rows where there were sales for Black products, you could use the following expression:
, {([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Black])})
If you wanted to see only the rows where there were sales for Black OR Silver products, you could use this expression:
, {([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Black])
,([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Silver])})
If you wanted to see only the rows where there were sales for Black AND Silver products you’d need to use two, nested NonEmpty functions:
, {([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Black])})
, {([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Silver])})
TOPCOUNT(), BOTTOMCOUNT(), TOPPERCENT(), BOTTOMPERCENT()
The TopCount() and related functions are, as you’ve probably guessed from their names, useful for doing top N style filters. If you wanted to see the top 10 dates for sales of Black products you
could use the following expression:
, 10
, ([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Black])
Notice here how the dates are displayed in descending order for the Black column, but no other – that’s how you can tell that the TopCount() function is doing what you want.
To get the top N dates that provide at least 5% of the total sales across all time for Black products, you can use the following expression:
, 5
, ([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Black])
The BottomCount() and BottomPercent() function (I’m always reminded of one of the old, old jokes here when I use the BottomCount() function…) do the opposite and return the bottom items in a set, but
you need to be careful using them because the bottom items in a set often have no values at all which is not very useful. So, for example, if you wanted to find the bottom 10 dates that have sales
for Black products you need to use the NonEmpty() function as well as the BottomCount() function as follows:
, {([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Black])})
, 10
, ([Measures].[Sum of SalesAmount], [DimProduct].[Color].&[Black])
Here I’m taking the set of all members on the FullDateAlternateKey level of the FullDateAlternateKey hierarchy, passing that to the NonEmpty() function to return only the dates that have values for
Black products and Sum of SalesAmount, and then getting the bottom 10 of those dates.
In part 5, I take a look at running MDX queries against a PowerPivot model. | {"url":"http://cwebbbi.wordpress.com/2012/11/page/2/","timestamp":"2014-04-16T13:03:20Z","content_type":null,"content_length":"49034","record_id":"<urn:uuid:1ed5fd10-ddd5-41b0-b109-f28d25727add>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00172-ip-10-147-4-33.ec2.internal.warc.gz"} |
Trigonometry: Area of Triangles
August 3rd 2012, 12:49 PM #1
Jul 2011
Trigonometry: Area of Triangles
Based on the books answers, (i) & (ii) are correct but I'm confused about (ii), seeing as how the answer is provided in the question!! Am I meant to prove the statement through the information
The book's answer to (iii) is provided below, but the confusion for me here is the application of tan, in defining the area of the triangle. The 1st attempt in (iii) is what I expected the answer
to be but, based on the books final answer, I made a second attempt to try and account for tan. Also, again based on the books answer, I'm assuming |ad| is meant to = r but I'm unsure as to how I
prove this. Can anyone help me out here?
Many thanks.
Q. A triangle is inscribed in a sector of a circle, center c, radius r, $\theta<90^o$. A right-angled triangle cad circumscribes the sector, as shown. If the area of a sector is $\frac{1}{2}r^2\
theta$, find the area of (i) $\triangle cab$, (ii) the sector cab, (iii) $\triangle$ cad. Hence show that $sin\theta<\theta<tan\theta$.
Attempt: (i) Area of $\triangle abc$: $\frac{1}{2}(a)(b)(sinC)$
|ac| = r = |bc|, where both lines are connected from the center to the circumference of the circle whose sector is cab
Thus, $\frac{1}{2}(a)(b)(sinC)$ = $\frac{1}{2}r^2sin\theta$
(ii) $\frac{1}{2}r^2\theta$
(iii) Area of $\triangle cad$: $\frac{1}{2}(d)(a)(sinA)$ = $\frac{1}{2}(r)(|cd|)(sin90^o)$ = $\frac{1}{2}(r)(a)(1)$ = $\frac{1}{2}ra$
$\frac{1}{2}(c)(d)(tan\theta)$ = $\frac{1}{2}(|ab|)(r)(tan\theta)$
Ans.: (From text book): (iii) $\frac{1}{2}r^2tan\theta$
Last edited by GrigOrig99; August 3rd 2012 at 12:55 PM.
Re: Trigonometry: Area of Triangles
note that $|ad| = r\tan{\theta}$
$A = \frac{1}{2}bh = \frac{1}{2} r \cdot r\tan{\theta}$
Re: Trigonometry: Area of Triangles
Ok, so if I'm to demonstrate $sin\theta<\theta<tan\theta$, and knowing that $\theta<90^o$, I can simply pick any degree from $0^0-89^0$ and display the results e.g. $\theta=60^0$ in which case
$sin\60^o<\60^o<tan\60^0$ = $0.86<60<1.73$. Although, this leaves me with the problem of the 60 in the middle.
Is there a better way to prove the statement?
Re: Trigonometry: Area of Triangles
Ok, so if I'm to demonstrate $sin\theta<\theta<tan\theta$, and knowing that $\theta<90^o$, I can simply pick any degree from $0^0-89^0$ and display the results e.g. $\theta=60^0$ in which case
$sin\60^o<\60^o<tan\60^0$ = $0.86<60<1.73$. Although, this leaves me with the problem of the 60 in the middle.
Is there a better way to prove the statement?
$\text{Area}(\Delta CAB)\le \text{Sector-area}(CAB)\le\text{Area}(\Delta CAD)$
Re: Trigonometry: Area of Triangles
Ok, thank you.
August 3rd 2012, 02:12 PM #2
August 3rd 2012, 03:21 PM #3
Jul 2011
August 3rd 2012, 03:43 PM #4
August 3rd 2012, 03:55 PM #5
Jul 2011 | {"url":"http://mathhelpforum.com/trigonometry/201698-trigonometry-area-triangles.html","timestamp":"2014-04-18T01:37:56Z","content_type":null,"content_length":"52354","record_id":"<urn:uuid:d69e34f0-fc17-4ee9-a73a-d8e9dcc41117>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00504-ip-10-147-4-33.ec2.internal.warc.gz"} |