content stringlengths 86 994k | meta stringlengths 288 619 |
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Overflow: Louisiana and Florida Underwater
Team: 59
School: Las Cruces High
Area of Science: Climatology
Problem Definition:
With the current global temperature and sea level rise, we must be wary of the drastic outcomes that may soon affect us. It is not just the “at risk” cities that should worry about a slowly incoming
flood, but surrounding cities as well. This is because there will be mass population movements away from flood zones, which will have huge effects on economy, pollution and other factors of a city.
Our planet will undoubtedly become more “water based” and we cannot change that. We can, however, predict where water will rise the most and plan accordingly.
Problem Solution:
Our simulation would be conducted: based off the branch of Statistics known as probability. To be more specific: estimation theory and stochastic differential equations. Due to uncertainty in actual
sea level rise, and the effects of factors like storm surges, we have no way of measuring the exact amount water will rise. We can however give a good estimate based off of past trends and the return
periods of large storms, which increase flooding. The mathematical uncertainty of these factors allows us to use Monte Carlo simulation to come up with a model that will accurately reflect
possibilities in sea level rise.
Progress to Date:
Currently we have played around with MatLab and GNU Octave to try and graph what the future water levels will look like. We have decided to model Louisiana and Florida because they seem to have the
most active water level rise and because they both have a very dense coastal city population. To top it all off, we have had more than 5 meetings via Skype with mentors who have helped guide us in
our project.
Expected Results:
After programming, testing, and refining of the mathematical simulation our system should give a good estimate on water level rise for the future. Our model could also be further developed by showing
the effect it might have in different geographic and socioeconomic areas. This will broaden the relevance of our model and ensure a broad range of uses.
Team Members:
Kevin Lee, Dante Laroche, and Andrew Lawendy.
Sponsoring Teachers: Elisa Cundiff and Lauren Curry
http://en.wikipedia.org/wiki/Kalman_filter http://en.wikipedia.org/wiki/Probability_density_function http://www.maths.manchester.ac.uk/~scotter/MATH20401/week10.pdf http://www.ipcc.ch/
publications_and_data/ar4/wg1/en/figure-spm-5.html http://www.math.dartmouth.edu/opencalc2/cole/lecture8.pdf http://en.wikipedia.org/wiki/Estimation_Theory http://en.wikipedia.org/wiki/GNU_Octave
Team Members:
Dante Laroche
Andrew Lawendy
Hoon Jeong Lee
Sponsoring Teacher: Elisa Cundiff
Mail the entire Team | {"url":"http://mode.lanl.k12.nm.us/get_interim1314.php?team_id=59","timestamp":"2014-04-16T10:47:46Z","content_type":null,"content_length":"8624","record_id":"<urn:uuid:aae8aa3d-dc2c-432d-81e8-b29095d8a7ae>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00210-ip-10-147-4-33.ec2.internal.warc.gz"} |
Facility L
Results 1 - 10 of 91
- Packing and Covering , 1974
"... - reproduced, stored In a retrieval system, or transmlt'ted, In any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the author.
Cited by 453 (2 self)
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- reproduced, stored In a retrieval system, or transmlt'ted, In any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the author.
, 2004
"... The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system
design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamenta ..."
Cited by 33 (0 self)
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The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design.
Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of
the contributions to the current state-of-the-art. In particular, continuous location models, network location models, mixed-integer programming models, and applications are summarized.
- SIAM Journal on Optimization , 1997
"... Abstract. In recent years rich theories on polynomial-time interior-point algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to
yield better complexity results for various applications. In this paper, the problem of minimizing a sum o ..."
Cited by 22 (4 self)
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Abstract. In recent years rich theories on polynomial-time interior-point algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield
better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By
transforming the problem into a standard convex programming problem in conic form, we show that an ɛ-optimal solution can be computed efficiently using interior-point algorithms. As applications to
this problem, polynomial-time algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree
topology. In particular, by solving the Newton equation in linear time using Gaussian elimination on leaves of a tree, we present an algorithm which computes an ɛ-optimal solution to the shortest
network under a given full Steiner topology interconnecting N regular points, in O(N √ N(log(¯c/ɛ)+ log N)) arithmetic operations where ¯c is the largest pairwise distance among the given points. The
previous best-known result on this problem is a graphical algorithm which requires O(N 2) arithmetic operations under certain conditions. Key words. polynomial time, interior-point algorithm,
minimizing a sum of Euclidean norms, Euclidean facilities location, shortest networks, Steiner minimum trees
, 1993
"... The problem of maximizing diversity deals with selecting a set of elements from some larger collection such that the selected elements exhibit the greatest variety of characteristics. A new
model is proposed in which the concept of diversity is quantifiable and measurable. A quadratic zero-one model ..."
Cited by 19 (2 self)
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The problem of maximizing diversity deals with selecting a set of elements from some larger collection such that the selected elements exhibit the greatest variety of characteristics. A new model is
proposed in which the concept of diversity is quantifiable and measurable. A quadratic zero-one model is formulated for diversity maximization. Based upon the formulation, it is shown that the
maximum diversity problem is NP-hard. 'Tho equivalent linear integer programs are then presented that offer progressively greater computational efficiency. Another formulation is also introduced
which involves a different diversity objective. An example is given to illustrate how additional considerations can be incorporated into the maximum diversity model. Subject Areas: Discnk hgmmming,
Linear Rvgmmming, and Mathematical hgmmming.
- Proc. International Conference on Operations Research and Management Science , 1990
"... this paper we briefly survey the most recent results in the area of facility location, concentrating on versions of the problem that are likely to be unfamiliar to the transportation and
management science community and we explore the interaction between facility location problems and the field of c ..."
Cited by 18 (3 self)
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this paper we briefly survey the most recent results in the area of facility location, concentrating on versions of the problem that are likely to be unfamiliar to the transportation and management
science community and we explore the interaction between facility location problems and the field of computational geometry. Such versions of the problem include the standard models of points as
customers and facilities but with geodesic rather than the traditional Minkowski metrics as measures of distance, as well as more complicated models of customers and facilities such as
- In Pacific Rim International Conference on Artificial Intelligence , 2000
"... General purpose and highly applicable clustering methods are usually required during the early stages of knowledge discovery exercises. k-Means has been adopted as the prototype of iterative
model-based clustering because of its speed, simplicity and capability to work within the format of very larg ..."
Cited by 16 (2 self)
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General purpose and highly applicable clustering methods are usually required during the early stages of knowledge discovery exercises. k-Means has been adopted as the prototype of iterative
model-based clustering because of its speed, simplicity and capability to work within the format of very large databases. However, k-Means has several disadvantages derived from its statistical
simplicity. We propose an algorithm that remains very efficient, generally applicable, multi-dimensional but is more robust to noise and outliers. We achieve this by using the discrete median rather
than the mean as the estimator of the center of a cluster. Comparison with k-Means, Expectation Maximization and Gibbs sampling demonstrates the advantages of our algorithm.
- Proc. of the IEEE ICIT’02 , 2002
"... Optimisation of feeder setup and component placement sequence are very important to the efficiency of surface mount placement machines. Much works have been conducted to solve this problem.
However, the technological characteristics of the placement machine influences the nature of the planning prob ..."
Cited by 13 (10 self)
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Optimisation of feeder setup and component placement sequence are very important to the efficiency of surface mount placement machines. Much works have been conducted to solve this problem. However,
the technological characteristics of the placement machine influences the nature of the planning problems to be solved and the formulation of the associated models. As a result, little consensus
exists as to what a suitable model should be for a given machine characteristics, and the formulations proposed by different authors tend to be difficult to compare. Hence, this paper will survey the
relation between models, assembly machine technologies and heuristic methods.
- SIAM Journal on Optimization , 1997
"... We study the problem of minimizing a sum of p-norms where p is a fixed real number in the interval [1; 1]. Several practical algorithms have been proposed to solve this problem. However, none of
them has a known polynomial time complexity. In this paper, we transform the problem into standard conic ..."
Cited by 13 (2 self)
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We study the problem of minimizing a sum of p-norms where p is a fixed real number in the interval [1; 1]. Several practical algorithms have been proposed to solve this problem. However, none of them
has a known polynomial time complexity. In this paper, we transform the problem into standard conic form. Unlike those in most convex optimization problems, the cone for the p-norm problem is not
self-dual unless p = 2. Nevertheless, we are able to construct two logarithmically homogeneous self-concordant barrier functions for this problem. The barrier parameter of the first barrier function
does not depend on p. The barrier parameter of the second barrier function increases with p. Using both barrier functions, we present a primal-dual potential reduction algorithm to compute an
ffl-optimal solution in polynomial time that is independent of p. Computational experiences of a Matlab implementation are also reported. Key words. Shortest network, Steiner minimum trees,
facilities location, po...
- Networks , 2000
"... We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of
the ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including --- but not li ..."
Cited by 10 (2 self)
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We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of the
ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including --- but not limited to --- Steiner trees for rectilinear (or isothetic) polygons, obstacle-avoiding Steiner trees,
group Steiner trees and prize-collecting Steiner trees. Also, the weighted region Steiner tree problem is shown to be solvable on the Hanan grid; this problem has natural applications in VLSI design
routing. Finally, we give similar results for other rectilinear problems. 1 Introduction Assume we are given a finite set of points S in the plane. The Hanan grid H(S) of S is obtained by
constructing vertical and horizontal lines through each point in S. The main motivation for studying the Hanan grid stems from the fact that it is known to contain a rectilinear Steiner minimum tree | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=238058","timestamp":"2014-04-16T13:30:07Z","content_type":null,"content_length":"36838","record_id":"<urn:uuid:7f84e3d4-7616-43d7-aaf7-c3c816d2d30b>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00427-ip-10-147-4-33.ec2.internal.warc.gz"} |
st: Testing weak instruments agter reg3
Ekrem Kalkan <
[hidden email]
> :
The advice on a first-stage F greater than 10 is about 7 years out of
date now (Stock and Yogo overturned in 2002 that advice from Staiger
and Stock from 1997, more or less), and applies only to a single
endogenous regressor, in any case. -ivreg2- (from SSC) will give you
relevant diagnostics (read the help file) if you estimate one equation
at a time; if you have a recursive system of equations, these will be
very good measures of the strength of your excluded instruments. If
not, they will be imperfect, but better than nothing.
On Thu, Dec 3, 2009 at 3:40 AM, Ekrem Kalkan <
[hidden email]
> wrote:
> Dear Stata users,
> I am estimating a 3SLS regression (for an Almost Ideal demand system)
> with reg3 command. I do not know how to test whether my "excluded
> instruments" are weak or not (correlation with endogenous regressors).
> I know that with "ivreg2" there are some diagnostics for weak
> instruments. For example, the partial F-statistic of the reduced form
> regression is required to be higher than 10. Do you think that these
> diagnostics are valid for system estimation like reg3?
* For searches and help try: | {"url":"http://statalist.1588530.n2.nabble.com/st-Testing-weak-instruments-agter-reg3-td4104845.html","timestamp":"2014-04-18T05:30:55Z","content_type":null,"content_length":"54224","record_id":"<urn:uuid:f8309d02-8ce3-41a9-8767-75781f9b76eb>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00526-ip-10-147-4-33.ec2.internal.warc.gz"} |
simple yet hard finding area + volume question
September 25th 2009, 07:27 PM #1
Apr 2009
Calculate the area bounded by the curves
y1= 1/x^1/2 and y2 = x^1/2+1/x from x=2 and x=1
ok... a simple sketch should indicate that the integral is
S 2-1 y2-y1 dx
doing all the necessary steps you should get (if im correct) [ln(x)]2-1
the answer is ln(2) * please check if im correct or not
part b) now the limits change.. its x=1 and x= infinity ....
the area is infinity because ln(infinity) = infinity ?? is that strong enough argument ??? * please check once again
part c) calculate the volume now of that region about the x axis...
well simple
volume is
pi S 2-1 (y2)^2 - (y1)^2 * is this correct
and if you do all the correct simplifying and integration you should get
pi(9/2-4/2^1/2) ... ?? * please check again xD
same thing as part b) let the new limits b x=1 and x= infinity... prove that the volume is infinity.. same arugment as part b ??
thanks guys | {"url":"http://mathhelpforum.com/calculus/104333-simple-yet-hard-finding-area-volume-question.html","timestamp":"2014-04-21T12:53:16Z","content_type":null,"content_length":"30733","record_id":"<urn:uuid:96d82885-ed54-451c-8ac9-bacec5873567>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00523-ip-10-147-4-33.ec2.internal.warc.gz"} |
188 helpers are online right now
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
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The class
The class Rectangle
The class Rectangle represents the drawing primitive of a parallelogram with optional rounded corners. When drawn interactively the object is an axis aligned rectangle. After applying affine
transformations the rectangle may not be a rectangle anymore, but it will remain a parallelogram.
In Sketch, a rectangle is represented by the affine transformation necessary to map the unit square (opposite corners (0, 0) and (1, 1)), to the parallelogram. This is an elegant and convenient but
also somewhat unusual representation.
The transformation is of course stored as a transformation object.
The rounded corners are defined by two floating point numbers radius1 and radius2. Both numbers define a radius of the ellipses that make up the rounded corners of the unit square, radius1 defines
the horizontal radius and radius2 the vertical radius. Valid values lie in the range 0.0 <= radius1,radius2 <= 0.5.
The class Compound The Document and Graphics Classes The Undo Mechanism | {"url":"http://www.nongnu.org/skencil/Doc/devguide-18.html","timestamp":"2014-04-17T18:24:18Z","content_type":null,"content_length":"2806","record_id":"<urn:uuid:a860f7af-dda6-4b7f-a9e9-412ccad7fdf2>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00237-ip-10-147-4-33.ec2.internal.warc.gz"} |
Holyoke, MA Math Tutor
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Multiplication Tricks Roundup
Should kids know their multiplication facts?
When teaching math, whether in the home or in the classroom, having the multiplication facts memorized speeds things up. If you allow a calculator, you’re still reducing the speed of doing other math
calculations or thinking.
But is there some way to memorize these and make them fun?
I scrounged the web and found some resources for multiplication tricks.
Poems as Multiplication Tricks
Inspired by @Leighbra at Hearth to Heart, I found this list of math poems that can help kiddos (and grownups) remember some of the facts. My favorite: I ate and ate and got sick on the floor. Eight
times eight is 64.
This is an impressive list of various ways to multiply by seven. Tricks to Multiply by Seven
4, 6, 9
In addition to tips on the 4s and 6s, this one shows a way to multiply 9s without using your fingers. Which means you don’t have to waste time putting down and picking back up your pencil! Easy Math
Tricks: Multiplying by 4, 6, 9 and Beyond
Using Fingers
This article gives three tricks to multiply using your fingers. Not my favorite, only because I hate to put down my pencil. But if you are doing purely mental multiplication, this article’s a great
resource! Three finger tricks for multiplying
Multiplication Tricks for Other Numbers
Here’s a list of various sites and articles that give multiplication tricks for many numbers:
The one I made up
I created this while in class one day. I have a hard time remembering the difference between 7 x 8 and 9 x 6. So I thought about it and saw that 56 = 7 x 8, which is 5, 6, 7, 8 in order!
Do you have one? Please share with us!
Related articles
This post may contain affiliate links. When you use them, you support us so we can continue to provide free content!
7 Responses to Multiplication Tricks Roundup
1. Love these! My son is tackling the multiplication tables & these new ones will come in handy.
I STILL remember my 3rd grade teacher teaching us “8 & 8 fell on the floor, when they got up they were 64″ & it wasn’t even about being sick!
Also, thanks for the linky love, going to go bother you on Twitter now!
□ Thanks for the comment, Eryn.
That is a much nicer way to remember it. I guess it depends on if the child likes gross things, too.
They’re all so different, we’ve got to keep creating things to make sure we engage them all.
2. My kids both like using TimezAttack (http://www.bigbrainz.com/) to help learn multiplication facts.
□ Looks crazy cool! I’m downloading the free version now.
Thanks for stopping by and sharing this, Shauna!
3. I may have missed it, but my favorite way to remember the 9′s multiplication table is different than what I saw in any of your links.
Every result multiplying 9 by the factors 1 to 9 always adds up to nine, so that is a quick check to make sure you got it right. But, the answer is always one less than the other factor (not
nine). And then you add whatever number makes it add up to nine. Easier to tell the example than to explain, but my son picked up on this right away. Example: 9 * 6 means the first digit is 5
(6-1) and the second digit is 4 (9-5 = 4). I always thought that this was kind of magical!
Love what you are doing – and if I can ever do something to support you in your math mission, please let me know!
4. I want multiplication tricks for bigger numbers like 9999,57927
you are teaching for single digits
□ Tricks for those would be fun, Anandarao, but I don’t imagine there’s much use in the real world for knowing how to do it. Although Arthur Benjamin makes a living with it!
Leave a reply | {"url":"http://mathfour.com/arithmetic/multiplication-tricks","timestamp":"2014-04-18T18:38:30Z","content_type":null,"content_length":"35883","record_id":"<urn:uuid:97a50f6c-e00d-444d-95fa-77a55381cb72>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00063-ip-10-147-4-33.ec2.internal.warc.gz"} |
Why math, chemistry, physics are important? - Homework Help - eNotes.com
Why math, chemistry, physics are important?
Mathematics is the subject that is of vital importance to any serious scientist, for true science is largely characterized by the ability to quantify one's observations. The mathematician solves
problems by means of logical deductions made from abstract models of concrete cases - by translating a practical problem into mathematics and solving it according to mathematical rules and systems, a
mathematical scientist can interpret the solution in the context of the original problem, thereby solving it, or using the result to derive new models. Careers in the physical sciences, engineering,
actuarial science and statistics are impossible without a sound preparation in Mathematics. Truly, mathematics is the ‘language of science’.
Chemistry is the study of matter, its composition and properties, and the way in which matter changes from one form to another. By studying and therefore knowing how to manipulate these changes
(chemical reactions), chemists can make an enormous contribution to improving our society's quality of life. Many of the world's most urgent needs, such as food supplies, materials for housing, and
cures for diseases like AIDS have answers that lie in the field of Chemistry. Career opportunities for graduate chemists have always been good, and include industrial plant production and control,
education, research and development, and technical sales and service.
Physics is the study of the phenomena that occur in the natural universe, from the infinitesimally small realms of fundamental particles and atoms through to the almost infinitely large in the realms
of astronomy and cosmology. Knowledge of physics is essential in understanding and in advancing every aspect of modern science and technology.
Join to answer this question
Join a community of thousands of dedicated teachers and students.
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Mathematica 7 compares to other languages
Bakul Shah bakul+usenet at bitblocks.com
Fri Dec 12 22:19:29 CET 2008
George Neuner wrote:
> On Thu, 11 Dec 2008 10:41:59 -0800 (PST), Xah Lee <xahlee at gmail.com>
> wrote:
>> On Dec 10, 2:47 pm, John W Kennedy <jwke... at attglobal.net> wrote:
>>> Xah Lee wrote:
>>>> In lisp, python, perl, etc, you'll have 10 or so lines. In C or Java,
>>>> you'll have 50 or hundreds lines.
>>> C:
>>> #include <stdlib.h>
>>> #include <math.h>
>>> void normal(int dim, float* x, float* a) {
>>> float sum = 0.0f;
>>> int i;
>>> float divisor;
>>> for (i = 0; i < dim; ++i) sum += x[i] * x[i];
>>> divisor = sqrt(sum);
>>> for (i = 0; i < dim; ++i) a[i] = x[i]/divisor;
>>> }
>> i don't have experience coding C.
> Then why do you talk about it as if you know something?
>> The code above doesn't seems to satisfy the spec.
> It does.
>> The input should be just a vector, array, list, or
>> whatever the lang supports. The output is the same
>> datatype of the same dimension.
> C's native arrays are stored contiguously. Multidimensional arrays
> can be accessed as a vector of length (dim1 * dim2 * ... * dimN).
> This code handles arrays of any dimensionality. The poorly named
> argument 'dim' specifies the total number of elements in the array.
> George
Only if the length in each dimension is known at compile time (or
in C99, if this is an automatic array). When this is not the case,
you may have to implement something like the following (not the only
way, just one way):
float** new_matrix(int rows, int cols) {
float** m = malloc(sizeof(float*)*rows);
int i;
for (i = 0; i < rows; i++)
m[i] = malloc(sizeof(float)*cols);
return m;
In this case normal() fails since matrix m is not in a single
contiguous area.
But I suspect Xah is complaining because the function doesn't
*return* a value of the same type; instead you have to pass in
the result vector. But such is life if you code in C!
More information about the Python-list mailing list | {"url":"https://mail.python.org/pipermail/python-list/2008-December/492357.html","timestamp":"2014-04-21T13:46:26Z","content_type":null,"content_length":"5062","record_id":"<urn:uuid:8bebab6b-697b-45b0-a794-6759da4ea527>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00388-ip-10-147-4-33.ec2.internal.warc.gz"} |
On the hardness of computing the permanent of random matrices
Results 1 - 10 of 28
, 1997
"... We present a randomized algorithm which takes as input n distinct points f(xi; yi)g n i=1 from F \Theta F (where F is a field) and integer parameters t and d and returns a list of all univariate
polynomials f over F in the variable x of degree at most d which agree with the given set of points in a ..."
Cited by 216 (16 self)
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We present a randomized algorithm which takes as input n distinct points f(xi; yi)g n i=1 from F \Theta F (where F is a field) and integer parameters t and d and returns a list of all univariate
polynomials f over F in the variable x of degree at most d which agree with the given set of points in at least t places (i.e., yi = f (xi) for at least t values of i), provided t = \Omega (
, 1998
"... Madhu Sudan y Luca Trevisan z Salil Vadhan x Abstract Impagliazzo and Wigderson [IW97] have recently shown that if there exists a decision problem solvable in time 2 O(n) and having circuit
complexity 2 n) (for all but finitely many n) then P = BPP. This result is a culmination of a serie ..."
Cited by 127 (20 self)
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Madhu Sudan y Luca Trevisan z Salil Vadhan x Abstract Impagliazzo and Wigderson [IW97] have recently shown that if there exists a decision problem solvable in time 2 O(n) and having circuit
complexity 2 n) (for all but finitely many n) then P = BPP. This result is a culmination of a series of works showing connections between the existence of hard predicates and the existence of good
pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of "hardness amplification" (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a
second derandomized XOR Lemma) that are composed with the Nisan-- Wigderson [NW94] generator. In this paper we present two different approaches to proving the main result of Impagliazzo and
Wigderson. In developing each approach, we introduce new techniques and prove new results that could be useful in future improvements and/or applications of hardness-randomness trade-offs. Our first
result is that when (a modified version of) the NisanWigderson generator construction is applied with a "mildly" hard predicate, the result is a generator that produces a distribution
indistinguishable from having large min-entropy. An extractor can then be used to produce a distribution computationally indistinguishable from uniform. This is the first construction of a
pseudorandom generator that works with a mildly hard predicate without doing hardness amplification. We then show that in the Impagliazzo--Wigderson construction only the first hardness-amplification
phase (encoding with multivariate polynomial) is necessary, since it already gives the required average-case hardness. We prove this result by (i) establishing a connection between the
hardness-amplification problem and a listdecoding...
"... We present a deterministic strongly polynomial algorithm that computes the permanent of a nonnegative n x n matrix to within a multiplicative factor of e^n. To this end ..."
Cited by 63 (8 self)
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We present a deterministic strongly polynomial algorithm that computes the permanent of a nonnegative n x n matrix to within a multiplicative factor of e^n. To this end
- In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science , 2000
"... The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of
pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, ..."
Cited by 55 (8 self)
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The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure
randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, we consider the problem of deterministically converting a weak source of randomness into an
almost uniform distribution. Previously, deterministic extraction procedures were known only for sources satisfying strong independence requirements. In this paper, we look at sources which are
samplable, i.e. can be generated by an efficient sampling algorithm. We seek an efficient deterministic procedure that, given a sample from any samplable distribution of sufficiently large
min-entropy, gives an almost uniformly distributed output. We explore the conditions under which such deterministic extractors exist. We observe that no deterministic extractor exists if the sampler
is allowed to use more computational resources than the extractor. On the other hand, if the extractor is allowed (polynomially) more resources than the sampler, we show that deterministic extraction
becomes possible. This is true unconditionally in the nonuniform setting (i.e., when the extractor can be computed by a small circuit), and (necessarily) relies on complexity assumptions in the
uniform setting. One of our uniform constructions is as follows: assuming that there are problems in���ÌÁÅ�ÇÒthat are not solvable by subexponential-size circuits with¦� gates, there is an efficient
extractor that transforms any samplable distribution of lengthÒand min-entropy Ò into an output distribution of length ÇÒ, whereis any sufficiently small constant. The running time of the extractor
is polynomial inÒand the circuit complexity of the sampler. These extractors are based on a connection be-
- In 36th ACM Symposium on the Theory of Computing , 2004
"... ..."
- SIAM Journal on Computing , 1994
"... We show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the very
few examples of a lower bound in circuit complexity whose proof hinges on the uniformity condition; it is ..."
Cited by 24 (10 self)
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We show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the very few
examples of a lower bound in circuit complexity whose proof hinges on the uniformity condition; it is still unknown if there is any set in Ntime #2 n O#1# # that does not have nonuniform ACC
, 1999
"... . We prove that if there is a polynomial time algorithm which computes the permanent of a matrix of order n for any inverse polynomial fraction of all inputs, then there is a BPP algorithm
computing the permanent for every matrix. It follows that this hypothesis implies P #P = BPP. Our algorithm w ..."
Cited by 20 (3 self)
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. We prove that if there is a polynomial time algorithm which computes the permanent of a matrix of order n for any inverse polynomial fraction of all inputs, then there is a BPP algorithm computing
the permanent for every matrix. It follows that this hypothesis implies P #P = BPP. Our algorithm works over any sufficiently large finite field (polynomially larger than the inverse of the assumed
success ratio), or any interval of integers of similar range. The assumed algorithm can also be a probabilistic polynomial time algorithm. Our result is essentially the best possible based on any
black box assumption of permanent solvers, and is a simultaneous improvement of the results of Gemmell and Sudan [GS92], Feige and Lund [FL92] as well as Cai and Hemachandra [CH91], and Toda (see
[ABG90]). 1 Introduction The permanent of an n \Theta n matrix A is defined as per(A) = X oe2Sn n Y i=1 A i;oe(i) ; where Sn is the symmetric group on n letters, i.e., the set of all permutations of
"... Solving stochastic optimization problems under partial observability, where one needs to adaptively make decisions with uncertain outcomes, is a fundamental but notoriously difficult challenge.
In this paper, we introduce the concept of adaptive submodularity, generalizing submodular set functions t ..."
Cited by 17 (2 self)
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Solving stochastic optimization problems under partial observability, where one needs to adaptively make decisions with uncertain outcomes, is a fundamental but notoriously difficult challenge. In
this paper, we introduce the concept of adaptive submodularity, generalizing submodular set functions to adaptive policies. We prove that if a problem satisfies this property, a simple adaptive
greedy algorithm is guaranteed to be competitive with the optimal policy. We illustrate the usefulness of the concept by giving several examples of adaptive submodular objectives arising in diverse
applications including sensor placement, viral marketing and pool-based active learning. Proving adaptive submodularity for these problems allows us to recover existing results in these applications
as special cases and leads to natural generalizations. 1
- In Proc. of FCRC , 1999
"... We survey some recent developments in the study of the complexity of lattice problems. After a discussion of some problems on lattices which can be algorithmically solved efficiently, our main
focus is the recent progress on complexity results of intractability. We will discuss Ajtai's worstcase /av ..."
Cited by 12 (1 self)
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We survey some recent developments in the study of the complexity of lattice problems. After a discussion of some problems on lattices which can be algorithmically solved efficiently, our main focus
is the recent progress on complexity results of intractability. We will discuss Ajtai's worstcase /average-case connections, NP-hardness and non-NPhardness, transference theorems between primal and
dual lattices, and the Ajtai-Dwork cryptosystem. 1 Introduction There have been some exciting developments recently concerning the complexity of lattice problems. Research in the algorithmic aspects
of lattice problems has been active in the past, especially following Lovasz's basis reduction algorithm in 1982. The recent wave of activity and interest can be traced in large part to two seminal
papers written by Miklos Ajtai in 1996 and in 1997 respectively. In his 1996 paper [1], Ajtai found a remarkable worstcase to average-case reduction for some versions of the shortest lattice vector
- J. Artificial Intelligence Research , 2011
"... Many problems in artificial intelligence require adaptively making a sequence of decisions with uncertain outcomes under partial observability. Solving such stochastic optimization problems is a
fundamental but notoriously difficult challenge. In this paper, we introduce the concept of adaptive subm ..."
Cited by 12 (4 self)
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Many problems in artificial intelligence require adaptively making a sequence of decisions with uncertain outcomes under partial observability. Solving such stochastic optimization problems is a
fundamental but notoriously difficult challenge. In this paper, we introduce the concept of adaptive submodularity, generalizing submodular set functions to adaptive policies. We prove that if a
problem satisfies this property, a simple adaptive greedy algorithm is guaranteed to be competitive with the optimal policy. In addition to providing performance guarantees for both stochastic
maximization and coverage, adaptive submodularity can be exploited to drastically speed up the greedy algorithm by using lazy evaluations. We illustrate the usefulness of the concept by giving
several examples of adaptive submodular objectives arising in diverse AI applications including management of sensing resources, viral marketing and active learning. Proving adaptive submodularity
for these problems allows us to recover existing results in these applications as special cases, improve approximation guarantees and handle natural generalizations. 1. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1340769","timestamp":"2014-04-18T11:51:57Z","content_type":null,"content_length":"38485","record_id":"<urn:uuid:0059532b-94f6-4bef-b3c1-07fd2b03fdd4>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00013-ip-10-147-4-33.ec2.internal.warc.gz"} |
DetlefsenFOM: The Aristotelian ideal and reverse mathematics
Robert S Tragesser RTragesser at compuserve.com
Sat Dec 13 17:50:30 EST 1997
Summaryt: It is highly implausible that the Aristotelian ideal can be
realized in its orginal terms (whatever exactly they might be). It is
surely a good historical exercise to think about what those original terms
might be, as Defletsen is gamely attempting. But there is no point in
blindly imposing them on reverse mathematics. The logic needed for
mathematics is just too powerful to realize that ideal in its original
terms, even if there were some point to doing so (it is curious that
Detlefsen didn't say what the point would be). The important question is
to determine the measure in which a particular mathematical genos is
self-founding, whether the proofs of all its theorems can be "elementary
proofs." Is any mathematical genos genuinely self-contained,
"elementary", composed entirley of "trivial" truths, dominated by
analyticity? Or must "transcendental methods" always be evoked
(=transcendental = techniques from another genos), so that unity rather
than separateness is characteristic of mathematics.
It is important to be more cautious than Detlefsen is being in
invoking and framing the terms of the Aristotelian ideal. For one thing,
the elaborate and failed attempts to characterize and realize the ideal in
mathematics in the Late and Post Renaissance, as exposed in Mancosu,
should serve as a serious warning that it is highly problematic. For
another, recent attempts to specify in the context of a logic adequate to
mathematics the class of proofs which are in some strong sense explanatory
(not to mention the wider and more dramatic failure to do so in the
philosophy of science) suggest that one can't even get of the ground. (The
very fact that any mathematical domain can be represented in very many very
different ways militates vigorously against the possibility of specifying a
priviledged explanatory frame for any domain!)
It was exactly because Aristotle's logic was a term logic which
rather limited vehicles of deductive inference that he could imagine the
possibility of specifying which deductive inference structures are
causal/explantory. And more importantly, recall that it was a necessary
condition for an "axiom" or fundamental "definition" that it not be
deducible. How was one to recognize when by backward deduction [which
essentially would involve inferences which were not causal!] one got to an
axiom? The crude idea [that is, the idea which would have to be
explicated rigorously] wsas that one got to an AisB for which there were,
and could be, no "middle terms" which would allow one to
explantorily/causally deduce AisB from something more fundamental (from
anything else at all, in fact). One had to be able to recognize that the
concrete term logic of the genos at issue made (explanatory) proofs of the
proposed axioms impossible (without committing metabasis and going outside
the terms of the genos.
THE ONLY IMPORTANT AND DO-ABLE ASPECT OF THE ARISTOTELIAN IDEAL:
setting things up, perhaps in the context of RevMath, to see which
mathematical subjects can be generated by "elementary" rather than
"transcendental" methods (where transcendental means, not necessarily
infinitary, but rather OTHER, that is, a transcendental technique is one
which is borrowed from another genos, as for example geometry to algebra
or algebra to geometry.
THE REALLY DEEP ASSUMPTION IN ARISTOTLE -- whose truth is most
importantly as issue in the context of our mathematics -- IN THE CONTEXT OF
PROPER OR NATURAL TO A GENOS IS WELL-DETERMINED, THAT IS, WHETHER OR NOT
Robert Tragesser
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Supermarket D uses 2 tracking systems, S-1 and S-2. Both
Question Stats:
0%0% (00:00)based on 0 sessions
gmatbull wrote:
Supermarket D uses 2 tracking systems, S-1 and S-2. Both tracking systems use codes, each of which is an
ordered sequence of 6 characters. Each code for S-1 consists of 2 digits, each chosen from 0 through 9,
followed by 4 letters, each chosen from A through Z but excluding vowels (A, E, I, O, U) and Y. Each code
for S-2 consists of 4 digits, each chosen from 0 through 9, followed by 2 letters, each chosen from A through
Z but excluding vowels and Y. What fraction of the possible number of S-1 codes is the possible number of S-2 codes?
(Note: There are 20 letters from A through Z, excluding vowels and Y.)
Am sorry, no options, but will appreciate your contributions.
If the 1st tracking system uses different combinations (2D 4L), how can we have some common elements with
the 2nd tracking system, which uses (4D 2L)?
Question is not asking for common elements and as you rightly mentioned there cant be common codes as one is 2D,4L and other 4L, 2D
Instead, question is asking for the fraction that you get as : number of possible codes for S-2 / number of possible codes for S-1
(eg, say if possible number of S-1 is 100 then what fraction of it is S-2 which is 20. re-reading the question now should help
S-1 = 10*10*20*20*20*20
(Question has no restriction on repeating the digit or letter)
S-2 = 10*10*10*10*20*20
Hence, Ans = S-2/S-1 = 1/4
Hope it helps.
Lets Kudos!!!
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Brief Intro to LOCBIF
LOCBIF is distributed as part of the Windows version of XPP thanks to the kind contribution of Alex Khibnik who parted with the source code. LOCBIF was originally a stand-alone DOS program and a
recent improved version, CONTENT . The version here is somewhat old but still seems to work OK. AUTO is the bifurcation package that I originally included in XPP but was unable to get the port to
work in Windows. Anyone who'd like to try should contact me.
LOCBIF vs AUTO
LOCBIF is easier to use and handles bifurcations of equilibria and fixed points of maps better than AUTO. On the other hand, AUTO handles boundary value problems and continuation of periodic orbits
better than LOCBIF. If you really need AUTO, you can get the native X version for Windows of XPPAUT or switch to Linux (the recommended solution).
LOCBIF can continue and find bifurcations for the following four types of problems depending on parameters p with state variables x .
1. Equilibria of ODEs :
dx/dt = f(x,p)
2. Fixed points of maps :
x(n+1) = f(x(n),p)
3. Periodic solutions of forced systems :
dx/dt = f(x,p,t) where f(x,p,t+T)=f(x,p,t)
4. Autonomous oscillations of ODEs:
dx/dt = f(x,p)
NOTE The last three types of curves are essentially the same since the periodically forced system is equivalent to a map taking the phase-space at t=0 to the phase-space at t=T . In the case of an
autonomous oscillation, the one forms the Poincare map and looks at that map. Fixed points of the latter two systems correspond to periodic solutions. To invoke LOCBIF, you must click on File Locbif
from the main WINPP menu. You will get a window like this:
The basic objects that LOCBIF computes are curves in the product space of the parameters and the phase space. For example, it will compute curves of equilibria F(x,p)=0 for ODEs. Along these curves,
LOCBIF keeps track of special points where bifurcations occur. These special points are obtained by evaluating certain bifurcation functions the zeros of which tell you something about the system.
For example, a zero determinant tells you that there is a zero eigenvalue. The equations for these functions along with the ODE/MAP form the curve-defining system. So, if you want to follow the set
of equilibria where there is a zero eigenvalue, you need two parameters since you must solve the n+1 equations:
F(x,p1,p2) = 0
det A(p1,p2) = 0
along some curve in (x,p1,p2) space. There are many different types bifurcations and I will refer the reader to a comprehensive book such as Wiggins , Guckenheimer and Holmes or Kuznetsov.
File menu
Nothing here works yet.
This invokes the numerics menu. There are many numerical parameters and their suggested settings. I will briefly describe them but a comprehensive description is best obtained from the hardcopy
documentation for the DOS version lbf.ps
Curve parameters
These are parameters for the continuation in phase-space.
• H0crv is the initial step size for continuation of the curve.
• Hmxcrv is the maximum step allowable; make this smaller to force smoother bifurcation curves.
• Angcrv is the desired angle between the tangent at the current point and the secant with the previous point. Smaller angles lead to more points per curve and smoother curves.
• DhCrv increment for numerical evaluation of Jacobian matrix for the curve-defining system
• DhJac increment for numerical evaluation of Jacobian matrix for the right-hand sides of the ODE/Map.
• Maxit maximal number of Newton corrections
• Modit number of Newton corrector steps above with Jacobian re-evaluation.
• Epscrv tolerance for Newton corrector. Set this or any of the tolerances to zero to ignore such points.
• Epscrs tolerance for self-crossing (eg transcritical and closed curves)
• Epszer tolerance for zeros of the curve-defining system
• Epsext tolerance for location of extremal values of the parameters. I often set this to 0 as it is annoying.
• Ipsrng ordering number(s) of a nongeneric parameter(s) for continuation of non-transveral curve types. Possibilities are 1,2,3,12,13,14,23,24,34.
• Algcrv type of step size
□ +/- 1: doubling/halving
□ +/- 2: explicit dependence on curvature
□ +/- 3: implicit dependence on curvature
Curvature test (see Angcrv ) is ignored if this parameter is negative.
Orbit parameters
These are parameters for the LOCBIF integrators.
• Itmap number of iterations for the map or periods of length Tint for the flows.
• Tint length of time to integrate for flows. It should equal the period for the periodically forced system.
• H0int initial integration step size
• Hmxint maximal step size for integration
• Dhint increment for numerical Jacobian for stiff integrator
• Epsint absolute tolerance
• Epsrel relative tolerance (make this smaller than Epsint )
• Solver there are five internal LOCBIF solvers, the first three are fifth order Dormund-Prince solvers, the fourth is a Runge-Kutta type solver, and the last is a stiff solver.
• Isec the undex of the variable used for the Poincare map. When the derivative of the Isecth variable vanishes, this is the Poincare map used. Setting this to 0 means to automatically compute the
secant plane.
• Irhs not yet implemented
• Iorbit ignored
These items let you redraw the diagram and set the plot parameters. Only two-dimensional plots are allowed at present. The Autoscale check box will scale the plot to fit the diagram.
• Copy copies the plot to the clipboard
• Edit lets you save the current curve, start a new curve, and choose from previous saved curves. Curves are presently saved in memory and are lost when you exit the program. Only the currently
chosen curve is plotted.
• Delete lets you delete curves.
Note that by defining different curves, you can look at a system as one parameter varies, then as a different one varies and keep them separated.
This command lets you find equilibria and other curves as one or more parameters varies. To start, you must have an equilibriaum point as the initial condition in WinPP or you must start on a point
you have grabbed from a prior calculation. This opens the dialog box:
There are many different choices that you can pick here. For ordinary equilibria, choose Equilibrium and LOCBIF will track the equilibria as a parameter varies. Note the number in the parenthese in
each point is the number of parameters you must choose for that curve. Suppose after computing a curve of equilibria, you find a Hopf bifurcation. Then you can do a two-parameter continuation finding
all equilibria that are points of a Hopf bifurcation. Once you have chosen the type of curve to follow, choose the parameters and the maximum and minimum ranges. Finally choose a direction (+/- 1)
and click on OK. If all goes well, the diagram will be computed. You can terminate the calculation by clicking on ABORT.
Fixed points of maps
This command lets you find fixed points of maps as a function of some parameters and lets you trace multi-parameters curves of
These examples are from the LOCBIF manual, adopted for WINPP. Click on the ecology model, eco.ode . This is a modification of the Lotka-Volterra model:
x'= x(1-beta x) - x y/(1 + alpha x)
y'= -y(1+delta y) + x y/(1+alpha x)
• Find an equilibrium point by drawing the nullclines ( Phaseplane Nullclines ) and then using the mouse to start nearby the fixed point and integrate. Click Run Last a few times to land on the
fixed point.
• Click on File Locbif to get the LOCBIF window. The rest of this example is within the LOCBIF window. Plots Parameters and set the X-axis to be the parameter delta and the Y-axis to be the
variable Y . Set the X limits between 0 and 1 and the Y limits between 0 and 10. Click on Run Equilibria. Choose Equilibrium as the Curve Type and set the 1st parameter to be delta and the range
to be between 0 and 10. Click OK and the curve of fixed points will appear. Notice that a thick line is drawn indicating a stable branch. An error message will occur saying computations
terminated. This is just because you exceeded the parameter bounds and is OK.
• Click on Traverse and use the arrow keys or the mouse to trace the curve. Info is given in the window above the graph and stability is given in the small window with a circle. Eigenvalues in the
circle are stable. Click on Home and then Grab to get the first point. Click on Run Equilibrium again and change the Direction to -1 and then click OK and the rest of the branch will be computed.
You will have this curve:
• Traverse it again, this time clicking on Jump to see the special points that have been marked with a cross. The info window tells you what they are. The three of interest are the two fold points,
indicated by the descriptor Zero Eigenvalue and the Hopf point. The other three special points are of no interest here. Note at the Hopf point, the info tells you that L1=-.28037E-2 Since it is
negative, this tells you there is a supercritical Hopf bifurcation but that it is nearly vertical since L1 is small. Now we can do a two parameter continuation of the fold points.
• Traverse the diagram and grab the first zero eigenvalue point at delta=.26 or so. Now click Edit Edit Curves and click on Add New This will be for two-parameter curves. Click on Choose and then
Plots Parameters Choose the Y axis to be Alpha and let the X axis range between 0 and 3 and the Y axis between 0 and 1. If you want, pick a color, like red. Now click OK Click on Numerics and set
the parameter EpsExt to zero to turn off extrema calculations. Click OK Now click on Equilibrium Run and choose Fold as the curve type, and Alpha OK A closed curve will be drawn (at least on my
computer). This is a curve of fold points.
• Among the special points are two Neutrality points and a pair of Cusp points. Neutrality points indicate a double zero eigenvalue as can be seen by looking at the stability circle. We can pick up
a Hopf bifurcation from these points. Traverse and grab the first one at delta=.006 and alpha=.86 (Use the Jump button to go directly to special points.) Change the plot color to say Blue.
• Click on Run Equilibria and choose Hopf as the curve type. Make the direction -1 and click OK This will draw the curve of Hopf points. You should have this diagram:
• The intersection of the Hopf and fold points is a Takens-Bogdanov point and there are homoclinics near by. Grab a TB point and then in WINPP try to find some homoclinics! The cusps are also
special points. Inside the closed curve there are three fixed points and outside, one.
Clicking on the Traverse button once you have computed a curve, allows you to move a little cross around and information about the points is given in the window above the graph. You can use the mouse
to move through the diagram by clicking on
• >> or the right arrow to move up the diagram
• << or the left arrow to move down the diagram
• Home to go to the start of the curve
• Jump to go to the next marked point or PgUp and PgDn
• Directly click near a point on the curve
Then click on Grab to load this point into the initial conditions and to set the parameters at the value corresponding to the point. Use this to start new curves and extend old ones. Click on Esc or
type Esc to end traversing the diagram without grabbing a point.
1. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, NY, 1990
2. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations, Springer-Verlag, NY, 1983.
3. Y. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, NY, 1995 (1998, New Edition). | {"url":"http://www.math.pitt.edu/~bard/classes/wppdoc/locbif.html","timestamp":"2014-04-18T13:07:24Z","content_type":null,"content_length":"14615","record_id":"<urn:uuid:a260fa1e-f809-4867-b478-1b5dc529dcac>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00557-ip-10-147-4-33.ec2.internal.warc.gz"} |
[SOLVED] Simple probability question
March 23rd 2009, 07:05 PM
[SOLVED] Simple probability question
2 events that are independent of each other.
p(a) = .5 and p(b) = .95
What is the probability that b or a, or both b and a?
I know b and a is p(a)*p(b) but my book doesn't seem to cover both.
March 24th 2009, 12:17 AM
Hello othnin
I think a probability tree is the easiest way to understand this. See the attached diagram.
The four end-points describe, with their respective probabilities, the possible outcomes. If you want to know the probability of arriving at more than one of these, just add their probabilities
So, for instance, the probability of one of a or b but not both is 0.5 x 0.05 + 0.5 x 0.95 = 0.5; and the probability of a or b or both is 0.5 x 0.95 + 0.5 x 0.05 + 0.5 x 0.95 = 0.975.
Does that make it clearer?
March 24th 2009, 05:09 AM
Thanks for the help and nice diagram! | {"url":"http://mathhelpforum.com/statistics/80296-solved-simple-probability-question-print.html","timestamp":"2014-04-20T21:28:24Z","content_type":null,"content_length":"5198","record_id":"<urn:uuid:d1617698-ec7d-410c-95d0-2687a4089071>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00427-ip-10-147-4-33.ec2.internal.warc.gz"} |
Symbolic analysis of crypto-protocols based on modular exponentiation
- In Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP’05), volume 3580 of LNCS , 2005
"... Abstract. Most of the decision procedures for symbolic analysis of protocols are limited to a fixed set of algebraic operators associated with a fixed intruder theory. Examples of such sets of
operators comprise XOR, multiplication/exponentiation, abstract encryption/decryption. In this paper we giv ..."
Cited by 18 (3 self)
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Abstract. Most of the decision procedures for symbolic analysis of protocols are limited to a fixed set of algebraic operators associated with a fixed intruder theory. Examples of such sets of
operators comprise XOR, multiplication/exponentiation, abstract encryption/decryption. In this paper we give an algorithm for combining decision procedures for arbitrary intruder theories with
disjoint sets of operators, provided that solvability of ordered intruder constraints, a slight generalization of intruder constraints, can be decided in each theory. This is the case for most of the
intruder theories for which a decision procedure has been given. In particular our result allows us to decide trace-based security properties of protocols that employ any combination of the above
mentioned operators with a bounded number of sessions. 1
- , Theoretical Computer Science , 2005
"... In security protocols, message exchange between the intruder and honest participants induces a form of state explosion which makes protocol models infinite. We propose a general method for
automatic analysis of security protocols based on the notion of frame, essentially a rewrite system plus a se ..."
Cited by 2 (0 self)
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In security protocols, message exchange between the intruder and honest participants induces a form of state explosion which makes protocol models infinite. We propose a general method for automatic
analysis of security protocols based on the notion of frame, essentially a rewrite system plus a set of distinguished terms called messages. Frames are intended to model generic cryptosystems. Based
on frames, we introduce a process language akin to Abadi and Fournet’s applied pi. For this language, we define a symbolic operational semantics that relies on unification and provides finite and
effective protocol models. Next, we give a method to carry out trace analysis directly on the symbolic model. We spell out a regularity condition on the underlying frame, which guarantees
completeness of our method for the considered class of properties, including secrecy and various forms of authentication. We show how to instantiate our method to some of the most common
crypto-systems, including shared- and public-key encryption, hashing and Diffie-Hellman key exchange.
, 2007
"... Design and verification of cryptographic protocols has been under investigation for quite sometime. However, most of the attention has been paid for two parties protocols. In group key
management and distribution protocols, keys are computed dynamically through cooperation of all protocol participan ..."
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Design and verification of cryptographic protocols has been under investigation for quite sometime. However, most of the attention has been paid for two parties protocols. In group key management and
distribution protocols, keys are computed dynamically through cooperation of all protocol participants. Therefore regular approaches for two parties protocols verification cannot be applied on group
key protocols. In this paper, we present a framework for formally verifying of group key management and distribution protocols based on the concept of rank functions. We define a class of rank
functions that satisfy specific requirements and prove the soundness of these rank functions. Based on the set of sound rank functions, we provide a sound and complete inference system to detect
attacks in group key management protocols. The inference system provides an elegant and natural proof strategy for such protocols compared to existing approaches. The above formalizations and rank
theorems were implemented using the Prototype Verification System (PVS) theorem prover. We illustrate our approach by applying the inference system on a generic Diffie-Hellman group protocol and
prove it in PVS.
"... Abstract. We study the algebra underlying symbolic protocol analysis for protocols using Diffie-Hellman operations. Diffie-Hellman operations act on a cyclic group of prime order, together with
an exponentiation operator. The exponents form a finite field: this rich algebraic structure has resisted ..."
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Abstract. We study the algebra underlying symbolic protocol analysis for protocols using Diffie-Hellman operations. Diffie-Hellman operations act on a cyclic group of prime order, together with an
exponentiation operator. The exponents form a finite field: this rich algebraic structure has resisted previous symbolic approaches. We define an algebra that validates precisely the equations that
hold almost always as the order of the cyclic group varies. We realize this algebra as the set of normal forms of a particular rewriting theory. The normal forms allow us to define our crucial notion
of indicator, a vector of integers that summarizes how many times each secret exponent appears in a message. We prove that the adversary can never construct a message with a new indicator in our
adversary model. Using this invariant, we prove the main security goals achieved by UM, a protocol using Diffie-Hellman for implicit authentication. Despite vigorous research in symbolic analysis of
security protocols, many
"... Abstract. We extend symbolic protocol analysis to apply to protocols using Diffie-Hellman operations. Diffie-Hellman operations act on a cyclic group of prime order, together with an
exponentiation operator. The exponents form a finite field. This rich algebraic structure has resisting previous symb ..."
Add to MetaCart
Abstract. We extend symbolic protocol analysis to apply to protocols using Diffie-Hellman operations. Diffie-Hellman operations act on a cyclic group of prime order, together with an exponentiation
operator. The exponents form a finite field. This rich algebraic structure has resisting previous symbolic approaches. We work in an algebra defined by the normal forms of a rewriting theory (modulo
associativity and commutativity). These normal forms allow us to define our crucial notion of indicator, a vector of integers that summarizes how many times each secret exponent appears in a message.
We prove that the adversary can never construct a message with a new indicator in our adversary model. Using this invariant, we prove the main security goals achieved by several different protocols
that use Diffie-Hellman operators in subtle ways. We also give a model-theoretic justification of our rewriting theory: the theory proves all equations that are uniformly true as the order of the
cyclic group varies. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1023195","timestamp":"2014-04-20T10:28:31Z","content_type":null,"content_length":"24763","record_id":"<urn:uuid:6c5e8539-110e-4ec3-98c1-ea1999ecc0eb>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00534-ip-10-147-4-33.ec2.internal.warc.gz"} |
Point of inflection difficulty
December 9th 2008, 06:32 AM #1
Nov 2008
Point of inflection difficulty
I have this graph <(x^2 - 8)/(x + 3)>, and I'm supposed to find the point(s) of inflection. So I used the second derivative test, got 2/(x + 3)^3 But when I put that in my calculator, set to to
zero, and tell the calculator to solve for x; it just says 'False'. As if the problem has an undefined answer or something. So now I don't know what to do...is there another way to find points of
inflection? Or does that 'False' just mean that there is no points of inflection?
... that's right. Points of inflection give you a zero second derivative. How big would x have to be here to bring the value of 2/(x+3)^3 down to zero? (!)
December 9th 2008, 07:39 AM #2
MHF Contributor
Oct 2008 | {"url":"http://mathhelpforum.com/calculus/64106-point-inflection-difficulty.html","timestamp":"2014-04-17T01:49:35Z","content_type":null,"content_length":"31382","record_id":"<urn:uuid:05c20773-23e5-405d-946f-22ae14e4c8a1>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00295-ip-10-147-4-33.ec2.internal.warc.gz"} |
simple derivative problem!
1. This is a term and not a function, so ... 2. Where are you stuck (and why?)? 3. I assume that you want to differentiate $f(x) = \sqrt{2x^2}$ If so: Re-write the equation of the function: $f(x) = \
sqrt{2x^2}~\implies~f(x) = \left \lbrace \begin{array}{rcl}x \cdot \sqrt{2}& if & x \ge 0 \\ -x \cdot \sqrt{2}& if & x < 0\end{array} \right.$ 4. Continue!
This is where I am stuck. I have to find the equation of a line tangent to the curve $f(x) = \frac{\sqrt{2x^3}}{2}$at the point (2,2) but when I use the quotient rule to find the derivative I get
stuck when differentiating $\sqrt{2x^3}$ Why? Does the $x^\frac{1}{2}$ (from simplifying the root) get distributed to $2$ and $x^3$? If you know a simpler way of doing this please show me.
I don't understand why you want to use the quotient rule because there isn't any variable in the numerator. Re-write the equation of the function: $f(x) = \frac{\sqrt{2x^3}}{2} ~\implies~f(x)=\frac12
\cdot \sqrt{2} \cdot x^{\frac32}$ Now apply the power rule to differentiate this function.
Quotient rule not necessary here. f(x)=(root2)/2 times x^3/2 When differentiating (root2)/2 stays in front and x^3/2 becomes (3/2)x^1/2 You want this when x=2 ( I get 3/2 ) | {"url":"http://mathhelpforum.com/calculus/197481-simple-derivative-problem-print.html","timestamp":"2014-04-18T11:49:09Z","content_type":null,"content_length":"9997","record_id":"<urn:uuid:77a6337f-d96e-48c3-9dac-45a1a9a24a00>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00276-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Is there a simple formula to calculate solar eclipse timings from any position on Earth?
I am interested in solar eclipse calculation and I already read Jean Meeus, Duffet and Mottenbruck book but I still cannot find the formula to calculate solar eclipse time from any position on earth.
Please let me know where can I get the formula.
There is no simple formula to calculate what you are interested in. The calculation of eclipses is tedious work requiring many observations or calculations of the positions of the Moon and the Sun.
The lunar orbit data must then be extrapolated to find the months when eclipses might occur and then the exact times determined to see where the eclipse will be visible. The eclipse path can then be
projected into the Earth's surface to find if a total eclipse will be visible from a given location at a particular time. This work requires a lot of patience and an understanding of the geometry
involved and cannot be reduced to a formula.
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A Sudoku Solver in Java implementing Knuths Dancing Links Algorithm
A Sudoku Solver in Java implementing Knuth s Dancing Links Algorithm
For the Harker Research Symposium
Version: 1.2
Date: 2006 April 20
Knuth s paper on Dancing Links can be found here or follow the credits links below.
Dr. Donald Knuth s Dancing Links Algorithm solves an Exact Cover situation. The Exact Cover problem can be extended to a variety of applications that need to fill constraints. Sudoku is one such
special case of the Exact Cover problem.
The Journey
In early 2006, I participated in the ACSL Competition. The prompt of the competition was to create a simple Sudoku solver. I had never solved a Sudoku puzzle so I researched on methods to approach
the puzzle. I came across several strategy guides, Sudoku forums, and computer solvers. Hinted amongst the computer programs was the dancing links algorithm. However, given the time and simplicity
required for the competition, I reverted to a simple brute force candidate elimination algorithm to solve the simple Sudoku given by the ACSL. However, I found that without a guessing and
backtracking algorithm, I could not solve anything beyond the simplest puzzles.
Then around came the Symposium. I was unsure of what the symposium really meant, but with the influence of my CS teacher, Mr. Feinberg, I entered with the notion of research the Dancing Links
algorithm and creating a Sudoku solver. In creating the program, I read Knuth s paper and watched his recorded lecture. Although there are several different versions of programs written in different
languages available online with source code, I did not reference them. Rather, I went ahead to discover the intricacies of the algorithm through my own exploration. I also created the simple
graphical interface, which displays the data structure uses: the matrix of doubly-linked nodes.
And so, I learned what the Exact Cover problem was, how the algorithm worked to extract the answer from the problem, how the Exact Cover problem can be adapted to Sudoku, and how I could build a
program to accomplish my goal to create the Sudoku Solver.
Over the past two months, I have ventured through my first real encounter with Computer Science and discovered its underlying potential and power. With my background in web design, I have gained
another facet of admiration for CS.
The rest of the paper will describe the Exact Cover Problem, the Dancing Links Algorithm, and the application to Sudoku.
What is Exact Cover?
Given a matrix of 1 s and 0 s the Dancing Links will find a set or more of rows in which exactly one 1 will appear for each column. For example, in Knuth s paper figure 3, a matrix is given as:
Rows 1, 4, 5 are a set that solves this Exact Cover Puzzle.
Knuth takes advantage of a basic principle of doubly-linked lists. When removing an object from a list, only two operations are needed:
x.getRight().setLeft( x.getLeft() )
x.getLeft().setRight( x.getRight() )
However, when putting the object back in the list, all is needed is to do the reverse of the operation.
x.getRight().setLeft( x )
x.getLeft().setRight( x )
All that is needed to put the object back is the object itself, because the object still points to elements within the list. Unless x s pointers are changed, this operation is very simple.
Dancing Links takes the Exact Cover matrix and puts it into a toroidal doubly-linked list. For every column, there is a special ColumnNode, which contains that column s Unique Name and the column s
size, the number of nodes in the column. Every 1 in the list, is a Node. Each Node points to another object up, down, left, right, and to its corresponding ColumnNode. A special ColumnNode h points
to the first ColumnNode on the left as a starting point for the algorithm. So Knuth s figure 3 would become:
The Algorithm
Given the ColumnNode h, the searching algorithm is then simplified to:
if( h.getRight() == h ) {
else {
ColumnNode column = chooseNextColumn();
for( Node row = column.getDown() ; rowNode != column ; rowNode = rowNode.getDown() ) {
solutions.add( rowNode );
for( Node rightNode = row.getRight() ; otherNode != row ; rightNode = rightNode.getRight() )
cover( rightNode );
Search( k+1);
solutions.remove( rowNode );
column = rowNode.getColumn();
for( Node leftNode = rowNode.getLeft() ; leftNode != row ; leftNode = leftNode.getLeft() )
uncover( leftNode );
uncover( column );
cover( ColumnNode c ) This function is the crux of the algorithm. It removes a column from the matrix as well as remove all rows in the column from other columns they are in. The code becomes:
Node column = dataNode.getColumn();
column.getRight().setLeft( column.getLeft() );
column.getLeft().setRight( column.getRight() );
for( Node row = column.getDown() ; row != column ; row = row.getDown() )
for( Node rightNode = row.getRight() ; rightNode != row ; rightNode = rightNode.getRight() ) {
rightNode.getUp().setDown( rightNode.getDown() );
rightNode.getDown().setUp( rightNode.getUp() );
Note that we first remove the column from the other columns. Then we go down a column and remove the row by traversing the row to the right.
Let s look at some illustrations. Here is the matrix before Column A is covered. All the links in bold are the links that are going to be affected by the cover function.
Now here is the matrix after Column A has been covered. Notice how Column A and rows 2 and 4 are now independently linked outside of the matrix. In effect, they have been removed. Also note that each
node that has been removed from the matrix still points to an element inside the matrix. This allows us to easily backtrack.
uncover( ColumnNode c ) This is the answer to easy backtracking. Taking advantage of the fact that every node that has been removed retains information about its neighbors, we can easily put the node
back into the matrix using the reverse operation of cover.
Node column = dataNode.getColumn();
for( Node row = column.getUp() ; row != column ; row = row.getUp() )
for( Node leftNode = row.getleft() ; leftNode != row ; leftNode = leftNode.getRight() ) {
leftNode.getUp().setDown( leftNode.getDown() );
leftNode.getDown().setUp( leftNode.getUp() );
column.getRight().setLeft( column.getLeft() );
column.getLeft().setRight( column.getRight() );
Notice that the traversal through the column and row are opposite to that of cover. We first put the rows back by traveling up the column and to the left of the row. Then we put the column back. In
effect, we undo the operation of cover.
Miscellaneous Functions
printSolution() takes all the rowNodes in the solution index, which is built by some data structure solutions and translates their positions in the matrix into the positions in the actual puzzle.
chooseNextColumn() advances the column pointer right or chooses the column with the least number of nodes. Choosing the column with the least number of nodes decreases the branching of the algorithm.
It can be ignored if this isn t needed.
Finding a Solution
A solution is found when all the columns have been removed from the matrix. This means that very row that we have added to the answer has one node in every column. All constraints have been satisfied
by the set of rows.
A complete run-through of this algorithm is shown in file: Exact.Cover.Runthrough.xls.
So, what is Sudoku?
Sudoku is a logic puzzle. On a 9x9 grid with 3x3 regions, the digits 1-9 must be placed in each cell such that every row, column, and region contains only one instance of the digit. Placing the
numbers is simply an exercise of logic and patience. Here is an example of a puzzle and its solution:
Images from http://www.nikoli.co.jp/puzzles/1/index_text-e.htm
To create the sparse matrix of Sudoku needed to convert the problem into an Exact Cover Problem, we need to recognize what the rows and columns represent.
The columns represent the constraints of the puzzle. In Sudoku, we have four:
• A position constraint: Only 1 number can occupy a cell
• A row constraint: Only 1 instance of a number can be in the row
• A column constraint: Only 1 instance of a number can be in a column
• A region constraint: Only 1 instance of a number can be in a region
Each number comes with its own set of constraints. Therefore there are SIZE^2 * 4 columns., where SIZE is the number of candidates/rows/cols there are in the Sudoku Puzzle. In a 4x4, this would be 64
columns. In a 9x9, this would be 324 columns.
The rows represent every single possible position for every number. Therefore, there are SIZE ^ 3 rows. In a 4x4, this would be 64 columns. In a 9x9, this would be 729 rows. Each row would represent
only one candidate position. Therefore, only 4 1s will be in the row, representing the constraints of that position.
The sparse matrix for a 4x4 Sudoku puzzle is seen in: 4x4.dlx.64x64.xls. The 9x9 sparse matrix is impractical to create by hand.
Given initial positions in the matrix, those rows will be included in the answer and covered. Then the Search algorithm will produce the solutions to the puzzle.
Having gone through 5 different revisions of the code, I have finally produced a program that can be presented. The following is a list of features of the program.
Currently, I have hardcoded Sudoku puzzles into 2D arrays. In the future, I may learn to made a GUI that allows the user to input numbers onto a Sudoku puzzle display.
By default, the program will display the initial input puzzle and then the solution.
The graphical element displays the doubly-linked lists in a linear fashion. Currently, it is only practical for 4x4 and 9x9 Sudoku puzzles.
Multiple Solutions
When this feature is enabled, the program can find multiple solutions to the puzzle given the input. It will continue to search until all possible attempts fail to find anymore solutions.
Alternate search method
As described in Knuth s paper, there are two methods to choose the next column in the search method. The first simply chooses the node to the right of ColumnNode h. The second minimizes the
branching of the algorithm by looking for the column with the least number of nodes in the column. The second method is much faster for solving 16x16 Sudoku puzzles. Both methods can be used in
this program
Verbose and Debug Messages
When verbose messages are enabled, the program prints out the number of iterations search has been called and a snapshot of the current Sudoku puzzle. Debug messages allows the user to see what
the program is doing.
A simply delay allows the graphical elements to catch up with the program, as well as slow down the output of the message. To see the real power of the program, set delay to 0 and do not enable
messages and graphics.
• Source Code (link disabled)
Dancing Links is a powerful algorithm that solves Exact Cover problems. These problems can lead to interesting algorithmic exercises such as the Pentominos problem, polyiamonds, tetrasticks, the N
queens, traveling knight, and other chessboard derivatives. However, Exact Cover problems aren t just theoretical mathematical puzzles though. They describe many real life problems: problems such as
making hotel room assignments given a group that has a many specific room requests, and organizing simple flight schedules for airports. Dancing Links is a backtracking, depth-first algorithm that
can find all possible solutions of the problem.
Dr. Donald Knuth and his lecture series: Computer Musings and his paper: Dancing Links
For his great work in Computer Algorithms and for discovering the art of Dancing Links.
Stanford University: [http://www.stanford.edu/]
For educational possibilities through excellent professors
Stanford Center for Professional Development [http://scpd.stanford.edu/scpd/default.htm]
For providing a link to Knuth s Lecture Series at: [http://scpd.stanford.edu/scpd/students/Dam_ui/pages/ArchivedVideoList56K.asp?Include=musings]
Wikipedia [http://en.wikipedia.org/wiki/Main_Page]
For providing basic information and useful links
Sudoku Programmer s Forum [http://www.setbb.com/phpbb/index.php]
For bringing together programmers to discuss various implementations of their programs
Stan Chesnutt [http://www.bluechromis.com:8080/stan/chesnutt.html]
For his example of his Java implementation of a Sudoku Solver using the Dancing Links algorithm
Ruud van der Werf and his program SudoCue [http://www.sudocue.net/]
For a simple visualization of the Dancing Links implementation and for his program to create random puzzles
Bob Hanson and his Sudoku Solver [http://www.stolaf.edu/people/hansonr/Sudoku/]
For the visualization of the sparse matrix implementation of the Dancing Links algorithm
The American Computer Science League (ACSL) [http://www.acsl.org/]
For providing the idea for this project
The Harker Research Symposium [http://web.harker.org/WiSTEM/]
For providing me the opportunity to explore my passion for Computer Science
And finally my teacher, Dave Feinberg
For pushing me to enter the symposium
© 2006
Document by: Jonathan Chu | {"url":"http://www.ocf.berkeley.edu/~jchu/publicportal/sudoku/sudoku.paper.html","timestamp":"2014-04-17T00:49:02Z","content_type":null,"content_length":"19481","record_id":"<urn:uuid:33d60577-f561-4d64-9037-c85959ad25aa>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00319-ip-10-147-4-33.ec2.internal.warc.gz"} |
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May 18, 2006
This Week's Finds in Mathematical Physics (Week 232)
John Baez
I'm at the Perimeter Institute now. It's great to see how it's developed since I first saw their new building back in 2004 (see "week208" for the story).
There's now a busy schedule of seminars and weekly colloquia, with string theorists and loop quantum gravity people coexisting happily. Their program of Superstring Quartets features some really hot
bands, like the Julliard and Emerson - unfortunately not playing while I'm here. The Black Hole Bistro serves elegant lunches and dinners, there are at least two espresso machines on each floor, and
my friend Eugenia Cheng will be happy to hear that they still have a piano available (after 6 pm).
But don't get the impression that it's overly sophisticated: there are also a couple of guys constantly playing foosball in the Feynman Lounge.
Since I'm here, I should talk about quantum gravity - so I will. But first, let's have the astronomy picture of the week.
This week it comes, not from outer space, but beneath the surface of the South Pole:
1) Steve Yunck / NSF, Cerenkov light passing through the IceCube neutrino detector, http://icecube.wisc.edu/gallery/detector_concepts/ceren_hires
This is an artist's impression of a huge neutrino observatory called "IceCube". (Maybe they left out the space here so the rap star of that name doesn't sue them for trademark infringement, or go
down there and shoot them.)
IceCube is being built in the beautifully clear 18,000-year old ice deep beneath the Amundsen-Scott South Pole Station. When a high-energy neutrino hits a water molecule, sometimes the collision
produces a muon zipping faster than the speed of light in ice. This in turn produces something like a sonic boom, but with light instead of sound. It's called "Cerenkov radiation", and it's the blue
light in the picture. This will be detected by an array of 5000 photomultiplier tubes - those gadgets hanging on electrical cables.
One thing the artist's impression doesn't show is that IceCube is amazingly large. The whole array is a cubic kilometer in size! It will encompass the already existing AMANDA detector, itself 10,000
meters tall, shown as a yellow cylinder here with a neutrino zipping through:
2) Darwin Rianto / NSF, Comparison of AMANDA and IceCube, http://icecube.wisc.edu/gallery/detector_concepts/icecubeencomp_300
Even the very top of IceCube is 1.4 kilometers beneath the snowy Antarctic surface, to minimize the effect of stray cosmic rays. The station on top looks like this - not very cozy, I'd say:
3) Robert G. Stokstad / NSF, South Pole Station, http://icecube.wisc.edu/gallery/antarctica/PC140287_300
I heard about IceCube from Adrian Burd, one of the old-timers who used to post a lot on sci.physics, a former cosmologist turned oceanographer who recently visited Antarctica as part of an NSF-run
field course. He ran into some people working on IceCube. It sounds like an interesting community down there! You can read about it in their newspaper, the Antarctic Sun. For example:
4) Ice Cube turns up the heat, The Antarctic Sun, January 29, 2006, http://antarcticsun.usap.gov/2005-2006/contentHandler.cfm?id=959
For more on IceCube and Amanda, these are fun to read:
5) Francis Halzen, Ice fishing for neutrinos, http://icecube.berkeley.edu/amanda/ice-fishing.html
6) Katie Yurkiewicz, Extreme neutrinos, Symmetry, volume 1 issue 1, November 2004, http://symmetrymagazine.org/cms/?pid=1000014
For some of AMANDA's results, including a map of the sky as seen in neutrinos, try this:
7) M. Ackermann et al, Search for extraterrestrial point sources of high energy neutrinos with AMANDA-II using data collected in 2000-2002, available as astro-ph/0412347.
For much more, try these:
8) AMANDA II Project, http://amanda.uci.edu/
9) Welcome to IceCube, http://icecube.wisc.edu/
And now, on to gravity.
You may have heard of the gravitational 3-body problem. Well, Richard Montgomery (famous from "week181") recently pointed out this movie of the 60-body problem:
10) Davide L. Ferrario, Periodic orbits for the 60-body problem, http://www.matapp.unimib.it/~ferrario/mov/index.html
60 equal masses do a complicated dance while always preserving icosahedral symmetry! First 12 groups of 5 swing past each other, then 20 groups of 3. If you want to know how he found these solutions,
read this:
11) Davide L. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. Math. 155 (2004), 305-362.
It's quite math-intensive - though just what you'd expect if you know this sort of thing: they use the G-equivariant topology of loop spaces, where G is the symmetry group in question (here the
icosahedral group), to prove the existence of action-minimizing loops with given symmetry properties.
Next, I'd like to say a little about point particles in 3d quantum gravity, and some recent work with Alissa Crans, Derek Wise and Alejandro Perez on string-like defects in 4d topological gravity:
12) John Baez, Derek Wise and Alissa Crans, Exotic statistics for strings in 4d BF theory, available as gr-qc/0603085.
13) John Baez and Alejandro Perez, Quantization of strings and branes coupled to BF theory, available as gr-qc/0605087.
(Jeffrey Morton is also involved in this project, a bit more on the n-category side of things, but that aspect is top secret for now.)
In "week222" I listed a bunch of cool papers on 3d quantum gravity, but I didn't really explain them. What we're trying to do now is generalize this work to higher dimensions. But first, let me start
by explaining the wonders of 3d quantum gravity.
The main wonder is that we actually understand it! The classical version of general relativity is exactly solvable when spacetime has dimension 3, and so is the quantum version. Most of the wonders I
want to discuss are already visible in the classical theory, where they are easier to understand, so I'll focus on the classical case.
A nice formulation of general relativity in 3 dimensions uses a "Lorentz connection" A and a "triad field" e. This is a gauge theory where the gauge group is SO(2,1), the Lorentz group for 3d
spacetime. If we're feeling lowbrow we can think of both A and e as so(2,1)-valued 1-forms on the 3-manifold M that describes spacetime. The action for this theory is:
∫[M] tr(e ^ F)
where F is the curvature of A. If you work out the equations of motion one of them says that F = 0, so our connection A is flat. The other, d[A]e = 0, says A is basically just the Levi-Civita
This is exactly what we want, because in the absence of matter, general relativity in 3 dimensions says spacetime is flat.
A fellow named Phillipp de Sousa Gerbert came up with an interesting way to couple point particles to this formulation of quantum gravity:
14) Phillipp de Sousa Gerbert, On spin and (quantum) gravity in 2+1 dimensions, Nuclear Physics B346 (1990), 440-472.
He actually did it for particles with spin, but I'll just do the spin-zero case.
The idea is to fix a 1-dimensional submanifold W in our 3-manifold M and think of it as the worldlines of some particles. Put so(2,1)-valued functions p and q on these worldlines - think of these as
giving the particles' momentum and position as a function of time.
Huh? Well, normally we think of position and momentum as vectors. In special relativity, "position" means "position in spacetime", and "momentum" means "energy-momentum". We can think of both of
these as vectors in Minkowksi spacetime. But in 3 dimensions, Minkowski spacetime is naturally identified with the Lorentz Lie algebra so(2,1). So, it makes sense to think of q and p as elements of
so(2,1) which vary from point to point along the particle's worldline.
To couple our point particles to gravity, we then add a term to the action like this:
S = ∫[M] tr(e ^ F) - ∫[W] tr((e + d[A]q) ^ p)
Now if you vary the e field you get a field equation saying that
F = p δ[W]
Here δ[W] is like the Dirac delta function of the worldline W; it's a distributional 2-form defined by requiring that
∫[W] X = ∫[M] (X ^ δ[W])
for any smooth 1-form X on W. This sort of "distributional differential form" is also called a "current", and you can read about them in the classic tome by Choquet-Bruhat et al. But the main point
is that the field equation
F = p δ[W]
says our connection on spacetime is flat except along the worldlines of our particles, where the curvature is a kind of "δ function". This is nice, because that's what we expect in 3d gravity: if you
have a particle, spacetime will be flat everywhere except right at the particle, where it will have a singularity like the tip of a cone.
A cone, you see, is intrinsically flat except at its tip: that's why you can curl paper into a cone without crinkling it!
So, our spacetime is flat except along the particles' worldlines, and there it's like a cone. The "deficit angle" of this cone - the angle of the slice you'd need to cut out to curl some paper into
this cone - is specified by the particle's momentum p.
Since delta functions are a bit scary, it's actually better to work with an "integrated" form of the equation
F = -p δ[W]
The integrated form says that if we parallel transport a little tangent vector around a little loop circling our particle's worldline, it gets rotated and/or Lorentz transformed by the element
in SO(2,1). This will be a rotation if the particle's momentum p is timelike, as it is for normal particles. Again, that's just as it should be: if you parallel transport a little arrow around a
massive particle in 3d gravity, it gets rotated!
If p is timelike, our particle is a tachyon and exp(p) is a Lorentz boost. And so on... we get the usual classification of particles corresponding to various choices of p:
There are other equations of motion, obtained by varying other fields, but all I want to note is the one you get by varying q:
d[A] p = 0
This says that the momentum p is covariantly constant along the particles' worldlines. So, momentum is conserved!
The really cool part is the relation between the Lie algebra element p and the group element exp(p). Originally we thought of p as momentum - but there's a sense in which exp(p) is the momentum that
really counts!
First, exp(p) is what we actually detect by parallel transporting a little arrow around our particle.
Second, suppose we let two particles collide and form a new one:
p p'
\ /
\ /
\ /
Now our worldlines don't form a submanifold anymore, but if we keep our wits about us, we can see that everything still makes sense, and we get momentum conservation in this form:
exp(p") = exp(p) exp(p')
since little loops going around the two incoming particles can fuse to form a loop going around the outgoing particle. Note that we're getting conservation of the group-valued momentum, not the
Lie-algebra-valued momentum - we don't have
p" = p + p'
So, conservation of energy-momentum is getting modified by gravitational effects! This goes by the name of "doubly special relativity":
15) Laurent Freidel, Jerzy Kowalski-Glikman and Lee Smolin, 2+1 gravity and doubly special relativity, Phys. Rev. D69 (2004) 044001. Also available as hep-th/0307085.
This effect is a bit less shocking if we put the units back in. I've secretly been setting 4πG = 1, where G is Newton's gravitational constant. If we put that constant back in - let's call it k
instead of 4πG - we get
exp(kp") = exp(kp) exp(kp')
or if you expand things out:
p" = p + p' + (k/2) [p,p'] + terms of order k^2 and higher...
So, as long as the momenta are small compared to the Planck mass, the usual law of conservation of momentum
p" = p + p'
almost holds! But, for large momenta this law breaks down - we must think of momentum as group-valued if we want it to be conserved!
I think this is incredibly cool: as we turn on gravity, the usual "flat" momentum space curls up into a group, and we need to multiply momenta in this group, instead of add them in the Lie algebra.
We can think of this group has having a "radius" of 1/k, so it's really big and almost flat when the strength of gravity is small. In this limit, multiplication in the group reduces to addition in
the Lie algebra.
I should point out that this effect is purely classical! It's still there when we quantize the theory, but it only depends on the gravitational constant, not Planck's constant. Indeed, in 3d quantum
gravity, we can build a unit of mass using just G and c: we don't need ħ. This unit is the mass that curls space into an infinitely skinny cone! It would be a bit misleading to call it "Planck mass",
but it's the maximum possible mass. Any mass bigger than this acts like a negative mass. That's because the corresponding group-valued momenta "wrap around" in the group SO(2,1).
We also get another cool effect - exotic statistics. In the absence of gravitational or quantum effects, when you switch two particles, you just switch their momenta:
(p, p') |→ (p', p)
But in 3d gravity, you can think of this process of switching particles as a braid:
\ /
\ /
/ \
/ \
and if you work out what happens to their group-valued momenta, say
g = exp(kp)
g' = exp(kp')
it turns out that one momentum gets conjugated by the other:
(g, g') |→ (gg'g^-1, g)
To see this, remember that we get these group elements by doing parallel transport around loops that circle our particles. When we move our particles, the loops get dragged along, like this:
Note that the left-hand red loop moves until it looks just like the right one did initially, but the right-hand one gets wrapped around the left one. If you ponder this carefully, and you know some
math, you can see it yields this:
(g, g') |→ (gg'g^-1, g)
So, the process of braiding two particles around each other has a nontrivial effect on their momenta. In particular, if you braid two particles around other twice they don't wind up in their original
Thus, our particles are neither bosons nor fermions, but "nonabelian anyons" - the process of switching them is governed not by the permutation group, but by the braid group. But again, if you expand
things out in powers of k you'll see this effect is only noticeable for large momenta:
(p, p') |→ (p' + k[p,p'] + higher order terms..., p)
Summarizing, we see quantum gravity is lots of fun in 3 dimensions: it's easy to introduce point particles, and they have group-valued momentum, which gives rise to doubly special relativity and
braid group statistics.
Now, what happens when we go from 3 dimensions to 4 dimensions?
Well, we can write down the same sort of theory:
S = ∫[M] tr(B ^ F) - ∫[W] tr((B + d[A]q) ^ p)
The only visible difference is that what I'd been calling "e" is now called "B", so you can see why folks call this "BF theory".
But more importantly, now M is an 4-dimensional spacetime and W is an 2-dimensional "worldsheet". A is again a Lorentz connection, which we can think of as an so(3,1)-valued 1-form. B is an so(3,1)
-valued 2-form. p is an so(3,1)-valued function on the worldsheet W. q is an so(3,1)-valued 1-form on W.
So, only a few numbers have changed... so everything works very similarly! The big difference is that instead of spacetime having a conical singularity along the worldline of a particle, now it's
singular along the worldsheet of a string. When I call it a "string", I'm not trying to say it behaves like the ones they think about in string theory - at least superficially, it's a different sort
of theory, a purely topological theory. But, we've got these closed loops that move around, split and join, and trace out surfaces in spacetime.
They can also braid around each other in topologically nontrivial ways, as shown in this "movie":
(By the way, all the math pictures this week were drawn by Derek for our paper.)
So, we get exotic statistics as before, but now they are governed not by the braid group but by the "loop braid group", which keeps track of all the ways we can move a bunch of circles around in 3d
space. Let's take our spacetime M to be R^4, to keep things simple. Then our circles can move around in R^3... and there are two basic ways we can switch two of them: move them around each other, or
pass one through the other, like this:
If we just move them around each other, they might as well have been point particles: we get a copy of the permutation group, and all we see are ordinary statistics. But when we consider all the ways
of passing them through each other, we get a copy of the braid group!
When we allow ourselves both motions, we get a group called the "loop braid group" or "braid permutation group" - and one thing Alissa Derek and I did was to get a presentation of this group. This is
an example of a "motion group": just as the motion group of point particles in the plane is the braid group, and motion group of point particles in R^3 is the permutation group, the motion group of
strings in R^3 is the loop braid group.
As before, our strings have group-valued momenta: we can get an element of the Lorentz group SO(3,1) by parallel transporting a little tangent vector around a string. And, we can see how different
ways of switching our strings affect the momenta. When we move two strings around each other, their momenta switch in the usual way:
(g, g') |→ (g', g)
but when we move one through the other, one momentum gets conjugated by the other:
(g, g') |→ (gg'g^-1, g)
So, we have exotic statistics, but you can only notice them if you can pass one string through another!
In the paper with Alejandro, we go further and begin the project of quantizing these funny strings, using ideas from loop quantum gravity. Loop quantum gravity has its share of problems, but it works
perfectly well for 3d quantum gravity, and matches the spin foam picture of this theory. People have sort of believed this for a long time, but Alejandro demonstrated this quite carefully in a recent
paper with Karim Noui:
15) Karim Noui and Alejandro Perez, Dynamics of loop quantum gravity and spin foam models in three dimensions, to appear in the proceedings of the Third International Symposium on Quantum Theory and
Symmetries (QTS3), available as gr-qc/0402112.
The reason everything works so nicely is that the equations of motion say the connection is flat. Since the same is true in BF theory in higher dimensions, we expect that the loop quantization and
spin foam quantization of the theory I'm talking about now should also work well.
We find that we get a Hilbert space with a basis of "string spin networks", meaning spin networks that can have loose ends on the stringy defects.
So, there's some weird blend of loop quantum gravity and strings going on here - but I don't really understand the relation to ordinary string theory, if any. It's possible that I can get a
topological string theory (some sort of well-defined mathematical gadget) which describes these stringy defects, and that would be quite interesting.
But, I spoke about this today at the Perimeter Institute, and Malcolm Perry said that instead of "strings" I should call these guys (n-2)-branes, because the connection has conical singularities on
them, "which is what one would expect for any respectable (n-2)-brane".
I will talk to him more about this and try to pick his, umm, branes. In fact I took my very first GR course from him, back when he was a postdoc at Princeton and I was a measly undergraduate. I was
too scared to ask him many questions then. I'm a bit less scared now, but I've still got a lot to learn. Tomorrow he's giving a talk about this:
17) David S. Berman, Malcolm J. Perry, M-theory and the string genus expansion, Phys. Lett. B635 (2006) 131-135. Also available as hep-th/0601141.
Addenda: Here's an email from Greg Egan, and my reply:
John Baez wrote:
> The really cool part is the relation between the Lie algebra
> element p and the group element exp(p). Originally we thought
> of p as momentum - but there's a sense in which exp(p) is the
> momentum that really counts!
Would it be correct to assume that the ordinary tangent vector p still transforms in the usual way? In other words, suppose I'm living in a 2+1 dimensional universe, and there's a point particle
with rest mass m and hence energy-momentum vector in its rest frame of p=me[0]. If I cross its world line with a certain relative velocity, there's an element g of SO(2,1) which tells me how to
map the particle's tangent space to my own. Would I measure the particle's energy-momentum to be p'=gp? (e.g. if I used the particle to do work in my own rest frame) Would there still be no upper
bound on the total energy, i.e. by making our relative velocity close enough to c, I could measure the particle's kinetic energy to be as high as I wished?
I guess I'm trying to clarify whether the usual Lorentz transformation of the tangent space has somehow been completely invalidated for extreme boosts, or whether it's just a matter of there
being a second definition of "momentum" (defined in terms of the Hamiltonian) which transforms differently and is the appropriate thing to consider in gravitational contexts.
In other words, does the cut-off mass apply only to the deficit angle, and do boosts still allow me to measure (by non-gravitational means) arbitrarily large energies (at least in the classical
I replied:
Greg Egan wrote:
>John Baez wrote:
>>The really cool part is the relation between the Lie algebra
>>element p and the group element exp(p). Originally we thought
>>of p as momentum - but there's a sense in which exp(p) is the
>>momentum that really counts!
>Would it be correct to assume that the ordinary tangent vector p
>still transforms in the usual way?
Hi! Yes, it would.
>In other words, suppose I'm living in a 2+1 dimensional universe,
>and there's a point particle with rest mass m and hence
>energy-momentum vector in its rest frame of p=m e_0. If I
>cross its world line with a certain relative velocity, there's
>an element g of SO(2,1) which tells me how to map the particle's
>tangent space to my own. Would I measure the particle's
>energy-momentum to be p'=gp? (e.g. if I used the particle to
>do work in my own rest frame) Would there still be no upper
>bound on the total energy, i.e. by making our relative velocity
>close enough to c, I could measure the particle's kinetic energy
>to be as high as I wished?
To understand this, it's good to think of the momenta as elements of the Lie algebra so(2,1) - it's crucial to the game.
Then, if you have momentum p, and I zip past you, so you appear transformed by some element g of the Lorentz group SO(2,1), I'll see your momentum as
p' = g p g^-1
This is just another way of writing the usual formula for Lorentz transforms in 3d Minkowski space. No new physics so far, just a clever mathematical formalism.
But when we turn on gravity, letting Newton's constant k be nonzero, we should instead think of momentum as group-valued, via
h = exp(kp)
and similarly
h' = exp(kp')
Different choices of p now map to the same choice of h. In particular, a particle of a certain large mass - the Planck mass- will turn out to act just like a particle of zero mass!
So, if we agree to work with h instead of p, we are now doing new physics. This is even more obvious when we decide to multiply momenta instead of adding them, since multiplication in SO(2,1) is
But, if we transform our group-valued momentum in the correct way:
h' = ghg^-1
this will be completely compatible with our previous transformation law for vector-valued momentum!
>I guess I'm trying to clarify whether the usual Lorentz transformation
>of the tangent space has somehow been completely invalidated for
>extreme boosts, or whether it's just a matter of there being a second
>definition of "momentum" (defined in terms of the Hamiltonian) which
>transforms differently and is the appropriate thing to consider in
>gravitational contexts.
Good question! Amazingly, the usual Lorentz transformations still work EXACTLY - even though the rule for adding momentum is new (now it's multiplication in the group). We're just taking exp(kp)
instead of p as the "physical" aspect of momentum.
This effectively puts an upper limit on mass, since as we keep increasing the mass of a particle, eventually it "loops around" SO(2,1) and act exactly like a particle of zero mass.
But, it doesn't exactly put an upper bound on energy-momentum, since SO(2,1) is noncompact. Of course energy and momentum don't take real values anymore, so one must be a bit careful with this
"upper bound" talk.
>In other words, does the cut-off mass apply only to the deficit
>angle, and do boosts still allow me to measure (by non-gravitational
>means) arbitrarily large energies (at least in the classical theory)?
There's some sense in which energy-momenta can be arbitrarily large. That's because the space of energy-momenta, namely SO(2,1), is noncompact. Maybe you can figure out some more intuitive way to
express this.
I was sitting in a chair in the patent office in Bern when all of a sudden a thought occurred to me. If a person falls freely, he will not feel his own weight. - Albert Einstein
© 2006 John Baez | {"url":"http://math.ucr.edu/home/baez/week232.html","timestamp":"2014-04-23T17:46:58Z","content_type":null,"content_length":"30607","record_id":"<urn:uuid:cad779f7-c4d4-4670-a6dd-fcd512643c19>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00295-ip-10-147-4-33.ec2.internal.warc.gz"} |
Evgeny Dantsin
Primary research area: computational complexity with a focus on the complexity of the Boolean satisfiability problem. Selected publications.
Undergraduate and graduate courses on theory of computation, the design and analysis of algorithms, discrete mathematics, databases, information retrieval. Spring 2014.
Education and Experience
Ph.D. in Mathematics from Steklov Institute of Mathematics, Russia.
M.S. in Mathematics from St. Petersburg State University, Russia.
I joined Department of Computer Science of Roosevelt University in 2001. My previous experience includes work in IT industry and academic positions in:
• Laboratory of Mathematical Logic, Steklov Institute of Mathematics, St. Petersburg, Russia, 1993-2001;
• Computing Science Department, Uppsala University, Sweden, 1996-1999;
• School of Computer Science, The University of Manchester, England, 1999-2001. | {"url":"http://cs.roosevelt.edu/~dantsin/","timestamp":"2014-04-18T15:37:44Z","content_type":null,"content_length":"3693","record_id":"<urn:uuid:6207a1b6-b36f-462a-8d84-dd3e38171b1d>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00071-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Comonad.Reader
Thu 9 Nov 2006
Posted by Edward Kmett under
Monads [7] Comments
If we take a look at the Haskell (.) operator:
(.) :: (a -> b) -> (e -> a) -> e -> b
and take a moment to reflect on the type of fmap
fmap :: Functor f => (a -> b) -> f a -> f b
and the unnamed Reader monad from Control.Monad.Reader
instance Functor ((->) r)
we see that fmap applied to the Reader functor rederives (.).
fmap_reader :: (a -> b) -> (e -> a) -> e -> b
So if we were willing to forgo ease of learning, and to bake in the Reader monad as a primitive, we could quite concisely redefine (.) to give it a more general signature:
module Dot where
import Control.Monad.Reader
import Prelude hiding ((.))
infixr 0 .
(.) :: Functor f => (a -> b) -> f a -> f b
(.) = fmap
In this context, existing code continues to type check. For instance,
((+2) . (*3)) 5 ==> 17
And the . above doubles as filling the role of the * map operator mentioned in Richard Bird's 1990 Calculus of Functions paper generalized to any Functor.
((+2) . (*3)) . [1..10] ==> [5,8,..32]
((+2) . (*3)) . Just 5 ==> Just 17
((+2) . (*3)) . Nothing ==> Nothing
I was able to test this with the just about every example golfed back and forth on the #haskell channel in the last 6 months.
I'm not advocating this as a practice for Haskell as it is somewhat terrifying to think of how to teach to new programmers, but I found the exercise to be enlightening.
7 Responses to “Generalizing (.)”
1. Edward Kmett Says:
November 9th, 2006 at 1:22 am
As an aside, the idea came from observing the fact that lambdabot’s pointfree conversion command @pl had taken to using `fmap` instead of (.) in a lot of places.
2. mgsloan Says:
February 1st, 2007 at 2:14 am
This was added to The Other Prelude (perhaps one of the reasons it was created).
Probably inspired by your irc golfings.
3. mgsloan Says:
February 1st, 2007 at 2:15 am
PS, I don’t think it’d be too hard for noobs to learn it. When I was a noob I wondered why this didn’t work!
4. Fritz Says:
July 24th, 2007 at 2:58 am
I think you need doubled parens when hiding (.) in the import (the outer ones are part of the hiding syntax, the inner ones refer to the infix composition symbol in parens for lexical reasons).
Both hugs and ghci seem to like it this way. (Perhaps something in your text chain snarfed the “redundant” parens?)
5. Edward Kmett Says:
July 24th, 2007 at 10:05 am
Thanks. Fixed.
I honestly, just typed that off the cuff.
Thats what I get for not actually compiling it before posting ;)
6. Paul Keir Says:
March 11th, 2011 at 7:44 am
Another great post and really neat idea. I tried
((+2) . (*3)) 5
and also
((+2) . (*3)) $ 5
but this
(+2) . (*3) $ 5
complains about a missing Num instance for the literal ‘5′. It makes me wonder why the 5 is accepted in any case, as it’s not (here at least) an instance of Functor?
7. Edward Kmett Says:
March 11th, 2011 at 3:02 pm
Back when I wrote this post I gave the updated (.) the wrong precedence, it should be
infixr 9 . | {"url":"http://comonad.com/reader/2006/generalizing-dot/","timestamp":"2014-04-19T04:57:03Z","content_type":null,"content_length":"26678","record_id":"<urn:uuid:5e05172f-35b4-47f6-b606-299a64ef8af4>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00065-ip-10-147-4-33.ec2.internal.warc.gz"} |
Minimization Problem.
November 10th 2010, 10:46 AM
Minimization Problem.
Problem A
What is the cheapest soup can that will hold 75in3 (cubic inches) of soup? Assume the can is cylindrical in shape and that the total cost of a can is proportional to the amount of material (tin)
measured as area (in other words, don’t worry about the thickness of the tin). Be sure to provide both the diameter (or radius) and height of the can and sketch what this cheapest can looks like!
Problem B
Repeat the above problem but for the case where, for whatever reason, the cost of manufacturing the two lids of the can is three times higher than manufacturing the sidewall. Give your answer in
the same form as above, including the sketch.
November 10th 2010, 10:53 AM
The volume of a cylinder is given by:
$V = \pi r^2h$
The surface area is given by:
$A = 2\pi r^2 + 2\pi rh$
Find the derivative of A, using the volume given to express either the radius in terms of height or vice versa.
Then, substitute back the radius or height in the volume formula to get the other dimension.
Can you do this part first?
November 10th 2010, 11:33 AM
You mean 4pir+pirh?
November 10th 2010, 08:11 PM
What do you mean? For the surface area of the cylinder?
No, I meant what I meant.
The area of a circile is $\pi r^2$
In a cylinder, there are 2 circles for two cross sections, hence $2 \pi r^2$
Then, the curved surface, if opened has the length h and the width equal to the circumference of a circle, which gives: $2\pi r h$
November 11th 2010, 06:03 AM
Im totally lost please help me.
November 11th 2010, 08:58 AM
What do you not understand?
November 11th 2010, 09:15 AM
You want me to find the der of A = 2 pi r^2 + 2 pi r h?
November 11th 2010, 09:22 AM
You want to have a volume of 75 in^3 in a can having the lowest possible surface area so that the cost of the can is minimised yes?
Then, you need to know how does this area varies as r and h vary, while keeping the volume constant.
Hence, you are looking for the minimum point of the graph showing how A varies with either h or r.
November 11th 2010, 11:41 AM
Can you please show me the steps.. i really am bad at match i need help need to submit this...
November 11th 2010, 07:48 PM
As I told you in my first post in this thread, the volume, V, of a cylinder is given by
$V = \pi r^2h$
and the area of the same cylinder is given by:
$A = 2\pi r^2 + 2\pi rh$
We know that the volume is 75 in^3. We are looking for the minimum A while r or h vary. Let's look for the value of r, the radius of the circular cross section of the cylindrical can.
So, from the information we got, we have:
$V = 75\ in^3$
$\pi r^2h = 75$
$h = \dfrac{75}{\pi r^2}$
In the Area equation, we get:
$A = 2\pi r^2 + 2\pi rh$
Substitute h by what we just got;
$A = 2\pi r^2 + 2\pi r\left(\dfrac{75}{\pi r^2}\right)$
Can you simplify this and find it's derivative? | {"url":"http://mathhelpforum.com/calculus/162780-minimization-problem-print.html","timestamp":"2014-04-20T02:15:07Z","content_type":null,"content_length":"10255","record_id":"<urn:uuid:2da6ff94-a71d-4f5a-9c08-ef8d9954f86f>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00590-ip-10-147-4-33.ec2.internal.warc.gz"} |
Functional Cells : A Spreadsheet in F#
Live code is available at https://bitbucket.org/ptrelford/spreadsheet-developerfusion/src
Functional programming is beginning to hit the mainstream in Java and .Net. The Java community has seen significant growth in languages like Clojure and Scala, while the use of LINQ, a feature based
heavily on functional programming concepts, in C# and VB.NET is now common place. And of course, Visual Studio 2010 now ships with the F# programming language - the first brand-new “out of the box”
language to the .NET ecosystem since its introduction of C# back in 2001.
If you’ve ever written formulas in a spreadsheet then you’re already familiar with some of the principles of functional programming. The key building blocks of a spreadsheet are cells containing
functions and values. To demonstrate functional programming at large, this article will be looking at the implementation of a functional spreadsheet written in F# and bound to the DataGrid control,
as shown in Figure 1.
Figure 1 : The spreadsheet application running in the browser from http://fssnip.net/4v
There are two parts to this application which we'll look at in turn.
1. A parser to interpret any formula entered into a cell.
2. A simple object model representing the sheets, rows and cell in a spreadsheet.
When we’re happy they are working we can use data binding from a DataGrid to the Object Model to view the cells. For this we can use either WPF or Silverlight.
To follow this article, you’ll need to install F#. If you have a non-Express version of Visual Studio 2010 Professional, you already have F# installed. If not and you run Windows, you can download
the Visual Studio 2010 Shell for free and install F# into it. The F# Developer center has the details. If you’re on Linux or Mac then you can use F# with Mono 2.6/2.8 with MonoDevelop or Emacs. You
can even edit F# code in your browser here.
Note that should you wish to try out any of the F# code fragments below in Visual Studio, just type the code into an open F# file, highlight it and press ALT + ENTER to have it run inside Visual
Studio’s F# Interactive window.
Parsing Formulae In A Spreadsheet Cell
Any cell in a spreadsheet may contain a formula to derive its value from the values of other cells in the spreadsheet instead of a simple value of its own. For example,
In order to interpret these formulae correctly, we’ll need to create a parser that:
• Takes a formula and splits it into its component parts: strings, operators, cell references and integers. We'll refer to these collectively as tokens and the process of conversion as tokenization
• Parses the sequence of tokens to build an Abstract Syntax Tree for the cell’s formula, that can later be evaluated
We’ll look at these in order.
The first step in the tokenization process is to define a set of possible tokens. In addition to the four types mentioned above - strings, operators, cell references and integers - we also include
whitespace and single symbol characters ($, % etc) and define the six types using a Discriminated Union. (For C# users, this works like a C# class hierarchy where token is an abstract base class.)
type token =
| WhiteSpace
| Symbol of char
| OpToken of string
| RefToken of int * int
| StrToken of string
| NumToken of decimal
Our object model will reference a cell by column and row numbers, so we'll also need a function to convert cell references, e.g. A1, C3, into column and row numbers. It's called toRef.
let toRef (s:string) =
let col = int s.[0] - int 'A'
let row = s.Substring 1 |> int
col, row-1
With those two items in place, we can use regular expressions to identify the tokens in a formula. To implement this, we'll define an Active Pattern (a function that can be used actively inside a
pattern matching expression) for matching regular expressions and a function called toToken to map regular expressions to tokens.
let (|Match|_|) pattern input =
let m = System.Text.RegularExpressions.Regex.Match(input, pattern)
if m.Success then Some m.Value else None
let toToken = function
| Match @"^\s+" s -> s, WhiteSpace
| Match @"^\+|^\-|^\*|^\/" s -> s, OpToken s
| Match @"^=|^<>|^<=|^>=|^>|^<" s -> s, OpToken s
| Match @"^\(|^\)|^\,|^\:" s -> s, Symbol s.[0]
| Match @"^[A-Z]\d+" s -> s, s |> toRef |> RefToken
| Match @"^[A-Za-z]+" s -> s, StrToken s
| Match @"^\d+(\.\d+)?|\.\d+" s -> s, s |> decimal |> NumToken
| _ -> invalidOp ""
Note that toToken only expects to match one prospective token at a time in order to categorize it. So we also need to define a function we'll call tokenize that works through the whole formula string
in a cell as input and uses toToken to return a List of tokens:
let tokenize s =
let rec tokenize' index (s:string) =
if index = s.Length then []
let next = s.Substring index
let text, token = toToken next
token :: tokenize' (index + text.Length) s
tokenize' 0 s
|> List.choose (function WhiteSpace -> None | t -> Some t)
The tokenize function invokes the inner recursive function tokenize’, marked with the rec keyword, before choosing all tokens from the list which are not whitespace. The recursive tokenize’ function
repeatedly tries to match the front of the string against a token until the end of the string is reached.
Now we can tokenize a formula, let’s try it in F# interactive. Select all the code written so far and click Alt+Enter. F# interactive will open and you can now call tokenize directly:
> tokenize "C9+10";;
val it : token list = [RefToken (2,8); OpToken "+"; NumToken 10M]
> tokenize "(1+1)=2";;
val it : token list =
[Symbol '('; NumToken 1M; OpToken "+"; NumToken 1M; Symbol ')'; OpToken "="; NumToken 2M]
Token Parsing
The second half of the formula parser takes the list of tokens, turn them into a formula it can understand and then evaluates it. Just like the tokenization code, we begin by defining the allowed
elements of a formula. They are:
• Negation operator
• The arithmetic operators for addition, subtraction, multiplication and division
• The logical comparison operators equal to, less than, greater than, less than or equal to, greater than or equal to, and not equal to.
• A cell reference (as row and column number)
• A range of cells (as row and column numbers specifying the range’s start and end)
• A function being applied to part of the formula.
We can use three discriminated unions to define these elements as follows:
type arithmeticOp = Add | Sub | Mul | Div
type logicalOp = Eq | Lt | Gt | Le | Ge | Ne
type formula =
| Neg of formula
| ArithmeticOp of formula * arithmeticOp * formula
| LogicalOp of formula * logicalOp * formula
| Num of decimal
| Ref of int * int
| Range of int * int * int * int
| Fun of string * formula list
With these in place, we can combine several mutually recursive functions starting with Term to define a Recursive Descent Parser for each of these operators. The code below shows a cut down version
of this which deals with just addition and subtraction along with the parse function which passes the original set of tokens into Term. You can find the other five functions – Factor, Product, Atom,
Tuple and Params - within the parser in the code download.
let rec (|Term|_|) = function
| Number(f1, OpToken "+"::Term(f2,t)) -> Some(ArithmeticOp(f1,Add,f2), t)
| Number(f,t) -> Some(f,t)
| _ -> None
and (|Number|_|) = function
| NumToken(n)::t -> Some(Num(n),t)
| _ -> None
// Full parser in download
let parse s =
tokenize s |> function
| Term(e,[]) -> e
| _ -> failwith "Failed to parse formula"
Running in F# Interactive:
> parse "1";;
val it : formula = Num 1M
> parse "1+1";;
val it : formula = ArithmeticOp (Num 1M,Add,Num 1M)
> parse "1+1+1";;
val it : formula = ArithmeticOp (Num 1M,Add,ArithmeticOp (Num 1M,Add,Num 1M))
Formula Evaluation
Now we can parse formulae into a format we can act upon and therefore evaluate, we need to write the function that does the evaluation. Happily this is quite straightforward, using a recursive
function to work through the list of arithmetic and logical operations tokens and doing the relevant calculation: addition for Add, subtraction for Sub etc.
The following evaluate function computes any specified formula, using a specified valueAt function to resolve the values of other cells:
let evaluate (valueAt:int * int -> string) formula =
let rec eval = function
| Neg f -> - (eval f)
| ArithmeticOp(f1,op,f2) -> arithmetic op (eval f1) (eval f2)
| LogicalOp(f1,op,f2) -> if logic op (eval f1) (eval f2) then 0.0M else -1.0M
| Num d -> d
| Ref(x,y) -> valueAt(x,y) |> decimal
| Range _ -> invalidOp "Expected in function"
| Fun("SUM",ps) -> ps |> evalAll |> List.sum
| Fun("IF",[condition;f1;f2]) ->
if (eval condition)=0.0M then eval f1 else eval f2
| Fun(_,_) -> failwith "Unknown function"
and arithmetic = function
| Add -> (+) | Sub -> (-) | Mul -> (*) | Div -> (/)
and logic = function
| Eq -> (=) | Ne -> (<>)
| Lt -> (<) | Gt -> (>)
| Le -> (<=) | Ge -> (>=)
and evalAll ps =
ps |> List.collect (function
| Range(x1,y1,x2,y2) ->
[for x=x1 to x2 do for y=y1 to y2 do yield valueAt(x,y) |> decimal]
| x -> [eval x]
eval formula
Again we can use F# interactive to test this:
> let eval s = s |> parse |> evaluate (fun (x,y) -> 0);;
val eval : string -> decimal
> eval "1+1";;
val it : decimal = 2M
Finally, in order to propagate changes between cells in a sheet we need to resolve the other cells that a particular formula references:
let references formula =
let rec traverse = function
| Ref(x,y) -> [x,y]
| Range(x1,y1,x2,y2) ->
[for x=x1 to x2 do for y=y1 to y2 do yield x,y]
| Fun(_,ps) -> ps |> List.collect traverse
| ArithmeticOp(f1,_,f2) | LogicalOp(f1,_,f2) ->
traverse f1 @ traverse f2
| _ -> []
traverse formula
And in F# interactive:
> Range(1,1,3,1) |> references;;
val it : (int * int) list = [(1, 1); (2, 1); (3, 1)]
The Object Model
A Spreadsheet comprises sheets that are composed from columns and rows. Rows are composed from cells. Finally the cells contain data in the form of either literals or formulas which are in turn
evaluated and displayed as values. Or in UML:
This hierarchy can be expressed using F# classes:
type Cell (x,y) =
let mutable data = ""
member cell.Data
with get () = data
and set text = data <- text
member cell.Value = data |> parse |> eval
type Row (index,columnCount) =
let cells = Array.init columnCount (fun x -> Cell(x,index))
member row.Cells = cells
member row.Index = index+1
type Sheet (columnCount, rowCount) =
let columns = Array.init columnCount (fun i -> ('A' + char i))
let rows = Array.init rowCount (fun i -> Row(i, columnCount))
member sheet.Columns = columns
member sheet.Rows = rows
However this is not enough on its own. When a cell’s value changes all cells that reference that cell must also be updated. To achieve this we can implement an Update event to signal a change to a
cell’s value. The following code implements an Updated notification:
type Cell (x,y) =
let mutable data = ""
let mutable value = ""
let updated = Event<_>()
member cell.Data
with get () = data
and set text =
data <- text
updated.Trigger value
member cell.Value = data |> parse |> eval
member cell.Updated = updated.Publish
Then by using the references function we defined earlier, it is possible for a cell to subscribe to updates from all the cells it references.
The Object Model is defined as a set of mutually recursive types where cells contain both raw text data and an implied value. Dependent cells are notified of updates via the Updated event:
type Cell (sheet:Sheet) as cell =
inherit ObservableObject()
let mutable value = ""
let mutable data = ""
let mutable formula : formula option = None
let updated = Event<_>()
let mutable subscriptions : System.IDisposable list = []
let cellAt(x,y) =
let (row : Row) = Array.get sheet.Rows y
let (cell : Cell) = Array.get row.Cells x
let valueAt address = (cellAt address).Value
let eval formula =
try (evaluate valueAt formula).ToString()
with _ -> "N/A"
let parseFormula (text:string) =
if text.StartsWith "="
try true, parse (text.Substring 1) |> Some
with _ -> true, None
else false, None
let update newValue generation =
if newValue <> value then
value <- newValue
updated.Trigger generation
cell.Notify "Value"
let unsubscribe () =
subscriptions |> List.iter (fun d -> d.Dispose())
subscriptions <- []
let subscribe formula addresses =
let remember x = subscriptions <- x :: subscriptions
for address in addresses do
let cell' : Cell = cellAt address
|> Observable.subscribe (fun generation ->
if generation < sheet.MaxGeneration then
let newValue = eval formula
update newValue (generation+1)
) |> remember
member cell.Data
with get () = data
and set (text:string) =
data <- text
cell.Notify "Data"
let isFormula, newFormula = parseFormula text
formula <- newFormula
formula |> Option.iter (fun f -> references f |> subscribe f)
let newValue =
match isFormula, formula with
| _, Some f -> eval f
| true, _ -> "N/A"
| _, None -> text
update newValue 0
member cell.Value = value
member cell.Updated = updated.Publish
Rows are constructed from cells:
and Row (index,colCount,sheet) =
let cells = Array.init colCount (fun i -> Cell(sheet))
member row.Cells = cells
member row.Index = index
Sheets are constructed from of columns rows :
and Sheet (colCount,rowCount) as sheet =
let cols = Array.init colCount (fun i -> string (int 'A' + i |> char))
let rows = Array.init rowCount (fun index -> Row(index+1,colCount,sheet))
member sheet.Columns = cols
member sheet.Rows = rows
member sheet.MaxGeneration = 1000
To signal changes in cells to WPF or Silverlight we implement the INotifyPropertyChanged interface:
and ObservableObject() =
let propertyChanged =
member this.Notify name =
propertyChanged.Trigger(this,PropertyChangedEventArgs name)
interface INotifyPropertyChanged with
member this.PropertyChanged = propertyChanged.Publish
Referencing WPF 4.0 Assemblies (64-bit Windows)
Assembly reference can be added to the project, alternatively if you are running with F# interactive they can be specified using the #r directive:
#r @"C:\Program Files (x86)\Reference Assemblies\Microsoft\Framework\.NETFramework\v4.0\PresentationCore.dll"
#r @"C:\Program Files (x86)\Reference Assemblies\Microsoft\Framework\.NETFramework\v4.0\PresentationFramework.dll"
#r @"C:\Program Files (x86)\Reference Assemblies\Microsoft\Framework\.NETFramework\v4.0\System.Xaml.dll"
#r @"C:\Program Files (x86)\Reference Assemblies\Microsoft\Framework\.NETFramework\v4.0\WindowsBase.dll"
Now we can use data binding to show a Sheet on a DataGrid with WPF using Data Binding. First we need a function to create a grid template column that binds to a column’s cells:
open System.Windows
open System.Windows.Controls
open System.Windows.Data
open System.Windows.Markup
let createGridColumn i =
let header = 'A' + char i
let col = DataGridTemplateColumn(Header=header, Width=DataGridLength(64.0))
let toDataTemplate s =
let ns = "http://schemas.microsoft.com/winfx/2006/xaml/presentation"
sprintf "%s" ns s
|> XamlReader.Parse :?> DataTemplate
let path = sprintf "Cells.[%d]" i
col.CellTemplate <-
sprintf "" path
|> toDataTemplate
col.CellEditingTemplate <-
sprintf "" path
|> toDataTemplate
Next we need a function to create the grid and bind the sheet’s column’s and rows to it:
let createGrid (sheet:Sheet) =
let grid = DataGrid(AutoGenerateColumns=false,HeadersVisibility=DataGridHeadersVisibility.All)
for i = 0 to sheet.ColCount-1 do createGridColumn i |> grid.Columns.Add
grid.LoadingRow.Add (fun e ->
let row = e.Row.DataContext :?> Row
e.Row.Header <- row.Index
grid.ItemsSource <- sheet.Rows
Finally we create a sheet and a Window, setting the content of the window as a grid bound to the sheet:
let sheet = Sheet(26,30)
let win = new Window(Title="Spreadsheet", Content=createGrid sheet)
do win.ShowDialog()
When you run the code you should now see a functioning Spreadsheet:
What Next?
Why not try extending the Spreadsheet script with:
• your own functions (inside the evaluate function)
• support for Dates and Times
• or even units of measure
• support for copy and paste to and from Excel
• or even support import and export of Excel files
To see features like these implemented checkout the Cellz project on CodePlex.
Read more about Functional Programming in .Net with Real-World Functional Programming with examples in C# & F# | {"url":"http://www.developerfusion.com/article/123830/functional-cells-a-spreadsheet-in-f/","timestamp":"2014-04-20T06:28:44Z","content_type":null,"content_length":"38213","record_id":"<urn:uuid:8c669c01-c632-44d3-a709-c84503203978>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00515-ip-10-147-4-33.ec2.internal.warc.gz"} |
Not Positive Definite Matrices--Causes and Cures
The seminal work on dealing with not positive definite matrices is Wothke (1993). The chapter is both reabable and comprehensive. This page uses ideas from Wothke, from SEMNET messages, and from my
own experience.
The Problem
There are four situations in which a researcher may get a message about a matrix being "not positive definite." The four situations can be very different in terms of their causes and cures.
First, the researcher may get a message saying that the input covariance or correlation matrix being analyzed is "not positive definite." Generalized least squares (GLS) estimation requires that the
covariance or correlation matrix analyzed must be positive definite, and maximum likelihood (ML) estimation will also perform poorly in such situations. If the matrix to be analyzed is found to be
not positive definite, many programs will simply issue an error message and quit.
Second, the message may refer to the asymptotic covariance matrix. This is not the covariance matrix being analyzed, but rather a weight matrix to be used with asymptotically distribution-free /
weighted least squares (ADF/WLS) estimation.
Third, the researcher may get a message saying that its estimate of Sigma (
Fourth, the program may indicate that some parameter matrix within the model is not positive definite. This attribute is only relevant to parameter matrices that are variance/covariance matrices. In
the language of the LISREL program, these include the matrices Theta-delta, Theta-epsilon, Phi (
"Not Positive Definite"--What Does It Mean?
Strictly speaking, a matrix is "positive definite" if all of its eigenvalues are positive. Eigenvalues are the elements of a vector, e, which results from the decomposition of a square matrix S as:
S = e'Me
To an extent, however, we can discuss positive definiteness in terms of the sign of the "determinant" of the matrix. The determinant is a scalar function of the matrix. In the case of symmetric
matrices, such as covariance or correlation matrices, positive definiteness wil only hold if the matrix and every "principal submatrix" has a positive determinant. ("Principal submatrices" are formed
by removing row-column pairs from the original symmetric matrix.) A matrix which fails this test is "not positive definite." If the determinant of the matrix is exactly zero, then the matrix is
"singular." (Thanks to Mike Neale, Werner Wothke and Mike Miller for refining the details here.)
Why does this matter? Well, for one thing, using GLS estimation methods involves inverting the input matrix. Any text on matrix algebra will show that inverting a matrix involves dividing by the
matrix determinant. So if the matrix is singular, then inverting the matrix involves dividing by zero, which is undefined. Using ML estimation involves inverting Sigma, but since the aim to maximize
the similarity between the input matrix and Sigma, the prognosis is not good if the input matrix is not positive definite. Now, some programs include the option of proceeding with analysis even if
the input matrix is not positive definite--with Amos, for example, this is done by invoking the $nonpositive command--but it is unwise to proceed without an understanding of the reason why the matrix
is not positive definite. If the problem relates to the asymptotic weight matrix, the researcher may not be able to proceed with ADF/WLS estimation, unless the problem can be resolved.
In addition, one interpretation of the determinant of a covariance or correlation matrix is as a measure of "generalized variance." Since negative variances are undefined, and since zero variances
apply only to constants, it is troubling when a covariance or correlation matrix fails to have a positive determinant.
Another reason to care comes from mathematical statistics. Sample covariance matrices are supposed to be positive definite. For that matter, so should Pearson and polychoric correlation matrices.
That is because the population matrices they are supposedly approximating *are* positive definite, except under certain conditions. So the failure of a matrix to be positive definite may indicate a
problem with the input matrix.
Why is My Matrix Not Positive Definite, and What Can I Do About It?
Properly, the question is, why does the matrix contain zero or negative eigenvalues. However, it may be easier for many researchers to think about why the determinant is zero or negative? Either way,
there are many possibilities, and there are different possible solutions that go with each possible cause.
Further, there are other solutions which sidestep the problem without really addressing its cause. These options carry potentially steep cost. They are discussed separately, below.
Linear Dependency
A not positive definite input covariance matrix may signal a perfect linear dependency of one variable on another. For example, if a plant researcher had data on corn (maize) stalks, and two of the
variables in the covariance matrix were "plant height" and "plant weight," the linear correlation between the two would be nearly perfect, and the covariance matrix would be not positive definite
within sampling error. It may be easier to detect such relationships by sight in a correlation matrix rather than a covariance matrix, but often these relationships are logically obvious.
Multivariate dependencies, where several variables together perfectly predict another variable, may not be visually obvious. In those cases, sequential analysis of the covariance matrix, adding one
variable at a time and computing the determinant, should help to isolate the problem. (I would use a spreadsheet program for this, like Microsoft (TM) Excel (TM), for convenience.)
Dealing with this kind of problem involves changing the set of variables included in the covariance matrix. If two variables are perfectly correlated with each other, then one may be deleted.
Alternatively, principal components may be used to replace a set of collinear variables with one or more orthogonal components.
In regard to the asymptotic weight matrix, the linear dependency exists not between variables, but between elements of the moments (the means and variances and covariances or the correlations) which
are being analyzed. This can occur in connection with modeling multiplicative interaction relationships between latent variables. Jöreskog and Yang (1996) show how moments of the interaction
construct are linear functions of moments of the "main effect" constructs. Their article explores alternative approaches for estimating these models
Error Reading the Data
If the problem is with your input matrix in particular, first make sure that the program has read your data correctly. Remember, an empty covariance matrix (with no variables in it) is always not
positive definite. Try reading the data using another program, which will allow you to validate the covariance matrix estimated by the SEM program. If you generated the covariance matrix with one
program, and are analyzing it with another, make sure that the covariance matrix was read correctly. This can be particularly problematic when the asymptotic weight matrix is the focus of the
Typographical Error
Whenever a covariance matrix is transcribed, there is a chance of error. So if you just have the matrix (say, from a published article, but not the data itself, double-check for transcription errors.
Also remember that journals are not perfect, so a covariance matrix in an article may also contain an error. In a recent case, for example, it appeared that the sign of a single (relatively large)
coefficient was reversed at some point, and this reversal made the matrix not positive definite. In that case, changing the sign of that one coefficient eliminated the problem.
Starting Values
The model-implied matrix Sigma is computed from the model's parameter estimates. Especially before iterations begin, those estimates may be such that Sigma is not positive definite. So if the problem
relates to Sigma, first make sure that the model has been specified correctly, with no syntax errors. If the proposed model is "unusual," then the starting value routines that are incorporated into
most SEM programs may fail. Then it is up to the researcher to supply likely starting values.
Sampling Variation
When sample size is small, a sample covariance or correlation matrix may be not positive definite due to mere sampling fluctuation. As most matrices rapidly converge on the population matrix,
however, this in itself is unlikely to be a problem. Anderson and Gerbing (1984) documented how parameter matrices (Theta-Delta, Theta-Epsilon, Psi and possibly Phi) may be not positive definite
through mere sampling fluctation. Most often, such cases involve "improper solutions," where some variance parameters are estimated as negative. In such cases, Gerbing and Anderson (1987) suggested
that the offending estimates could be fixed to zero with minimal harm to the program.
Estimators of the asymptotic weight matrix converge much more slowly, so problems due to sampling variation can occur at much larger sample sizes (Muthén & Kaplan, 1985, 1992). Using an asymptotic
weight matrix with polychoric correlations appears to compound the problem. Where sampling variation is the issue, Yung and Bentler (1994) have proposed a bootstrapping approach to estimating the
asymptotic weight matrix, which may avoid the problem.
Missing Data
Large amounts of missing data can lead to a covariance or correlation matrix not positive definite. With simple replacement schemes, the replacement value may be at fault. With pairwise deletion, the
problem may arise precisely because each element of the covariance matrix is computed from a different subset of the cases (Arbuckle, 1996). To check whether this is the cause, use a different
missing data technique, such as a different replacement value, listswise deletion or (perhaps ideally) a maximum likelihood/EMCOV simultaneous estimation method.
My Variable is a Constant!
Sometimes, either through an error reading data or through the process of deleting cases that include missing data, it happens that some variable in a data set takes on only a single value. In other
words, one of the variables is actually a constant. This variable will then have zero variance, and the covariance matrix will be not positive definite. Simple tabulation of the data will provide a
forewarning of this. If this is the problem, either the researcher must choose a different missing-data strategy, or else the variable must be deleted.
Polychoric Correlations
Programs that estimate polychoric correlations on a pairwise basis--one correlation at a time--may yield input correlation matrices that are not positive definite. Here the problem occurs because the
whole correlation matrix is not estimated simultaneously. It appears that this is most likely to be a problem when the correlation matrix contains large numbers of variables. Try computing a matrix
of Pearson correlations and see whether the problem persists.
If the problem lies with the polychoric correlations, there may not be a good solution. One approach is to use a program, like EQS, that includes the option of deriving all polychoric correlations
simultaneously, rather than one at a time (cf., Lee, Poon & Bentler, 1992). But be warned--Joop Hox reports that the computational burden is enormous, and it increases exponentially with the number
of variables.
Ed Cook has experimented with an eigenvalue/eigenvector decomposition approach. If a covariance or correlation matrix is not positive definite, then one or more of its eigenvalues will be negative.
After decomposing the correlation matrix into eigenvalues and eigenvectors, Ed Cook replaced the negative eigenvalues with small (.05) positive values, used the new values to compute a covariance
matrix, then standardized the resulting matrix (diving by the square root of the diagonal values) so that the result was again was a correlation matrix. Ed reported that the bias resulting from this
process appeared to be small.
No Error Variance
Sometimes researchers specify zero elements on the diagonals of Theta-delta or Theta-epsilon. A zero here implies no measurement error. While it may seem unlikely, on reflection, that any latent
variable could be measured without error, nevertheless the practice is common, when a construct has only a single measure. Single measures often lead to identification problems, and analysts may
leave the parameter fixed at zero by default. If a diagonal element is fixed to zero, then the matrix will be not positive definite. However, since this is precisely what the researcher intended to
do, there is no cause for alarm. The only problem is that these values may cause the solution to fail an "admissibility check," which may lead to premature termination of the iterative estimation
process. In such cases, it is merely a matter of disabling the admissibility check. In LISREL, for example, this is done by adding AD=OFF to the OUtput line.
Negative Error Variance
Negative values on the diagonal are another matter. Since the diagonal elements of these matrices are variance terms, negative values are unacceptable. Further, since these error variances represent
the "left-over" part of some variable, a negative error variance suggests that the regression has somehow explained more than 100 percent of the variance. In my own experience, these values are
symptoms of a serious fit problem. Comprehensive fit assessment will help the researcher to isolate the specific problem.
Sidestepping the Problem
As with many problems, there are ways to sidestep this problem without actually trying to discern its cause. Besides simply compelling the program to proceed with its analysis, researchers can make a
ridge adjustment to the covariance or correlation matrix. This involves adding some quantity to the diagonal elements of the matrix. This addition has the effect of attenuating the estimated
relations between variables. A large enough addition is sure to result in a positive definite matrix. The price of this adjustment, however, is bias in the parameter estimates, standard errors, and
fit indices. Partial least squares methods may also proceed with no regard for the determinant of the matrix, but this involves an entirely different methodology.
Anderson, J. C., & Gerbing, D. W. (1984). The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis.
Psychometrika, 49(2--June), 155-73.
Arbuckle, J. L. (1996). Full information estimation in the presence of incomplete data. In G. A. Marcoulides & R. E. Schumacker (Eds.), Advanced structural equation modeling: Issues and techniques
(pp. 243-78). Mahwah, NJ: Lawrence Erlbaum.
Gerbing, D. W., & Anderson, J. C. (1987). Improper solutions in the analysis of covariance structures: Their interpretability and a comparison of alternate respecifications. Psychometrika, 52
(1--March), 99-111.
Jöreskog, K. G., & Yang F. [now Fan Yang Jonsson] (1996). Nonlinear structural equation models: The Kenny-Judd model with interaction effects. In G. A. Marcoulides & R. E. Schumacker (Eds.), Advanced
structural equation modeling: Issues and techniques (pp. 57-88). Mahwah, NJ: Lawrence Erlbaum.
Lee, S.-Y., Poon, W.-Y., & Bentler, P. M. (1992). Structural equation models with continuous and polytomous variables. Psychometrika, 57(1--March), 89-105.
Muthén, B. & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables. British Journal of Mathematical and Statistical Psychology, 38, 171-89.
Muthén, B. & Kaplan, D. (1992). A comparison of some methodologies for the factor analysis of non-normal Likert variables: A note on the size of the model. British Journal of Mathematical and
Statistical Psychology, 45, 19-30.
Wothke, W. (1993). Nonpositive definite matrices in structural modeling. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 256-93). Newbury Park, CA: Sage.
Yung, Y.-F., & Bentler, P. M. (1994). Bootstrap-corrected ADF test statistics in covariance structure analysis. British Journal of Mathematical and Statistical Psychology, 47, 63-84.
http://www.gsu.edu/~mkteer/npdmatri.html Return to the SEMNET FAQ home page.
Return to Ed Rigdon's home page.
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Solutions Using the Discriminant
10.10: Solutions Using the Discriminant
Difficulty Level:
At Grade
Created by: CK-12
Practice Solutions Using the Discriminant
What if you were given a quadratic equation like $x^2 - 3x + 1 = 0$discriminant of a quadratic equation like this one.
Watch This
CK-12 Foundation: 1010S The Discriminant
In the quadratic formula, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ discriminant. The discriminant can be used to analyze the types of solutions to a quadratic equation without actually solving the
equation. Here’s how:
• If $b^2-4ac>0$
• If $b^2-4ac<0$
• If $b^2-4ac=0$ double root.
Find the Discriminant of a Quadratic Equation
To find the discriminant of a quadratic equation we calculate $D=b^2-4ac$
Example A
Find the discriminant of each quadratic equation. Then tell how many solutions there will be to the quadratic equation without solving.
a) $x^2-5x+3=0$
b) $4x^2-4x+1=0$
c) $-2x^2+x=4$
a) Plug $a = 1, \ b = -5$$c = 3$$D=(-5)^2-4(1)(3)=13$$D > 0$ two real solutions.
b) Plug $a = 4, \ b = -4$$c = 1$$D=(-4)^2-4(4)(1)=0$$D = 0$ one real solution.
c) Rewrite the equation in standard form: $-2x^2+x-4=0$
Plug $a = -2, \ b = 1$$c = -4$$D=(1)^2-4(-2)(-4)=-31$$D < 0$ no real solutions.
Interpret the Discriminant of a Quadratic Equation
The sign of the discriminant tells us the nature of the solutions (or roots) of a quadratic equation. We can obtain two distinct real solutions if $D > 0$$D < 0$$D = 0$$x-$$D > 0$$x-$$D = 0$$D < 0$
Example B
Determine the nature of the solutions of each quadratic equation.
a) $4x^2-1=0$
b) $10x^2-3x=-4$
c) $x^2-10x+25=0$
Use the value of the discriminant to determine the nature of the solutions to the quadratic equation.
a) Plug $a = 4, \ b = 0$$c = -1$$D=(0)^2-4(4)(-1)=16$
The discriminant is positive, so the equation has two distinct real solutions.
The solutions to the equation are: $\frac{0 \pm \sqrt{16}}{8}=\pm \frac{4}{8}=\pm \frac{1}{2}$
b) Re-write the equation in standard form: $10x^2-3x+4=0$
Plug $a = 10, \ b = -3$$c = 4$$D=(-3)^2-4(10)(4)=-151$
The discriminant is negative, so the equation has two non-real solutions.
c) Plug $a = 1, \ b = -10$$c = 25$$D=(-10)^2-4(1)(25)=0$
The discriminant is 0, so the equation has a double root.
The solution to the equation is: $\frac{10 \pm \sqrt{0}}{2}=\frac{10}{2}=5$
If the discriminant is a perfect square, then the solutions to the equation are not only real, but also rational. If the discriminant is positive but not a perfect square, then the solutions to the
equation are real but irrational.
Example C
Determine the nature of the solutions to each quadratic equation.
a) $2x^2+x-3=0$
b) $5x^2-x-1=0$
Use the discriminant to determine the nature of the solutions.
a) Plug $a = 2, \ b = 1$$c = -3$$D=(1)^2-4(2)(-3)=25$
The discriminant is a positive perfect square, so the solutions are two real rational numbers.
The solutions to the equation are: $\frac{-1 \pm \sqrt{25}}{4}=\frac{-1 \pm 5}{4}$$x = 1$$x=-\frac{3}{2}$
b) Plug $a = 5, \ b = -1$$c = -1$$D=(-1)^2-4(5)(-1)=21$
The discriminant is positive but not a perfect square, so the solutions are two real irrational numbers.
The solutions to the equation are: $\frac{1 \pm \sqrt{21}}{10}$$x \approx 0.56$$x \approx -0.36$
Solve Real-World Problems Using Quadratic Functions and Interpreting the Discriminant
You’ve seen that calculating the discriminant shows what types of solutions a quadratic equation possesses. Knowing the types of solutions is very useful in applied problems. Consider the following
Example D
Marcus kicks a football in order to score a field goal. The height of the ball is given by the equation $y=-\frac{32}{6400}x^2+x$
Define: Let $y =$
Let $x =$
Translate: We want to know if it is possible for the height of the ball to equal 10 feet at some real distance from the goalpost.
$&\text{Write the equation in standard form:} && -\frac{32}{6400}x^2+x-10= 0\\&\text{Simplify:} && -0.005x^2+x-10 = 0\\&\text{Find the discriminant:} && D =(1)^2-4(-0.005)(-10)=0.8$
Since the discriminant is positive, we know that it is possible for the ball to go over the goal post, if Marcus kicks it from an acceptable distance $x$
To find the value of $x$quadratic formula:
$x=\frac{-1 \pm \sqrt{0.8}}{-0.01}=189.4 \ feet \ \text{or} \ 10.56 \ feet$
What does this answer mean? It means that if Marcus is exactly 189.4 feet or exactly 10.56 feet from the goalposts, the ball will just barely go over them. Are these the only distances that will
work? No; those are just the distances at which the ball will be exactly 10 feet high, but between those two distances, the ball will go even higher than that. (It travels in a downward-opening
parabola from the place where it is kicked to the spot where it hits the ground.) This means that Marcus will make the goal if he is anywhere between 10.56 and 189.4 feet from the goalposts.
Watch this video for help with the Examples above.
CK-12 Foundation: 1010 The Discriminant
In the quadratic formula, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ discriminant. The discriminant can be used to analyze the types of solutions to a quadratic equation without actually solving the
equation. Here’s how:
• If $b^2-4ac>0$
• If $b^2-4ac<0$
• If $b^2-4ac=0$ double root.
Guided Practice
Emma and Bradon own a factory that produces bike helmets. Their accountant says that their profit per year is given by the function $P=-0.003x^2+12x+27760$$x$
We want to know if it is possible for the profit to equal $40,000.
Write the equation in standard form: $-0.003x^2+12x-12240=0$
Find the discriminant: $D=(12)^2-4(-0.003)(-12240)=-2.88$
Since the discriminant is negative, we know that it is not possible for Emma and Bradon to make a profit of $40,000 this year no matter how many helmets they make.
Find the discriminant of each quadratic equation.
1. $2x^2-4x+5=0$
2. $x^2-5x=8$
3. $4x^2-12x+9=0$
4. $x^2+3x+2=0$
5. $x^2-16x=32$
6. $-5x^2+5x-6=0$
7. $x^2+4x=2$
8. $-3x^2+2x+5=0$
Determine the nature of the solutions of each quadratic equation.
9. $-x^2+3x-6=0$
10. $5x^2=6x$
11. $41x^2-31x-52=0$
12. $x^2-8x+16=0$
13. $-x^2+3x-10=0$
14. $x^2-64=0$
15. $3x^2=7$
16. $x^2+30+225=0$
Without solving the equation, determine whether the solutions will be rational or irrational.
17. $x^2=-4x+20$
18. $x^2+2x-3=0$
19. $3x^2-11x=10$
20. $\frac{1}{2}x^2+2x+\frac{2}{3}=0$
21. $x^2-10x+25=0$
22. $x^2=5x$
23. $2x^2-5x=12$
24. Marty is outside his apartment building. He needs to give his roommate Yolanda her cell phone but he does not have time to run upstairs to the third floor to give it to her. He throws it straight
up with a vertical velocity of 55 feet/second. Will the phone reach her if she is 36 feet up? (Hint: the equation for the height is $y = -32t^2 + 55t + 4$
25. Bryson owns a business that manufactures and sells tires. The revenue from selling the tires in the month of July is given by the function $R=x(200-0.4x)$$x$
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Mathematically speaking in Wellington, Numeracy is District Goal
Jessica Bedwell sets up her own equation.
"Numeracy is the ability to manipulate numbers, perform computation, reason mathematically, and to be fluent and confident when working with numbers," according to Robin Hussey the district secondary
math specialist. Achieving numeracy is as important in mathematics instruction as literacy is in language arts instruction. To be competent in this century, students need to achieve fluency in
mathematics. Teachers throughout the district are working hard to achieve this goal with their own students and within their own grade levels. One of the best ways to reach the goal of numeracy is to
use an approach known as hands-on math or manipulatives in mathematics.
The state goals are for learners to value mathematics, communicate mathematically, and reason mathematically. Further, learners will become confident in their ability to do mathematics and be
mathematical problem solvers.
If you have not watched this process, you do not know what you are missing. "Kids are not afraid of math," says Kathy Railsback, sixth grade math teacher at Wellington Elementary. "They are engaged
in their learning when they use objects to understand concepts. They talk like mathematicians, now."
Hands On Math
What is hands-on mathematics? It is a lot of things according to Jill Pappas, district elementary math specialist, but mostly it involves helping students learn abstract concepts with concrete
objects. First a math concept is introduced to students using concrete objects such as dies, cubes, dominos, tangrams, or cards. Then, students advance to pictorial representations of the objects;
they are asked to sketch their understanding of the concept. They also discuss, ask questions, answer questions, and write about the concept. Finally, they move to the abstract form of the concept.
Pappas says, "Using manipulatives to teach math may take longer at first, but long-term, children learn math faster. Using manipulatives correctly increases students understanding of math and
accelerates retention. Students know why they are doing something. They no longer simply memorize rules; they must show why an operation is performed. Students discuss math differently than they used
to talk about it. Students are forced to think mathematically, to show and explain their work and reasoning."
Staff Development
Teachers are meeting in grade levels in elementary across the district. They are now grade level teams, not just a school's grade level. They discuss their own students' progress in terms of what
their students mastered on core assessments as well as what concepts were weak. Teachers share their most successful practices and participate in inservice to strengthen challenging areas.
Teachers also discuss strategies for instructing students in heterogeneous groups. Traditionally, elementary teachers taught students in homogeneous or ability groups. Students were divided according
to high, medium, and low groups within the grade level. Now, students stay with their teacher and are taught the same concept. They participate in cooperative learning groups versus traditional
leveled groups. When instruction is aimed at a high level, the pace of learning is increased and accelerated. Children are not left behind because the instruction is tiered in terms of difficulty.
One advantage of heterogeneous grouping is that teachers can receive feedback from core assessments for their entire class. In the past, teachers did not necessarily instruct math to their homeroom
class. Teachers can now more accurately self-assess their instruction.They are able to pace their instruction more accurately, monitor practice of concepts, and use new methods where needed.
Elementary teachers from across the district align the core curriculum and their math textbooks.
Staff development workshops focus on helping teachers use objects, manipulatives, and activities to teach math concepts. The literature connection is such an activity. Books that deal with concepts
personalize the concept for the student. It engages them more forcefully with their learning.
How to use the textbook is another staff development topic. The textbook is a tool for teaching the core curriculum. The core dictates what to teach, and the text is one of the tools teachers use to
reach the goal.
State and National Standards in Math
Carbon teachers are using a pacing guide developed by the Utah State Office of Education for the elementary level. The pacing guide is the recommended movement through the curriculum for the year in
the K-12 Mathematics Standards. The math standards focus on both content and process. Content skills are number and operations, algebra, geometry, measurement, and data analysis/probability. Process
skills are problem solving, reasoning/proof, communication, connections, representation. These standards reflect the standards of the National Council of Teachers of Mathematics, which are a national
consensus of the concepts students should know in mathematics.
According to NCTM professional standards, as teachers shift toward the vision of teaching presented by the standards, one would expect teachers to be asking questions such as, "Why do you think that?
Why is that true? How did you reach that conclusion? Can you make a model to show that? What assumptions are you making? Do you see a pattern? Can you predict the next one? How did you think about
the problem?"
Try this
Here is an activity to increase your number sense and establish a literature connection.
How much is a million? David M. Schwartz, illustrated by Steven Kellogg, Lothrop, New York, 1985. Marvelosissimo the Mathematical Magician says that if one million kids stood on one another's
shoulders, they would be farther up than planes can fly. If you wanted to count to a million, it would take you about 23 days. How big would a gold fish bowl need to be to hold a million gold fish?
If you started counting today, how long would it take you to count to a billion? The answers to these questions and a lot more are found in the back of the book where Marvelosissimo tells readers how
he arrived at his answers. Hint, large numbers like 347,996,268 require at least 2 seconds to say. So, do the math.
March 15, 2014
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Posts by
Posts by bob
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Additional information: Ksp= (CaSO4)(KCl)^2 / (CaCl2)(K2SO4) Ksp= 2.4 x 10^ -5
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elementary math
when finding the factors for 841 what is the largest factor you would have to test?
6 over x to the 10th times x to the 17th over 15
6 over x to the 10th times x to the 17th over 51
The ______ of a transverse wave is the distance between 2 crests or 2 troughs.
32. Bucharest, Romania 33. Budapest, Hungary
How do you calculate the value of z* and t* on the TI-84 and TI-89 (the number used in confidence intervals)?
I know that's the first thing that comes to mind, but my teach said that it can be done. He said that the answer is $0.05, so if someone can work backwards somehow and explain to me how he arrived at
that answer I would appreciate it. :)
If a bottle and a stopper cost $1.10. How much does the stopper cost?
X= 1
Where did you get .55? I don't need the slope I don't think. This is a pretty straightforward Q, I don't need to graph it.
another word problem!
iN 1993, THE LIFE EXPECTANCY of males was 68.6 years. In 2000, it was 71.1 years. Let E represent the life expectancy in year t and t represent the number of years since 1993. The linear function E
(t) that fits this data is: E(t)=_t + _ (round to the nearest tenth) Use the fun...
please help! Examples in book are different!
train A and b are traveling in the same direction on parallel tracks. Train A is traveling at 40 mph, and B is at 60 mph. Train A passes the station at 12:25 am, and train b passes the station at
12:40 am, what time will train b catch up to train A?
word problem
train A and b are traveling in the same direction on parallel tracks. Train A is traveling at 40 mph, and B is at 60 mph. Train A passes the station at 12:25 am, and train b passes the station at
12:40 am, what time will train b catch up to train A?
I need help! Solve by hte elimination method: 7r-9s=-12 9r+7s=96 When I do this, I get a crazy fraction, and I know the answer is a point. (r,s) But what I'm getting is ridiculous. I got -564/130=s!
That has to be totally wrong!
how do I solve this? I know it's common math, but I cannot remember how! -4x> 1/11 I think I multiply both sides by 11, but..... I can't remember how!
Help! These are fractions, so bear with me: -5/8 x= -9/10 Solve with the multiplicaation principle. How do I do this? The fractions throw me off every time!!!
math substitution method
I know this! Put it in for x. So 7(28-6y) +5y=-26 use that to get y, then plug your answer for y into one of the equations to get x
Help! These are fractions, so bear with me: -5/8 x= -9/10 Solve with the multiplicaation principle. How do I do this? The fractions throw me off every time!!!
algebra/common math
how do I solve this? I know it's common math, but I cannot remember how! -4x> 1/11 I think I multiply both sides by 11, but..... I can't remember how!
I need help! Solve by hte elimination method: 7r-9s=-12 9r+7s=96 When I do this, I get a crazy fraction, and I know the answer is a point. (r,s) But what I'm getting is ridiculous. I got -564/130=s!
That has to be totally wrong!
3rd grade
Unscramble cioponam
6th grade - English
what is the antonym for ndrvr
MATH - Can someone please help!
What is the solution of the interval notion? x less than 6 and x greater than -3
How do I solve this with the elimination method? 5x-10y=4 5x-10y=5 I need x=, and y= to get an ordered pair, but every time I mult. by a neg, everything goes away instead of just one variable! please
At 300.K and 1.00 atm, assume that 25 mL of NO gas reacts with 22 mL of oxygen gas and excess water to produce nitric acid according to the following equation: O (g) H O(l) 2 HNO (g) 2 2NO(g) 3 + 2 +
2 ⎯⎯→ 3 . If all of the nitric acid produced by this reacti...
What is the solution of the interval notion? x less than 6 and x greater than -3
how do I break this down? 4x-7 <-4 over 6 the 4x-7 is all over 6 (like a fraction) I thought I just multiply by 6 to get rid of the fraction?
6th grade math
answear is x>54(11)+ 20 what is x,=81
What are the probably products of... a) an aqueous solution of two ionic compounds b) a single compound c) two elements d) oxygen and a compound of carbon and hydrogen ?
Environmental Science
very helpful :)
How do I solve the equations using the elimination method: .3x-.2y=4 and .4x +.3y= 49/23 Normally I would multiply the decimals by 10 to make them whole numbers, but that fraction is really throwing
me off. What do I multiply to eliminate that fraction? Please help!
1. Which of the following matters would an auditor most likely consider to be a significant deficiency to be communicated to the audit committee? A. Management's failure to renegotiate unfavorable
long-term purchase commitments. B. Recurring operating losses that may indic...
Looks right to me
algebra 1
Where did you get the (L-5) and the (W-4)?
When a lady faints, what number do you need to bring her?
A gas at a pressure of 1 atm, a volume of 1 , and a temperature of 25 C. What would happen to the pressure if the volume were reduced to 0.5 and the temperature increased to 250 C ?
Diacetyl, a substance contributing to the characteristic flavor and aroma of butter, consists of 55.8% C, 7.03% H, and 37.17% O. In the gaseous state at 100 degrees celsius and 747 torr, a 0.30 gram
sample occupies a volume of 111 ml. What is the molecular formula of diacetyl?
A mixture of gasses at a total pressure of 95 kPa contains N2, CO2, and O2. The partial pressure of the CO2, is 24 kPa and the partial pressure of the N2 is 48 kPa. What is the partial pressure of
the O2?
A gas storage tank has a volume of 350,000 when the temperature is 27 degrees celcius and the pressure is 101 kPa. what is the new volume of the tank if the temperature drops -10 degrees celcius and
the pressure drops to 95 kPa? please show work.
u r wight and ask ur teacher if u are confused it is the best way...
You are 100% correct...I think...
A gas occupies a volume of 140 mL at 35 degrees celsius and 97 kPa. What is the volume of the gas at STP?
A 10 gram of mass of krypton occupies 15.0 L at a pressure of 210 kPa. What is the volume of the krypton when the pressure is increased to 790 kPa?
College physIcs
"You don't seem to be willing to take step one in answering these problems yourself. " What good is just knowing a formula?
ethics 125
· Consider the following statements: (a) Asian Americans and Pacific Islanders are a model minority, and (b) Asian Americans and Pacific Islanders belong to a single Asian American culture. If
someone said one of these statements to you, how would you explai...
4th grade math
I think Erik meant to say that Judy had 15, not 12. Then, it solves that 27+ (27-12)=42, right? My SIL just called asking us this same question...must be at the same place in the book!
4.76 kg
Choose the correct command form. No __________ uds.esos zapatos sucios en la casa! (llevar) A. llevan B. lleven C. lleve D. lleva
If 2500 ml container is filled with 175 grams of argon at -48 degrees celsius, what is the pressure?
chemistry 2
Then your answer is B
chemistry 2
im confused i cant set this problem up correctly? I need a lot of help
chemistry 2
chemistry 2
chemistry 2
chemistry 2
A gas diffuses only 0.5 times as fast as an oxygen molecule at the same temperature. What is the molecular weight of the gas?
How do you convert from mass percentage to molality? I get 0.084 m but that is incorrect.
how many grams of propane are in a 35L container at 40 degrees and 900 torr?
how many grams of propane are in a 35L container at 40 degrees and 900 torr?
a^2 + b^2 = c^2
2sin(3x-3.14)-1 find the inverse form of this equation
Can I simplify this further SQRT( 2 - SQRT(2) )
need help with this problem 9-10x2 divided by 5+7
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A Treatise On The Theory Of Invariants
by Oliver E. Glenn
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Foundations of k-theory for c*-algebras :: Tulane University Theses and Dissertations Archive
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Title Foundations of k-theory for c*-algebras
Author Hilgert, Joachim
School Tulane University
Academic Mathematics
Let X be a compact space and Y a closed subset of X. For M(,k), the complex k x k-matrices, consider the C*-algebra of continuous functions f : X (--->) M(,k) with the property that f(x)
is a diagonal matrix for all x (ELEM) Y. We shall study the K-theory of this C*-algebra and some closely related C*-algebras for various spaces X and Y. The tools used in this study are a
Mayer-Vietoris Sequence and a Puppe Sequence for K-theory of C*-algebras, both of which reduce to the respective sequence in K-theory of locally compact spaces if the involved C*-algebras
Abstract are commutative First we set up K-theory of unital C*-algebras, following the approach of Karoubi. We define relative K-groups K(,(alpha))((phi)) for unital C*-morphisms (phi) and prove
two excision theorems, which will allow us to define K-theory of non-unital C*-algebras. Moreover, we show that the K-functors do not distinguish between homotopic C*-morphisms. This will
enable us to define K(,n) of a C*-algebra for all n (ELEM) and to establish a long exact sequence in K-theory associated to a short exact sequence of C*-algebras. We also define a cup
product in K-theory of C*-algebras, which will be a (,2)-graded bilinear map K(,*)(A) x K(,*)(B) (--->) K(,*)(A(' )(CRTIMES)(' )B), give some of its basic properties, and use it to define
module structures on the K-groups Finally we prove a non-commutative splitting principle which generalizes the well known splitting principle for vector bundles over compact spaces
Language eng
Degree Date 1982
Degree Ph.D
Publisher Tulane University
Publication 1982
Source 123 p., Dissertation Abstracts International, Volume: 43-03, Section: B,
Identifier See 'reference url' on the navigation bar.
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Symbolic Transformations
Table of Contents
Symbolic Transformations
Parts of an expression may be replaced by other expressions using subst(a,b,c): a is substituted for b in c. This is a powerful function with many uses.
First, it may be used to insert numbers for variables, in the example $3$ for $x$ in der formula $2\sqrt{x}\cdot e^{-x^2}$.
>> syms x
>> a=2*sqrt(x)*exp(-x^2);
>> subst(3,x,a)
ans = 4.275E-4
Second, one can replace a symbolic variable by a complex term. The expression is automatically updated to the canonical format. In the following example $z^3+2$ is inserted for $x$ in $x^3+2x^2+x+7$.
>> syms x,z
>> p=x^3+2*x^2+x+7;
>> subst(z^3+2,x,p)
ans = z^9+8*z^6+21*z^3+25
Finally, the term b itself may be a complex expression (in the example $z^2+1$). Jasymca then tries to identify this expression in c (example: $\frac{z\cdot x^3}{\sqrt{z^2+1}}$). This is accomplished
by solving the equation $a = b$ for the symbolic variable in b (example: $z$), and inserting the solution in c. This does not always succeed, or there may be several solutions, which are returned as
a vector.
>> syms x,y,z
>> c=x^3*z/sqrt(z^2+1);
>> d=subst(y,z^2+1,c)
d = [ x^3*sqrt(y-1)/sqrt(sqrt(y-1)^2+1)
-x^3*sqrt(y-1)/sqrt(sqrt(y-1)^2+1) ]
>> d=trigrat(d)
d = [ x^3*sqrt(y-1)/sqrt(y)
-x^3*sqrt(y-1)/sqrt(y) ]
Simplifying and Collecting Expressions
The function trigrat(expression) applies a series of algorithms to expression.
• All numbers are transformed to exact format.
• Trigonometric functions are expanded to complex exponentials.
• Addition theorems for the exponentials are applied.
• Square roots are calculated and collected.
• Complex exponentials are backtransformed to trigonometric functions.
It is often required to apply float(expression) to the final result.
>> syms x
>> trigrat(sin(x)^2+cos(x)^2)
ans = 1
>> b=sin(x)^2+sin(x+2*pi/3)^2+sin(x+4*pi/3)^2;
>> trigrat(b)
ans = 3/2
>> trigrat(i/2*log(x+i*pi))
ans = 1/4*i*log(x^2+pi^2)+(1/2*atan(x/pi)-1/4*pi)
>> trigrat(sin((x+y)/2)*cos((x-y)/2))
ans = 1/2*sin(y)+1/2*sin(x)
>> trigrat(sqrt(4*y^2+4*x*y-4*y+x^2-2*x+1))
ans = y+(1/2*x-1/2)
trigexpand(expression) expands trigonometric expressions to complex exponentials. It is the first step of the function trigrat above.
>> syms x
>> trigexp(i*tan(i*x))
ans = (-exp(2*x)+1)/(exp(2*x)+1)
>> trigexp(atan(1-x^2))
ans = -1/2*i*log((-x^2+(1-1*i))/(x^2+(-1-1*i))) | {"url":"http://jwork.org/scavis/wikidoc/doku.php?id=man:jmathlab:symbolic","timestamp":"2014-04-20T00:57:35Z","content_type":null,"content_length":"45051","record_id":"<urn:uuid:0aaf8dab-2ec3-44fc-ae69-93d318a34358>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00223-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Annales Academiæ Scientiarum Fennicæ
Series A. I. Mathematica
Volumen 20, 1995
The articles are stored in DVI format and also in compressed (with gzip) PostScript format, so you can view them on your screen if your WWW browser supports it. You can also download the articles in
compressed PostScript format for printing. Note that some of the figures are not included in the electronic versions.
Arés Gastesi, Pablo: Coordinates for Teichmüller spaces of b-groups with torsion, pp. 279-300.
Astala, K., and M. Zinsmeister: Abelian coverings, Poincaré exponent of convergence and holomorphic deformations, pp. 81-86.
Baker, I.N., and Anand P. Singh: Wandering domains in the iteration of compositions of entire functions, pp. 149-153.
Balogh, Z., and A. Volberg: Normalization of almost conformal parabolic germs, pp. 109-121.
Dodson, M.M., M.V. Melián, D. Pestana, and S.L. Velani: Patterson measure and ubiquity, pp. 37-60.
Freden, Eric M.: Negatively curved groups have the convergence property I, pp. 333-348.
Ghamsari, Manouchehr: Quasiconformal groups acting on B^3 that are not quasiconformally conjugate to Möbius groups, pp. 245-250.
Heinonen, Juha, and Shanshuang Yang: Strongly uniform domains and periodic quasiconformal maps, pp. 123-148.
Herron, David A., and Pekka Koskela: Locally uniform domains and quasiconformal mappings, pp. 187-206.
Järvi, Pentti: On the behavior of meromorphic functions around some nonisolated singularities II, pp. 373-378.
Kim, Kiwon: Necessary and sufficient conditions for the Bernstein inequality, pp. 419-432.
Kizuka, Takashi: On the movement of the Poincaré metric with the pseudoconvex deformation of open Riemann surfaces, pp. 327-331.
Korenblum, B., P.J. Rippon, and K. Samotij: On integrals of harmonic functions over annuli, pp. 3-26.
Krushkal, Samuel L.: Teichmüller spaces are not starlike, pp. 167-173.
Krushkal, Samuel L.: Exact coefficient estimates for univalent functions with quasiconformal extension, pp. 349-357.
Krzyz, Jan G.: Universal Teichmüller space and Fourier series, pp. 387-400.
Kurki, Jukka: Invariant sets for A-harmonic measure, pp. 433-436.
Kuusalo, T., and M. Näätänen: Geometric uniformization in genus 2, pp. 401-418.
Masur, Howard A., and Michael Wolf: Teichmüller space is not Gromov hyperbolic, pp. 259-267.
Matsuzaki, Katsuhiko: Teichmüller spaces with variable bases in the universal Teichmüller space, pp. 27-36.
Mizuta, Yoshihiro: Tangential limits of monotone Sobolev functions, pp. 315-326.
Murphy, Gerard J.: Translation-invariant function algebras on compact abelian groups, pp. 175-178.
Perry, Peter A.: A trace-class rigidity theorem for Kleinian groups, pp. 251-257.
Rickman, Seppo: Nonremovable Cantor sets for bounded quasiregular mappings, pp. 155-164.
Rickman, Seppo: Defect relation and its realization for quasiregular mappings, pp. 207-243.
Rossi, John: A sharp result concerning cercles de remplissage, pp. 179-185.
Sastry, Swati: A converse defect relation for quasimeromorphic mappings, pp. 61-79.
Stratmann, Bernd: A note on counting cuspidal excursions, pp. 359-372.
Vainio, Juhani V.: Properties of real sewing functions, pp. 87-95.
Wang, Shupei: A note on the oscillation theory of certain second order differential equations, pp. 379-385.
Wolf, Michael: On the existence of Jenkins-Strebel differentials using harmonic maps from surfaces to graphs, pp. 269-278.
Ye, Zhuan: On Nevanlinna's secondary deficiency, pp. 97-108.
Zhong, Li: A note on geodesics in infinite-dimensional Teichmüller spaces, pp. 301-313. | {"url":"http://www.kurims.kyoto-u.ac.jp/EMIS/journals/AASF/Vol20/vol20.html","timestamp":"2014-04-17T15:37:43Z","content_type":null,"content_length":"5283","record_id":"<urn:uuid:8462d994-0ec3-4e3d-8d39-b791b813144b>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00550-ip-10-147-4-33.ec2.internal.warc.gz"} |
Spontaneous Patterns in Disk Packings
Boris D. Lubachevsky Ron L. Graham Frank H. Stillinger
Lucent Technologies University of California Lucent Technologies
Bell Labs Innovations at San Diego Bell Labs Innovations
700 Mountain Avenue La Jolla, CA 92093 700 Mountain Avenue
Murray Hill, NJ 07974 Murray Hill, NJ 07974
bdl@bell-labs.com graham@ucsd.edu fhs@bell-labs.com
Using a computational procedure that imitates tightening of an assembly of billiard balls, we have generated a number of packings of n equal and non-equal disks in regions of various shapes for n in
the range 10 to 10,000. Our experiments revealed various geometric patterns and regular features, including polycrystalline textures with "rattlers'' typically trapped along the grain boundaries. We
also observed series of packings with similar patterns for n taking on increasing sequences of values, n = n(1), n(2),... n(k),....
Billiards algorithm
Packings of non-overlapping disks or cylinders can be mechanically generated by tightening the container boundary [7]. A variant of this procedure has been implemented on a computer [4] as a discrete
event "billiards'' simulation algorithm [3]. Rather than shrinking the container, in the billiards simulations we uniformly expand the disks. Fig. 1 illustrates the work of the billiards simulation
algorithm while packing 2000 equal disks in a square with periodic boundary (torus). The initial stage at time t = 0 is represented on the top square in the figure where 2000 points are randomly
scattered. To each point an initial velocity vector is randomly assigned (not shown). Some points lie outside the square; they are periodic images of the corresponding points inside. When t > 0, the
points grow into disks, and all disks at each time t have common diameter d = Et. The growth continues until the configuration "jams,'' at which time we expect to have a packing.
At t = 0 disks do not overlap because their sizes are zero. For t > 0 disks move along straight lines with given velocities; their motions may conflict with each other. At an instance of such a
two-disk conflict, we simulate an elastic collision of these disks assuming their masses are equal. At a collision both disks change their velocity vectors so as to exchange momenta and energies
according to known mechanical laws.Note that the evolution of the disk configuration does not change if we proportionally change both the disk expansion speed E and linear disk velocities.We
normalize the simulation input by assuming the unity mean initial disk velocity.
Figure 1: Successive stages in an instance of disk packing by billiards algorithm.
Figure 2: A packing of 2000 disks in a square with periodic boundary. The packing is obtained under a fast disk expansion, E = 100.The packing consists of crystalline grains with many rattlers
represented as unshaded disks concentrated along the grain boundaries. Monovacancies occur within the hexagonally packed grains.
A number of packings have been produced by randomly varying initial configuration of points, their initial velocity, and the expansion speed E. Packings generated under a high E (in Fig. 2), a
moderate E (in Fig. 3), and a low E (in Fig. 4) were obtained. For high E the pattern is a combination of hexagonally packed fragments, "grains,''with each grain having generally different
orientation. Irregularity concentrates along the grain boundaries, as do the "rattlers,'' the disks that are not rigidly fixed in their positions but trapped in the cages formed by their fixed rigid
neighbors or boundary walls if any.
Figure 3: A packing of 2000 disks in a square with periodic boundary. The packing is obtained under a moderately fast disk expansion, E = 3.2, and consists of grains that are larger than those in
Fig. 2.As in Fig. 2, the rattlers concentrate along the grain boundaries but their number and that of the monovacancies is smaller than in Fig.2.
The size of the grains increases (and their number for a fixed n decreases) as the E decreases. For a sufficiently small E, a single hexagonal "crystal'' emerges. The array of hexagonally packed
disks contains interesting structural deviations and insertions.Sometimes the symmetry of the packing coupled with a high value of its density suggests that perhaps we have achieved the optimum,
i.e., the packing of the largest possible disk diameter and density. This might be the case in Fig. 4. (The density is the fraction of the region area covered by disks.)
Figure 4: A packing of 2000 disks in a square with periodic boundary obtained under a slow disk expansion, E = 10^-3. If the monovacancy near the center is filled with the 2001-st disk, the obtained
packing seemingly becomes perfectly symmetric. Might that be the optimum packing of 2001 equal disks in a square with periodic boundary? Its experimentally computed density (when the 2001-st disk is
inserted) is 0.901635...
Frustrated packings
56 equal disks stacked hexagonally in 8 alternating columns, 7 disks in a column, constitute the optimum packing in a rectangle with periodic boundary conditions if the ratio of the height of the
rectangle to its width is (1 + 1/48)^1/2 = 1.01036... We frustrate the packing by increasing the width of the rectangle and making it equal to the height. In the new shape, previously jammed disks
become loose and can grow further to exploit the extra space. As the disks grow, a new jamming is achieved with a larger disk diameter.
Figure 5: The best found packing of 56 disks in a square with periodic boundary conditions. (The square perimeter is dashed). Pairs of disks that are in contact are indicated by line segments
connecting their centers. Triangles formed by such lines are shaded. The shading helps to see the structure of the packing: the plane is tiled with 4×7 blocks, each block having pattern " 'Z," shaded
on it.
In the example of a frustration in Fig. 6 we begin with a perfectly hexagonally packed configuration of 10,800 equal disks. Note that we can always perfectly pack 3k^2 equal disks in a regular
hexagon with periodic boundary conditions, as explained in the figure. k disks will fit along each hexagon side. Here k = 60.
We uniformly decrease the size of all the disks but one in the center. The diameter of the latter is increased. As a result all the disks become loose. We tighten the configuration by simulating
expansion with very small rate E. During the expansion, while diameters of all disks are increasing, we keep constant the ratio of the diameter of the large disk to that of other disks.
Figure 6: Displacement vectors of centers of 10800 disks packed in a regular hexagon. Boundary conditions are periodic as shown in the right top corner box where the same letter, a or b, marks
identical points. If all disks were of the same size ("pure'' crystal), the packing would be perfectly hexagonal. Displacements from this perfect order are caused by the central "impurity'' disk
being 20% larger than the others. Displacement vector of the central disk is 0 by definition, each other displacement vector begins at the perfect order position and its length is magnified 20 fold
to enhance the resolution.
As a result, a frustration with a "crack'' centered at the larger disk is developed. To enhance visually the structure of the frustration, we display disk displacements. Each disk has the "ideal''
position of its center p[0] in the original non-frustrated packing and the final position of the center p[1] in the frustrated packing.Vector p[1] - p[0] is the displacement. Accumulation of the
round off error may cause the entire configuration to shift during the expansion. We "calibrate'' the displacement vectors by subtracting from each of them the displacement of the larger disk. The
latter displacement becomes zero by definition.
Repeated patterns in packings in equilateral triangle,
square, and circle with hard walls
The optimum packing of D(k) = k(k+1)/2 equal disks in an equilateral triangle is the hexagonal arrangement [6] and the optimality holds for all k = 1, 2, .... Are the triangle numbers D(k) the only
such lucky sequence? We conjecture the existence of an infinite number of sequences so that each sequence has its own well-defined pattern of optimal packings.
Figure 7: The conjectured densest packing of n[p](k) = 256 disks inside an equilateral triangle, where p = 5 and k = 3 and n[p](k) is defined by formula (1). The densest packings of n =n[p](k) disks
for all checked values of p and k, have this pattern consisting of one triangle of side (k+1)p-1 and 2p+1 alternating triangles of side k with p-1 rattlers that are "falling off'' the larger
For each p = 0, 1, 2,... consider sequence
n[p](k) = D((k+1)p-1) + (2p+1)D(k), k = 1, 2, ... (1)
For p = 0 sequence n[p](k) is identical with the sequence of triangle numbers D(k) with known optimal packings. For each p > 0, we conjecture the optimal packing of n[p](k) disks as the pattern that
consists of n-p+1 solid disks and p-1 rattlers; it includes one triangle of side (k+1)p-1 and 2p+1 alternating triangles of side k each as shown in Fig. 7 for the case p = 5 and k = 3.
A k×k orthogonal arrangement is a "natural'' square packing in the same spirit as hexagonal disk arrangement is a natural packing in an equilateral triangle. This square arrangement becomes
non-optimal for large k, though. What we observe in the best found packings in the square is an interplay between two patterns: square and hexagonal. For a sufficiently large number of disks n the
hexagonal pattern becomes dominant.With the billiards simulation algorithm we were able to examine details of this interplay for small n.
The following sequences were recently identified [1, 5] as candidates for pattern repetitions: k^2-3, k^2 -2,k^2-1. These are "frustrations'' of the "natural'' square pattern. Some members of these
sequences are presented in Fig. 8. In this and the following figures black dots identify contacts (between two disks in a pair or between a disk and the boundary). Sometimes, disks are seemingly in
contact, like disks 2 and 15 in the packing of 22 disks in Fig. 8, but no dot is present in the contact site. This indicates a positive gap between the disks. The gap may be not discernible on the
picture, but can be verified computationally. Because a pair of rows and a pair of columns are "frustrated'' in best packings of k^2-2 disks (middle row in Fig. 8), the two pairs of indices can
identify the packing, like (3,5; 2,4).
Sequences k(k+1) and k^2+ [k/2] were also identified (where [x] denotes largest integer that is not exceeding x); they can be considered as "frustrations'' of a hexagonal pattern adjusted to the
square boundaries. Some of their members are presented in Fig. 9. Note that the best packing of 10 disks does not follow the common pattern of sequence k^2+ [k/2].
As in the case of a square, a circle boundary shape does not conform to the hexagonal disk arrangement. Unlike the case of a square or equilateral triangle, no obviously "natural'' packing of equal
disks has been proposed. Perhaps, the curved hexagonal packings [2] can be taken as such. They are presented in the left column in Fig. 10. Three different equally dense curved hexagonal packings of
61 disks are known. Fig. 10 presents one of them. Similarly, 12 different equally dense curved hexagonal packings of 91 disks are known and Fig. 10 presents one of them.The curved hexagonal packings
of 3k(k+1)+1 disks exist for arbitrary large k but they become not the densest ones for k = 6, 7,... (n = 127, 169, ...). The right column in Fig. 10 presents best found packings for the sequence 3k(
k+3)+1. These demonstrate a different way of adjusting hexagonal packing to the circular-wall boundary.
Figure 8: Best found packings of equal disks in a square. Top row: first (k = 5) and last (k = 8) members of sequence k^2-3. The packings consist of a heavier shaded (k-3)×(k-3) square packing in the
bottom left corner and three lighter shaded alternating rows and columns and one unshaded rattler. Middle row: two out of four existing best packings of 34 disks, a member of sequence k^2-2 for k =
6. Each packing consists of a (k-2)×(k-2) heavier shaded square pattern with two lighter shaded hexagonally arranged rows and two columns. Two pairs of insertion indices identifies a packing. Bottom
row: two out of three existing best packing of 35 disks, a member of sequence k^2-1 for k = 16. Each packing consists of a (k-1)×(k-1) heavier shaded square pattern with one lighter shaded inserted
row andone column. A pair of insertion indices identifies a packing.
Figure 9: Members of sequences k(k+1) (56 disks) and k^2+[k/2] (5, 10, 18, and 52 disks). Optimal packing of 10 disks (middle row, left) does not follow the common pattern of the latter sequence. The
inferior packing of 10 disks that follows the pattern is also shown (top row, right)
Figure 10: Left: best found packings of 3k(k+1)+1 equal disks in a circle for k = 1, 2, 3, 4, and 5. Right: best found packings of 3k(k+3)+1 equal disks in a circle for k = 2, 3, and 4 and the
highest obtained density configuration of 121 disks (k = 5).
VisMath HOME | {"url":"http://vismath5.tripod.com/lub/","timestamp":"2014-04-18T08:02:27Z","content_type":null,"content_length":"37372","record_id":"<urn:uuid:bf2881fc-e659-4361-a71b-9cfc0a0ceb35>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00576-ip-10-147-4-33.ec2.internal.warc.gz"} |
• FOLIATIONS I, Book coauthored with L. Conlon.
• FOLIATIONS II, also with L. Conlon, has been published by the AMS in August 2003. Take a look at the Table of Contents and some sample pages.
Preprints and Reprints
By default, preprint files are in PDF format. The PDF plug-in is available from Adobe. At the end of each item there is also the choice of DjVu format, of comparable quality but smaller file size.
The DjVu plug-in is available from Lizardtech for PC's and Mac's, and from DjVuLibre for Unix/Linux systems.
Lecture Notes
Just that! Lecture Notes from several courses and seminars that I have taught, offered without warranty.
Contact Info
Alberto Candel
Office: SN 430
Office Hours: Wed 4-5pm, Thurs 3-4pm
Phone: 677-6512
Acknowledgment and Disclaimer
Some of the research results posted on this Web site were supported by grants from the Division of Mathematical Sciences of the National Science Foundation. Opinions, findings, or recommendations
expressed in those results are those of the author(s) and do not necessarily reflect the views of the NSF.
Alberto Candel
Last modified: Fri Aug 5 16:21:52 PDT 2011
Web Design by ACM | {"url":"http://www.csun.edu/~ac53971/research/index.html","timestamp":"2014-04-20T16:17:05Z","content_type":null,"content_length":"8121","record_id":"<urn:uuid:bc0810ea-dafc-4acf-b2e3-088f34b2747d>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00359-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathML and SVG in HTML 5 with Firefox
I’ve been using MathML for a while now for some of my documentation work on 3D graphics. Unfortunately the only way at the moment is to use XHTML 1.1 modular doctype to include either or both of
MathML and SVG. In HTML 5 these have become embedded content parts of the specification. So for example, using MathML would be as simple as doing:
[html]<!DOCTYPE HTML>
<meta charset="utf-8">
<title>MathML test</title>
Unfortunately the only browser to support either MathML or (parts of) HTML 5 at this moment is Firefox 3.5. However, the MathML or SVG embedded content did not render under 3.5. After reading John
Resig’s post about a new HTML parsing engine in Mozilla’s Gecko engine I set out to test this engine’s support by downloading the latest nightly and setting html5.enable to true in about:config and
‘lo and behold, it renders as expected.
3 thoughts on “MathML and SVG in HTML 5 with Firefox”
1. Just one thing I don’t entirely understand: If Firefox 3.5 is the only browser with sufficient HTML5 support, but Firefox also and for some time now supports XHTML1.1, why would one need MathML
in HTML5? Is it just about saving that few characters for the namespace declaration?
2. Well, the problem lies that XHTML 1.1 needs to be sent as application/xhtml+xml and any malformed document tends to trigger very scary, for an end-user, messages about wellformedness. Add to that
that now the XHTML 2 work group will be disbanded later this year, the only new HTML specification will be HTML 5. So the only option you have in the near future will be HTML 5 for MathML.
The best part is that SVG and MathML are integral parts of the specification. So this means that we can finally see some more widespread implementations. My Opera contact says that there’s work
done, so maybe Opera will add more support for it in the 10.x release series.
I also have good hopes that Chrome/Chromium will add this support.
3. Re Fredo’s comment, I think the issues are that (1) xhtml failed to be widely adopted, so it’s seen as a dead end, and (2) people want to be able to write a single web page, rather than a
nonstandard one for IE and a standard one for every other browser. If I write html5+mathml, then (in the very near future), firefox will render it correctly, and IE users will at least get some
rendering of the page that will be somewhat readable. | {"url":"http://www.in-nomine.org/2009/07/08/mathml-and-svg-in-html-5-with-firefox/","timestamp":"2014-04-19T06:53:15Z","content_type":null,"content_length":"18664","record_id":"<urn:uuid:2a5ec4ad-3af2-4405-a455-8621f844073a>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00616-ip-10-147-4-33.ec2.internal.warc.gz"} |
Programming science
Mr. Squirrel
Programming science
Hi. I have a question relating to science, but I need the answer to write a program. So, let's see if you have an answer.
Say we are in a place with very little friction. Now, we take a hill and put a car at the top. We also put a car on a crain, holding it up at the same height as the hilltop.
Without friction, all objects on earth fall at 9.8m/s(squared). Saying we let the car on the hill roll (remember, very little friction), what would be the change in velocity. It would not fall at
the same rate. Is there a formula for this?
I am making a roller coaster sim.
searching memory for any traces of A level Physics.......
You need to also take into account the air resistance as it is this that dicates the speed of fall, though the friction generated will also have a minor effect.
the change in its velocity will depend on the mass of the body (ie : the car) and the vertical and horizontal distances,
remember bodies accelerate by 9.8ms every second. There is almost certainly a formula for this - but i cannot think of it at the time as it has been over a year since i was doing a level physics.
is one of the only ones i remember
i will have a think and post back any more if i remember...
Given: Acceleration due to gravity = 9.8 m/s
>>Now, we take a hill and put a car at the top. We also put a car >>on a crain, holding it up at the same height as the hilltop.
I think we need to know how steep the incline is... however, with little friction, the car will be accelerating at 9.8 m/s, eventually if it keeps going on the same incline, it will reach a
maximum speed. Unfortunately, even if you give us the angle, I'm still not sure how to find the velocity ;/
sorry - forgot to explain the equation
d = delta and means 'change in'
V = velocity
tt = total time taken
dti = correction for external effects (though i doint remember how to calc them )
Assuming no friction or air resistance, the the object will accelerate at g*sin(theta), where theta is angle of the incline. Testing this for two extremes, a flat surface (theta = 0, sin(0) = 0,
and a vertical surface (no surface at all, theta = 90, sin(theta) = 1), shows that it works in both extreme conditions.
Talk to Newton
use this formula
where t=time passed
i=initial velocity (if something falling, use 0)
h=initial height
c=current height
This does not include area resistance or friction but it should do.
As you don't have friction I think you can use
mgh = (1/2)mv^2
At least you will find the change in velocity, if you meant acceleration than just use SilentStrikes formula.
You are lucky cus i just took a Physics 1 College Exam:)
You can use the kinematic equations all ready posted on this board, but you can also use Conservation of Energy to help you out.
Kenetic energy is equal to 1/2 of the mass time the velocity squared
KE = 1/2 m * v^2;
Potential energy is equal Mass * Gravity (9.8) * Height.
PE = m * g * h
So if you have a car at the top of the track it is not moving but has PE.
Then it rolls down a hill and PE transfers into KE.
So if a car rolls from the top of a hill to a bottom of a dip, the speed can be found from those 2 equations
1/2 * m * v^2 = m * g * h
v = sqrt( 2 * g * h )
Look for a physics book and look up motion with constant acceleration. Conservation of momentum and energy will also help you out.
Im just curious what classes have you taken in physics/ what is your current level of education? | {"url":"http://cboard.cprogramming.com/cplusplus-programming/7151-programming-science-printable-thread.html","timestamp":"2014-04-19T09:38:41Z","content_type":null,"content_length":"11217","record_id":"<urn:uuid:6fbb5cec-b2ab-4e41-9b34-cb212c74e1d9>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00469-ip-10-147-4-33.ec2.internal.warc.gz"} |
Who wins when we compare polynomials and logarithmic functions? Look at a picture.
Eventually, after not too long, the polynomial will pull ahead of the logarithmic function. This makes sense, because the polynomial is curved upwards, while the logarithmic curve looks like it's
flattening out.
The logarithmic curve never flattens out, it has no horizontal asymptotes, and grows without bound, but it does so slowly.
If we take some limits, we find
Think of these three types of functions as if they are racing. Power functions are like powerful race horses; polynomials (Polly want a cracker?) are like parrots fluttering along; and logarithmic
functions are like logs, plodding and slow. The category a function belongs to is determined by its leading term, that is, the individual term that "grows the fastest'."
Identify the "leading term" of the function. That is, find the term whose magnitude grows the fastest.
Identify the "leading term" of the function. That is, find the term whose magnitude grows the fastest.
Identify the "leading term" of the function. That is, find the term whose magnitude grows the fastest.
Identify the "leading term" of the function. That is, find the term whose magnitude grows the fastest.
Identify the "leading term" of the function. That is, find the term whose magnitude grows the fastest.
Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞.
Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞.
Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞.
Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞.
Find the limit. Remember to consider signs when determining whether a limit is ∞ or -∞. | {"url":"http://www.shmoop.com/functions-graphs-limits/comparing-polynomial-log-help.html","timestamp":"2014-04-20T13:43:53Z","content_type":null,"content_length":"46515","record_id":"<urn:uuid:e5937c30-60b8-4cc6-9f5d-aff37bf1d184>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00574-ip-10-147-4-33.ec2.internal.warc.gz"} |
Random polynomials, random matrices, and L-functions
- MR MR2442050
"... We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree
N is naturally associated to a subset of R Nā 1. We calculate the volume of this set, prove the set is hom ..."
Cited by 2 (0 self)
Add to MetaCart
We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree N is
naturally associated to a subset of R Nā 1. We calculate the volume of this set, prove the set is homeomorphic to the N ā 1 ball and that its isometry group is isomorphic to the dihedral group of
order 2N. 1
, 2011
"... of all real zeros for random polynomial ..." | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=9040891","timestamp":"2014-04-20T18:04:57Z","content_type":null,"content_length":"13449","record_id":"<urn:uuid:f5d64aea-389a-4379-a960-71b40a72defa>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00062-ip-10-147-4-33.ec2.internal.warc.gz"} |
Journal of the Southern African Institute of Mining and Metallurgy
Services on Demand
Related links
Print version ISSN 0038-223X
J. S. Afr. Inst. Min. Metall. vol.112 no.5 Johannesburg May 2012
JOURNAL PAPERS
Uncertainty assessment for the evaluation of net present value: a mining industry perspective
Ö. Erdem; T. Güyagüler; N. Demirel
Department of Mining Engineering, Middle East Technical University, Ankara, Turkey
The investment decisionmaking process in the insurance and finance industries is affected by new advances such as simulation techniques. These advances have improved the discounted cash flow method
(DCF). In DCF, a dynamic and flexible model construction is not possible because it does not consider uncertain conditions. Each project should be evaluated taking into account the related
uncertainties because the related uncertainties determine the characterization of the project. Related uncertainties can be processed in dynamic DCF, which is applicable to both financial and
non-financial industries with different kind of uncertainties, such as power generation and petroleum projects. The dynamic DCF method can estimate net present value (NPV) while managing related
project uncertainties with a simulation method like Monte Carlo Simulation. The simulation method is preferred because its output is unbiased. Therefore, a more realistic financial evaluation of the
project can be concluded. In spite of the improvement of dynamic DCF, this project evaluation method is not used frequently in mining industry. However, the mining industry is ideally suited to this
concept because it is highly dependent on estimations such as orebody size and ore grade. During the project evaluation stage, these uncertainties can be included with the dynamic DCF method. This
study aims to contribute to increasing the usage of this method in mining projects. A copper reserve in Turkey is selected as a case study to apply uncertainty assessment for the evaluation of NPV.
Keywords: uncertainty, Monte Carlo simulation, net present value (NPV), probability.
The mining industry is a very risky industry compared to other industries because it depends on orebody estimations and decisionmakers must consider many uncertain inputs. The uncertainties have an
important impact on project investment decisions. Identifying the potential sources of uncertainties is very important in order to obtain accurate results. Therefore, each uncertainty and its impact
on the project should be analysed carefully. Snowden et al. (2002) stressed the importance of communicating and compiling all related mining uncertainties and their likelihood and distribution of
occurrence in order to obtain reliable results for better decision making. If the upside at Sunrise Dam in Western Australia had not been considered, the deposit may never have been mined. The
company produced 60 per cent more gold than the estimated value. Managing uncertainty does not mean minimizing risk, because this may result in loss of opportunities. However, there are so many mines
where planning has been applied on the basis of the most optimistic estimates but in the end the companies encountered financial disaster. For example, Morley et al. (1999) indicated that the 70
percent of small mining companies in South Africa failed during the 1980s mainly because of over-estimation of the reserve tonnage and grade.
Analysis of uncertainty can help decision-makers and prevent possible errors. The modern approach in the evaluation of uncertainty in mineral resource estimation is based on determining the frequency
distribution of each variable involved in the calculations and arriving at the result within some confidence interval. Managing of uncertainty related to mining can be done with the aid of the output
of simulations like the Monte Carlo Simulation (MCS). Stochastic permutations of uncertainties are investigated and unbiased and consistent estimation can be obtained using MCS. The MCS procedure
consists of generating random numbers according to assumed probabilities associated with sources of uncertainties. The estimated outcomes related to the random draws are then analysed to determine
the possible results and associated risks. The MCS technique is widely used for dealing with uncertainty in many aspects of operations (Chance, 2008). In MCS, single-output estimation is represented
as one iteration. The number of iterations is determined taking into account the project size and the importance of risks. It could be stated that as the number of runs increases, many more
stochastic scenarios are evaluated in the solution space(Rezaie et al., 2007). This simulation method can be applied to estimate the Net Present Value (NPV) of mineral deposits. NPV will be obtained
as a probability distribution from an output of simulation, so a decisionmaker can decide the probability of a mining project's success. The lower limit and upper limit of the NPV can also be
indicated by the distribution.
There are many factors affecting the feasibility of mining investment. Traditionally, until now the effective factors have been taken as constant, although some uncertainties are involved in these
factors. Since fixed values are considered as inputs in the estimations, the risks involved in the estimation cannot be defined. For a beter decision, the risk involved in the project should be
determined before an investment has been made. In this study, such a model, which uses a simulation technique, has been prepared.
In this study, the importance of utilizing uncertainties in the estimation of NPV of orebody is underlined. A copper deposit (Derekoy copper deposit, Kirklareli, Turkey) is selected as a case study
and it is modelled in a mine design software environment to estimate its size and copper grade. MCS method is applied to estimate NPV of the deposit while considering related uncertainties. The model
has been successfully applied to a low-grade copper deposit that was regarded as mineral resources having no economic value. In the study, not only the uncertainties in the reserve estimation, but
also the uncertainties involved in the economic analysis, were considered.
Dereköy copper deposit
Reserve estimation is the process that defines which part of the resource can be economically extracted (Morley et al. 1999) and in mining, the main uncertain source is the orebody, because knowledge
of the orebody is based largely on estimates(Snowden et al. 2002). The importance of the reserve estimation for calculating the value of the mining project is emphasized by several researchers such
as Dimitrakopoulos (1998), Yamamoto (1999), Morley et al. (1999), Snowden et al. (2002), Dominy et al. (2002), Rendu (2002), Ross (2004) and Emery et al. (2006). Dominy et al. (2002) and Morley et
al. (1999) indicated that mineral resources and ore reserve reports generally contain a single tonnage and grade value. The tonnage and grade values do not contain any reference to the potential
uncertainties in the estimations. 'Any resource and reserve estimation is guaranteed to be wrong. Some, however, are less wrong than others' (Morley et al. 1999). Variability of an ore reserve can
significantly affect the critical decisions. Therefore, reserve estimation should be conducted using technological advances such as mine design software. Estimation of ore reserve size and average
grade using mine design software can reduce the estimation errors. If the estimation risk of the reserve amount is low, the variance of NPV is reduced. In this study, to reduce the estimation errors,
the Micromine 10 mine design software was used.
Derekoy copper deposit, a porphyry copper deposit in Kirklareli, Turkey, was selected as a case study to estimate the NPV of the deposit while considering related uncertainties. The location of the
deposit is shown in Figure 1. The deposit was explored by the General Directorate of Mineral Research and Exploration (MTA) who drilled 25 boreholes with the total length of 8,776 m. Grade data of
these drillholes was supplied from drillholes. The orebody was modelled in 3D with the data in a mine design software environment. The average grade of the deposit, reserve amount, and overburden
amount were estimated as 0. 244% Cu, 210 Mt, and 140 million m3 respectively. The output of the software is presented in Table I.
Developing an uncertainty model for the deposit
Main uncertain inputs and defined distributions
A financial model must be constructed to estimate value of a mining project. The economic value of a mining project can be determined by evaluation of cash flow. The aim of evaluation of the cash
flow is to investigate the profitability of the project with related uncertainties. The economic value of a mining project is determined by the NPV (Nasuf and Orun, 1990). The main variables for the
estimation of NPV in mining projects are defined. They number 20, and only three of them are defined as constant. These are mine life (20 years), ore grade after processing (20% Cu), and grade of the
blister copper (99.99% Cu). The list of the all defined variables in the estimation of NPV is presented in Table II.
In the created model, independent variables are defined as a probability distribution function (PDF). Dependent variables are computed using independent and constant variables in the constructed
model. PDF properties of independent variables are presented in Table III. Ore grade data was estimated by Micromine 10 by the block modelling technique, using each block grade. The distribution of
ore grade is determined in Best Fit application of Palisade. Also, the PDF of the copper selling price and interest rate were determined by Best Fit application using historical data. The PDFs of the
others were defined as normal distributions as seen in Table III. In this study, annual production amount is defined as a fixed value. The constructed model in @Risk environment selects a random
value from the PDF of reserve amount, which is divided by mine life (20 years). Therefore, the fixed annual production value is estimated. This fixed value is used for estimation of a single NPV
value. This cycle is repeated for each NPV value estimation.
Developed model
An uncertainty assessment model was created to simulate the NPV of the deposit. The model selects the uncertain variables from the related PDF randomly. In other words, the model is based on the
Monte Carlo Simulation method. The principle behind the model is estimating revenue and cost per year. Estimation of both involves constants, independent and dependent variables. Annual cost
estimation is more difficult than estimation of annual income because it includes more uncertain variables. Therefore, annual cost is divided into three main parts; namely annual mining cost, annual
processing cost, and annual metallurgical cost. Estimations of the three costs were done independently in the model. After the estimations, their summation gives the annual cost. The estimation
equations for annual costs and annual revenue in are illustrated in Equations [1]-[4]. In these equations, D is density in ton/m3, GAM is grade after metallurgy (It was defined as 99.99%), i is
annual interest rate in percent, MC is mining cost in $/ton, Met. C is metallurgical cost in $/ton, Met. R is metallurgical recovery in percent, ML is mine life (it was defined as 20 years), MR is
mining recovery in percent, n is number of year, OG is ore grade in percent (in situ), OGAP is ore grade after processing in percent (it was defined as 20 percent), PC is processing cost in $/m^3, PR
is processing recovery in percent, SC is stripping cost in $/m3, SP is selling price in $/ton, TO is total overburden in m^3, and TOVis total ore volume, m^3 (in situ).
i = annual interest rate, %
MC = mining cost, $/ton
Met. C = metallurgical cost, $/ton
Met.R = metallurgical recovery, %
ML = mine life (defined as 20 years)
MR = mining recovery, %
n = number of year
OG = ore grade, % (in situ)
OGAP = ore grade after processing, % (defined as 20%)
PC = processing cost, $/m3
PR = processing recovery, %
SC = stripping cost, $/m3
SP = selling price, $/ton
TO = total overburden, m3
TOV = total ore volume, m^3 (in situ).
The profitability of the project can be determined by the NPV of the cash flow, which was found by the summation of the present value (PV) of the annual gross profit (AGP). The model estimates AGP
independently for each year during the mine life. In a part of the model, Equation [5], the PVs of annual gross profits of the deposit are estimated independently, considering each year separately.
In other words, to increase the accuracy of the model, selection of each input value in Equation [5] is not affected by the other years's selected values. For example, in the same scenario (in a
single iteration) the first year's interest rate may be selected as 4.06 percent while the second year's interest rate is selected as 3.98 percent from the distributions. Annual gross profit, annual
income, and annual operating costs were estimated by the model in the similar manner. Another important step in the model is the time value of money. Estimated annual gross profits are used to
calculate NPV of the cash flow using a randomly selected discount rate from the related PDF.
Two important assumptions were made in the construction of the model. The inflation rate was assumed to be zero. It was also accepted that the deposit will be operated by the government and the
annual gross profit will be net profit because there will be no related tax paid to the government, since MTA, which is a government institute, is the holder of the mining licence. Administration,
environmental, and plant costs are not included in this study.
NPV estimation of the deposit under uncertainty
As mentioned previously, the MCS method is used by the model and 10 000 successive iterations were done. This means that 10 000 random scenarios were evaluated for the established uncertainty
assessment model by @Risk 4.5.7 of Palisade. This number of iterations is selected because it is the maximum limit of the software. When iteration was conducted, the input variables were selected
randomly from the related PDF and one output was estimated. In the estimation, 10 000 iterations are conducted and the results of iterations are saved by the @Risk 4.5.7 software. Probability
distribution of NPV of the deposit is estimated using the results obtained by iterations. Therefore, all reliable information for the NPV was gathered.
Each year's net profit was estimated separately from the other years. Therefore, the cash flow of the project could also be observed from the model as seen in Table IV. After estimating the cash
flow, the model finally, estimates 10 000 random NPVs for the Derekoy copper deposit. Using these random values, @Risk 4.5.7 established a probability distribution and cumulative density curve to
indicate the probability of the profit for the deposit. The probability distribution and cumulative density curve are presented in Figure 2 and Figure 3 respectively. The type of the NPV distribution
was checked with 'Fit Distribution' module of @Risk 4.5.7. The output of the fit distribution indicates that the NPV probability distribution is a normal distribution.
After the checking the type of the distribution, the properties of it were investigated. The mean of the NPV probability distribution is $197 126 000 and standard deviation (ó) of the distribution
was found to be $120 709 600. One standard deviation interval (÷±ó = 68. 27%) and two 'standard deviation interval (÷±2ó = 95.45%) were evaluated on the probability distribution. In this case the NPV
will be in the range of $77.97 million and $318.78 million with the probability of 68.27 percent as it seen in Figure 2. Considering the ÷±2ó, the range of NPV will be in the range of -$45. 37
million to $443.54 million with 95. 45 percent probability as seen in Figure 3. The probability of achieving positive NPV (profit) is 94.95 percent and probability of loss of money is only 5. 05
percent as indicated in Figure 4.
IRR estimation of the cash flow under uncertainty and sensitivity analysis
Apart from the NPV assessment of the project, the Internal Rate of Return (IRR) value of the investment analysis with uncertainty assessment was also conducted to check the profitability of the
investment. In the model, a part is developed to estimate the IRR values of the project with the MCS method. The model calculates the possible IRR values and then creates a probability distribution
with these values. The probability distribution for IRR is shown in Figure 5. As presented in the figure, the mean value of the distribution is 19. 2 percent. IRR value is higher than defined Maximum
Allowable Rate of Return(MARR) value, 15 percent, with 62.73 percent probability. In other words, the IRR method indicates that the project is profitable with 62.73 percent probability.
Sensitivity analysis is applied to the project to determine which variables affect the estimation of the NPV of the deposit. It is found that NPV was the most sensitive to grade, and secondly it was
sensitive to the selling price of copper. The results of the sensitivity analysis are shown in Figure 6. This analysis illustrates that grade data should be updated when new data are available during
the operation. The market conditions have also a significant impact on the NPV estimation because of the effect of the selling price.
NPV estimation with certain inputs
Apart from the NPV estimation with considering related uncertainties using the MCS method, NPV of the project was also calculated using the average values of the related variables. In the literature
this type of estimation is called the classical NPV estimation method. The model also calculates this type of estimation by evaluating just one scenario. The same assumptions were used as in with the
previous estimation technique. The average values are ore grade 0.244 percent Cu, density as 2.7 ton/m3, selling price of copper $7,434/ton, interest rate 4.68 percent, mining, processing and
metallurgical recovery 90 percent, 90 percent and, 93 percent respectively, mining, stripping, processing and metallurgical cost as $3.25/ton, $3.25/ton, $4.50/ton of ore, and $100/ton of
concentrate, annual ore mining 9 439 520 ton and yearly stripping cost 7 021 899 m3. Averages of historical data were applied for calculating the selling price and interest rate. It was also assumed
that mine life is 20 years and annual ore production and stripping amount are constant.
The model estimated the NPV as $260 402 962. An IRR value was also estimated as 22.2% for the project. There is a big difference between the simulation and classical method. The output of the
simulation is a PDF. Therefore, the uncertainty of the project can be evaluated. However, the output of the classical method is just a number and it cannot say anything about the estimation errors.
Results and discussion
There are many uncertain variables in the evaluation of the mineral reserves. A good financial model should be created to evaluate the ore reserves. Each uncertainty related to mining should be
assessed carefully. In this case, accurate results can be obtained by the financial model. The decision on the mining investment is mostly related to the NPV of the project. A financial model
construction needs accurate estimations of income and costs. Estimation of the revenue and costs includes many uncertainties. Therefore, a simulation method is the best tool to estimate them.
Simulation can provide many scenarios related to the project. The success of the financial modeling simulation depends on the estimation of the uncertainties accurately. In other words, uncertainty
assessment of the investment is not considered in the classical method. Therefore, the investor cannot answer questions such as, what is the probability of NPV exceeding $100 000 000?, or what is the
probability of losing money? Mining is ae risky operation. When a mining project is evaluated, related uncertainties should be investigated and they should be included in the calculations and
In this study, Derekoy copper deposit was evaluated considering related uncertainties. After the evaluation of the deposit, the probability distribution of NPV was estimated instead of a fixed value
and the type of the distribution was investigated as a normal distribution. The mean of the distribution is found as $197 126 000 and the probability of financial loss is found to be only 5.05
percent and the probability of the NPV exceeding $150 000 000 (capital cost at present) is 65.05 percent as seen in Figure 7.
In classical estimation, a certain value was estimated. This is not a accurate approach to mining with high risks in different stages. The probability of the realization of a NPV equal to or more
than $260 402 962 (estimated by model without uncertainty assessment) is 29. 44 percent as indicated in Figure 8. When the IRR values are analysed, the classical method estimates more IRR value than
the other method. It means that the defined uncertainties affected the estimation of the NPV because the classical model made an overestimation. The overestimation may cause problems, such as losing
money, for the investor because future of the investment is defined by the result of the NPV estimation. Therefore, NPV estimations should be conducted by including related uncertainties.
BADIOZAMANI, K. 1992. Computer Methods. SME Mining Engineering Handbook, (598-625). Colorado. Society for Mining, Metallurgy and Exploration. [ Links ]
CHANCE, D.M. 2008. Lecture notes: Monte Carlo Simulation. Louisiana State University, E. J. Ourso Collage of Business http://www.bus.lsu.edu/academics/finance/faculty /dchance/Instructional/
TN96-03.pdf (accessed July 2008). [ Links ]
DIMITRAKOPOULOS, R. 1998. Conditional simulation algorithms for modelling orebody uncertainty in open pit optimization. International Journal of Surface Mining, Reclamation and Environment, vol. 12.
pp. 173-179. [ Links ]
DOMINY S. C., NOPPE M.A., and ANNELS A.E. 2002. Errors and uncertainty in mineral resource and ore reserve estimation: the importance of getting it right. Exploration Mining Geology, vol. 11. pp.
77-98. [ Links ]
EMERY, X., ORTIZ, J.M., and RODRIGUEZ, J.J. 2006. Quantifying uncertainty in mineral resources by use of classification schemes and conditional simulations. Mathematical Geology, vol. 38. pp.
445-464. [ Links ]
ERDEM, Ö. and GÜYAGÜLER, T. 2009. REServe estimation of Derekoy copper deposit considering the Feasibility of the Project. Third Balkan Mining Congress, Izmir, Turkey, 1-3 October 2009. [ Links ]
MORLEY, C., SNOWDEN, V., and DAY, D. 1999. Financial impact of resource/reserve uncertainty. Journal of the South African Institute of Mining and Metallurgy, vol. 99. pp. 293-301. [ Links ]
NASUF, E. and ORUN, E.S. 1990. Madencilik Projelerinde Mikro-Bilgisayar Destekli Risk Analizi (in Turkish). Madencilik Dergisi, vol. 3. pp. 19-30. [ Links ]
RENDU, J.M. 2002. Geostatistical simulations for risk assessment and decision making: the mining industry perspective. International Journal of Surface Mining, Reclamation and Environment, vol. 16.
pp. 122-133. [ Links ]
REZAIE, K., AMALNIK, N.S., GEREIE, A., OSTADI, B., and SHAKHSENIAEE, M. 2007, Using extended Monte Carlo simulation method for the improvement of risk management: Consideration of relationships
between uncertainties. Applied Mathematics and Computation, vol. 190. pp. 1492-1501. [ Links ]
ROSS, J.G. 2004. Risk and Uncertainty in Portfolio Characterization. Journal of Petroleum Science and Engineering, vol. 44. pp. 41-53. [ Links ]
SNOWDEN, D.V., GLACKEN, I., and NOPPE, M. 2002. Dealing with demands of technical variability and uncertainty along the mine value chain. Paper presented at Value Tracking Symposium.. Brisbane,
Queensland, Australia. [ Links ]
YAMAMOTO, J.K. 1999. Quantification of Uncertainty in ore-reserve estimation: applications to Chapada copper deposit, State of Goias, Brazil. Natural Resource Research, vol. 8. [ Links ]
Paper received Jun. 2011
Revised paper received Sep. 2011 | {"url":"http://www.scielo.org.za/scielo.php?script=sci_arttext&pid=S0038-223X2012000500011&lng=en&nrm=iso","timestamp":"2014-04-20T22:03:14Z","content_type":null,"content_length":"49225","record_id":"<urn:uuid:2293121b-19f7-4d65-8f92-477374374055>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00353-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is ORDINAL DATA IN SPSS?
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All the definitions and meanings found are from third-party authors, please respect their copyright.
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A New Kind of Science: The NKS Forum - Call to coders, a new type of fractal.
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Posts: 712
If you have Mathematica 6, you can program this system and display the results, interactively, in two lines, as follows -
SinFract[start_, anglefactor_, angleadd_, steps_] :=
NestList[Sin[# * anglefactor + angleadd] &, start, steps]
ListLinePlot[SinFract[start, anglefactor, angleadd, steps],
PlotMarkers -> Automatic, AxesOrigin -> {0, 0}, Frame -> True,
PlotRange -> All],
{{start, 0.9}, -1, 1}, {{anglefactor, 1.3}, .5,
6.3}, {{angleadd, .1}, -[Pi]/2, [Pi]/2}, {{steps, 10}, 1, 100, 1}]
The first line is your function, with four parameters. NestList says do this nestedly steps times and leave a trail of the results.
The second line creates an interactive plot with controls for your variables on "sliders".
Even if you don't have Mathematica 6 to make this sort of illustration in a few minutes, you can still use the result, because I used our new "Publish for Player" feature, here -
You can get Mathematica Player, free, from this link -
I will attach this specific interactive notebook below. All you need is Player to use it.
It is a reasonably interesting little system, incidentally. It reminds me closely of the kneading route to chaos system described on 307 of the NKS book. The whole section starting at page 304 is
Report this post to a moderator | IP: Logged | {"url":"http://forum.wolframscience.com/showthread.php?postid=5128","timestamp":"2014-04-16T10:35:29Z","content_type":null,"content_length":"34995","record_id":"<urn:uuid:9a48e045-9e5e-4449-89c7-9aab26f09f82>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00604-ip-10-147-4-33.ec2.internal.warc.gz"} |
RE: [SI-LIST] : Oscillation in lumped circuits and transmission l
Peterson, James F (jfpeterson@space.honeywell.com)
Fri, 5 Feb 1999 13:26:38 -0500
Hey all,
Seems like this question is getting different answers from our group (how
shocking!). I believe Fred makes a good case for saying they're not the same
(see the end of this e-mail).
He said :
"LCR parameters can approximate a transmission line once enough RLC
segments are used. However such a model is NOT dynamic and will
oscillate once the stimulus is made fast enough."
thanks again,
-----Original Message-----
From: Fred Balistreri [mailto:fred@apsimtech.com]
Sent: Friday, February 05, 1999 12:53 PM
To: dima@tdasystems.com
Cc: Peterson, James F (FL51); si-list@silab.Eng.Sun.COM
Subject: Re: [SI-LIST] : Oscillation in lumped circuits and transmission
l ines
Dima Smolyansky wrote:
> Although, it could be noted that the propagation delay of a transmission
line Pd is related to L and C as
> Pd = (LC)^1/2. In addition, the factor of 2*pi is just a conversion
between linear and angular frequency, and therefore for even-handed
comparison must be dropped. Then, the factor of 4 is apparently due to
assumption of quarter
> wavelength oscillation. But then, there will be a similar high-order
oscillation for an LC circuit. I think this implies good correlation between
the two models. The choice between the two comes down to a standard
> lumped-distributed type analysis.
> Hope this helps.
> --
> Thanks,
> -Dima
> =========================
> Dima Smolyansky
> TDA Systems, Inc.
> 7465 SW Elmwood St.
> Portland OR 97223
> 503-977-3629
> 503-245-5684 (FAX)
> dima@tdasystems.com
> http://www.tdasystems.com
> Peterson, James F (FL51) wrote:
> > Good question Arani,
> > I have often wondered the same thing (every time someone uses the term
> > ringing and reflection to describe the same phenomenon). The more I
> > the SI area of engineering the more I notice that people can be
> > things differently based on different perspectives but still be correct.
> > That said, I believe that a lumped LRC circuit's behavior is not related
> > a transmission line reflection. One good reason is that the period of
> > lumped circuit's oscillation will be 2*pi*(LC)^1/2, whereas the period
of a
> > reflection will always be 4*Pd (where Pd is the 1 way prop. time of the
> > signal). In fact this is a great way to tell the difference. If the
> > of a measured signal is much larger than 4 * Pd, then it's a lumped LRC
> > problem (add some series r to get rid of it).
> > Hope this helps,
> > Jim
> >
> >
> > -----Original Message-----
> > From: Arani Sinha [mailto:sinha@poisson.usc.edu]
> > Sent: Friday, February 05, 1999 3:02 AM
> > To: si-list@silab.Eng.Sun.COM
> > Cc: sinha@poisson.usc.edu
> > Subject: [SI-LIST] : Oscillation in lumped circuits and transmission
> > lines
> >
> > Hi,
> >
> > I have the following question.
> >
> > We can model an interconnect as either a lumped circuit or a
> > transmission line. By means of lumped modeling, we can say that
> > it has an oscillatory response if its damping factor is less
> > than 1. By means of transmission line modeling, we can say that
> > it has an oscillatory response if the signal reflection
> > co-efficients at source and load satisfy certain conditions.
> >
> > My question is whether oscillation in a lumped circuit and
> > signal reflection in a transmission line are actually the same
> > phenomenon. If so, there should be a correlation between
> > conditions for oscillation in a lumped circuit and those for
> > oscillation in a transmission line.
> >
> > After many discussions and much thought, I have not been able
> > to determine a correlation. I am also ambivalent about whether
> > they are the same phenomenon.
> >
> > I understand that the damping factor in a lumped circuit is
> > equivalent to the attenuation constant in a transmission line
> > and that condition of no reflection is equivalent to the
> > maximum power transfer theorem.
> >
> > I will really appreciate help in this regard.
> >
> > Thanks,
> >
> > Arani
> >
> > **** To unsubscribe from si-list: send e-mail to
> > majordomo@silab.eng.sun.com. In the BODY of message put: UNSUBSCRIBE
> > si-list, for more help, put HELP. si-list archives are accessible at
> > http://www.qsl.net/wb6tpu/si-list ****
> >
> > **** To unsubscribe from si-list: send e-mail to
majordomo@silab.eng.sun.com. In the BODY of message put: UNSUBSCRIBE
si-list, for more help, put HELP. si-list archives are accessible at
> **** To unsubscribe from si-list: send e-mail to
majordomo@silab.eng.sun.com. In the BODY of message put: UNSUBSCRIBE
si-list, for more help, put HELP. si-list archives are accessible at
It should be noted that Oscillation due to LC lumped parameters and
transmission line behavior are NOT the same thing and there is NO
correlation between the damping resistor used in lumped LC circuits and
attenuation in transmission lines. Theoritical (lossless) transmission
lines (distributed, not lumped) will NOT oscillate when properly
terminated with the known characteristic impedance. This is independent
of the length used since this case is theoritical. Lumped circuit
parameters should NOT be used in place of transmission lines. Lumped
LCR parameters can approximate a transmission line once enough RLC
segments are used. However such a model is NOT dynamic and will
oscillate once the stimulus is made fast enough. Then more RLC
segments will be needed. But the important fact is that the resistance
used to dampen the oscillation bears no relationship to the attenuation
due to copper losses at DC or skin effect losses at higher frequencies.
This is a different issue altogether.
Distributed RC lines used at the IC level work because of the geometries
involved and the fact that aluminum has been used which is not as good
a conductor as copper. That is changing. Even as shrinking occurs the
IC sizes and level of integration is dramatically increasing. This means
the RC approximation is already breaking down. Transmission lines will
have to be used in the not so distant future even for IC's especially
systems on a chip.
Best Regards,
Fred Balistreri
**** To unsubscribe from si-list: send e-mail to majordomo@silab.eng.sun.com. In the BODY of message put: UNSUBSCRIBE si-list, for more help, put HELP. si-list archives are accessible at http://www.qsl.net/wb6tpu/si-list **** | {"url":"http://www.qsl.net/wb6tpu/si-list2/0104.html","timestamp":"2014-04-19T02:27:56Z","content_type":null,"content_length":"10424","record_id":"<urn:uuid:29801fa6-9bc3-4a27-b41d-7f515b8463ec>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00197-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
Prove tht cos A + cos B + cos C <= 2
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A B n C r angles of a triangle
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any one pleease help
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ny hint ?
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is it ever equal to 2 ? i'd say the max occurs when a=b=c i.e. max value is 3/2 ? isnt it ? you can verify from hit and trial ?
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hmn... cn we prove ny inequality for cos a + cos b + cos c
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i like this question
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Inputting A=60,B=90,C=30 cos 60+cos 90+cos 30=1/2+0+[(3)^1/2]/2=[1+(3)^1/2]/2=(1+1.7)/2=2.7/2=1.35 which is less than 2
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@ashwinjohn3 That doesn't count as a proof. =/
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@ParthKohli Then prove it by Mathematical induction.....
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can you use the law of cosines /
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i would argue that since this is symmetric in \(A,B,C\) the max must occur when \(A=B=C=\frac{\pi}{3}\)
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but A , B n C r angles of a triangle.
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How can each angle be pi/2 ? A + B + C = 180 @satellite73 sir
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(its a three not a two)
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Sorry :(
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yea, the same triangle and therefore you cannot tell \(A,B,C\) apart, which is what i meant when i said it is symmetric. you label the triangle one way, i label it another, we have the same thing
which is why the max occurs when they are equal
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But can this hep us to proceed to get the proof ?
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|dw:1356530855111:dw|\[\cos(x) \le 1\]\[then \ \ \cos(90 - x) = \sin(x) \le 1\]\[also \ \ \cos(90) = 0\]Adding all,\[\cos(x) + \cos(90 - x) + \cos(90) \le 2\]BANG
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\[A = x, B = 90 - x, C = 90\]
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but this proof assumes that the triangle is right angled, which may not be the case.
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i dont think we can assume a right angle
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Oh, well.
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yeah right we shall assume for any triangle
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cos(A) + cos(B) + cos(C) = \(\large \cos\frac{A+B}{2}\cos\frac{A-B}{2}+\cos(C)\) = \(\large \sin\frac{C}{2}\cos\frac{A-B}{2}+\cos C\) Let us fix angle C and try to maximise the sum. This will
only happen when \(\large \cos\frac{A-B}{2} = 1\) i.e. \(A=B\). Similarly, if we fix B and maximise the sum, then A=C fix A => sum is maximum when B=C so as long as any two angles are unequal,
the sum can be increased further. => that the maximum occurs when all three are equal. The sum cant be maximised further when A=B=C. So the max. occurs when A=B=C and the max value = 3/2 <2
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but we are also to prove for "equal to" condition
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@mathslover \( n < r\) automatically satisfies \(n \le r\).
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@mathslover indirectly, what I am trying to do here is to show that, if any two angles are unequal, the sum can be increased further. So, if we were trying to find the max value, we would wish to
make all pairs of angles equal so that the sum cannot be further maximised and will have attained its maximum value.
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oh.. right
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is replying to Can someone tell me what button the professor is hitting...
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This is the testimonial you wrote.
You haven't written a testimonial for Owlfred. | {"url":"http://openstudy.com/updates/50daff3ee4b0d6c1d5429dfc","timestamp":"2014-04-18T03:49:22Z","content_type":null,"content_length":"99018","record_id":"<urn:uuid:445f10c4-4f5b-4dce-89bc-a06fddb8ad90>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00137-ip-10-147-4-33.ec2.internal.warc.gz"} |
Combating Keystoning
we have
i.e. the throw distances for the top and bottom of the image are known. Given this information and the beam angle we can calculate the width of the image at top and bottom:
The ratio of the bottom width to the top width gives us the fractional increase in image size over its full height; subtracting this from 1 and multiplying by 100 gives the percentage change in the
size of the top: | {"url":"http://freespace.virgin.net/tom.baldwin/keyst_deriv.html","timestamp":"2014-04-19T22:13:16Z","content_type":null,"content_length":"1995","record_id":"<urn:uuid:b0d602ea-8c06-49bb-a1fc-39db81faee54>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00398-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Haskell-cafe] help understanding lazy evaluation
Derek Elkins derek.a.elkins at gmail.com
Wed Aug 22 19:43:45 EDT 2007
> > [*] Which notation do you use for functions in text? is f() ok?
> Sure, although a little unusual for Haskell where f() means f applied
> to the empty tuple. Some people use |f| (generally those who use
> latex), but generally it can be inferred from the context what is a
> function
Neil's answer was complete I just want to elaborate on this last point.
f is the most logical (and for Haskell things appropriate) notation.
It's an odd historical quirk of mathematical notation that using f(x)
for f is still so common.
More information about the Haskell-Cafe mailing list | {"url":"http://www.haskell.org/pipermail/haskell-cafe/2007-August/030900.html","timestamp":"2014-04-21T08:00:41Z","content_type":null,"content_length":"3286","record_id":"<urn:uuid:ac13127a-8409-4dac-90a0-9c8cc1e2f293>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00361-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: st: list x matrix
Notice: On March 31, it was announced that Statalist is moving from an email list to a forum. The old list will shut down at the end of May, and its replacement, statalist.org is already up and
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Re: st: list x matrix
From richard boylan <richardtb25@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: list x matrix
Date Sun, 28 Mar 2010 13:12:34 -0500
Ok, got it.
The problem was as follows.
The regression is y = x b + e. (1)
However, to estimate it (b/c of a variety of issues such
autocorrelation, system of equation with correlated errors), the model
that I end up estimating is
yt = xt b + z c + v, (2)
where yt is a transformation of y, xt is a transformation of x, and z
are variables from the other regressions.
So, if I request the R^2, I am going to the r-squared for (2), while
the r-squared which I am interested in reporting is the R^2 for (1).
So, what I need to do is to get the estimates of b from (2) and plug
back into (1) to compute my R^2.
On Sun, Mar 28, 2010 at 11:11 AM, Maarten buis <maartenbuis@yahoo.co.uk> wrote:
> --- On Sun, 28/3/10, richard boylan wrote:
>> I saw the documentation for predictnl, and I can see how xb
>> is an option after running an estimation.
>> However, I did not understand how I could use it to compute
>> xb without an estimation.
> The point I made was that you almost never have to reinvent
> that operation yourself, there are many tools inside Stata
> that do it for you. Which tool is right for you depends on
> your situation. I used -predictnl- as an example because it
> is a common situation where one might be tempted to perform
> that operation. If you tell us why you want to perform that
> computation, we can tell you if there is such a tool
> available that is appropriate for your situation.
> -- Maarten
> --------------------------
> Maarten L. Buis
> Institut fuer Soziologie
> Universitaet Tuebingen
> Wilhelmstrasse 36
> 72074 Tuebingen
> Germany
> http://www.maartenbuis.nl
> --------------------------
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2010-03/msg01727.html","timestamp":"2014-04-16T16:17:07Z","content_type":null,"content_length":"9598","record_id":"<urn:uuid:90a08724-46ff-46d9-920d-70097e6fc7aa>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00147-ip-10-147-4-33.ec2.internal.warc.gz"} |
-Ground State
3. ^3Σ-Ground State Molecules
The O[2] and SO molecules are the only diatomic molecules in this compilation which possess a ^3Σ electronic ground state. Since the energy level calculations differ quite markedly from those in
section 2, a detailed description of the calculations will be given here. Although a number of authors have treated this problem in slightly different manners than that discussed below, for
uniformity we have chosen the formulation which corresponds closest to that employed in the previous section.
In order to describe the rotational spectra of this class, Hund's coupling case (b) was chosen as the starting point. The rotational levels are characterized by the rotational angular momentum
quantum number, N, and the resultant angular momentum quantum number, J, which includes the total electron spin angular momentum. If the molecule has nuclei with non-zero nuclear spin, I, these are
coupled to J to form the total angular momentum quantum number F, whereby coupling case (b[βJ]) is assumed here. For pure case (b[βJ]) the electric dipole transitions occur with the selection rules:
ΔN = ± 1, ΔF = 0, ± 1, and ΔJ = 0, ± 1, in the absence of external fields. Since an intermediate coupling case is actually observed, transitions are allowed for ΔN = ± 3. The magnetic dipole
transitions occur with the selection rules: ΔN = 0, ± 2 and ΔJ = 0, ± 1.
a. Molecular Parameters and Energy Level Formulation
The rotational energy levels may be described with the Hamiltonian [8]: λ(3S[z]^2 - S^2) + γ(N · S) + BN^2 whereby a molecule fixed cartesian coordinate system is employed with the z-axis along the
molecular axis. The first term describes the spin-spin interaction, the second term refers to the spin-rotation interaction and the last term describes the rotational kinetic energy. Since the
coefficients λ, γ and B are functions of the internuclear distance, r, centrifugal distortion and vibration-rotation interactions arise. If we define the coefficients as follows:
where the Dunham coefficients, Y[lj ], are defined in section 2 and
The centrifugal distortion terms are defined as:
With these definitions, the rotational energy levels are given in the form [9]:
The sextic terms, H[υ], of the rotational energy are neglected because they cannot be determined from the data presently available for the spectral observations on ^3Σ electronic ground state
molecules. The energy equations are utilized with the selection rules stated above to allow the determination of the molecular constants[ ] B[υ], λ[υ], γ[υ], D[υ], ρ[υ], and δ[υ], for vibrational
state υ. Combining the data available for various vibrational states allows the derivation of potential coefficients, a[i], and the expansion parameters of λ and γ.
Magnetic hyperfine structure has been described by Frosch and Foley [10] in terms of the determinable parameters, b and c. The nuclear electric quadrupole hyperfine structure is described by Amano,
et al. [11] and results in determination of the constant, eQq[υ], as defined in the discussion of ^1Σ ground electronic state molecules.
b. List of Symbols
Symbols (See section 2b for additional definitions.)
a[i] Dunham potential coefficients.
λ[υ]^ Spin-spin coupling parameter in the υ^th vibrational state (MHz).
α[λ]^ Spin-spin vibrational constant (MHz).
γ[υ]^ Spin-rotation coupling parameter in the υ^th vibrational state (MHz).
α[γ]^ Coefficient in the power series expansion of γ[υ].
ρ[υ] Centrifugal distortion correction to λ[υ] (MHz).
δ[υ] Centrifugal distortion correction to γ[υ] (MHz) .
λ[e], λ[(1)], λ[(2)] Expansion coefficients of λ in a power series of ξ.
γ[e], γ[(1)] Expansion coefficients of γ in a power series of ξ.
b, c Magnetic hyperfine coupling constants:
where µ[B] is the Bohr magneton, µ[N] the nuclear magneton and g[N], the nuclear g-valve. | {"url":"http://physics.nist.gov/PhysRefData/MolSpec/Diatomic/Html/sec3.html","timestamp":"2014-04-17T18:38:24Z","content_type":null,"content_length":"12142","record_id":"<urn:uuid:ace4c923-4927-4519-85fb-0a7f35f8787f>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00057-ip-10-147-4-33.ec2.internal.warc.gz"} |
Proof by contradiction
February 27th 2011, 02:40 PM
Proof by contradiction
I am having difficulty with the following question: ( see attachment)
I tried expanding it and i still didnt seem to quite get it.
Any help would be apriciate, thanks in advance. :)
February 27th 2011, 02:56 PM
I am having difficulty with the following question: ( see attachment)
I tried expanding it and i still didnt seem to quite get it.
Any help would be apriciate, thanks in advance. :)
Induction on n is a straighforward way, but you can do as follows as well:
$3^{4n+2}+1=9\cdot (3^4)^n+1=9\cdot (81)^n+1$ , and you're done since 9 times a
number ending in 1 ends in 9...
February 27th 2011, 03:23 PM
If you want to do it by induction here is a trick.
February 27th 2011, 03:54 PM
February 27th 2011, 04:33 PM
February 28th 2011, 10:06 AM
Wonder why the topic says "Proof by contradiction " ?? By induction you mean ? | {"url":"http://mathhelpforum.com/number-theory/172860-proof-contradiction-print.html","timestamp":"2014-04-20T07:40:27Z","content_type":null,"content_length":"7827","record_id":"<urn:uuid:19a5b883-60f1-488c-81f0-9573e6ad6646>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00285-ip-10-147-4-33.ec2.internal.warc.gz"} |
Experimental determination of the linear and nonlinear dynamic
ASA 130th Meeting - St. Louis, MO - 1995 Nov 27 .. Dec 01
2pPA7. Experimental determination of the linear and nonlinear dynamic moduli of rock from quasistatic measurements.
L. Zhu
Phys. Dept., New Mexico State Univ., Las Cruces, NM 88003
R. A. Guyer
Univ. of Massachusetts, Amherst, MA 01003
K. R. McCall
Los Alamos Natl. Lab., Los Alamos, NM 87545
G. N. Boitnott
New England Res., Inc., White River Junction, VT 05001
L. B. Hilbert, Jr.
Univ. of California, Berkeley, CA 94720
T. J. Plona
Schlumberger-Doll Res., Ridgefield, CT 06877
The central construct of a new theory of the elastic behavior of consolidated materials is the density in Preisach--Mayergoyz (PM) space. PM space is an abstract space in which the response of the
mechanical units in the material to changes in stress state can be tracked. The theory provides a recipe for using quasistatic data to determine (rho)[inf PM], the density of mechanical units in PM
space. This recipe has been applied to quasistatic stress/strain data on three sandstones samples: (a) Berea I, (b) Berea II, and (c) Castlegate. The density of mechanical units (rho)[inf PM] was
found for each sample. From (rho)[inf PM] the dynamic behavior of the samples can be predicted. Using the experimentally determined (rho)[inf PM] for each of the three samples the strain response to
complicated stress protocols is predicted and the linear and nonlinear dynamic moduli of the samples are found as a function of pressure. The predictions agree well with experiments that test them. | {"url":"http://www.auditory.org/asamtgs/asa95stl/2pPA/2pPA7.html","timestamp":"2014-04-17T12:49:27Z","content_type":null,"content_length":"2086","record_id":"<urn:uuid:ed4ad8e0-0e50-4bc9-8570-e27211cdac80>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00213-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mechanics – A car
May 3rd 2006, 09:19 AM
Mechanics – A car
Any help with this would be most appreciated. Thanks in advance
The EMF (Effective Motive Force) for a car is the engine driving force minus friction and air resistance.
a)A car of mass 1000Kg travels from rest with constant acceleration to 25 m/s in 30 seconds. Use the impulse-momentum relationship to find the EMF?
b)Now suppose that instead the EMF is made by the driver to increase uniformly during the 30 seconds (by gradually increasing foot pressure on the accelerator). Use the impulse-momentum
relationship to find the final speed reached?
c)Compare a) and b)?
May 3rd 2006, 12:05 PM
Originally Posted by Natasha1
Any help with this would be most appreciated. Thanks in advance
The EMF (Effective Motive Force) for a car is the engine driving force minus friction and air resistance.
a)A car of mass 1000Kg travels from rest with constant acceleration to 25 m/s in 30 seconds. Use the impulse-momentum relationship to find the EMF?
b)Now suppose that instead the EMF is made by the driver to increase uniformly during the 30 seconds (by gradually increasing foot pressure on the accelerator). Use the impulse-momentum
relationship to find the final speed reached?
c)Compare a) and b)?
The impulse-momentum theorem is:
$\overline{\sum F} \Delta t = \Delta p$ where $\overline{\sum F}$ is the average net force applied to the object.
a) The acceleration is constant, so the net force applied is constant. I will call the net force F for simplicity.
$F \Delta t = \Delta p = m \Delta v$
$F = m \frac{ \Delta v}{ \Delta t} = 1000 \frac{25}{30} \, N$
Thus F = 833 N.
b) The acceleration is now a linear function of time over the first 30 seconds of motion. Thus the net force is also a linear function of time. Using the Calculus version of the impulse-momentum
$\int_{t_0}^t F \,dt = \int_{v_0}^v m \, dv$
The problem I am having here is that the acceleration is a linear function of time, and we don't have the constant: $a(t) = ct$, giving $F(t)=ma(t)=mct$.
In terms of c:
$\int_0^{30}mct \, dt = \int_0^v m \, dv$
$\frac{1}{2}mct^2|_0^{30} = mv|_0^v$
So $v = 450c \, \, m/s$. | {"url":"http://mathhelpforum.com/calculus/2802-mechanics-car-print.html","timestamp":"2014-04-17T23:24:57Z","content_type":null,"content_length":"7757","record_id":"<urn:uuid:ff9ad188-8d25-467e-aea0-8d938005e352>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00392-ip-10-147-4-33.ec2.internal.warc.gz"} |
Voting Paths
Abraham, David J and Kavitha, Telikepalli (2010) Voting Paths. In: Siam Journal On DiscreteMathematics, 24 (2). pp. 520-537.
VOTING_PATHS.pdf - Published Version
Restricted to Registered users only
Download (278Kb) | Request a copy
We consider a variant of the popular matching problem here. The input instance is a bipartite graph $G=(\mathcal{A}\cup\mathcal{P},E)$, where vertices in $\mathcal{A}$ are called applicants and
vertices in $\mathcal{P}$ are called posts. Each applicant ranks a subset of posts in an order of preference, possibly involving ties. A matching $M$ is popular if there is no other matching $M'$
such that the number of applicants who prefer their partners in $M'$ to $M$ exceeds the number of applicants who prefer their partners in $M$ to $M'$. However, the “more popular than” relation is not
transitive; hence this relation is not a partial order, and thus there need not be a maximal element here. Indeed, there are simple instances that do not admit popular matchings. The questions of
whether an input instance $G$ admits a popular matching and how to compute one if it exists were studied earlier by Abraham et al. Here we study reachability questions among matchings in $G$,
assuming that $G=(\mathcal{A}\cup\mathcal{P},E)$ admits a popular matching. A matching $M_k$ is reachable from $M_0$ if there is a sequence of matchings $\langle M_0,M_1,\dots,M_k\rangle$ such that
each matching is more popular than its predecessor. Such a sequence is called a length-$k$ voting path from $M_0$ to $M_k$. We show an interesting property of reachability among matchings in $G$:
there is always a voting path of length at most 2 from any matching to some popular matching. Given a bipartite graph $G=(\mathcal{A}\cup\mathcal{P},E)$ with $n$ vertices and $m$ edges and any
matching $M_0$ in $G$, we give an $O(m\sqrt{n})$ algorithm to compute a shortest-length voting path from $M_0$ to a popular matching; when preference lists are strictly ordered, we have an $O(m+n)$
algorithm. This problem has applications in dynamic matching markets, where applicants and posts can enter and leave the market, and applicants can also change their preferences arbitrarily. After
any change, the current matching may no longer be popular, in which case we are required to update it. However, our model demands that we switch from one matching to another only if there is
consensus among the applicants to agree to the switch. Hence we need to update via a voting path that ends in a popular matching. Thus our algorithm has applications here.
Item Type: Journal Article
Additional Information: Copyright of this article belongs to Society for Industrial and Applied Mathematics.
Keywords: matchings;bipartite graphs;one-sided preference lists
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 12 Jul 2010 07:22
Last Modified: 22 Feb 2012 10:14
URI: http://eprints.iisc.ernet.in/id/eprint/29187
Actions (login required) | {"url":"http://eprints.iisc.ernet.in/29187/","timestamp":"2014-04-19T04:50:08Z","content_type":null,"content_length":"29504","record_id":"<urn:uuid:eafaea6c-8784-4e6e-9224-d7beb975cb37>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00136-ip-10-147-4-33.ec2.internal.warc.gz"} |
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Any of the numbers or symbols in mathematics that when multiplied together form a product. For example, 3 is a factor of 12, because 3 can be multiplied by 4 to give 12. Similarly, 5 is a factor
of 20, because 5 times 4 is 20 (cf What are Multiples Discussion).
A set of numbers formed by adding the last two numbers to get the next in the series: 0, 1, 1, 2, 3, 5, 8, 13. Named for Leonardo of Pisa, an Italian mathematician of the Middle Ages, who called
himself Fibonacci, short for filius Bonacci which means "son of Bonacci". The original problem he investigated in1202 A.D. was about how fast rabbits could breed under ideal circumstances. His
research led to the construction of this unique set of numbers (cf Recursion Discussion).
Term coined by Benoit Mandelbrot in 1975, referring to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some
piece of the object is a scaled down version of the previous iteration (cf Properties of Fractals Discussion, also Plane Figure Fractals Discussion).
A rational number of the form a/b where a is called the numerator and b is called the denominator (cfFractions Discussion, Percents Discussion).
The number of items occurring in a given category.
An approach taken by mathematicians and scientists to determine the chances of an event happening by repeating the experiment many times and using the results to calculate the probability. See
theories of probability (cf Probability vs. Statistics Discussion).
A function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important (cf Functions as
Processes or Rules Discussion).
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z | {"url":"http://www.shodor.org/interactivate1.0/dictionary/f.html","timestamp":"2014-04-17T21:24:02Z","content_type":null,"content_length":"4818","record_id":"<urn:uuid:9e852f19-0d92-45c3-bd4e-45748fe686fa>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00285-ip-10-147-4-33.ec2.internal.warc.gz"} |
The multi-tree approach to reliability in distributed networks
Results 1 - 10 of 55
- SIAM Journal on Computing , 1993
"... Two tasks of communication in a multi-hop synchronous radio network are considered: point-to-point communication and broadcast (sending a message to all nodes of a network). Efficient protocols
for both problems are presented. Even though the protocols are probabilistic, it is shown how to acknowled ..."
Cited by 69 (1 self)
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Two tasks of communication in a multi-hop synchronous radio network are considered: point-to-point communication and broadcast (sending a message to all nodes of a network). Efficient protocols for
both problems are presented. Even though the protocols are probabilistic, it is shown how to acknowledge messages deterministically. Let n, D, and Δ be the number of nodes, the diameter and the
maximum degree of our network, respectively. Both protocols require a setup phase in which a BFS tree is constructed. This phase takes O ((n + Dlogn)logΔ) time. After the setup, k point-to-point
transmissions require O ((k +D)logΔ) time on the average. Therefore the network allows a new transmission every O (logΔ) time slots. Also, k broadcasts require an average of O ((k +D)logΔlogn) time.
Hence the average throughput of the network is a broadcast every O(logΔlogn) time slots. Both protocols pipeline the messages along the BFS tree. They are always successful on the graph spanned by
the BFS tree. Their probabilistic behavior refers only to the running time. Using the above protocols the ranking problem is solved in O (nlognlogΔ) time. The performance analysis of both protocols
constitutes a new application of queueing theory.
, 1986
"... [LEC-67] linear time serial algorithm for testing planarity of graphs uses the linear time serial algorithm of [ET-76] for st-numbering. This st-numbering algorithm is based on depth-first
search (DFS). A known conjecture states that DFS, which is a key technique in designing serial algorithms, is n ..."
Cited by 42 (2 self)
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[LEC-67] linear time serial algorithm for testing planarity of graphs uses the linear time serial algorithm of [ET-76] for st-numbering. This st-numbering algorithm is based on depth-first search
(DFS). A known conjecture states that DFS, which is a key technique in designing serial algorithms, is not amenable to poly-log time parallelism using "around linearly" (or even polynomially) many
processors. The first contribution of this paper is a general method for searching efficiently in parallel undirected graphs, called ear-decomposition search (EDS). The second contribution
demonstrates the applicability of this search method. We present an efficient parallel algorithm for st-numbering in a biconnected graph. The algorithm runs in logarithmic time using a linear number
of processors on a concurrentread concurrent-write (CRCW) PRAM. An efficient parallel algorithm for the problem did not exist before. The problem was not even known to be in NC. 1. Introduction We
define the problems ...
, 1995
"... A certificate for the k connectivity y of a graph G = (V; E) is a subset E 0 of E such that (V; E 0 ) is k connected iff G is k connected. Let n = jV j and m = jEj. A certificate is called
sparse if it has size O(kn). We present a distributed algorithm for computing sparse certificate for k co ..."
Cited by 16 (1 self)
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A certificate for the k connectivity y of a graph G = (V; E) is a subset E 0 of E such that (V; E 0 ) is k connected iff G is k connected. Let n = jV j and m = jEj. A certificate is called sparse if
it has size O(kn). We present a distributed algorithm for computing sparse certificate for k connectivity whose time complexity is O(k(D+n 0:614 )) where D is the diameter of the network. A new
algorithm for identifying biconnected components is also presented. This algorithm is significantly simpler than many existing algorithms and can be implemented in a distributed environment to run in
O(D+n 0:614 ) time. Both algorithms improve on the previous best known time bounds. Our main focus in this paper is the time complexity. However, no more than a polynomial number of messages, each of
size O(log n), are generated by the algorithm. 1 Introduction Connectivity is an important property of graphs with many applications in computer science. We study the distributed time complexity o...
- Elsevier COMNET , 2005
"... Abstract — Multipath routing (MPR) is an effective strategy to achieve robustness, load balancing, congestion reduction, and increased throughput in computer networks. Disjoint multipath routing
(DMPR) requires the multiple paths to be link- or node-disjoint. Both MPR and DMPR poses significant chal ..."
Cited by 14 (5 self)
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Abstract — Multipath routing (MPR) is an effective strategy to achieve robustness, load balancing, congestion reduction, and increased throughput in computer networks. Disjoint multipath routing
(DMPR) requires the multiple paths to be link- or node-disjoint. Both MPR and DMPR poses significant challenges in terms of obtaining loop-free multiple (disjoint) paths and effectively forwarding
the data over the multiple paths, the latter being particularly significant in IP datagram networks. This paper develops a two-disjoint multipath routing strategy using colored trees. Two trees, red
and blue, that are rooted at a designated node called the drain are formed. The paths from a given source to the drain on the two trees are link- or node-disjoint. Such an approach requires every
node to maintain only two preferred neighbors for each destination, one on each tree. This paper (1) formulates the problem of colored-trees construction as an integer linear program (ILP); and (2)
develops the first distributed algorithm to construct the colored trees using only local information. We demonstrate the effectiveness of the distributed algorithm by evaluating it on grid and random
topologies and comparing to the optimal obtained by solving the ILP. I.
- IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATION , 2003
"... In this paper, we study quality-of-service (QoS) and quality-of-protection (QoP) issues in redundant tree based preplanned recovery schemes for a single-link failure in two-edge connected graphs
and for a single-node failure in two-connected graphs. We present schemes (to be called G-MFBG schemes) ..."
Cited by 13 (0 self)
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In this paper, we study quality-of-service (QoS) and quality-of-protection (QoP) issues in redundant tree based preplanned recovery schemes for a single-link failure in two-edge connected graphs and
for a single-node failure in two-connected graphs. We present schemes (to be called G-MFBG schemes) that generalize the schemes (to be called MFBG schemes) developed by Médard et al. to construct a
pair of redundant trees, called red and blue trees, which guarantees fast recovery from any single-link/node failure, as long as the failed node is not the root node. Using the G-MFBG schemes, we
study QoS issues relating to red/blue trees. We present effective heuristics for computing a pair of redundant trees with low average delay or small total cost. We develop an optimal algorithm for
computing a pair of red/blue trees with maximum bandwidth. Furthermore, a pair of red/blue trees guarantees fast recovery from simultaneous multiple failures if it satisfies certain properties. This
leads us to define the concept of QoP of a pair of red/blue trees. We present an effective heuristic to construct a pair of red/blue trees with high QoP. The paper concludes with a discussion of
computational results that demonstrate the effectiveness of the different algorithms presented.
, 1996
"... A k-path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing k-path
queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
Cited by 11 (2 self)
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A k-path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing k-path queries,
with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes k-path queries
in output-sensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar
graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
"... Let G = (V, E) be a directed graph and n denote |V|. We show that G is k-vertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k- I)-dimensional space
Rk-l, ~ : V ~Rk-l, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f( ..."
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Let G = (V, E) be a directed graph and n denote |V|. We show that G is k-vertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k- I)-dimensional space Rk-l, ~
: V ~Rk-l, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f(v) is in the convex hull of {~(w) I (v, W) G E}. This result generalizes to directed graphs the
notion of convex embedding of undirected graphs introduced by Linial, LOV6SZ and Wigderson in ‘Rubber bands, convex embedding and graph connectivity, ” Combinatorics 8 (1988), 91-102. Using this
characterization, a directed graph can be tested for k-vertex connectivity by a Monte Carlo algo-rithm in time O((M(n) + nkf(k)). (log n)) with error probability < l/n, and by a Las Vegas algorithm
in ex-pected time O((lf(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (Al(n) = 0(n2.3755)). Our Monte Carlo algo-rithm improves on the best
previous deterministic and randomized time complexities for k> no. *9; e.g., for k = @, the factor of improvement is> n0.G2. Both al-gorithms have processor efficient parallel versions that run in O
((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (logn) times the respective sequential time complexi-ties. Our Monte Carlo parallel algorithm improves on
the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of s-t
numberings, we give a combinatorial construction of a directed s-t nulmberiug for any 2-vertex connected directed graph.
- Discrete Applied Mathematics , 1997
"... The problem of finding an implicit representation for a graph such that vertex adjacency can be tested quickly is fundamental to all graph algorithms. In particular, it is possible to represent
sparse graphs on n vertices using O(n) space such that vertex adjacency is tested in O(1) time. We show he ..."
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The problem of finding an implicit representation for a graph such that vertex adjacency can be tested quickly is fundamental to all graph algorithms. In particular, it is possible to represent
sparse graphs on n vertices using O(n) space such that vertex adjacency is tested in O(1) time. We show here how to construct such a representation efficiently by providing simple and optimal
algorithms, both in a sequential and a parallel setting. Our sequential algorithm runs in O(n) time. The parallel algorithm runs in O(log n) time using O(n=log n) CRCW PRAM processors, or in O(log n
log n) time using O(n = log n log
- IN PROCEEDINGS OF IEEE INFOCOM , 2005
"... ... elegant recovery scheme (known as the MFBG scheme) using redundant trees. Xue, Chen and Thulasiraman extended the MFBG scheme and introduced the concept of quality of protection (QoP) as a
metric of multifailure recovery capabilities for single failure recovery schemes. In this paper, we present ..."
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... elegant recovery scheme (known as the MFBG scheme) using redundant trees. Xue, Chen and Thulasiraman extended the MFBG scheme and introduced the concept of quality of protection (QoP) as a metric
of multifailure recovery capabilities for single failure recovery schemes. In this paper, we present three linear time algorithms for constructing redundant trees for single link failure recovery in
2-edge connected graphs and for single node failure recovery in 2-connected graphs. Our first algorithm aims at high QoP for single link recovery schemes in 2-edge connected graphs. The previous best
algorithm has a running time of O(n 2 (m + n)), wherenand m are the number of nodes and links in the network. Our algorithm has a running time of O(m + n), with comparable performance. Our second
algorithm aims at high QoS for single link recovery schemes in 2-edge connected graphs. Our algorithm improves the previous best algorithm with O(n 2 (m + n)) time complexity to O(m + n) time
complexity with comparable performance. Our third algorithm aims at high QoS for single node recovery schemes in 2-connected graphs. Again, our algorithm improves the previous best algorithm with O(n
2 (m + n)) time complexity to O(m + n) time complexity with comparable performance. Simulation results show that our new algorithms outperform previously known linear time algorithms significantly in
terms of QoP or QoS, and outperform other known algorithms in terms of running time, with comparable QoP of QoS performance.
- Mathematical Programming , 1998
"... Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in
combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems ..."
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Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in
combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems on matroids: finding an optimum matroid basis and packing disjoint matroid bases.
Applications of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts. An optimum matroid basis is typically found by a greedy algorithm that grows an
independent set into an the optimum basis one element at a time. This continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm. We
simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a given fixed basis is optimum, showing that the two problems can be
solved in roughly the same ... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=413456","timestamp":"2014-04-17T23:02:38Z","content_type":null,"content_length":"42374","record_id":"<urn:uuid:98f99d59-fc77-4ad8-9368-a30c30581aa3>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00526-ip-10-147-4-33.ec2.internal.warc.gz"} |
[BioC] Limma and Affymetrix ?
Stan Smiley swsmiley at genetics.utah.edu
Wed Feb 11 23:39:31 MET 2004
I have been looking for some examples of limma with Affymetrix data. There
is a simple example in the documentation, but not a case study as with the
dye swap experiments, and I am confused as to how to proceed with using
limma on my data.
I need to eventually analyze a 2^4 factorial experiment, but I'd like to
start with a smaller example so I can get the idea first. I know what
contrasts are of interest to me, but I am unclear on how to use them with
ie. For a 2^3 factorial experiment:
3 Factors A,B,C ; each at two levels, high and low.
Treat Factors
ments A B C
c low low low
a high low low
b low high low
c low low high
ab high high low
ac high low high
bc low high high
abc high high high
If I'm thinking about this correctly, the seven effects I'm looking for
would be:
Single: For A,B or C alone
2x interactions: AB, AC and BC
3x interaction : ABC
With corresponding contrasts of:
4A = -c + a - b + ab - c + ac - bc + abc
4B = -c - a + b + ab - c - ac + bc + abc
4C = -c - a - b - ab + c + ac + bc + abc
4AB = +c - a - b + ab + c - ac - bc + abc
4AC = +c - a + b - ab - c + ac - bc + abc
4BC = +c + a - b - ab - c - ac + bc + abc
4ABC = -c + a + b - ab + c - ac - bc + abc
I'm unsure how to proceed in getting significant genes
from each contrast.
Any help and/or references to examples etc. would be greatly
Thanks for reading this far!
Stan Smiley
stan.smiley at genetics.utah.edu
More information about the Bioconductor mailing list | {"url":"https://stat.ethz.ch/pipermail/bioconductor/2004-February/003774.html","timestamp":"2014-04-19T19:43:23Z","content_type":null,"content_length":"3848","record_id":"<urn:uuid:8c815c4c-7697-49c7-9fcc-f361c325f986>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00185-ip-10-147-4-33.ec2.internal.warc.gz"} |
Elliptic curves and algebraic stacks
up vote 3 down vote favorite
I am a student almost without background on algebraic geometry (but I do know basic graduate algebra and topology). Now I am trying to understand something about algebraic stacks.
I want to start with this short AMS article first: http://www.ams.org/notices/200304/what-is.pdf
This article tries to define algebraic stacks by introducing the notion of the moduli space of elliptic curves. However, I do not know anything about elliptic curves and there are just too many
possible books.
Can someone give some suggestions about how to proceed to learn enough elliptic curves so as to understand something about algebraic stacks? I also welcome other suggestions. Thank you.
17 Learn algebraic geometry first... – zeb Jul 20 '10 at 16:12
12 The answers of zeb and Ryan Reich may seem a little harsh, but I think they are also realistic -- it is unlikely that you will be able to acquire a good working knowledge of algebraic stacks
without a rather strong background in algebraic geometry. On the other hand, is it possible that you don't actually need a good working knowledge but just have some specific question or issue
which is phrased in the language of stacks and you need to be "decoded"? If so, try asking that here. – Pete L. Clark Jul 20 '10 at 16:27
To paraphrase Pete, it might help focus the answers if you told us your motivation for trying to learning stacks. The subject arose out of the needs of algebraic geometers, and that's the
3 traditional path to it, but in principle there could be others. Perhaps for what you want, it might be sufficient to look up the definition of an orbifold, which arose independently in geometry/
topology. On the other hand, elliptic curve theory is beautiful subject, and learning it would be time well spent. However, it's going to be a long road to stacks, be patient. – Donu Arapura Jul
20 '10 at 21:24
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5 Answers
active oldest votes
You should really learn algebraic geometry first. After you're done with that, try reading Mumford's paper "Picard Groups of Moduli Problems", which is a fascinating window into the mind of
up vote one of the people who later invented algebraic stacks as he was himself figuring out what you are also trying to figure out. From that, you should have a sketch of how the properties of
8 down elliptic curves contribute to the properties of their moduli stack; make sure you work out or look up the various things he says about them. Then you can read a regular treatise on stacks.
1 Telling someone "you should learn algebraic geometry" is like telling them "you should learn math." Obviously you don't mean all of it, but it's not clear which parts are necessary/
relevant, or where the stopping point should be. – Charles Staats Jul 20 '10 at 22:06
That's true. It's not clear which parts are necessary, but I think in fact that just about everything is relevant since so much of the theory of schemes has word-for-word generalizations
1 to stacks. However, simply to understand the definition of an algebraic stack, you need at least to really appreciate the concept of base change (i.e. fibered product) and know what a
smooth morphism is. To understand what is going on, you also have to love the functor of points. This is experiential knowledge; you can't research it, you can only assimilate it. Hence,
"learn algebraic geometry first". – Ryan Reich Jul 20 '10 at 22:20
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It is rather simple to study elliptic curve themselves. But since your objective is stacks, you are going to have to be much more abstract than is usual. If you want to avoid algebraic
geometry as much as possible, you could take the analytic approach. For this, you would need:
1. A sound footing in Complex Analysis(which I assume you have)
2. And you would need to know some very basic things about compact Riemann surfaces. These aren't that hard, provided you know complex variables. Compact Riemann surfaces are like
manifolds; just that the local model is that of an open set in a complex plane and the transition maps are analytic.
3. Next is the fact to keep in your mind that "smooth algebraic curves over the complex numbers" and "Compact Riemann surfaces" are one and the same thing. Here you are to think of
algebraic curves as curves specified by algebraic equations, and you are to define smoothness by the Jacobian criterion.
4. Elliptic curves can then be studied using the analytic approach. They are nothing but the complex plane modulo some lattice, ie 1 dimensional complex tori, which is also the same thing
as genus $1$ algebraic curves with one marked point.
up vote 5 5. Then, there is the notion of an analytic space. The local model here is that approximately of the zero set of some finite number of analytic functions, and you glue such up using
down vote analytic transition maps.
6. An elliptic curve $E$ over an analytic space $S$ can be thought of as a morphism $E \to S$ with a section $0$, with some properties being satisfied, in order that each fiber would then
be an elliptic curve.
And I suppose you could then read the AMS notices article you cite.
But I strongly suggest that you do not follow the above approach, as you would miss out on all the interesting algebraic geometry. I suggest that you first learn it well(boosted by some
background of compact Riemann surfaces). This is not easy and is best done through a course given by some teacher. And then you can read stacks with the illuminating examples which would
all appear trivial to you once you know the requisite background.
For elliptic curves themselves, you could try Knapp's book for an introduction.
1 Since nobody has mentioned it yet, I would suggest Frances Kirwan's book "Complex Algebraic Curves" for an introduction to algebraic curves and Riemann Surfaces. – Adrián Barquero Jul
21 '10 at 4:15
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I think it is possible to go quite a few steps towards an understanding of stacks, and even that of elliptic curves, without mastering lots of Algebraic Geometry:
There is a wonderful, friendly written, undergraduate readable book which starts out introducing moduli spaces (and later gives a rough idea abut stacks as well): It is Kock/Vainsencher: An
Invitation to Quantum Cohomology
Chapter 1-2, maybe 3, can give you a feeling of how you can work with moduli objects, and are a pleasant reading experience. You will get to know moduli stacks/spaces of curves, close
enough to elliptic curves for a start, maybe (those latter are trickier to compactify, but first first you have to get far enough to want to to do that, and this book is wonderful for that
A very readable and short introductory source for Algebraic Stacks is then Gomez: Algebraic Stacks
up vote 5 You say you have no background in Algebraic Geometry - maybe you should know that the concept of stack is not limited to the world of Algebraic Geometry. If you are more comfortable with
down vote Topology or Differential Geometry, maybe this helps: Heinloth: Some notes on Differentiable Stacks
If you are categorically minded and can read french, there is a very nice master course on Stacks by Bertrand Toen. It is very abstract - you can fill in your favourite geometric context -
and will leave a considerable gap to the study of the concrete moduli stack of elliptic curves, but it explains very well the other, non-moduli, motivations for introducing stacks and
algebraic spaces, e.g. "bad quotients". The good thing about is that you won't feel a lack of algebro-geometric background, even "scheme" is defined - but you need categorical background.
Finally, again thinking of your non-Algebraic-Geometry background, the moduli stack of elliptic curves is used in Algebraic Topology (maybe this is your motivation to study it?) and there
are some introductions (quite high-level though) aimed at people with an according background. You can look around here
Have fun exploring this nice topic!!
add comment
The article of Richard Hain "Lectures on moduli spaces of elliptic curves", arXiv:0812.1803, is a very good introduction to the subject, which contains a lot of motivations. Here you
up vote 5 down can learn many things about orbifolds, and there is also an appendix on stacks.
I want to second Hain's lecture notes. They're very gentle and concrete. – Andy Putman Jul 20 '10 at 19:56
add comment
I basically agree with Anweshi on this one.In fact,one of the traditional routes for breaking slowly into algebraic geometry is to learn elliptic curves first as one the critical classes of
algebraic curves. From there, you would learn thier generalization as varieties,then schemes,then stacks.
The definitive beginning text on elliptic curves is Joseph Silverman's The Arithmetic of Elliptic Curves-it only requires basic graduate algebra and complex analysis. After that,you could
try Fulton's classic on algebraic curves-still probably the single best book on classical algebraic geometry.
up vote 0 My best recommendation for you,however,is Rick Miranda's Algebraic Curves And Riemann Surfaces,available through the AMS. Miranda's book is an attempt to introduce modern algebraic geometry
down vote to graduate students by beginning with a complete development of complex algebraic curves and then using them as concrete examples of modern algebraic geometric structures like varieties
and schemes. The last part of the book develops all the machinery of AG in this context, all the way up to sheaves and deformation theory.
If you want a bridge from basic algebra and analysis into modern algebraic geometry,I can't think of a better source.
4 While I do like the three books you mentioned, your description of them is bizarre (have you actually tried to read any of them?). Silverman is a much more advanced book than Fulton... –
Andy Putman Jul 20 '10 at 19:34
Andrew might be referring to earlier editions of Fulton. As far as I can tell, the earlier editions incorporated a lot more material than the current one. – Qiaochu Yuan Jul 20 '10 at
A comment on Miranda: it's very gentle, and consequently not very concise. – Charles Staats Jul 20 '10 at 22:09
@Qiaochu : Are you sure? Looking at the prefaces to the 1989 and 2008 versions (available on Fulton's webpage here : math.lsa.umich.edu/~wfulton/CurveBook.pdf ), it appears that he only
performed minor revisions. @Charles : Yes, I agree. It's a very nice place to learn standard material about compact Riemann surfaces (Riemann-Roch, Abel's thm, etc), but the second half
is verbose enough that I haven't read much of it. – Andy Putman Jul 20 '10 at 22:35
add comment
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The definite integral and applications to area, volume, work, differential equations, etc. Sequences and series, vectors and analytic geometry in 2 and 3-space, polar coordinates, and parametric
equations. 4 hours discussion. (005507)
MATH 121X Calculus Problem Session 1.0 FS
Prerequisites: Concurrent enrollment in MATH 121, faculty permission.
Designed to supplement MATH 121 with additional applications and expanded explanations of concepts encountered in second-semester calculus. Provides the student with the opportunity for additional
assistance in coming to an understanding of the concepts of calculus. 3 hours independent study. Credit/no credit grading. (005511)
MATH 195 Project MATH Seminar Year 1 1.0 FS
The Project M.A.T.H. Seminar - Year 1 is a biweekly seminar for students in their first year of Project M.A.T.H., an innovative program for students interested in becoming secondary mathematics
teachers. Students work with mentor teachers, prepare and present lessons, and participate in a structured early field experience. Completion of the seminar series satisfies the Credential Program's
Early Field Experience requirement. 1 hour seminar. You may take this course more than once for a maximum of 2.0 units. Credit/no credit grading. (020431)
MATH 198 Special Topics 1.0 -3.0 FS
Prerequisites: Completion of ELM requirement.
This course is for special topics offered for 1.0-3.0 units. Typically the topic is offered on a one-time-only basis and may vary from term to term and be different for different sections. See The
Class Schedule for the specific topic being offered. 3 hours discussion. (005528)
MATH 199 Special Problems 1.0 -3.0 FS
This course is an independent study of special problems offered for 1.0-3.0 units. 9 hours supervision. You may take this course more than once for a maximum of 6.0 units. Credit/no credit grading. (
MATH 217 Discrete Mathematical Structures 3.0 FA
Prerequisites: Completion of ELM, MATH 119 (or equivalent), CSCI 111.
A rigorous introduction to discrete mathematical structures for computer science majors (fulfills a requirement for the minor in math). Topics include propositional and predicate calculus; basic
proof methods; sets, functions, and operations with them; algorithms and their complexity; applications of number theory to computer science and computer security; matrices and matrix arithmetic;
mathematical induction, recursive definitions and algorithms; combinatorics and counting techniques; relations and their representation by matrices and digraphs, applications to databases;
equivalence relations and partitions of sets; partially ordered sets, lattices, and Boolean algebras; Boolean functions and circuits; graphs, trees, and their applications; formal languages and
grammars; finite-state automata and language recognition, regular languages. 3 hours discussion. (005550)
MATH 220 Analytic Geometry and Calculus 4.0 FS
Prerequisites: MATH 121.
Vector functions and space curves. Functions of several variables, partial derivatives, and multiple integrals. Vector calculus line integrals, surface integrals, divergence/curl, Green's Theorem,
Divergence Theorem, and Stokes' Theorem. 4 hours discussion. (005508)
MATH 220X Calculus Problem Session 1.0 FS
Corequisites: MATH 220.
Designed to supplement MATH 220 with broader and deeper applications of calculus, providing students with opportunities for additional problem-solving skill building. Twenty hours activity minimum
for credit, but 40 hours are available to students. 3 hours independent study. Credit/no credit grading. (020358)
MATH 230 An Introduction to Computational Mathematics 3.0 SP
Prerequisites: MATH 121, no previous computer experience required.
An introduction to the use of mathematical computer software. This course provides an introduction to a programming environment, preparing math majors to use computers to explore and solve varied
math problems. The software used in this class depends on the instructor and may be chosen from Mathematica, GP/PARI, GAP, SAS, R, etc. This course satisfies the computer literacy requirement for
mathematics majors. 3 hours discussion. You may take this course more than once for a maximum of 9.0 units. (005526)
MATH 235 Elementary Linear Algebra 3.0 FS
Prerequisites: MATH 121.
Matrices, determinants, cartesian n-space (basis and dimension of a subspace, rank, change of basis), linear transformations, eigenvalues. Numerical problems will be emphasized. 3 hours discussion. (
MATH 235X Linear Algebra Problem Solving Session 1.0 FS
Corequisite: MATH 235.
This is a 1-unit supplement to Linear Algebra, MATH 235. This is structured as a workshop to complement MATH 235 students with broader and deeper applications of the subject, providing students with
opportunities for additional problem-solving and skill-building in a student-centered collaborative environment. 3 hours independent study. You may take this course more than once for a maximum of
2.0 units. Credit/no credit grading. (021281)
MATH 241 Secondary Math Early Field Experience 1.0 FS
This seminar and the associated CAVE field experience give prospective teachers early exposure to issues relevant to the profession of teaching secondary mathematics. In particular, the experience
helps these future teachers develop a deeper understanding of the K-12 mathematics curriculum, understand connections between their university subject matter preparation and K-12 academic content,
and reflect on developmental and social factors that affect K-12 students' learning of mathematics. 1 hour seminar. You may take this course more than once for a maximum of 4.0 units. Credit/no
credit grading. (020432)
MATH 260 Elementary Differential Equations 4.0 FS
Prerequisites: MATH 121.
First order separable, linear, and exact equations; second order linear equations, Laplace transforms, series solutions at an ordinary point, systems of first order linear equations, and
applications. 4 hours discussion. (005509)
MATH 260X Elementary Differential Equations Problem Session 1.0 FS
Corequisites: MATH 260.
Designed to supplement MATH 260 with broader and deeper applications of differential equations, providing the student with opportunities for additional problem-solving skills. A minimum of 20 hours
of activity are required to earn credit for the class; forty hours are available. 3 hours independent study. You may take this course more than once for a maximum of 3.0 units. Credit/no credit
grading. (020315)
MATH 290 Mathematics and Statistics Tutoring 1.0 FS
Corequisites: Concurrent enrollment in a course offered through the Dept of Mathematics & Statistics at CSU, Chico.
This course provides supplemental mathematics & statistics tutoring. 3 hours independent study. You may take this course more than once for a maximum of 8.0 units. Credit/no credit grading. (020823)
MATH 295 Project MATH Seminar Year 2 1.0 FS
Prerequisite: MATH 195.
The Project M.A.T.H. Seminar - Year 2 is the continuation of a biweekly seminar for students in Project M.A.T.H., an innovative program for students interested in becoming secondary mathematics
teachers. Students work with mentor teachers, prepare and present lessons, and participate in a structured early field experience. They also take on a leadership role in the seminar. Completion of
the seminar series satisfies the Credential Program's Early Field Experience requirement. 1 hour seminar. You may take this course more than once for a maximum of 2.0 units. Credit/no credit grading.
MATH 302 Science and Strategy in War and Peace 3.0 SP
Prerequisites: Completion of the FE Pathway Foundation Quantitative Reasoning, Mathematical Concepts.
The study of the scientific and strategic principles underlying war and peace in the 21st Century. Concepts from the physical sciences. The strategic theories of Sun Tzu and Clausewitz. Warfare in
the 20th Century, especially the development and effects of nuclear weapons. The world after Sept. 11, 2001. This course cannot be used for credit toward a mathematics major or minor, a mathematics
emphasis under the Liberal Studies major, or any credential. 3 hours lecture. (004876)
MATH 304 Statistical Tests for Inequalities 3.0 FS
Prerequisites: Completion of GE Pathway Foundation Quantitative Reasoning.
The study of statistical principles. Descriptive statistics, regression analysis, sampling theory, statistical inference for population means and proportions. Economic systems. Ideologies related to
economic systems. Quantitative and statistical measures of economic inequality, including the Gini index, Lorenz curve, Income Disparity Index, and the Kuznets curve. Applications to social and
economic inequality. This course may not be used to fulfill requirements for a major or minor in mathematics. 3 hours lecture. (005557)
MATH 305 Conceptual and Practical Statistics 3.0 SP
Prerequisites: MATH 120 or MATH 109 (may be taken concurrently).
Design of statistical experiments, graphing, sampling techniques, probability, and common probability distributions will be discussed, with an emphasis on practical applications. Uses and misuses of
statistics, misrepresentation of data, and proper and improper statistical analyses will be discussed. 3 hours discussion. (005532)
MATH 310 Patterns and Structures in Mathematics 3.0 FS
Prerequisites: MATH 110, MATH 111.
An intuitive investigation into mathematics to find recurrent themes. Problem-solving techniques and a search for patterns will be used throughout a discussion of topics such as modular arithmetic,
complex numbers, mathematical sequences and their applications. Not acceptable for a mathematics major or minor. 3 hours discussion. (005542)
MATH 311 Intuitive Foundations of Geometry 3.0 FS
Prerequisites: MATH 110, MATH 111.
An intuitive approach to problem-solving in coordinate geometry, motion geometry, and space geometry. Concrete models will be used for analyzing abstract ideas. Not acceptable for a mathematics major
or minor. 3 hours discussion. (005543)
MATH 314 Probability and Statistics for Science and Technology 4.0 SP
Prerequisites: MATH 121.
Basic concepts of probability and statistics with emphasis on models used in science and technology. Probability models for statistical estimation and hypothesis testing. Confidence limits. One- and
two-sample inference, simple regression, one- and two-way analysis of variance. Credit cannot be received for both MATH 350 and MATH 314. 4 hours discussion. (005533)
MATH 315 Applied Statistical Methods I 3.0 FA
Prerequisites: Completion of ELM requirement. Introduction to common procedures used to analyze data.
Single and two sample inference, analysis of variance, mulitple regression, analysis of co-variance, experimental design, repeated measures, nonparametric procedures, and categorical data analysis.
Examples are drawn from biology and related disciplines. Statistical packages are introduced. Appropriate for biology, agriculture, nutrition, psychology, social science and other majors. 3 hours
discussion. (005568)
MATH 330 Methods of Proof 3.0 FS
Prerequisites: MATH 121.
A survey of elementary principles of logic, emphasizing the nature of proof. Standard methods of proof will be illustrated with examples from various branches of mathematics, including set theory and
the theory of functions and relations. Other possible sources of examples include the calculus, number theory, theory of equations, topology of the real line. 3 hours seminar. (005530)
MATH 333 History of Mathematics 3.0 SP
Prerequisites: MATH 220 and at least one upper-division mathematics course.
MATH 330 is recommended. Study of the historical development of mathematics, with particular emphasis on the relationship between mathematics and society. 3 hours discussion. (005531)
MATH 337 Introduction to the Theory of Numbers 3.0 FA
Prerequisites: MATH 121, MATH 330.
Basic properties of the integers, division algorithm, fundamental theorem of arithmetic, number-theoretic functions, Diophantine equations, congruences, quadratic residues, continued fractions. 3
hours discussion. (005585)
MATH 341 Mathematical Topics for the Credential 3.0 FA
Prerequisites: MATH 121.
This course is designed to supplement the mathematical background of the candidate for the single subject credential in mathematics. The mathematical topics will be discussed from the student's and
the teacher's points of view to aid the candidate in making the transition to secondary school mathematics. Topics include mathematical problem-solving, conceptual ideas using algebra, geometry, and
functions, incorporating technology into the mathematics curriculum, and finite systems. 3 hours seminar. (005544)
MATH 342 Math Topics for the Credential 3.0 SP
Prerequisites: MATH 341.
This course focuses on having students examine mathematical pedagogy and the understanding and evaluations of students as mathematical learners as it analyzes secondary mathematics curriculum from an
advanced standpoint. Students will have opportunities to be involved in the facilitation of mathematical learning. Topics include: history of mathematics education, contemporary mathematics
curricula, problem solving, mathematical reasoning and methods of proof, mathematical learning theories, communication, assessment and collaborative learning communities. 3 hours discussion. (005545)
MATH 344 Combinatorial Mathematics and Graph Theory 3.0 FA
Prerequisites: MATH 121, MATH 330.
The analysis of mathematical and applied problems through the use of permutations and combinations, generating functions and recurrence relations. Directed graphs, trees, connectivity, and duality. 3
hours discussion. (005591)
MATH 346 College Geometry 3.0 SP
Prerequisites: MATH 220, MATH 330.
An exploration of axioms and models for Euclidean and non-Euclidean geometries focusing on the independence of the Parallel Postulate. Additional topics will be chosen from Euclidean plane geometry,
transformation geometry, and the geometry of polyhedra. 3 hours discussion. (005561)
MATH 350 Introduction to Probability and Statistics 3.0 FA
Prerequisites: MATH 121.
Basic concepts of probability theory, random variables and their distributions, limit theorems, sampling theory, topics in statistical inference, regression, and correlation. 3 hours discussion. (
MATH 351 Introduction to Probability and Statistics 3.0 SP
Prerequisites: MATH 350.
Continuation of MATH 350. 3 hours discussion. (005535)
MATH 360 Ordinary Differential Equations 3.0 SP
Prerequisites: MATH 260.
Systems of first order linear equations, existence and uniqueness theorems, stability, Sturm separation theorems, power series methods. 3 hours discussion. (005538)
MATH 361 Boundary Value Problems and Partial Differential Equations 3.0 FA
Prerequisites: MATH 260.
Partial differential equations, separation of variables, orthogonal sets of functions, Sturm-Liouville problems, Fourier series, boundary value problems for the wave equation, heat equation, and
Laplace equation; Bessel functions, Legendre polynomials. 3 hours discussion. (005540)
MATH 398 Special Topics in Math 1.0 -3.0 FS
Prerequisites: At least one 100- or 200-level mathematics course appropriate to the subject, faculty permission.
This course is for special topics offered for 1.0-3.0 units. Typically the topic is offered on a one-time-only basis and may vary from term to term and be different for different sections. See the
Class Schedule for the specific topic being offered. 9 hours supervision. (005559)
MATH 399 Special Problems 1.0 -3.0 FS
This course is an independent study of special problems offered for 1.0-3.0 units. You must register directly with a supervising faculty member. MATH 399 cannot be used to fulfill major requirements
without prior approval of the advisor and department chair. You may take this course more than once for a maximum of 6.0 units. Credit/no credit grading. (005560)
MATH 401 CMP Institute - Summer 1 2.0 SM
2 hours discussion. You may take this course more than once for a maximum of 4.0 units. (005578)
MATH 403 Math Project Implementation 1.0 SX
1 hour discussion. Credit/no credit grading. (005551)
MATH 405 Cmp Institute-Sp 1.0 SP
1 hour lecture. You may take this course more than once for a maximum of 6.0 units. Credit/no credit grading. (005552)
MATH 407 CMP Institute - Summer 2 1.0 SM
1 hour discussion. You may take this course more than once for a maximum of 2.0 units. (005579)
MATH 420 Advanced Calculus 3.0 FS WP
Prerequisites: ENGL 130 or JOUR 130 (or equivalent) with a grade of C- or higher, MATH 220, MATH 330, upper-division standing.
Limits, continuity, uniform continuity, the definite integral, series, convergence, uniform convergence, and metric spaces. Differentiation and integration of functions of several variables.
Transformation of multiple integrals. 3 hours discussion. This is an approved Writing Proficiency course; a grade of C- or better certifies writing proficiency for majors. (005575)
MATH 421 Advanced Calculus 3.0 SP
Prerequisites: MATH 420.
Continuation of MATH 420. 3 hours discussion. (005576)
MATH 426 Projective Geometry 3.0 INQ
Prerequisites: MATH 220, MATH 330.
Axiomatic geometry of the projective plane. Duality, collineations, correlations, polarities, and conics. 3 hours discussion. (005564)
MATH 428 Differential Geometry 3.0 INQ
Prerequisites: MATH 220, MATH 330.
The geometry of curves and surfaces in Euclidean 3-space. 3 hours discussion. (005566)
MATH 435 Linear Algebra 3.0 FA
Prerequisites: MATH 220, MATH 330, MATH 335.
Vector spaces, linear operators, bilinear forms and scalar products, unitary spaces; matrix polynomials, eigenvalues, and Jordan normal form. 3 hours discussion. (005581)
MATH 437 Topology 3.0 S2
Prerequisites: MATH 220, MATH 330.
Metric spaces, continuous functions, homeomorphisms, separation, and covering axioms, connectedness. 3 hours discussion. (005563)
MATH 441 Math Topics for the Credential 4.0 FS
Prerequisites: MATH 342.
Corequisites: Assignment as a Mathematics Department intern.
Supervised internship in teaching mathematics with accompanying seminar. Guidance in facilitation of mathematical learning. Topics include contemporary mathematics curriculum topics, mathematical
learning theories, communication, and assessment. 3 hours seminar, 3 hours supervision. You may take this course more than once for a maximum of 8.0 units. Credit/no credit grading. (005546)
MATH 442 Mathematics and the Teaching of Mathematics 3.0 FA
Prerequisites: MATH 342.
Completes a three course series, started with two semesters of Mathematics for the Credential, MATH 341 and MATH 342. Students compare instructional strategies and explore the role content and
pedagogical content knowledge has in these strategies. Central to the class is a lesson study project which entails a cycle of lesson development, implementation, reflection and revision, and
implementation again. Students concurrently enrolled in EDTE 535A, Teaching Practicum I for Blended Math Candidates, are able to implement their lesson as part of the practicum, and have a real
context for the full content of the course. 3 hours lecture. (020978)
MATH 449 Modern Algebra 3.0 SP
Prerequisites: MATH 220, MATH 235, MATH 330.
Introduction to basic algebraic structures such as groups, ring, and fields. The fundamental concepts of homomorphism, subgroup, normal subgroup and factor group of a group as well as subring, ideal
and factor ring of a ring; permutation groups and matrix groups. 3 hours discussion. (005582)
MATH 450 Mathematical Statistics 3.0 FA
Prerequisites: MATH 220, MATH 330, MATH 351.
A rigorous theoretical treatment of the following topics: transformations of random variables, estimation, Neyman-Pearson hypothesis testing, likelihood ratio tests, and Bayesian statistics. 3 hours
discussion. (005562)
MATH 456 Applied Statistical Methods II 3.0 S2
Prerequisites: MATH 315.
Advanced topics in applied statistics including multiple regression, multivariate methods, nonparametrics, analysis of covariance, bootstrap methods and others as appropriate. Statistical computer
packages are introduced and used. Appropriate for biology, agriculture, nutrition, business, psychology, social science and other maojrs. 3 hours discussion. (005570)
MATH 458 Sampling Methods 3.0 S1
Prerequisites: One course chosen from MATH 105, MATH 305, MATH 350, or MATH 315.
The theory and application of survey sampling techniques. Topics include simple random sampling, stratified sampling, systematic sampling, and cluster sampling. Appropriate for mathematics, computer
science, psychology, social science, agriculture, biology, and other majors. 3 hours discussion. (005573)
MATH 460 Numerical Analysis 3.0 INQ
Prerequisites: MATH 220, MATH 335, completion of the computer literacy requirement.
Error analysis; numerical solution of algebraic and transcendental equations; systems of linear and non-linear equations; matrix inversion; matrix eigenvalues and eigenvectors. 3 hours discussion. (
MATH 461 Numerical Analysis 3.0 SP
Prerequisites: MATH 260 (may be taken concurrently), MATH 335, completion of the computer literacy requirement. MATH 460 is not a prerequisite for MATH 461.
Approximation; numerical integration; numerical solution of ordinary and partial differential equations; interpolation and extrapolation. 3 hours discussion. (005584)
MATH 465 Introduction to Complex Variables 3.0 FA
Prerequisites: MATH 220.
Algebra of Complex Numbers, Cauchy-Riemann Equations, the exponential, trigonometric, and logarithmic functions, complex integration and Cauchy integral formula, Taylor and Laurent series, the
residue theorem, conformal mapping, and applications. 3 hours discussion. (005577)
MATH 472 Introduction to Chaotic Dynamical Systems 3.0 F1
Prerequisites: MATH 260; MATH 360 and MATH 335 are recommended.
An introduction to the study of non-linear dynamical systems. Both discrete and continuous systems will be studied using classical analysis combined with geometric techniques and computer simulation.
Areas of application include fractal geometry, coding theory, fluid turbulence, population fluctuation, and chaotic vibrations of structures and circuits. 3 hours discussion. (005588)
MATH 475 Calculus of Variations 3.0 F2
Prerequisites: MATH 260; MATH 361 is recommended.
Classical problems in the calculus of variations. Euler-Lagrange equations. Isoperimetric problems, Fermat's principle. Lagrangian and Hamiltonian mechanics of particles. Two independent variables.
Applications to physics and engineering. 3 hours discussion. (005590)
MATH 480 Mathematical Modeling 3.0 SP
Prerequisites: MATH 260, MATH 335.
The translation of real world phenomena into mathematical language. Possible applications include population and competing species models, mathematical theories of war, traffic flow, river pollution,
water waves and tidal dynamics, probabilistic and simulation models. 3 hours discussion. (005592)
MATH 495H Honors Reading Course 3.0 FS
Prerequisites: Admission to the Department Honors Program, completion of MATH 420 with a grade of B or higher.
Directed reading in an advanced topic under the guidance of an Honors thesis supervisor. The course exceeds the usual level of difficulty associated with undergraduate work. It provides the
background necessary to write an Honors thesis. 9 hours supervision. (005595)
MATH 498 Advanced Topics in Mathematics 1.0 -3.0 FS
Prerequisites: At least one 300- or 400-level mathematics course appropriate to the subject, faculty permission.
This course is for special topics offered for 1.0-3.0 units. Typically the topic is offered on a one-time-only basis and may vary from term to term and be different for different sections. See the
Class Schedule for the specific topic being offered. 3 hours supervision. (005593)
MATH 499 Special Problems 1.0 -3.0 FS
Prerequisites: Faculty permission.
This course is an independent study of special problems offered for 1.0-3.0 units. You must register directly with a supervising faculty member. 3 hours supervision. You may take this course more
than once for a maximum of 6.0 units. Credit/no credit grading. (005594)
MATH 499H Honors Thesis 3.0 FS
Prerequisites: Completion of MATH 495H with a grade of B or higher, and approval by the department Honors advisor and thesis supervisor of the proposed thesis topic.
Preparation of written thesis in mathematics under supervision of Honors thesis advisor. The thesis, based on studies begun in MATH 495H, will require original work beyond that normally required in
undergraduate work. Completed written thesis must be approved by the thesis supervisor and Honors advisor. A summary of the thesis will be presented by the student in public lecture. Successful
completion of MATH 495H and MATH 499H is one of the requirements for being designated as an Honors graduate in mathematics. 9 hours supervision. (005596)
MATH 610 Topics in Mathematics for Secondary Teachers: Analysis 3.0 INQ
Prerequisites: Admission to the master's program in mathematics education or instructor permission.
Through an array of pedagogical strategies, secondary mathematics teachers explore analysis topics appropriate for the secondary school curriculum. These topics and strategies provide a basis for
reflective analysis and deepening knowledge of analysis. 3 hours seminar. You may take this course more than once for a maximum of 6.0 units. (005599)
MATH 615 Statistical Methods for Graduate Research 3.0 FA
Prerequisites: MATH 105, MATH 350, MATH 315, or MATH 305 (only one is required).
Introduction to common procedures used to analyze data. Single and two-sample inference, analysis of variance, multiple regression, analysis of co-variance, experimental design, repeated measures,
nonparametric procedures, and categorical data analysis. Examples will be drawn from Biology and related disciplines. Statistical computer packages will be introduced. Appropriate for biology,
agriculture, nutrition, psychology, social science, and other majors. 3 hours discussion. (005597)
MATH 620 Topics in Mathematics for Secondary Teachers: Geometry 3.0 INQ
Prerequisites: Admission to the master's program in mathematics education or instructor permission.
Through an array of pedagogical strategies, secondary mathematics teachers explore geometry appropriate for the secondary school curriculum. These topics and strategies provide a basis for reflective
analysis and deepening knowledge of geometry. 3 hours seminar. You may take this course more than once for a maximum of 6.0 units. (005602)
MATH 630 Topics in Mathematics for Secondary Teachers: Foundations of Mathematics 3.0 INQ
Prerequisites: Admission to the master's program in mathematics education or instructor permission.
Through an array of pedagogical strategies, secondary mathematics teachers explore the foundations of mathematics topics appropriate for the secondary school curriculum. These topics and strategies
provide a basis for reflective analysis and deepening knowledge of the foundations of mathematics. 3 hours seminar. You may take this course more than once for a maximum of 6.0 units. (005601)
MATH 633 Topics in Mathematics for Secondary Teachers: Number Theory 3.0 INQ
Prerequisites: Admission to the master's program in mathematics education or instructor permission.
Through an array of pedagogical strategies, secondary mathematics teachers explore number theory appropriate for the secondary school curriculum. These topics and strategies provide a basis for
reflective analysis and deepening knowledge of number theory. 3 hours seminar. You may take this course more than once for a maximum of 6.0 units. (005605)
MATH 635 Topics in Mathematics for Secondary Teachers: Discrete Mathematics 3.0 INQ
Prerequisites: Admission to the master's program in mathematics education or instructor permission.
Through an array of pedagogical strategies, secondary mathematics teachers explore discrete mathematics topics appropriate for the secondary school curriculum. These topics and strategies provide a
basis for reflective analysis and deepening knowledge of discrete mathematics. 3 hours seminar. You may take this course more than once for a maximum of 6.0 units. (005600)
MATH 637 Topics in Mathematics for Secondary Teachers: History of Mathematics 3.0 INQ
Prerequisites: Admission to the master's program in mathematics education or instructor permission.
Through an array of pedagogical strategies, secondary mathematics teachers explore the history of mathematics appropriate for the secondary school curriculum. These topics and strategies provide a
basis for reflective analysis and deepening knowledge of the history of mathematics. 3 hours seminar. You may take this course more than once for a maximum of 6.0 units. (005603)
MATH 640 Topics in Mathematics for Secondary Teachers: Modern Algebra 3.0 INQ
Prerequisites: Admission to the master's program in mathematics education or instructor permission.
Through an array of pedagogical strategies, secondary mathematics teachers explore modern algebra topics appropriate for the secondary school curriculum. These topics and strategies provide a basis
for reflective analysis and deepening knowledge of modern algebra. 3 hours seminar. You may take this course more than once for a maximum of 6.0 units. (005598)
MATH 650 Topics in Mathematics for Secondary Teachers: Probability and Statistics 3.0 INQ
Prerequisites: Admission to the master's program in mathematics education or instructor permission.
Through an array of pedagogical strategies, secondary mathematics teachers explore probability and statistics appropriate for the secondary school curriculum. These topics and strategies provide a
basis for reflective analysis and deepening knowledge of probability and statistics. 3 hours seminar. You may take this course more than once for a maximum of 6.0 units. (005606)
MATH 660 Topics in Mathematics for Secondary Teachers: Mathematical Modeling 3.0 INQ
Prerequisites: Admission to the master's program in mathematics education or instructor permission.
Through an array of pedagogical strategies, secondary mathematics teachers explore mathematical modeling appropriate for the seconday school curriculum. These topics and strategies provide a basis
for the reflective analysis and deepening knowledge of mathematical modeling. 3 hours seminar. You may take this course more than once for a maximum of 6.0 units. (005604)
MATH 697 Independent Study 1.0 -3.0 FS
This course is a graduate-level independent study offered for 1.0-3.0 units. You must register directly with a supervising faculty member. 3 hours supervision. You may take this course more than once
for a maximum of 6.0 units. (005616)
MATH 698 Grad Advanced Topics in Math 1.0 -3.0 FS
This course is for special topics offered for 1.0-3.0 units. Typically the topic is offered on a one-time-only basis and may vary from term to term and be different for different sections. See the
Class Schedule for the specific topic being offered. 9 hours supervision. (005615)
MATH 699P Master's Project 1.0 -3.0 FS
This course is offered for 1.0-6.0 units. You must register directly with a supervising faculty member. 9 hours supervision. You may take this course more than once for a maximum of 6.0 units. Credit
/no credit grading. (005622)
MATH 699T Master's Study 1.0 -3.0 FS
This course is offered for 1.0-6.0 units. You must register directly with a supervising faculty member. 9 hours supervision. You may take this course more than once for a maximum of 6.0 units. Credit
/no credit grading. (005620)
SUBJ NUM Title Sustainable Units Semester Offered Course Flags
MTHE 601 Research in Mathematics Education 3.0 SM
Prerequisites: Admission to a master's degree program in mathematics education or permission of instructor.
This course will examine research in mathematics education that includes areas of teaching, learning, curriculum, and socio-cultural context. Selected research will be critically reviewed for
research design and claims. In a culminating project, students will conduct a review on a specific topic in math education research literature. This is a required course in the MA and MS programs in
mathematics education. 3 hours seminar. (005929)
MTHE 680 Research Methods in Science Education. 3.0 INQ
Prerequisites: Admission to master's degree program in science teaching, NSCT 687, faculty permission.
This course is also offered as NSCT 680 .
The course focuses on quantitative and qualitative methods to conduct research in science education that informs and strengthens their classroom practice. Successful completion of the course requires
students to develop a research proposal. 9 hours supervision. (005930)
MTHE 690 Thesis/Project Writing Seminar 1.0 -3.0 FS
Prerequisites: MTHE 680.
Formulation and pursuit, with supervision, of advanced projects and theses. The emphasis is on planning, reading, discussing, and evaluating student's manuscript-in-progress. This is a required couse
in the MA and MS programs in mathematics education. 3 hours seminar. You may take this course more than once for a maximum of 3.0 units. (005931) | {"url":"http://catalog.csuchico.edu/viewer/12/MATH/offerings.html","timestamp":"2014-04-18T05:33:24Z","content_type":null,"content_length":"145251","record_id":"<urn:uuid:0dfd8815-91c0-4daf-82f2-7cc6ec9fcccb>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00396-ip-10-147-4-33.ec2.internal.warc.gz"} |
Irreducible Polynomials:
March 3rd 2008, 09:37 AM #1
Junior Member
Aug 2007
Irreducible Polynomials:
Find all irreducible polynomials of the form $x^2 + ax +b$, where a,b belong to the field $\mathbb{F}_3$ with 3 elements.
Show explicitly that $\mathbb{F}_3(x)/(x^2 + x + 2)$ is a field by computing its multiplicative monoid.
Identify [ $\mathbb{F}_3(x)/(x^2 + x + 2)$]* as an abstract group.
any suggestions please?
Just list all of them $x^2 + x+1,x^2+x+2,...$ there are only $9$ or them. Now, just check which ones have zeros (there are only three zeros to check). And this tells you which are irreducible and
which are not.
Show explicitly that $\mathbb{F}_3(x)/(x^2 + x + 2)$ is a field by computing its multiplicative monoid.
Identify [ $\mathbb{F}_3(x)/(x^2 + x + 2)$]* as an abstract group.
Any element in $\mathbb{F}_3[x]/(x^2+x+2)$ has form $a+bx+(x^2+x+2)$. Now show that these elements form a field.
March 3rd 2008, 10:07 AM #2
Global Moderator
Nov 2005
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Lectures in Homological Algebra
A co-publication of the AMS and CBMS.
             
CBMS Regional This volume constitutes a record of the course in homological algebra given at the Virginia Polytechnic Institute in July 1970 under the auspices of the National Science
Conference Series Foundation's Regional Conference project. The nature of the audience required that the course begin with an introduction to the notion of modules over a unitary ring, but permitted
in Mathematics rapid development of the theory from that starting point. The first three chapters may be regarded as containin g material essential to any introductory course in homological
algebra, while the later chapters reflect the choices actually made by the audience among many possible special topics accessible to those who had mastered the early material. Thus
1971; 74 pp; it may be claimed that the course achieved depth of penetration on a narrow front, while it is admitted that breadth of coverage of the entire domain of homological algebra was
softcover neither attempted nor achieved.
Number: 8 "There obviously is a need for a short smooth introduction to homological algebra. This is a contribution in that direction."
ISBN-10: -- S. Moran, Mathematical Reviews
List Price: US$20
Member Price:
All Individuals:
Order Code: CBMS/ | {"url":"http://ams.org/bookstore?fn=20&arg1=cbmsseries&ikey=CBMS-8-D","timestamp":"2014-04-19T15:37:16Z","content_type":null,"content_length":"15020","record_id":"<urn:uuid:b2520b1f-ab47-4868-9a86-7b980a0f2498>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00457-ip-10-147-4-33.ec2.internal.warc.gz"} |
STA 6444, Fall 2010
Mathematical Probability
Dr. Hongwei Long, office: SE 268, phone: 297-0810, e-mail: hlong@fau.edu
Course homepage: http://math.fau.edu/long/STA6444F10.htm
Time and Place:
TR 12:30-1:50pm in BU 411.
Office Hours:
TR 2:00-4:30 pm in SE 268.
Other times by appointment or just stop by the office.
A Probability Path, by Sidney I. Resnick, Birkhauser, Boston, Basel, Berlin, 1999.
Course Description:
This course will provide an introduction to the rigorous mathematical treatment of probability theory. Topics covered include construction of probability spaces, random variables, independence,
integration and expectation, convergence concepts, laws of large numbers, convergence in distribution, characteristic functions and central limit theorem (Chapters 1-9 of the textbook).
Prerequisites: STA 4442.
Take-home midterm Tentatively October 5-12
Final Thursday, December 2, 10:30am-1:00pm, location: BU 411. Closed book exam.
There will be about six homework assignments. These will involve using methods presented in class to solve problems from the textbook. Assignments should be handed in on the due date. Late
assignments will not be accepted.
Grading will be based on the following weighting:
30% Assignments
30% Midterm exam
40% Final exam
There will be no make-up midterm. If a student has an acceptable excuse for missing the midterm, the weight of the midterm will be shifted to the final. Make-up final exam will be given only under
exceptional circumstance, and written, verifiable excuses must be provided.
Grading Scale:
The grading scale will be A: 90-100%, B: 80-89%, C: 70-79%, D: 60-69%, F: 0-59%.
Honor Code:
Students at Florida Atlantic University are expected to maintain the highest ethical standards. Academic dishonesty, including cheating and plagiarism, is considered a serious breach of these ethical
standards, because it interferes with the University mission to provide a high quality education in which no student enjoys an unfair advantage over any other. Academic dishonesty is also destructive
of the University community, which is grounded in a system of mutual trust and places high values on personal integrity and individual responsibility. Harsh penalties are associated with academic
dishonesty. For more information, see http://www.fau.edu/regulations/chapter4/4.001_Honor_Code.pdf.
Students with Disabilities:
In compliance with the Americans with Disabilities Act (ADA), students who require special accommodations due to a disability to properly execute coursework must register with the Office for Students
with Disabilities (OSD) located in Boca Raton-SU 133 (561-297-3880), in Davie-MOD I (954-236-1222), in Jupiter- SR 117 (561-799-8585), or at the Treasure Coast- CO 128 (772-873-3305), and follow all
OSD procedures.
Word and PDF files
· Homework Assignments
Last modified: November 3, 2010 | {"url":"http://math.fau.edu/long/STA6444F10.htm","timestamp":"2014-04-18T13:08:43Z","content_type":null,"content_length":"16073","record_id":"<urn:uuid:2c6d8c37-878f-43e4-80f6-42bf78ae98c0>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00075-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculus Derivatives and Limits Math Sheet
An Engineers Quick Calculus Derivatives and Limits Reference
Limits Math Help
The limit is a method of evaluating an expression as an argument approaches a value. This value can be any point on the number line and often limits are evaluated as an argument approaches infinity
or minus infinity. The following expression states that as x approaches the value c the function approaches the value L.
The following expression states that as x approaches the value c and x > c the function approaches the value L.
The following expression states that as x approaches the value c and x < c the function approaches the value L.
The following expression states that as x approaches infinity, the value c is a very large and positive number, the function approaches the value L.
Also the limit as x approaches negative infinity, the value of c is a very large and negative number, is expressed below.
Given the following conditions:
The following properties exist:
Limit Evaluation at +-InfinityReturn to Top
If f(x) is continuous at a then:
Continuous Functions and CompositionsReturn to Top
If f(x) is continuous at b:
Derivatives Math Help
The derivative is way to define how an expressions output changes as the inputs change. Using limits the derivative is defined as:
This is a method to approximate the derivative. The function must be differentiable over the interval (a,b) and a < c < b.
If there exists a derivative for f(x) and g(x), and c and n are real numbers the following are true:
The product rule applies when differentiable functions are multiplied.
Quotient rule applies when differentiable functions are divided.
The power rule applies when a differentiable function is raised to a power.
The chain rule applies when a differentiable function is applied to another differentiable function.
These are some examples of common derivatives that require the chain rule. | {"url":"http://www.eeweb.com/toolbox/calculus-derivatives-and-limits-reference-sheet","timestamp":"2014-04-19T09:24:45Z","content_type":null,"content_length":"43223","record_id":"<urn:uuid:a0e82302-538d-4d9e-a589-a0b8ef447809>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00110-ip-10-147-4-33.ec2.internal.warc.gz"} |
Trace formula for nonnuclear perturbations of selfadjoint operators
Milutin Dostani\'c
Matematicki fakultet, Beograd, Yugoslavia
Abstract: The trace formulas for the operator $\varphi(H_1)-\varphi(H_0)$ are deduced when $H_1-H_0$ is a nonnuclear operator and $\varphi$ is an enough wide class of functions.
Classification (MSC2000): 47A55
Full text of the article:
Electronic fulltext finalized on: 1 Nov 2001. This page was last modified: 16 Nov 2001.
© 2001 Mathematical Institute of the Serbian Academy of Science and Arts
© 2001 ELibM for the EMIS Electronic Edition | {"url":"http://www.emis.de/journals/PIMB/068/12.html","timestamp":"2014-04-21T00:03:44Z","content_type":null,"content_length":"3124","record_id":"<urn:uuid:15d7cd50-ddd2-43b0-862f-e965ec8b058d>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00643-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
March 18th, 2013, 05:36 AM
Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
I am really new to java and wanted to know how I can print an int which is both negative and odd and if so, needs to be put into the formula (n^2+1) . I am not looking particularly for answers
(although I would appreciate them greatly) I just generally want to better my knowledge of java. Thanks in advanced.
March 18th, 2013, 05:56 AM
Re: Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
how I can print an int
Can you use the System.out.println() method to print it?
March 18th, 2013, 03:18 PM
Re: Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
Yes I can use println
March 18th, 2013, 03:24 PM
Re: Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
That answers the question. Did you have any other questions?
March 18th, 2013, 03:28 PM
Re: Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
I was wondering how I could go about printing a number both negative and odd using "if" statements and (I'm assuming but not entirely sure) boolean?
March 18th, 2013, 03:35 PM
Re: Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
Are you asking how to determine if a number is odd and negative?
A test for even or odd is to compare the results of theNumber % 2 : 1 is odd, 0 is even
A test for negative is if theNumber < 0
March 18th, 2013, 05:00 PM
Re: Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
Thank you I will try this and come back
--- Update ---
That was helpful (i now understand the modulus operator) but I am getting problems in my code where it is not converting to boolean. See I'm trying to get the code to run with two requirements
but I can't seem to do that. This is my code for the int so far:
public static void main(String[] args){
int n=11;
if (n<0){
if (n%2);
March 18th, 2013, 05:03 PM
Re: Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
I am getting problems
Please copy the full text of the error messages and paste it here.
Please edit your post and wrap your code with
<YOUR CODE HERE>
to get highlighting and preserve formatting.
March 18th, 2013, 05:04 PM
Re: Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
Sure! Hold on...
Code java:
package variables;
public class SimpleCalc {
public static void main(String[] args){
int n=11;
if (n<0){
if (n%2); //Error Message occurs here: Mismatch - cannot convert to boolean
if (n>=0) { //The ones after this are fine
if (n>=0) {
March 18th, 2013, 05:49 PM
Re: Beginner at Java: I need to figure out how to print a number which is odd and negative and fits in a formula.
The expression in the if statement must use a boolean operator that returns a boolean result. % returns a numeric value.
See the tutorial for a list of operators that return a boolean result:
Equality, Relational, and Conditional Operators (The Java™ Tutorials > Learning the Java Language > Language Basics)
The if statement requires a boolean value, not a numeric value.
a == b is a boolean value
a + b is a numeric value
(a + b) > 3 is a boolean value | {"url":"http://www.javaprogrammingforums.com/%20java-theory-questions/25879-beginner-java-i-need-figure-out-how-print-number-odd-negative-fits-formula-printingthethread.html","timestamp":"2014-04-19T03:13:43Z","content_type":null,"content_length":"14889","record_id":"<urn:uuid:fe6b3fe9-a757-45b4-9632-bb4c60c57318>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00464-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Jan van den Heuvel
Talks at conferences and seminars by Jan van den Heuvel
Back to my my homepage.
Presentations I've given since "recently"
Below is a list of presentations I have given in recent years. More will be added in the future, once I come round to adding the files and so.
(A small remark about the meaning of "edited": Presentations given using data-projection usually involve using overlays, where whole slides are not shown all at once, but appear bit by bit. This has
the drawback that the files become very large, since one page can have several overlays, each of which is a separate page in the PDF file. To make the file-size manageable for the versions you can
download from here, I removed (as far as possible) this overlay option.)
Back to my my homepage.
Copyright © Jan van den Heuvel & London School of Economics and Political Science 2007-10
Last modified: Fri Aug 31 16:25:22 BST 2012
Send comments to webmaster | {"url":"http://www.maths.lse.ac.uk/Personal/jan/talks.html","timestamp":"2014-04-20T11:55:06Z","content_type":null,"content_length":"12545","record_id":"<urn:uuid:22a28de1-ac2b-4def-a6b4-851ffbe67cbf>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00545-ip-10-147-4-33.ec2.internal.warc.gz"} |
Why is it your job to teach your kid math? - Macleans.ca
Why is it your job to teach your kid math?
Parents are being forced to hit the books and help tutor their kids through a confusing curriculum.
When mother of two Anna Stokke began digging into the elementary school math curriculum last year, she was flabbergasted by what she found. Instead of teaching the standard methods of arithmetic, the
emphasis had shifted to a wide range of alternative methods, such as using grids, blocks, or strips of paper to multiply. Stokke is a professor of math at the University of Winnipeg, but even she
found the methods confusing. “It was shocking,” she recalls. “We’re talking about adding, subtracting, multiplying and dividing. It shouldn’t be so overly complicated that even parents can’t
understand it. It’s absolutely ridiculous.”
Stokke began speaking out and soon parents from all over Canada were sending her similar stories of discontent: kids who couldn’t do their homework without help, parents who couldn’t make heads or
tails of the assignments so they were hiring tutors, or spending hours looking up math sites on the Internet because the textbooks are so vague. She heard from teachers who felt pressured not to
teach the traditional methods. Stokke and her husband, Ross—who is also a math professor at the University of Winnipeg—started up a biweekly math club for their daughter and 11 of her friends to pick
up the educational slack. Out of concern for math education in general, Stokke, along with three colleagues, also co-founded the Western Initiative for Strengthening Education in Math (WISE Math), a
coalition lobbying to improve K-12 math education. Parents (and teachers) from all over Canada have flooded their online petition with support. “I don’t have a problem with alternate strategies,”
Stokke says. “But I fear they’re learning so many, that in the end they’re not mastering any.”
It was never supposed to be this way. Changes in the curriculum—which have rolled out cumulatively over the past decade and intensified in recent years—had sounded so promising: instead of stifling
them with rote memorization and rigid methods, children are to use their own learning style to explore mathematical knowledge and conceptualize innovative solutions to complex problems, preparing
them for an ever-changing tech-based economy. But the execution of this vision hasn’t been so idealistic. Instead of building a generation of math whizzes, it’s creating a Tower of Babel, where
teachers can’t understand textbooks, students can’t understand teachers, and parents and children have no idea what the other is talking about. The confusion, some critics say, lays the groundwork
for innumeracy that sticks with students through the grades, into post-secondary school and out into the workforce—this week business leaders reported that a growing chunk of Canadian kids simply
don’t have what it takes to fill skilled positions.
To sort out the discord, some boards are hiring cadres of costly numeracy consultants to facilitate workshops for families and teachers, and developing online tools for parents to access. In many
districts, families are left to grapple with the mess themselves. Either way, more parents are starting to speak out about the increasing amount of time, money and stress required to teach their kids
what they should be able to master at school. “Kids spend six hours a day there—I think the schools should be able to teach math to children themselves,” says Stokke. “It’s completely wrong-headed.
And the moment you say parents should play a significant role in public education, you have a two-tiered system.”
This month, in Hamilton, the Catholic school board is launching one of the more intensive initiatives to support parents—SuccessMaker, an online math program intended for the classroom, redirected
for home use. Consisting of animated games and word problems, for the program to be effective it requires that parents commit 20 minutes a night, for a minimum of four nights a week, over a span of
eight weeks. “We recognized that parents were having difficulty supporting their kids, so we wanted to give them this tool,” says Sandie Pizzuti, assistant superintendent of academic programs and
services at the Hamilton-Wentworth Catholic District School Board. The flyer marketing the initiative uses language that goes directly to the core of parental fears: “Concerned about your child’s
math scores? Does your child say he or she hates math? Finding it difficult to help with math homework? Wish you had the time or money to have your child tutored in math?” So far, 200 parents have
signed up for the information session and they anticipate they’ll register about 500 parents; the ministry is covering the initial $12,500 in licensing fees.
Spending 20 minutes on math would qualify as a really great night for Jane Snider, a Saskatoon mother who asked to use a pseudonym for her daughter’s sake. When her daughter entered Grade 7, she
began asking for help with homework. Snider tried to step in, but was blindsided by the methods they were supposed to employ, such as using graph paper to show how you can divide fractions and strips
of paper to demonstrate ways to multiply them. “I never expected to run into problems at this grade level, and I knew I was making a mess of it,” says Snider, so she turned things over to her
husband, an engineer. Accustomed to solving equations with formulas, her husband was spending up to two hours after work learning the new strategies and terminology himself, then teaching them to his
daughter. There were times, says Snider, when he couldn’t understand the assignment at all. “It just became a blur of stress and frustration,” she says, adding that her daughter has lost all
confidence with math and now professes to hate it. “We’ve had to spend so many nights dealing with this mess when we could be doing other things. It makes me so angry.”
Giving parents a better sense of the subject is one reason more schools are offering Math Nights. The theme of the workshops can vary, but in some cases parents are taught the same math strategies as
their children. But with only an hour, one Toronto mother who attended a Math Night at her Grade 3 son’s school left confused. “But I’m glad I went—at least now I know how not to help my son,” she
says, explaining that she was told that stacking two-digit numbers and carrying the one isn’t a good method for addition because kids don’t understand what it means to carry the one, that they’re
really adding 10. Instead she learned alternate methods that yielded deep understanding, which the parents were told is more important than arriving at the correct answer. “But that seems wrong. Why
does it matter so much for someone to always realize they’re adding 10? You get the right answer, and I do think that’s important.”
As one of the designers of the math curriculum for the western provinces, Debbie Duvall now works as a math consultant for the Elk Island public school district near Edmonton, where she gives
workshops to parents. If a student can’t remember what 9 x 6 is, she says, they’re lost. But with a deeper understanding of multiplication, students realize they can multiply 10 by 6—an easier
calculation—and subtract 6 to arrive at the answer. “We want to provide options for kids,” says Duvall, adding that at least one parent comes up to her after her workshops to say they wished they
were taught this way. Taking into account all learners’ styles and capabilities is another big part of the approach. Back in Hamilton, numeracy consultant Kerry Dwyer-Mitchell recently divided a
Grade 4 class into groups to teach multiplication through a pizza game (how many pieces are needed if 27 kids want three slices each?). Each group, she says, brainstormed a different strategy, such
as repeated addition, adding partial products, or using a manipulative, like Base Ten Blocks. “Traditionally, you’d get your algorithm and you would just do it,” says Dwyer-Mitchell. “But they’re
free to use whatever works best for them.”
Yet it’s that very freedom that, some say, is at the root of much of the confusion. “They’re really creative—but they don’t know what to do with it,” says Kim Langen, CEO and co-founder of the
after-school math enrichment program Spirit of Math, based in Toronto. “The rigour of the procedure is being lost and the bar is getting set far too low.” While students need a minimum B+ to apply
for Spirit of Math, Langen says that even these achievers can possess a worrying level of innumeracy—Grade 5 students who don’t know multiplication facts, have never encountered division, and just
look at you blankly when you ask them what 23 + 7 is. In order to build students’ math facts, the first 10 minutes of the 90-minute session is dedicated to drills—then, explains Langen, because
they’re not bogged down on simple calculations, they can handle the high-level conceptual work. “We had one vice-principal who saw the incredible difference drills made,” she says. “But he said, ‘I
can’t do drills—the board won’t let me.”
And it’s not just students struggling, says Langen. When a parent came to her in tears because she couldn’t help her daughter who was struggling with math, Langen met with the teacher. At the
meeting, the teacher sought Langen’s advice—she couldn’t get her students to understand metric conversions, in this case, converting 110 cm into metres. She showed Langen the blackboard filled with
an elaborate matrix that neither adult really understood. Langen reached for a metre stick. “I told her she could use this to show there are 100 centimetres in one metre, and she just lit up.” Though
she was taken aback by the encounter, Langen is quick to add that the teacher was bright, just uncomfortable with math—and this highly conceptual approach is often toxic for teachers without a solid
grounding in math. “It’s so overcomplicated now. We use the old-fashioned methods. They’re fast, they work and it’s part of the reason parents come to us.” Nathalie Foy of Toronto has enrolled her
Grade 5 son in the program. “I wish I didn’t have to supplement their education,” says Foy, “For one, it’s expensive. But this is a safety net—and now he knows his math facts cold.” At more than
$1,800 per school year, it’s a pricey measure, but with more than 30 per cent of Canadians supplementing their children’s education, it’s one that more parents are investing in, especially as their
kids move up the grades.
Christian Mihaila is a Grade 6 teacher who also works for Vancouver-based Academic Advantage, tutoring Grades 9 to 11 math. He says that a different set of problems can arise when students make the
leap into high school math. From kindergarten to Grade 8, students become accustomed to using visual methods like grids and physical manipulatives like blocks to solve problems. When they hit high
school, it’s suddenly more abstract, more pencil and paper, more . . . old school. But by then, the math has become too advanced for most parents to effectively help with and they either have to hire
a tutor or drop their kids down to an easier stream. Or hope the school’s no-fail policy will continue to sail them through the grades and into university, where professors from coast to coast are
vociferously complaining that first-year students are woefully unprepared in math. WISE Math co-founder Robert Craigen at the University of Manitoba was appalled to discover that two of his students
had never learned long division, a necessary skill for his course. “Math is a universal language, but our kids aren’t learning how to speak it,” he says.
Back in Winnipeg, this domino effect is one of Anna Stokke’s biggest concerns. If they don’t know their math facts, she says, they won’t be able to do fractions, which means they won’t be able to
handle algebra, which means calculus is out, which means they can’t be engineers, doctors, pharmacists, economists, programmers, or any discipline that requires math, including skilled tradeswork.
But one thing they can become? Teachers, who can go through the system with minimal math training and arrive in class expected to inspire children to create and conceptualize their own mathematical
knowledge—and relying on a new set of parents to fill the gap. “This is a never-ending cycle of innumeracy,” says Stokke. “And we have an obligation to speak up for the kids.”
Posted on:
Tuesday, March 13, 2012
Why is it your job to teach your kid math?
1. My multiplication table was drilled into my head by rote in grade three. It is the one piece of information from my school years that I use every day.
□ Quick, what’s 54 x 93?
☆ 160 – 13…147…I think?
In my day I would have multiplied 54 by 93…what’s the diff really?
○ Tables don’t do you any good unless you understand the principles at work in them. Because unless you’re some kind of savant, you simply can’t hold big enough tables in your head.
■ Tables up to 10×10 should be by rote in addition to whatever conceptual exercises are needed to show the meaning – then the algorithm for using this table to find all other
multiplications is simple. The fact that the table to 10×10 is not drilled in at all any more is a problem. it’s all well and good to show a 3×9 piece of graph paper and count
your way to 27 squares to show what multiplication represents. But at some point you need to cut away from the graphical tools because they don’t scale and learn to just work
with the numbers.
■ The presence of manipulatives is not mutually exclusive to memorizing your multiplication tables.
■ that’s why they taught us the easiest method to calculate that type of problem, strips of paper, grids etc. are not necessary if you use the time tested methods.
■ The problem is that the easiest method in the short run is not always the easiest method in the long run. For example, having a deep understanding that 37×41 is really (30+7)
(40+1) and a visual way to make sense of this will have incredible benefits when kids are faced with (x+3)(x-5)
■ Oh, I disagree, Thwim. Take the times tables, for instance. Sure, I learned 1×1 to 12×12 when I was a kid. But I’ve been a college math teacher for decades now, and it seems to
me that if people know up to 9×9, that’s plenty – x10 and x11 are easy. And since the x0, x1 and x2 are also really easy, it’s only 3×3 to 9×9 that people need to memorize.
Within that batch of 49 combinations, almost half are symmetrical – 3×5 and 5×3 are the same. So there are only 28 that need to be memorized. You don’t need to be a savant to
memorize this many, and, once you have done so, everything else is way easier: you can start to see the principles, because you’ve got the foundation.
■ really – quick what is 9×9? it’s called building blocks
■ 81
I think it’s easier for me since my first language is cantonese and I learned it in catonese. All the numbers 1-9 have one syllable (unlike say, seven), there is more of a
syllable-digit-spacing pattern that helps with memory? Even better, even though the proper way to say 81 is eight-ten-one, I just boil it down to digits. 9 9 81, 4 7 28.
I had a very large and very colourful multiplication table when I was younger than 8, that I like looking at, because it was colourful.
Rote is okay, as long as you don’t screw up and have kids associate it with stress and feeling stupid. Which was piano for me, I dreaded going because it was hot and the teacher
always sounded like she was one step away from smacking me even though she didn’t.
■ 9×9 = 9×10 – 9
You see, multiplication tables are pretty lousy building blocks if you don’t understand what’s going on behind them.
However, if you *do* understand, then you can do things that let you figure out 54×93 = 54×100-54×7=54×100-50×7-4×7=5400-350-28=5022
■ I don’t understand what is wrong with learning the basics by rote, if it’s backed up with understanding. In the early 80s, by Gr 1 or 2 we were learning “grouping arrays” (ie, 12
objects on the paper, with the direction to draw circles to make four groups of three)…the basic “groups of” idea behind multiplication and division. Then by Gr 3, we were
starting to learn the Times Tables up to 12 (with math drills and flash cards).
With those basic building blocks in place, students could then move on to long multiplication and division…
■ Nothing wrong with them at all, but the point I was making to marjory — which perhaps I wasn’t clear enough about — it’s not just the memorization which is used every day. It’s
the understanding beneath it.
■ I agree with what you have addressed…
○ Oh lord I got that wrong didn’t I? Got my dunce cap right with me.
☆ try using the current methods to anser that “quick”. that’s the problem with those of you that think the change has been benificial ( assuming you do) no one ever had the 54 times tables
memorized. the entrenched buraeaucracy howver needs to attempt to justify their existence somehow though.
☆ 4500
150 = 4650
for me this is the first way i came up with but i’m sure there are better ones. i can keep track of numbers this way.
○ 54×100 = 5400
54×7 = 350 + 28 = 378
5400 – 378 = 5022
That’s my preferred method.
■ 54
Effortless and brief — nothing wrong with your breakup, Owen, but it masks something: many people complain over multi-digit subtractions that require “borrowing”, like the one in
your last step. It relies heavily on memorized basics, another strong argument for drills to reinforce foundational skills in early education.
Anyone who has attended a workshop run by the consultants promoting the “no algorithms approach” will see something like the following example:
512 – 497
In the OLD method, you’d line them up and start on the right. Can’t take 7 from 2, so you have to borrow. Change the 1 to 0 and the 2 to 12. 7 from 12 is 5 (IF You remember
this). Now you have to subtract 9 from 0 . Borrow again … anyone lost yet? And we’re only halfway done!
Then the consultant will say “Here’s how WE teach your children how to do this — it’s called a strategy”. Observe that 497 is 3 less than 500 subtracting 500 from 512 is easy:
that’s 12. Then we correct by adding the 3 back on: the answer is 15.
And the triumphal pitch line: ”Which method do you wish YOU were taught? Which way do you think your children should be taught?”
There are several problems with the conclusion this example is supposed to be forcing. I’ll mention three.
First, It is a STRATEGY, not an algorithm. It works in a teensy minority of problems. Yes, there are other strategies taught, but none of them are general. Altogether the
collection of strategies only works in a minority of small “toy” problems. They are not well suited to, say, multiplying two randomly chosen 4 digit numbers (especially if one or
both include a decimal point) or to adding five randomly selected 3 digit numbers. The standard algorithms work EVERY TIME. Further, they don’t work only on toy or small
problems. With only a larger investment of effort, one can solve larger problems. The “strategies” fundamentally break down when numbers are too big, for example, to draw a
rectangle diagram for them. Such as … numbers in real world problems. The strategies tend to be used for numbers up to 2 or at most 3 digits, and even then not all cases fit
them “comfortably”. This is why, throughout the curriculum framework documents, you will repeatedly read phrases like “…for larger numbers students should use a calculator”.
Second, it is claimed that, while the traditional methods do not teach understanding, strategies are all about understanding. But it should be obvious that, when one learns a
hodge-podge of processes and you’re left to sort out which one to use on your own, there is no overriding understanding whatsoever, no systematic framework into which one fits the
general picture. Knowledge and understanding come by having tools that allow us to organize information in our world into a “big picture”. That is what the standard algorithms
do. In order for students to “understand” arithmetic they must be exposed to methods that, IN PRINCIPLE, work in all cases. If a child has this there is no harm in using
calculators to preserve effort in large problems. But without it, turning to calculators is a crutch that reinforces lack of understanding.
Finally, there is nothing “new” about this or similar strategies. I have gone through the same example with groups of adults having no advanced education. Halfway through the
deliberate muddling of digit-work, I ask if someone already has the answer. Someone does. Then I ask for a show of hands and — by golly — practically the entire room does! I
ask how they got it, and — lo and behold — they all used “the strategy”.
So then I ask “At what grade level were you taught this strategy?” They weren’t. They weren’t taught the strategy, you see, because it was unnecessary for anyone to teach it to
them. When students are exposed to enough variation in early arithmetic, they develop obvious mental shortcuts in places where they recognize problems with a certain
characteristic. Familiarity breeds fluidity. And this is a typical trick that people develop.
Trouble is, along with eliminating standard algorithms, the current approach de-emphasizes drill-work and repetitive exercises, the very place where students develop such
strategies ON THEIR OWN. It is said that drill- work destroys understanding. You’ll often hear the pejorative phrase “drill and kill” in this context. But nothing could be
further from the truth. Familiarity through repeated exposure and variation is absolutely essential for understanding. Insight is like oil — you have to drill for it. In
reality, there is no need whatsoever to TEACH these tricks (“strategies”) because if students are given proper levels of exposure they develop them themselves. PLUS they go home
with a method that works EVERY TIME in their back pocket, and understand processes that apply in EVERY CASE.
■ That’s how I would figure it out (the multiplication problem)…it’s so much quicker!
I agree with all you’ve said. I’ve witnessed my 12 yr old son use “strategies” to answer 3-digit addition and subtraction in seconds. Pretty impressive, but did he understand how
to do it the “long way” too? (Turns out he did…so I’m comfortable letting him continue to use his “strategies” when he can).
■ Well said!
■ This is the smartest comment I’ve seen on this entire thread. Thank you for writing it.
■ And you can use your fingers and toes to add up six, five and two!
■ I’m with you Robert & I’m curious about your age. I’m 65 & was taught by rote – tedious at the time but well worth it from my current perspective. I was good at arithmatic – still
am… but never any good at math. Algebra might as well have been Greek but I di eventually pass & Geometry was a breeze by comparison. I took math till the end of gr. 11 and passed
it, with limited understanding but some invaluable tools used with ease to this day. I can use those times tables efficiently – I even teach them; it’s like having a calculator in
your head. And basic arithmatic skills will get you through most day to day calculations – anything more complicated – use a calculator. Beyond that hire someone else to do it. I
write for other people & get paid for it so I’m quite willing to pay someone to do complicated math for me - at least I undestand the basic principles of it. Thank you Miss
Rosenberg & Mrs Bain and all the other grade school teachers who drilled it into us… and to my dad who taught me how to add long columns of figures – by ‘bunching’ later called
something else; 7+3 = 10, 6+4= 10, 5+5 = 10 etc… a tool I still use.
I my day they taught grammar much the same way – quite useful in my long & checkered career. Just listen to people today – on the street, on the radio, on TV, in the ads & it’s
plain that method got tossed aobut the same time as the times tables! People today don’t even understand prepositions! never mind verb tenses. ‘To’ cannot be swicthed for ‘of” or
‘on’ or ‘under’…
■ How is this different than students’ attitude towards math, and their success (or lack thereof) from years past? When I was in high school, most of the students who were good at
math were the students who would have been good regardless of the teacher and their style. The mistake is attributing success in math then to the use of rote and algorithms. The
students who “owned the math” then were going to own it no matter what strategy was used
Let’s keep some semblance of perspective here, the amount of students who hate math now is no higher than the amount of students who hated math when I was in school. Students
weren’t lining up to take calculus then, so why are we making it sound like those were the “glory days” of math and we have to do whatever it takes to get back there?
Multiple strategies to responding to a particular problem in math is supposed to increase students’ flexibility, NOT replace efficient algorithms. I’m disappointed that people
believe that we don’t need to understand why we “carry the 1″. It is that very attitude that created the general dislike towards math in the last generation or two. Ironically,
those generations are todays teachers who generally lack a passion and/or understanding for math.
I believe efficiency and understanding in math should go hand-in-hand. I’m disappointed that ignorance still persists in this area. It seems to be all about people trying to take
credit for “saving the day”. We need to stop trying to save the day and save our students’ education; all of them, not just the students who “get it”, but also those who don’t.
To think algorithms will solve all our students’ woes is ignorant, and to think multiple strategies will with no acknowledgement of efficiency is equally ignorant. Let’s please
try to find a balance so that ALL students can begin to enjoy math the way we all want them to
☆ Hey, that’s not one anyone’s memorized from a table. What’s your point?
○ That’s exactly my point.
■ But if you’ve memorized the tables, you’d have the basic single-digit multiplication available to solve the algorithm fairly simply.
☆ What’s 11×72 – and if you can’t write down the answer without thinking you have a problem.
☆ I would go 50 x 90 = 4500, then 50 x 3 = 150, then 90 x 4 = 360, then add 12. Answer is 5022.
This is why we show kids different methods.
□ But do you understand why the multiplication table is why it is? I think that the blatent memorization that the article promotes is wrong – understanding is the key to knowledge, and that is
what today’s curriculum is rightly endorsing. Math does not exist in a vaccuum. As a undergrad student at the prestigious school with an above A average, as well as A+ in my math courses, I
can tell you that understanding the math has brought me success.
2. This article is misleading, and it’s unfortunate that it frames public school teachers as simpletons who have no idea what it is they are doing or why they are doing it. I did some “digging” of
my own through the Ontario mathematics curriculum document, and I found expectations such as, “evaluate expressions that involve integers,including expressions that contain brackets and
exponents, using order of operations” (pg 112), and “measure the circumference, radius, and diameter of circular objects, using concrete materials (Sample Problem: Use string to measure the
circumferences of different circular objects” (pg 113). These expectations don’t seem particularly mystifying to me. How is it that parents ‘from all over Canada’ are so confused? Do you have
any data to support your claims, or is your evidence just anecdotal?
□ Paintchips- don’t confuse the curriculum, which demonstrates the desired learning outcomes, with instructional methods, which this article is about.
I’m a math teacher, and I very rarely use the instructional methods taught to me in university. They are ineffective for the vast majority of students. Granted, I teach in junior high, so I
can’t speak to higher or lower levels, but I use drills and rote learning for a slim majority of my practice time.
For instance, when the area of a rectangle is to be determined, a student could draw the object on graph paper and count the squares, or use Base 10 blocks to build the object and count the
squares, but that is very time-consuming (and, I expect, a little demeaning) and does not reflect the reality of the tasks expected of students in high school, university, or the real world.
My students are expected to know the formula, how that formula works, and execute the equation.
□ You’ll probably count this as anecdotal, but while I was good at math in school, use basic math daily and can often add subtract multiply or divide in my head as fast as I can do it using a
calculator, I have often been baffled by my daughter’s elementary and middle-school math. The methods they are expected to follow are overly complicated and take way longer than the way we
were taught.
☆ Sure, and I completely understand the necessity for competency with basic math skills. Especially as children progress to high school, something as simple as having a solid grasp of
multiplication facts can become the difference between passing or failing grade 9 math. Higher concepts become unattainable without a solid foundation. Moreover, a student’s success in
grade 9 math is often an indicator of how successful their overall high school career will be.
I teach grade 8, and mathematics is of course a daily focus. Both myself and the other intermediate teachers in my school are dedicated to making sure that our students are prepared for
high school, and our focus is on building foundational competencies. Maybe I’m not in touch with what is going on in the rest of the country, but this article does not ring true to my
own experiences as an educator.
○ Glad to hear it is not true of your experience. My own experience lies somewhere between yours and what is expressed in the article. My ex is a teacher; I have a teaching degree and
taught for thre years. So I know how edicts from above can run counter to common sense and good teaching practices. (My degree is in English and I well remember the “whole language”
nonsense wherein kids were expected to learn language skills by osmosis and any teacher caught giving instruction in spelling or grammar would face the wrath of the system.)
☆ There’s an assumption that if it takes a “long time” that it’s a bad thing. When, in a “real life” job were you timed on how fast you could get an answer to a mathematical equation? I do
understand that there are deadlines to meet and timelines to meet goals in “real life” jobs.
A university Math professor would have learned through the wrote process and they clearly had an interest in Math and thus would have been successful in learning the rote process and
would now say that if I don’t understand it then it’s not right for children. Just as an english professor who learned how to spell phonetically may say that if they don’t understand the
whole word process would say that it is not right for children.
What I’ve seen in my classroom is that more students are understanding math and using or coming up with strategies that make them successful. Some of them even use the rote process. Now I
have students who would not be successful with the rote process who are successful even though their parents don’t understand the strategy they’ve learned.
Yes, I know that 9×9 is 81….but I also know at least 3 other strategies in which to solve this….some take longer than others, some are concrete (strips of paper or grids) some
abstract…..but they all get to 81….then again…I’m just an inexperienced simpleton when it comes to math because I’m a teacher.
○ I don’t think anyone is saying that teachers are inexperienced simpletons.
The issue as I see it (as a parent of school-aged children) is that in SOME districts (across the country, it would appear) teachers’ hands are tied when it comes to how they deliver
the curriculum. Some teachers are told they MUST teach and evaluate ALL students on EVERY strategy, when it might be more effective to teach all the strategies and let each student
use what they are comfortable with.
○ Every time I wait for a teenager to give me a looney instead of 95 cents change I can hear the clock ticking in my head as I watch them stair blankly at the change I hand over. Yes
there are real life times where math has to be done on a timer and patterns don’t appear to be helping the youth I deal with daily. Just because you teach it and you like it doesn’t
mean it’s working.
☆ Look at factoring a quadratic function and you will understand why it is important to know how to factor small integers in your head.
□ It is all over Canada. I live in NB and students are having problems with the “new math strategies” here including my daughter in grade 3. Not all teachers understand it and not all
teachers can teach it. My daughter understands math if we get her to do it the old way.
☆ I agree Jim, my daughter (also in gr 3) was having difficulty understanding multi-digit addition and subtraction. I showed her the carry/borrow 1 system, and it was a success, she got it
– and can do the equations without difficulty.
3. …and tomorrow is Pi day
4. Or you could just watch the 60 Minutes section on the Khan Academy
□ Excellent link, Emily!
☆ Then you might enjoy his original TED talk too
□ My kids use the Khan Academy. It’s brilliantly done. Straightforward, friendly, and (this is the best part) never impatient. Kids can & do pause, rewind and repeat until the concepts are
clear. Then they do exercises until proficient. Probably the best practical application of the internet I’ve seen yet.
I am a student there too. When my high school kids ask for help with trigonometry, I go to Khan Academy to brush up on my Soh Cah Toa.
☆ Thanks for letting me know….I’ve never heard from anybody that actually uses it!
I’m so tickled that we’ve got a new way of teaching kids so that they understand a concept and feel confident with it!
□ The problem on this site is that my daughter has been told that she is not allowed to stack numbers and borrow. Is this the case with others?
□ I checked out the Khan Academy and it is good, but it doesn’t show the new “strategies” that my daughter is being shown in grade 3. When we helped her using the “old” way, she picked it up
quickly, but when she took the work back to school, she was told to show her work using the new strategy and that she is not allowed to stack the number and borrow or carry numbers. She is
only allowed to use the one method that her current teacher is using. To be blunt, this is bullshit. Other teachers are using other “new strategies”, whatever the Hell they are. There is
no standard across the system at all. No wonder children are so confused and having problems.
☆ Oh I agree that it’s BS….it was when my dad showed me how to do something too, and the teacher wouldn’t accept it because it wasn’t the ‘approved’ method of the time.
Noop, there is no ‘standard’. There never was. And I agree, we are confusing our kids …it’s a dumb way to run a school system. The whole thing needs an overhaul
5. This article is mis-leading. Just because the curriculum allows for multiple styles of instruction does not mean that a teacher is required to use all of them in the classroom. All parents should
be spending time with their children at night to help them review their homework but that doesn’t mean the parents must adhere to the teacher’s style of instruction. I am a grade 2 and 6 math
teacher in Mississauga and have a very high success and engagement rate with my students. I will be the first to say that many of the instruction styles the curriculum provides us are confusing
at best but they do work for specific students who are having a hard time grasping concepts the “mainstream” way. It is the responsibility of an effective teacher to not use many of the styles
provided in the curriculum and teach using a style that most of the students can understand. This article makes it sound as if math teachers are just throwing concepts at their students without
support or instruction. This article reads as if all students are missing the concepts when in fact their students examples could be those students who are not grasping concepts through a variety
of teaching styles and learning methods. The example of Robert Craigen’s two students not learning long-division may not be the fault of their teachers, in fact I am fairly certain that their
educational sphere did everything in their ability to teacher them long-division through many different ways however some people for whatever reason cannot grasp mathematics and that is not the
fault of their educators.
□ You are right – it is not the fault of the teachers that kids aren’t learning long division. It’s the fault of the WNCP math curriculum that Manitoba has adopted. Long division has been
completed eliminated in the curriculum and in many divisions teachers are explicitly told not to teach it. Over the next few years, as more and more students enter universities having
learned under the new curriculum, the majority of students will not have been taught long division.
☆ Why should they have to learn long division when calculators can do it for them? The curriculum should change to reflect advancements in technology. Math should teach people to
think…not to memorize.
○ Because the exercise of doing long divisions by hand builds in a feel for what the result should be allowing you to catch when your calculator does something funky, and a backup
method when the calculator’s not handy. A blind trust in technology is not a step forward.
■ I learned long division in the 3rd/4th grade. In the 9th/10th grade we learned how to divide algebraic expressions from one another in a very similar way to divide we were taught
in elementary with regular numbers.
■ Things can be more complicated than that. While I do not know how computers do these operations in binary (I am only a math major, not a CS major), I do know that computers use
calculus to do trig. For example, if you want sin (pi / 3), it would need to use repeated differentiation to approximate an answer 0.8660……so you do not expect people to do trig
for a weird value like tan (3pi/10) or something. Sometimes, you have to trust technology.
○ Hi, I’m the Rob Craigen mentioned in the piece, and someone alerted me to these comments. I’d like to thank Mr. B and SPW for providing what we like to call in the profession a
“teachable moment”.
First, Mr. B, I blame neither the students nor their teachers for the lack of long division. To expand on the teensy factoid that appeared in this article from the 45 minute
interview I gave, the reason I asked the question in the first place is that, as a university representative on our province’s Math Curriculum steering committee, I was watching for
this very thing because, as mrmath says above, long division is completely excised from the current revision of the WNCP curriculum that affects all of the Western Provinces. In
fact, all four standard algorithms of elementary arithmetic are GONE. The curriculum has been implemented grade by grade and this was the first year we expected the “early
implementers” to graduate students taught with the “no algorithms” approach. From here, if nothing changes, the number of such students will increase until the ONLY public school
graduates knowing long division will be those whose teachers serve them contraband or whose parents school them in the dark art at home.
Second, SPW, your comment reveals an ignorance of the whole point of learning math. Why learn anything you can pay someone to do or obtain a device to do for you? Taking your
statement to the extreme, we could raise a pampered but severely ignorant generation of adult infants, completely dependent and completely incompetent.
But that is not really what’s wrong with the comment. It’s merely why it is silly.
What’s wrong is the fantasy that understanding can be had in the absence of competence and skill. The problem is that the developers of the current curriculum have bought into
precisely that fantasy. Cognitive developmental psychologists tell us what every parent knows: understanding and skill form a mutual scaffolding — you cannot have one without the
other. Want to teach understanding? Then it is ESSENTIAL that you provide a framework of skills along with it.
The four standard algorithms are critical skills upon which hang some of the most important elements of understanding in the framework of the mathematical education students need to
advance. This is an inappropriate place to attempt a full development, so I’ll just assert what most educators know: the modern concept of number, which lies at the heart of
mathematical UNDERSTANDING, turns on a key fact learned in the traditional curriculum before grade 8: that rational numbers have repeating decimals whereas irrational numbers do not.
The ancients would have been astonished that children could understand this, for they found the presence of irrational numbers deeply puzzling, even disturbing. The development of a
theoretical tool that assists understanding to this level is one of the great accomplishments of western society.
What is that tool? Ask yourself, SPW — I’m sure you are familiar with the above fact — how do you know that rational numbers have repeating decimals?
It so happened that, in the calculus class where I did that little survey, this factoid came up in our lesson and I asked my students who were able to provide a simple proof with just
two words: ”Long Division”. Except, of course, for the two who had never been taught this essential tool.
Calculators are wonderful things. But they will never replace the essential insights gained from learning the four standard algorithms. If you care about teaching understanding in
math, you ought to care about those algorithms.
■ Wrong. Go read the current curriculum. You have not seen one single child that had graduated from that curiculum, as it is not yet offered in grade 12 (next year will be year 1)
To make it simple, so you can understand, not one University Student is a product of the math program currently in use. Not one. The students you see are products of the LAST
program, and if you were truly involved in THAT debacle, then this article should be about your irresponsibility.
The logical conclusion here is that if you would lie about one thing, then you would lie about everything.
As for long division, it is indeed in the new curriculum.
Go fish.
■ Lorne, Lorne. You didn’t read my comment very carefully.First, I HAVE read the curriculum. I have sat on the Manitoba Provincial Committee, whose job it is to implement the
curriculum, since 2005.Second, I was quite clear that the whole point of identifying those students in my class who had not seen long division was to detect the presence of
students from schools that were EARLY IMPLEMENTERS of the new WNCP curriculum. Navigate, if you will, to the Manitoba Ministry of Education website, where you will find a
timeline for changing to the new curriculum, and verify the following:System-wide implementation is to be complete by 2013. Early-implementing schools were to produce their first
high-school graduates in the Spring of 2011. Thus, a FEW such students are now in university; soon to be a LOT.You say that long division is in the new curriculum — are you
talking about the WNCP curriculum, used across Western Canada, as I am? Well, Lorne, that is news to me. I invite you to publicly shame me on this point. Here’s how: Go to the
official website for the curriculum, wncp.caNavigate to the current framework documents (which describe every learning outcome in the curriculum). Handily, these are pdf files
that you can text-search at your convenience.Search for “long division”.I look forward to you reporting back to us what you find. I predict that you’ll find exactly one place
where long division is mentioned: In the Grade 12 Precalculus stream, where teachers are told that students can understand the remainder theorem in terms of long division of
polynomials (an interesting prospect, given that long division of numbers does not appear in the learning outcomes or achievement indicators for ANY grade level).Happy hunting!
■ Wow, you are one rude knowitall.
■ Wow! Looks like I touched a nerve with that calculator comment. As it turns out I am high school math teacher with 30 years of experience. Just playing devil’s advocate here
to stir things up.
○ Yes, let’s just teach them to use calculators and computers, they will never have a need to think on their own. WTF are you smoking? A huge part of learning is memorizing. Do you
think Doctors know everything, and when faced with a problem, they just automatically figure it out? No, they learned and memorized. They know what medication is good for what
disease, because the remember their education and training. If you can’t remember or memorize, you can’t learn.
□ It seems that there is a disproportionately large number of “people for whatever reason cannot grasp mathematics” and although you may not think that is the fault their educators, I tend to
disagree. I believe the people who teach mathematis don’t have enough tricks in their bag…they don’t know enough about math to teach it in enough different ways to overcome people who don’t
grasp the subject easily. I had trouble as a child with mathematics. Later, I had a teacher who had her PhD in mathematics and I soared. I believe math teachers should have a degree in
mathematics and then a post degree in education so that they can assit the many people who don’t grasp the subject readily. Afterall, it isn’t really those that grasp it easily that need the
teacher the most, is it?
□ Wrong. My daughter was told to only use the method that her teacher showed her, not to stack numbers or use the borrow or carry methods. This is why it’s so frustrating.
□ Sounds like you are a teacher the way you are defending them. There are some good teachers, but there are also some bad teachers, lazy teachers, etc… I have dealt with several of them and
our education system in NB is suffering very badly. Our government keeps making cuts to the system and it is hurting the students.
6. One thing about the new math is for students to be able to figure our their own algorithms when figuring out things like adding two 2-digit numbers, subtracting, dividing, or adding fractions.
If you want to figure out what it means to design your own algorithm, here is a problem:
It takes 3 hours 52 minutes and 47 seconds to to go from Toronto to Sudbury. It takes another 2 hours 14 minutes and 38 seconds to go from Sudbury to Sault Ste. Marie. How long does it take to
go from Toronto to Sault Ste. Marie? ( Don’t worry about gas breaks.) Clue: you are not using units of 10′s or 100′s.
Part of the problem with the new math is that students are not taught particular algorithms by the teacher unless that teacher chooses to demonstrate a particular way. There are good points in
having students work to create their own algorithms. They can hopefully understand the meanings of the values in their responses. For example 35 + 65 does not equal 910. Through estimation,
the students should be able to figure out that their responses should be anywhere between 90 and 110. Then, they should be able to solve the problem by either adding the one’s first or do some
other method like adding 30 + 60, then 5 + 5. The answer will be 100.
One concern about teaching the new math is working with children with special needs. They will struggle until someone helps them with an algorithm. The Khan Academy and Jump programs help
students step by step.
Another concern has to do with different cultural groups learning math in different ways. For example, Chinese and Korean students will learn problem solving in some rote fashion which they
excel. However, the new math requires students to explain their understanding of how they solved the problem. For example, “What’s one way to make two dollars in change?”
Student: “40 nickels!”
Teacher: “How did you get that answer?”
Student: “I don’t know, I just did it in my head.”
7. “instead of stifling them with rote memorization and rigid methods,
children are to use their own learning style to explore mathematical
knowledge and conceptualize innovative solutions to complex problems”
Or in other words, we teach them complete bullshit. Rather than spending hours every night (this is just math – parents also have to teach grammar and spelling because the schools don’t do that
anymore either), it would be far less confusing for the kids to just be home-schooled in the first place.
My wife teaches accounting, payroll and finance at a private college, and at least half of her students, high school grads all, who have chosen this field, do not know how to calculate sales tax,
and do not understand that 1/2 and 0.5 and 50% are equivalent. This is what the “new math” has wrought…… but they have good self-esteem!
8. I’m an Engineer in Alberta and my brother who is in Grade 3 has outrageous math homework. One of his assignments was to “estimate” for multiplication. What’s 9 x 9? Use 9 x 10 and write down the
answer to that instead. For a whole sheet of homework! So basically if you got the question wrong, it was right, and if you put down the actual answer, IT WAS MARKED WRONG. He now goes to
tutoring for 45 minutes each week, and his marks have doubled from failing to honours. Yeah, alternative methods are great tools to use, but when you are FORCED to use these methods, even when
they don’t make sense, you’re penalized for them.
□ That, I think, is the bigger concern. The point of these tools is to give children the freedom to figure out how to do things in a way that works for them. If you force them to use a
particular tool, however, you’re taking away the very point of using these tools in the first place.
□ If your brother had estimated 9×9 as 45 he could be employed to give “cost analysis” to the government
9. Why do parents (and university professors) continue to insist that the way they learned was the best? My grade eight son has a better understanding of basic mathematical concepts and problem
solving than most adults. He doesn’t memorize facts but knows how to find answers.
□ I agree that they have a better understanding of maths concepts, but unfortunately application of concepts and transferal of this understanding to real problems is where I find their
knowledge lacking. I remember doing two main strains of maths, Pure and Applied and it is Appllied that is suffering now.
Also I can remember sitting down with my parents doing my homework and using them as a resource when I found a concept too difficult to get my head around at school. I thought parents being
involved in the home part of their child’s education was normal; I didn’t realise that this was passed off in its entirety to the state.
Could be another case of people wanting kids but not wanting to do the work that that entails.
10. Just go to
and this will help just about everyone with their math skills and so much more.
11. Maybe if teachers were spending less time constantly complaining about their pay and benefits, they could spend some time being, I don’t know, capable at their jobs?
□ You are right there, and don’t forget about complaining that they don’t get a day off every time it snows, or how much it sucks to have snow on the weekend because it is a waste of a snow
12. Most teachers are too lazy to teach anyone but the top 5 or 10 percent of their students. The rest are a waste of the teacher’s time. This story is nothing new,, it has been going on for years.
Thats why the teaching methods have degenerated into gross obfusication. No one knows whats going on and the teacher can give high marks to whoever the teacher wants to put in the top 5 or 10
percent of the class.
The unfortunate part of this story is that the top 5 or 10 percent are not necessarily the smart students, just the kids whose parents have money or a high social status in the community.
It doesn’t take a degree and someone being paid (I won’t say earn) 50 grand a year to put a few examples on the blackboard, tell the kids to do all the odd questions from page 125 to page 130,
never check the homework and call the students who fail the next test stupid.
□ Another critic who’s evidently never taught but yet knows all about lazy teachers. How bout lazy uninformed criticism?
☆ “.How bout lazy uninformed criticism?”
Yeah, I get it all the time.
13. My daughter gets 90′s in math but the other students in her class are struggling. Even though she is a very good student, the teacher screamed at her in front of the whole class and said she was
doing it wrong even though she got the right answer. I was not impressed, she had a formula as to how she got the answer and figuered it out when the teacher couldn’t.
□ Exactly the problem. My daughter has had correct answers, but has been told she found them the wrong way, and to show her work the way the teacher showed her. That is bullshit.
14. I would’ve said quite the opposite – the pace of school education is tailored to the bottom 50%. The top 5-10% are being held back because the entire room needs to learn at the same pace.
15. As a sixteen year old 10th grader and I could not agree more with the results I see in the class room around me and the story written. I see other students using Ipods or printout multiplication
charts, exasperated teachers telling kids time and time again how to do grade 3 work. I appreciate that this was put on the front page, because really, it’s about time people realized it.
16. This is irresponsible, one-sided and negative reporting that has, no doubt, insulted many gifted, thoughtful and effective math teachers across the country. I am an elementary school teacher,
and I have had the pleasure of teaching problem-based math to students for a number of years. The reporter who wrote this story neglected to get a classroom teacher’s opinion which, to me, shows
a lack of balance in telling the story. Yes, it is true that parents are often frustrated when faced with something unfamiliar when asked to help with homework, but that is a problem that can
arise in any subject. Teaching rote math skills serves to strengthen one ability – memorization. Students need to understand the meaning behind the numbers, problems and algorithms before they
can apply them with accuracy and efficiency. Problem-based math does this by encouraging students to apply what they already know, to use their own invented strategies such as manipulatives,
charts, diagrams, slips of paper, and yes, even calculators, and to work with peers and teachers to understand real-world problems in order to develop a strong number sense before just having
them learn and apply a formula. When they understand the meaning behind what multiplication and division problems are actually asking them to do, and they have been able to apply this in a way
that makes sense to them, then most teachers also allow and encourage students to apply this rich foundation of understanding to a formula or algorithm. Math teaching at its best is thoughtful,
nuanced and rich. This article painted the work of today’s teachers as disorganized and groundless. There is plenty of research backing up today’s methods. Check out Van De Walle, Trevor Brown,
and David DeCoste to name a few. I am particulary offended by the last sentence of the article that paints teachers as less-than educated professionals who can’t fill the ‘real’ jobs out there.
We have one of the most important jobs in society and this article is disrespectful to those who dedicate their lives to loving and teaching your children.
□ What drivel.
I have had two children go through the public school system. And I am a professor of mathematics at a “big research-intensive” university in Canada (26 years experience, research active and
my teaching evaluations are in the top 5 percentile).
In my experience, the answer to the question that forms the title of this article is clear: Because math teachers, generally speaking, in the K-12 system do not themselves understand math,
are not trained in math and are afraid of math.
And I know. More than once I would read over a marked assignment written by my son, for example, and it was clear that the teacher didn’t understand the solution presented. It has
also certainly happened that the solution the teacher though was correct was in fact simply wrong. When confronted, the inevitable result would be that the teacher got defensive and angry.
Teachers should be subjected to annual peformance review like everybody else and the bad ones fired and parent-teacher associations whould have the right to sit on these evaluation
And I will go further, having been involved in an off and on way in university administration over the years and having seen the data from across Canada, I can say with absolute certainty
that the academic records of students that enter faculties of education are among the weakest at any university and the pass rates within faculties of education are striking high compared to
all other faculties. Yes, students with weak academic skills enter the teaching so-called profession and that these are the people that teach our kids in K-12. It’s a joke.
The fact remains, the reason why little johnny or suzy isn’t learning math properly in K-12 is because the teachers themselves don’t understand it.
It’s the blind leading the blind.
□ I am also a teacher, and I agree with this article 100%. Van de Walle is a theorist whose research always supports his own biased agenda. Too many superintendents and education directors
fall easily for the twisted theories of a Van de Walle. Public education is in a crisis situation. It’s about time there was some dialogue on how to fix this very broken system.
☆ Have you tried using his methods?
□ As a parent of a Grade 3 student I have seen multiplication strategies being done over a period of 2 weeks (not even every day) – an insufficient time for an 8-year-old to grasp what is asked
of her to grasp. During that time she’s been bombarded with various strategies and, I am sure, confused with some. Then, at the end of that 2 week period, she is supposed to “think outside
the box” and do an art project (draw groups of candy) in order to achieve Level 4 (exceeding provincial curriculum expectations). If, however, she solves the problem without the “crutches”,
she is (just) meeting the provincial curriculum expectations for that grade and is discouraged to use the method that suits her best.
I am sorry, but I do not understand! Those strategies are supposed to be the training wheels used just until you achieve the necessary skill level needed for the two-wheel ride.
As educators aren’t we supposed to give our students the tools so that one day they will become autonomus learners? Isn’t it our job to observe and adapt to our students’ needs, even if they
are above the grade level not just below?
□ Concerend, hate to tell you but you’re completely wrong here. Most grade school teachers in the public system have no background in math and really don’t understand it.
My kid brother is quite a bit younger than me and when I came home for a weekend my parents would regularly ask me to remark his math assignments and test – without fail, the teacher’s
answers would be wrong (and generally worse than his). And they’d tell me that whenever they went in for parent teacher conferences the teachers would pull out my comments and be very upset
because I was daring to question their authority – my parents had a very simple answer – get it right and we won’t ask his brother to review your comments – get it wrong and he won’t stop.
That’s why my kids have never seen the inside of a public school and never will.
□ I so agree with this statement especially about how teachers are being given no support. Thank you. There’s alot of ignorance and I wish that people are more involved or are at least aware of
the development in the curriculum. They should be asking, who’s writing this stuff and why? And then ask how much is this gonna cost us. Unfortunately, the teachers are just pawn as someone
said in the thread.
17. It wouldn’t be fair to say that teachers are lazy or incompetent and that’s why the kids aren’t learning math. Seems these days they are only allowed to teach the board approved curriculum, one
that takes “into account all learners’ styles and capabilities”. I wouldn’t necessarily call this dumbing it down but the main focus is trying to accommodate everyone, which rarely ever works.
As parents we should better understand our kids’ capabilities and help them learn accordingly.
18. So… new-fangled teaching methods that parents don’t understand leads to more parents doing their kid’s homework for them? Colour me unconvinced. I think the problem is that mom didn’t like
getting a B in primary school math, when the real lesson for mom should be: stop doing your kid’s freakin’ homework. It’s bad for you (you have better things to do), it’s bad for your kids (they
won’t learn), and it’s bad for other kids (grade inflation).
We don’t celebrate this enough in Canada, but we have a great education system – in a large part because we teach principles well, rather than rote memorization. In the PISA we consistently score
well, and there hasn’t been the decline in math scores that this article seems to presume exists. Yet in a typically Canadian fashion we like to assume that the American story of declining
quality of education applies to us as well. Step aside healthcare, our education system is our real success story.
19. I picked up this issue solely because of the cover story, and I hadn’t bought a Maclean’s magazine in well, forever. Math education is an interesting and often polarizing topic in education, with
no easy answers. I like the discussion in the comments here, and Robert Craigen thanks for expanding upon what you said in the article.
Also, I’d like to add that while people like SPW sound ignorant when they say things like “Calculators can do long division quicker,” ignoring any impact of the understanding of place value,
rational and irrational numbers, multiplication, and division etc (as mentioned by Mr. Craigen) gained in the process of doing those long division questions, there is some truth to those
seemingly ignorant claims. It’s obviously important to focus on the process and methods of multiplication and division, but we can’t forget that we have some amazing technologies that can help us
out for certain types of questions.
From what I’ve seen in classrooms, I’ve seen teachers at all levels afraid to let students use calculators for anything, and I’d say it hurts more than helps in most cases.
I’ve written up a post about this article on my blog (http://www.macroeducation.org/response-macleans-article-why-is-it-your-job-teach-kid-math/) if anyone cares to read it. It was too much to
post here, and I have to warn that it’s a bit long.
20. Let me address some of your points.
While I agree that teachers should have been interviewed, the individuals who should really have been asked to provide a statement should have been the students. It’s about them after all.
Yes, it is true that parents are often frustrated when faced with something unfamiliar when asked to help with homework, but that is a problem that can arise in any subject. ”
Unfortunately, math isn’t exactly a subject where there’s significant divergence in meaning. If international students can come to Canadian universities and exceed and outperform Canadian
students with no Canadian educational experience then it’s fair to assume that parents should be able to do the same with their kids in primary education. The point is addition, subtraction,
multiplication, division, etc, shouldn’t BE unfamiliar. If the systems used make no sense to engineers or math professors who are capable of using numbers then there is obviously a problem. ”
Teaching rote math skills serves to strengthen one ability – memorization. ”
I’m sorry, but knowing that 9 x 7 is 63 is not memorization in my opinion. It’s something that should be an automatic response in the same way as knowing the difference between left and right. No
adult takes their thumb and index finger and tries to figure out which hand forms the “L”. There should be no reason kids should be multiplying 7 x 10 and then subtracting to get to the right
answer. And yes, it’s wonderful that kids understand that 2 x 5 is actually 2 +2 +2 +2 + 2 (and they SHOULD know this), but if they’re REQUIRED to do that to answer the question, then there’s a
significant problem. But by your account, such a system should be utilized by our children, even promoted as being superior to anything else.
“Problem-based math does this by encouraging students to apply what they already know, to use their own invented strategies such as manipulatives, charts, diagrams, slips of paper, and yes, even
calculators, and to work with peers and teachers to understand real-world problems in order to develop a strong number sense before just having them learn and apply a formula.”
Calculators? Really? You realize that most university calculus don’t allow any calculators. In fact, what I remember from Canadian elementary schools is the fact that we spent half the time
trying to see what swear words we could spell upsides on our calculators (A$$HOLE, HELL etc). I guess that I helped me expand my vocabulary, but it didn’t improve my math skills.
I don’t understand how kids are supposed to apply “their knowledge” if at this point, by almost your own admission, you haven’t taught them anything because they’ve been busy using “inventive
strategies” to explore numbers on their own.
Not all teachers are bad. There are some great teachers out there. But the reality is, there are also some that are complete rubbish. What Anna Stokke wants to address is the fact that many
teachers receive poor math training and curricula in both public schools and education faculties is cheating the kids of a real education.
21. Today’s math curriculum ensures that there will be plenty of work for me to do for years to come… This article makes some very good points. Deep understanding is good, but basic understanding
needs to come first. Without an understanding of how to add fractions or do long division or carry numbers, there is no foundation for deeper understanding to be built on. Creative teaching
methods have their place, but when it comes to building a mathematical foundation, the tried and true old fashioned methods should not be discounted – including memorizing that times table.
22. I was educated first by the British system from grades 1 to 8, and then ended up in Canada for the rest. I’m now teaching at a university.
While it is true that I had to do rote learning from early on, at a certain level we were required to understand the principles behind these algorithms that we are learning. When I got to grade
9 in Canada one of the things that amazed me was that the material was mostly things that I have learned in grade 4 or 5. I went from a B+ to A- math student to being labelled a “math genius” in
my school simply because I was competent for material that I have learned much earlier. I later found out that some of the material I already learned in grade 8 (and struggled through) would not
appear until grade 12 here. My experience as I was going through high school is that most people have such a weak foundation and can’t remember facts that I consider basic. When they are
finally taught advanced topics, they struggle even more. I struggled too, but I would be struggling on the new concepts that are taught to me. My classmates would often be struggling on the
foundations as well even if they may not realize it.
I was lucky in grades 11 and 12 to have math teachers who actually had a real mathematics degree (e.g. honours degree), but this seems to be a dying breed. At least they encouraged me to look
elsewhere for more advanced topics. If I wasn’t so bored I probably wouldn’t have learned so much more and ended up doing well in much harder topics in university.
The “old math” worked for me, but is this “old system” really that great? Many people got bad marks and didn’t understand anything. And maybe it is very stressful for the kids (it was certainly
for me). That is very true. But at least we are honest with them about their skills.
Maybe some people would have benefited from the “new math”. But let’s not kid ourselves here. Many people now “understand math better” and not “struggling so much” because we evaluate them
differently under the new system. They still don’t understand any real math. Take someone with 80% in grade 12 “old math” and someone with the same grade for “new math”, throw them a “real
world” problem, and you will see the “old math” student consistently solving problems that “new math” students cannot solve but not the other way around.
“Discovery-based” education is great for those relatively few students who actually try to discover things. The average student can’t (or won’t) be doing more than just memorizing the
discovery. For them, it is better to memorize the rules and the boring stuff because these things are more applicable. In the old days we have average people who don’t understand very deeply
but at least can do basic tasks. Now the average person neither understand very deeply, nor can they do these tasks.
23. Parents should help their kids, after all they should be the one who know their kids learning style the best. I have a 20 year old daughter in university and I remember sitting with her for up to
2 hours a day from the time she was in kidergarden all the way to the 8th grade helping her with her school work. Along the way we noticed the “new math” simply did not work for her so we
adapted. After she learned the basics, she never had to study for aa math exam. Has has to put in the time with your child. You have to know your child and you have to decide that your child and
thier education is more important than sitting on the couch drinking a beer while watching grown men play. There are no shortcuts, you really reap what you sow and so will your kids
24. It sounds like the cirriculum hasn’t changed much, since my son was in elementary school in Alberta.
The math cirriculum at the time was a confusing nightmare that focused on “estimating” from Grade 1 to 3. When my son entered Grade 5, we paid for a tutor, and enrolled him in night classes
(along with about a 1/3 of the class) for the latter part of Grade 6. One teacher told us our son wasn’t “wired for math”. He managed to scrape by with a C+ and was promoted to Grade 7. For the
record, the rest of his marks where average – Bs and Cs. We were told by various teachers that he was an average student at best, who possibly had ADD. It was recommended that he should consider
going into a general studies type program in middle school.
Over the summer, we moved to Maryland. He was tested by the County’s school board to make sure that he was, in fact, eligible to move on to Grade 7. He failed miserably in math and English. He
tested at a Grade 2 level in math and a Grade 3 level in English! Obviously, the school wasn’t going to send him all the way back to Grade 2 or 3, but he did have to repeat Grade 6. His math
teacher went over all his work. She determined that the way he’d been taught was hugely confusing – made no sense. A similar cirriculum had been used in MD, but had been dumped, because it was so
bad. Anyway, she stated that he would understand math by the end of the year.
Not only did he know his math, he was offered a spot in the Johns Hopkins gifted program. He was in the 99th percentile (PSATs) by the end of the year, and excelled in all his subjects. He
continued to do extraordinarly well for the rest of his school years in MD and DC.
25. During any time of significant change there is always discomfort and criticism from the old for the new. The old often is reluctant to give the new time to prove itself and because it does not
make sense from their experience it is not wanted. You have to consider that the people so in favour of the old methods were successful using those methods. Learners come in many styles and
teaching only one method will not enable as many students to be successful as will teaching several different methods. Because something was good for you does not make it good for everyone.
Understanding is key for any subject beyond mere memorization. As for the old method being “time tested” therefore successful that is a myth. The whole reason the curriculum changed is because
ressearch showed that students were not able to
26. be successful in high school and post secondary math because their understanding was lacking. Armed only with algorithms that applied to specific circumstances they were not flexible in their
understanding of numbers and became frustrated. You wonder why the person at the till can’t make change when you give her 20.85 and the item cost is 12.85, it’s because the algorithm didn’t
specify. It said to cross out the next digit, add a one and subtract. Students who understand numbers and have worked in many wways with them see the interconnections and can provide change.
Don’t short change our children because it isn’t they way you did it. For every person who is successfull with the old way there is at least one person who was not, one person who dropped math as
fast as they could because it became too complicated without understanding, one person who hated math in school and made different career and life choices because there was only the right way or
the wrong way.
□ Yes! Well said- I have many friends who are worried about retraining and returning to university due to math. When I taught them some math conceptually, concretely and pictorially, they asked
“why didn’t we ever learn it this way??” The assumption that the old way worked is an assumption and is very far from factual.
27. We’ve switched to using the Jump Math program and it’s been glorious. All the students can do the fundamentals, their confidence is sky high, and they’re all suceeding. I’d reccomend that program
to anyone.
28. Canada keeps following American bandwagons even though their education system is generally much, much worse than ours on any number of fronts. Decades ago, it was New Math. Now it is New-New
Math, which is really foo-foo dust…squared. The Math curriculum seems to want to create more pure mathematics professors, even though there are a limited number of jobs in that field. Anything
practical or needed for business, like money, is taught for a few days at most in the elementary grades. Now those skills are considered applications and are insulted with the phrase “rote
skills”. In most provinces, there are five strands to be taught each year from Grade 1 – 8, so in Ontario, we are teaching Algebra (no kidding), Probability (yep!) with as much or more time and
emphasis as Arithmetic and Finance. Because it is in the Elementary curriculum, this is what is tested.
29. I have had two children go through the public school system. And I am a professor of mathematics a “big research-intensive” university in Canada. In my experience, the answer to the question that
forms the title of this article is clear: Because math teachers, generally speaking, in the K-12 system do not themselves understand math, are not trained in math and are afraid of math.
30. Last week I had an intervention at my daughter’s school regarding her multiplication test. I wanted to know what was required of her in order to get Level 4 (which she always used to get). I am
an educator myself (a teacher trainer), my husband an engineer (taught math at the University) and we come from Europe where curriculum implementation is narrowed down to syllabus (used to ensure
consistency between schools and that all teachers know what must be taught and what is not required) and where there is a prescribed and logical flow of the lesson material. So, I had a talk with
my daughter’s Grade 3 teacher, and was completely disgusted with what I had learned. My daughter solved the multiplication problem (5 containers each containing 25 candies) using both
multiplication algorithm (5×25) and addition (25+25+25+25+25). She even checked her answer by drawing 5 groups of 25 tallies.I was told that for Level 4 she should have thought “outside the box”
and drawn 125 candies!? Furthermore, she should have shown as she did her addition that 5 ones and 5 ones make 10 by drowing lines and writing the numbers on the side of her addition. And that
would’ve been “thinking outside the box”! I told the teacher that I want my daughter not to use the “crutches” (I’ve been teaching her to be more effective- that’s what math is all about) and I
argued that her not using all those ridiculous and time consuming “strategies” IS thinking outside the box because she’s one step ahead of her classmates who wasted who knows what time drawing
125 candies! I don’t even want to mention that my daughter’s class did the multiplication only for two weeks (and not even every day) and after that went on the division for which, you’ll all
agree, you need a sound understanding and knowledge of the multiplication (tables). I just want to know why education here in Canada is not as transperent as everywhere else? Why does the quality
of our children’s education depend so much on one (omniscient) individual and their interpretation of the Curriculum? I want to be able to take a proactive role in my children’s education and to
know what methods are being (ab)used in their classrooms.I want to know when they are being tested and what is expected of them BEFORE they take the test. I want to know that if we move to a
different neighbourhood and change schools,they won’t do the same material they’ve already done just because there is no prescribed order of the material taught.
I am disappointed and worried. As an educator, I have already seen the reprecussions of this system; as a parent I want it changed.
□ I agree with you re: having to draw 25 candies for full marks. I’ve come across the same issue. Some teachers are very particular…a square with 4 circles as wheels didn’t look enough like
a “car”, for example (sheesh!). Students should NOT be graded down if they have the right answer. Perhaps the students who “think outside the box” and complete the “pictures, numbers and
words” segments should get bonus points, rather than penalizing the students who figure out the correct answer using only one.
Not ALL teachers (in fact, very few) are as “particular” as the example I gave above, but not ALL teachers are great, either.
I respect teachers, and for the most part we’ve been very lucky with the quality of teaching our children have received at our Ontario public school (as an aside: It’s a lower-middle class
school in a large city…not a “rich” suburban school). Perhaps it’s because my children are “easy” learners? They “get” the math fairly quickly. How much of that is their teachers, how much
is curriculum, and how much is just innate, intuitive math ability?
Sorry for veering a bit off topic…my mind was wandering and my fingers followed.
31. There is a simple premise that has worked for me my entire life. If it ain’t broke, don’t fix it. Now that i have children and I grow increasingly frustrated to see how the “grand thinkers” at
the various ministries of education believe we should be constantly changing the curriculum to address some idealogical ideal. The very idea that we should completely change how math has been
taught for years simply because they believe that a new, but entirely untested, idea is better is just wrong. Don’t get me started about how the schools decided that phonics was no longer any
good – and now we have a generation of kids that can’t read properly. We’ve had to struggle to teach our kids phonics to undo the damage caused by the schools early reading program…
32. As a public school student, I completely agree with this article. The curriculum teaches us an inordinate number of equations in a very short time when one method would suffice. This causes a
great deal of confusion and frustration for all parties involved. The Ministry of Education needs to revise the mathematical curriculum, and soon.
33. To every parent struggling with math who can’t afford or would rather not pay thousands of dollars, I’d recommend the JUMP math series of workbooks. At $20 a year that’s a lot cheaper. I’ve been
using them with my children since my older child was in grade 5, and they’re quite different in their gifts and needs. The workbooks operate step by step building on both skills and concepts in
an orderly way. The home page is: http://jumpmath1.org/
34. One of the biggest problems in Ontario is that the ministry keeps changing the expectations and students in grade 3 are trying to learn concepts that we learned in grade 5 or 6. Also, there has
been so much emphasis placed on literacy that math has gone by the wayside and the ministry knew this would happen. Math has been put on the back burner and the ministry and the boards knew this.
They caught teachers and students in their mess.
35. We are dealing with this in NB. What exactly is the job of a teacher now? They get paid pretty well and only work 39 weeks per year at the most, but they send home bag after bag of work for
“parents to teach”. WTF is up with this? Do I not pay enough taxes to have my children taught at school? Have the unions ruined careers such as teaching?
36. I would like to thank today’s teachers for all the uneducated, disrespectful bullies that I have had the displeasure of working with in industry. I have worked with illiterates and innumerates in
the field of industrial instrumentation. Some of these people have the smarts of people who should not have even graduated, or been given a licence. The skill level in my occupation has gone down
to a disgusting low level.
□ The article does point out “no-fail” policies, which are in place in many schools across Canada. As teachers we are not allowed to fail students. Even if we give them a passing mark, we are
not surprised to see students advanced to the next level the following year.
37. I am a mathematician educated at some of the most prestigious places. I have hard time understanding my kids math. It looks like re-engineered mathematics. I find so many inaccuracies in the
manuals and the material that is handed to the kids. I wrote the Ontario Ministry of Education. After several weeks I got a useless response from a policy person. I reported to them concrete
errors in a concrete book used in 9th Grade. The same book is still used with the same errors served to the students. We are already way behind many Third Word countries. Very soon we will be
irrelevant in science.
38. I am a public school teacher in Manitoba, and I agree with much that is said in this article. I know many K-8 teachers, for example, who cannot subtract fractions with unlike denominators. These
teachers are shaping our young students. The math situation is a disgrace and parents and anyone concerned better wake up very quickly.
I know many teachers who feel the same way as me, yet we are afraid to speak up for fear of losing our jobs or being forced to teach courses we don’t want to teach. The whole public system is
very rotten. Principals and superintendents know the least of what happens in the classroom, yet these well paid careerists are making the decisions that are damaging many students. Teachers are
just pawns who are expected to agree with the latest fad or bite their tongue so that real debate is stymied. Parents who know the value of education should be VERY concerned.
□ Dear Silenced teacher,
Thank you for sharing your story. I know that you want to remain anonymous. Would you consider setting up a fake email account and contacting one of the co-founders of WISE Math?
☆ There are several issues that this post highlights: (1) elementary teachers are under prepared to teach mathematics and there are a number of contributing factors (e.g., not enough
mathematics education in their teacher training program or prior), (2) school boards are not investing in hiring subject specialists to do this specialized teaching at the elementary
panel, and (3) professional development is helpful but not the solution.
39. Add to this that if the child does not get it, the child has a “problem”. I have three children being exposed to this math teaching. In each case although each has very different skill levels –
they all have been accused of having problems in math because they can’t understand what is in the text book ironically called “Math makes sense”. It’s a real travesty. No need to go back to
basics. There’s lots of interesting approaches; see John Mighton’s writings. My experience is that if you complain there is a circling of the wagons as the teachers and schools see it as an
attack. Very frustrating.
40. Awesome… Without math you can’t become anything useful… But you can be a teacher and teach kids to be even stupider than you are!!! Classic!
41. I think leading kids through all the creative, hands-on, active ways of learning about how numbers work is great. And at the same time they also need to have the basic number facts down. Both
can be worked on in parallel.
Imagine building a house: we make sure the foundation is strong and level; we can also be imagining how colourful and different we’re going to make the exterior walls look, or the upstairs
bathroom, but without that strong, reliable foundation, we won’t be able to bring our imagined ideas to life.
42. Part, maybe most, of the problem is that we are teaching calculus type mathematics to ver younger students when 99% or more of us will never use that in our lives. For example, our kids spent
hours on problems that required the use of the symbols, yet at the age of 52 with years of trade and commerce behind me, I have never in my life used these symbols. Ditto for complex graphing,
etc. Beyond the deep imprinting of the 10×10 multiplication square, and the vital importance of understanding natural fractions- 1/2, 1/4/ 1/8 & the decimal equivalents- 1/2=.500, 1/4=.250, 1/16
=.0625, etc. most of the math I was taught in school has been wasted.
In trade school, we were required to learn advanced trigonometry which was vital to the trade. Learning it in high school would have been pointless, yet, as a tradesman, I encountered many
would-be apprentices who excelled at math, yet could not over the course of several months grasp the concept of 1/16 equals 6.25%, or a 1/8″ depth of cut meaning .250″ off the diameter, or even
begin to grasp the meaning of the small lines on a tape measure and see the 7/8″ mark at a glance.
Today, I am often required to calculate, in my head, the temp rise in a given container of water based on the number of BTU’s per hour applied. In my hobby, we discuss and understand fairly
esoteric mathematical equations on a regular basis. The key is that it’s applied math. It’s useful, and that’s the key.
It is pointless to try and teach 12 year olds the importance of recognizing the when they will never-NEVER- likely use it, and even more pointless if we haven’t deeply embedded the understanding
of the 10×10 square and the significance of natural fractions. It’s pointless to teach calculus to teenagers, when we haven’t skilled them at sound, basic math such as how to calculate the
volume of a cylinder, the area of a triangle, or the weight of a given volume of a given material.
We’ve quit teaching hard, solid skills in reading, writing, and arithmetic, and then wonder why kids can’t read, write, or do math.
43. There are very few teachers in the elementary panel with teaching qualifications in mathematics. Moreover, it should not be assumed that even a mathematics consultant has credentials in
mathematics education. The public should be asking more questions about subject specialization and what it means to children’s learning.
□ That is because there is a growing tendency to force all elementary homeroom teachers to teach their own math, whether they are comfortable doing so or not. Meanwhile teachers who have
expertise in math are only allowed to teach math to their own class and are forced to teach English whether they like it or not.
44. Look. The whole point of a public school is to produce semiliterate “TLDR”-s. Which is why the little lab mice will be able to recite talking points, follow instructions and remember that the
only defence from bullies is whatever petty authority is given to them. Conversely, they should not be capable of forming an informed opinion on anything or standing for themselves.
Why did you expect basic math to be an exception? I genuinely fail to understand such a specific concern.
45. I learned my multiplication tables up to 10×10 (possibly 12×12, but I don’t remember that clearly) by rote. I do math visually, but feel that knowing the multiplication tables gave me a solid
foundation for that. There’s no reason it shouldn’t continue to be used in conjunction with the frou frou strips of paper and graphs, IMO.
46. My son is in Grade 2 in Castlegar, BC – I have been trying to find a way to understand this “NEW” MATH. I love math & was eager & willing to help him nightly to learn to ‘love’ it as much as
me. After the 1st assignment that i didn’t understand and the “manipulatives” that he was to use to “maybe” reach a correct answer – I became frustrated and hated these assignments myself – when
I complained that I wanted him taught the “proper” way to add & subtract – the teacher told me to not teach him that way because when he gets to Grade 8 he won’t understand algerbra! She also
said that math use to be taught with “rules” – now there are no “rules” – it’s teaching kids to find a solution in different ways and not with pencils/paper – with charts, etc…..It’s crazy – if
they can’t come up with the right answer (they might get close ) she gives them the answer after they play with “manipulatives” for 15 minutes….What the hell has happened to MATH. There’s rules
to guide us in everything in life – learning to drive a car – we teach our kids the “rules” – what has changed? So together we fight through these ridiculous methods & my son continues to hate
MATH – as do I – pls. help us in BC we are in big trouble.
□ The manipulatives are meant to be used as an introduction, so that when the procedure is introduced it will actually makes sense. The procedures are important in math, but they are not all
that is important in math. The mistake the government of BC made is not changing the curriculum (the changes are based on sound brain based research and pedagogical research), but rather,
that they changed it without re-training the teachers. If all educators were properly trained in teaching the new curriculum, these issues wouldn’t be arising. I say this as a math teacher
in BC, who has had a huge amount of success with the new curriculum (far more than with the old).
47. You’re right, without a solid accurate foundation, you will never be able to build on anything. You have to put in the work to get to the fun stuff!
48. The public (citizens of canada) needs to take back the education system from the “educators”. Not teachers in general but people who are in control of education and whose main goal is making
their own life easier, with no realistic realization of the outcome of thier actions. I went to school in the 80′s and 90′s and I feel I learned substantially less than my parents who were
educated in the 60′s and 70′s. Talking to my younger cousins who graduated in the last couple years it is obvious they know even less. Just look at our public knowledge of our government systems
for an example!
Our governments seem to have less and less control over the education system, either by choice or by neglect, and its time we made them step up and take back control. Educate kids so that they
will be prepared for life and living and working in society. Our democracy and future freedom depends on it whether we like it or not.
49. what? the ‘old’ system worked so well they needed to replace it?
Why do these people come up with this crap … and who in hell is wasting public money by paying them?
50. Jerking the curriculum around is never good. Incremental improvement would be best since it will take time for the teachers and parents to absorb it.
I can say as a post secondary teacher it is astounding how poor some students are at manipul…. uhh rearranging equations. I figured this was because they never worked with fractions… uhh
rational numbers. Fractions are first mentioned in grade 6. It’s been so long since I took math I don’t remember which grade I learned fractions in.An anecdote; I was working a problem with a
student and we came to 5 x 12 and the student grabbed for their calcula….. uhh technology. I was so startled I boomed “DON’T TOUCH THAT CALCULATOR! I apologized immediately, it can be
scarring to be spoken to in loud tones.
I guess all I wanted to say is balance the approach please. There are at least 8 modes of learning and a good teacher will try to use as many of these modes as possible.
Khan Academy is amazing, free, comprehensive, wide ranging and good. Imagine having a tutor repeat a presentation 20 times until you get it. I use Khan Academy in the classroom when I think he
explains something better than I would. There are teachers that routinely use Khan Academy to give their students the main instruction and class time is mainly for doing “homework” and helping
students one on one. I can’t believe a major publication hasn’t made a story of The Khan Academy.
51. Thanks for posting about this. I was a teacher and found that the Math Makes Sense curriculum…didn’t make sense and was very confusing to students. I stopped using it. I do find it useful for
students to understand the concept behind the math but just as a introduction, not as something they continually need to do.
I have created a website to help parents access products and free info about the education system, how to teach reading to their children and other fantastic resources. Please take a look to get
some advice on teaching your children at home from a young age.
52. The very basic building blocks of knowledge, multiplication tables and periodic tables, needed to be drilled in by rote – until you have those blocks, you don’t have anything to be creative with!
Trust me kiddos, the beginning is the hardest, learning music notes, learning basic formulas, but once you have enough of it in your head to be fluent, it’s yours.
…and I’m sick and tired of social experiments being tried out on general populations of kids.
53. I find this terrifying. There’s a reason why people learned rote mathematics and it WORKED – it’s because by memorizing, it, it removes cognitive burden and allows your brain to focus on other
more important things – say, actually SOLVING the problem. If you spend more time trying to figure out what 9 x 6 is, as opposed to the context of the math problem, then I guarantee you that
you’ll hate math.
These articles are inspiring me to homeschool my kids. Fortunately, I have a pure mathematics minor. But what about parents who don’t? These stupid policies developed most likely by math
illiterates are furthering the inequality between the educated vs. uneducated and rich/poor.
□ You are absolutely right! It is time to return to basics in mathematics. The curriculum should be fixed immediately.
Currently I teach in the intermediate division. While 1/3 of my class is well ahead of the curriculum, at least half are very far behind. I feel I need to spend time teaching concepts that
should have been learned in Grade 3 or 4 (such as multiplication and division) before I can even begin to teach the intermediate level curriculum. What a mess!
Parents are well advised to make sure their own kids have learned how to add, subtract, multiply, and divide without the use of calculators or manipulatives before they get to Grade 7 and 8.
54. “It’s completely wrong-headed. And the moment you say parents should play a significant role in public education, you have a two-tiered system.”
Professor Stokes Have you ever visited an inner city school in this country?? We already have a two-tiered system.
55. Maybe the powers to be really want to dumb things down. After all, a dumb population is an easily controlled population; a population that will need to rely more and more on government. How else
do you explain the great emphasis on social justice and saving the whales in schools, as compared to being expected to perform challenging math?
Less social justice and making kids feel guilty about the environment, and more back to basics education is what is needed.
56. Dear everyone, multiplication, simply, is not repeated addition….
57. The problem is not that the teachers are not qualified, or good. The main problem is that there is a relentless campaign to produce stupid people. The teachers are not allowed to do their job.
There is a continuous dilution of the curricula, not only for math. The word discipline is not allowed in schools. If you do not give the forming minds a disciplined way of thinking and
behaving they will be not prepared for the real life. In real life if you don’t do the “homework” you are not being forgiven or given a second chance. Continuously lowering the the standard was
the only way to “keep” marks going up or stay the same. I think we reached a threshold when a chain reaction is imminent and the quality of the education is in dire straits.
We should thank for that to our Ministry of Education a School Boards which are stuffed with incompetent people holding on their chairs and who invent useless policies just to justify their
time. In fact these policies, which try to reinvent the wheel, do more damage than helping education.
In addition the use of technologies prevents the use of children’s brains when they need it most. Even pocket calculators should be banned. No child has any incentive to learn multiplication
table when it is much easier to click a few buttons.
58. My daughter is in grade 5, and I love all the different strategies she’s been learning the last couple of years and how she can figure out fairly complicated problems in her head using these
tools. BUT she does get stuck on basic multiplication facts, and that does require drilling. Like everything – the answer is probably somewhere in the middle – a combination of traditional math
skills, along with the much better understanding of how you can manipulate numbers.
The key is making sure teachers are well trained with these new skills — it’s not always easy for a teacher who has been teaching older methods for many years to switch over to the new methods.
They may need a lot of support.On the other hand, my other daughter graduated from high school last year, and struggled with her top level math classes from grade 10 onwards. We hired a tutor
each year. That math seems far, far more complicated than the grade 12 algebra I excelled in. And I don’t believe that’s even the latest curriculum that she was learning — that’s being switched
over in the next few years in our high school as kids come up through it. Not sure what the differences are. I guess I’ll find out when daughter #2 goes through it.
□ I disagree. Please read my reply to Nikki. These new methods just make things worse. There is no wonder that your daughter has problems with basic things. The teachers are well trained, but
they are forced to use these new methods, which have produced only failures.
The math is not more complicated then 10-20 years ago, on the contrary, Calculus was thinned out, many things have been taken out. Nobody learns integrals in high school anymore. The
struggles you mention are a result of implementing these new and wonderful methods instead using the methods which produced good scholars for generations. In addition taking out the authority
from teachers there are no incentives for children to learn anymore. Read the book I mentioned above, our brain learns only through coercion, which is the only way to create “new synapses” =
☆ Good scholars for generations? I think that’s optimistic. I may have done very well in algebra, but many didn’t. Traditional methods work very well for some, but not all.
○ I never claimed that everyone did well in the past, but there were no chronic problems with math as they are now with the toy play which takes the abstraction out of the equation.
Math was more difficult than what is taught now in schools. People had more time to study as there were less distractions in the class and at home. Teachers did not have to deal with
cell phones, pocket calculators, mp3 players and many other disrupting tools. We had less gadgets and more brain. Traditional methods worked because they used well known and tested
learning strategies. Now there are many people at the Board level who need to justify their time and position by changing things, which did not need to be changed. Why change
something that works? I am very sorry that you are not able to see all those things.
☆ Teachers are NOT forced to use these new methods. Many teachers choose to use these ‘methods’. Rather, they choose to use a different philosophy of learning…one not focused on
reductionistic behaviorism.
What authority loss are you referring to Bogdan? Is it the loss of authority due to everybody and their dog thinking that they know best about how to teach a specific set of students in
a specific context?
59. The problem is not that the teachers are not qualified, or good. The main problem is that there is a relentless campaign to produce stupid people. The teachers are not allowed to do their job.
There is a continuous dilution of the curricula, not only for math. The word discipline is not allowed in schools. If you do not give the forming minds a disciplined way of thinking and
behaving they will be not prepared for the real life. In real life if you don’t do the “homework” you are not being forgiven or given a second chance. Continuously lowering the the standard was
the only way to “keep” marks going up or stay the same. I think we reached a threshold when a chain reaction is imminent and the quality of the education is in dire straits.
We should thank for that to our Ministry of Education a School Boards which are stuffed with incompetent people holding on their chairs and who invent useless policies just to justify their
time. In fact these policies, which try to reinvent the wheel, do more damage than helping education.
In addition the use of technologies prevents the use of children’s brains when they need it most. Even pocket calculators should be banned. No child has any incentive to learn multiplication
table when it is much easier to click a few buttons.
60. I have been teaching math using manipulatives and focusing on both conceptual understanding and procedural fluency for 6 years (and taught using “traditional” methods for 5 years before that). I
have also been working with teachers and teaching them how to use the manipulatives effectively in the classroom. What I have noticed is this: what is challenging for adults (including teachers
and parents) is often easy for a student. Most adults have never seen math concepts taught concretely and must divert away from their procedural knowledge to understand it differently. It feels
confusing, and overly complicated. When a student encounters the concrete application FIRST, it’s not so confusing or complicated- they get it and they often like it. Furthermore, they don’t
forget it. Before teaching fractions concretely, students forgot how to add or subtract very quickly after “learning” them- because they only had a rote procedure. Now my students know that they
need the same size pieces (common denominator) because fractions have to have equal sized parts in order to name them, so they know WHY they need a common denominator and as a result, they don’t
forget it just like they don’t forget how to add 5 + 3. The other point I’d like to add is that there is very little mention in this article about HOW the brain learns. The new curriculum is
based on sound brain and pedagogical research. All change takes time and will have obstacles. We need to be educating our teachers better, and our parents. I have seen in my own classroom, and
heard from countless other teachers who I’ve worked with, the power of teaching using methods that work for many different styles of learners and understanding the concepts in math. I would NEVER
go back to the old method!
□ Sorry, I disagree. Because of these “new” and “innovative” methods the level of our children goes down every year. Don’t get caught in Board propaganda! The sound brain and
pedagogical research is pure BS. Have you seen those researches? Read the book “The brain that changes itself” and you may change your opinion.
It is well known that the brain needs repetition to build synapses. The brain functions on the principle “use it or lose it”. By making things “easier” you just lower the bar and make them
think less. By repetition fundamentals are well established and using those they can build on. There is a certain age when people can learn fast: languages, math etc. They need to learn
certain work ethic and principles at early age.
☆ I have done the research; I focused my major project for my Masters around making math more meaningful in the middle school years. I have also read “The brain that changes itself” and use
those principles as well. The students FIRST learn through making meaning of the math, THEN they practice the skills using procedures until fluent. They absolutely still need the practice
and “drills” but they also need some understanding as well. Times tables should be memorized but first allow the student to understand what multiplication is so that can easily use other
mental math strategies to estimate and solve problems.
○ I am glad that you are not focused on “thinking only”. Thinking undoubtedly is very important, however, you can’t think without basic knowledge. We should not put the carriage
before the horse.
I din not say that memorization is all, but in our schools there is too much emphasis on thinking and memorization is belittled. Memorization is very important. We need to train to
build muscles and the same goes with the brain.
Another brain formation fact is discipline which is not enforced in schools. A young forming brain needs to be guided and not let “discover” everything. It may discover some wrong
things too.
I know what is taught in OISE and other educational institutions, but I believe that they have a wrong approach to education. We can see the results already.
■ There is a difference between discipline and support. I would argue that it is support from parents and support for students that makes the difference, not ‘discipline’.
○ The problem is that too often we never get around to the times tables being memorized or other operations being consolidated without the use of manipulatives.
Now we are beginning to reap the rewards!
□ When I hear about doing things based on “sound brain and pedagogical research” all I really hear is “someone wants to turn their master’s thesis into a book and sell it to every school board
possible”. I’ve been to enough workshops and been handed enough “resources” (ie dust collecting doorstops) to know. Understanding the why is vital, I agree. You don’t need new pedagogies
and manipulatives to explain the why. I use pictures of pizza: if I have 1/2 and 1/4 of 2 pizza’s leftover and I want to save room by putting them in the same box in my fridge, how much
pizza do I have? (hint:to figure this out the pieces have to be the same size) Do a few examples like this and they get it. Want one for equivalant fractions? Here’s a joke: a guy walks
into a pizza store and orders a whole pie. The waiter asks “would you like it cut into 4 or 8 pieces? Guy says “I’m not that hungry today, please cut it into 4″. Kids get the joke and
remember a very important lesson: it doesn’t matter how many slices you cut it into, the amount of pizza is the same. I do all this on a CHALKBOARD (practically a bad word in teaching now)
and they all get it. These are grade 7 and 8′s and they all have the same reaction: somebody finally explained this to me.
□ Well said Nikki! I agree 100%
61. The problems both my sons have encountered with the new math is its need for the increased reading comprehension and writing skills required to achieve any degree of success with the new math
programmes. Both boys have learning deficits in those areas and although skilled in areas requiring visual spacial amd hands on skills they are not achieving success in math because they cannot
understand what is being asked and explain the solutions to problems presented. They also were not drilled in the basic math facts to assist them in computations. We have had to drill basic
facts at home and paid a private tutoring service to help them get basic numeracy skills so they can get enough math to get high school credits to enrol in community college programmes to achieve
their career choices in the construction industry.
62. The heading below this box where I am invited to make a comment reads, “Showing 118 of 116 comments”. I wonder where this writer learned his/her math?
63. Knowing the times table, by memory, is a very useful and necessary tool in studying Math. Gives you number recognition – the ability to estimate, judge if an answer is reasonable, figure out all
kinds of math questions quickly and accurately without relying on a calculator - whatever method you use – even some of Thwims convoluted ones. Easier to UNDERSTAND math concepts once you’ve
memorized the timestables, and learned operations by rote.
Knowing that 1/2 x 1/3 means that you divide a figure into three equal parts, and then take half of one of them, makes a lot more sense after you’ve practiced the rote method many times, and you
don’t get the kid to figure out 7/9 x 14/23 using the “visual” method!
You talked to a lot of “experts” in this article, but few teachers.
Thanks to the teachers that wrote comments on this article.
□ I agree. Better to learn the basic facts and computational methods for math operations and then go back to investigate why they work with simple examples and concrete materials when needed.
That’s the way I learned (way back in the 1950′s) and I was much more able to teach the subject to my students when I became a teacher in the elementary school system way back in the 1970′s.
But even I found the new methods difficult and confusing for most students I taught in my last years of teaching in the early 2000′s. Too much material was presented and too much was
expected much too early before students were really cognitively ready for the concepts presented. It was almost impossible to assign homework as most parents found the concepts even more
incomprehensible than the students did in most units presented. And I found that classroom teachers having to present the material were not really consulted before the new math programme was
64. Math tables were drilled into us by Grade 3! I don’t see what’s so wrong about using rote learning. Anatomy is memorized in the same fashion. Once you know it’s stuck with you.
65. In BC, the new math students are in grade 11. If college students are having trouble, it was with the old curriculum.
□ In Manitoba, the new math began in the early 90s with 2 math series called Quest 2000 and Interactions. They were terrible math programs, yet most Manitoba schools implemented them.
Students who used these math programs are definitely in university as we speak, and have been for a few years, so I can understand how university students don’t have proper math skills.
Those 2 useless math programs have recently been replaced with the equally useless Math Makes Sense, Math Focus, and Math Links programs.
Kids need to spend time completing math computations and learning the basics….not spending an hour cutting strips of paper to confuse them even more when dividing fractions.
☆ I should also add that if anyone has ever looked through the Quest 2000 or Interactions programs, they will very quickly understand why a number of students have such poor math sense.
I remember telling my principal that I didn’t want to use these programs in my classroom. He told me I had no choice because the division invested a pile of money buying them. I told him
they were making math very confusing for my students, and other students in other grades. He didn’t care about that.
○ Quest 2000 was at least preferable to the Nelson Math programme which is the one implemented in Ontario and Alberta in the early 2000′s. I’m not sure if it used in other provinces.
It is too heavily laden with confusing language and an emphasis on multiple solutions to problems (also confusing to kids who want to just find the answer). Concepts are also
introduced way to early for children to understand.
□ Hi Pete. I refer you to my earlier reply to someone who mentioned this. Sorry, it’s not so simple. Silenced Teacher here makes another good point — the current “reform” trend began a long
time ago. ”No algorithms” began to be vigorously sold to schools with the NCTM “Standards” report over 20 years ago, which also marked the divorce of mathematical curriculum development from
the professional mathematical community — one of the root causes of the problems we’re seeing today). But ever since the tinkering began in the 1970s serious harm has been done, and
continues to be done to the teaching of mathematics. Critics, by the way, are not simply calling for the clock to be turned back. In recent years serious improvements have been made (at
least on the policy side). For example, in 44 states the “no algorithms” approach to elementary arithmetic has been tossed out. For a good idea of what we’re promoting at wisemath dot org,
navigate to our site, download the NMAP report and read (at minimum) the 45 recommendations there.
☆ Why should the ‘professional mathematical community’ who teaches less than 10% of the students who attend K-12 schools determine what is best for the 90% who THEY NEVER WORK WITH?
Oh, and thank you very much for excluding professional teachers from your hierarchy. I forgot that they weren’t professionals and are not part of the mathematical community.
I would argue that the ‘professional teaching community’ knows more about those 90%.
Lastly, it is not about an either or approach. It is rather about approaching things in more than one way.
I teach math and science in high school and teach it well, new curriculum and old. I would suggest that you are more concerned about your own identity than the 90% of students who you
will never see.
66. I can understand these alternative systems, but believe we should start with basic memorizing of tables, followed by universal algorithms. The visual concepts should be added later to permit
operations in other bases and dimensions. Don’t forget the old abacus and slide rules either. In earning my science and engineering degrees in both math and chemistry, we were not allowed
calculators because only the rich kids could afford those modern gadgets. I can still beat the typical sales clerk, calculator in hand, doing the same computation in my head.
□ Same here. Basic knowledge is essential to be learned to create a strong brain. Calculators , GPSes, etc are good, but for the young mind they can slow down their development.
67. As a BC public school teacher, I think that the problem is due in large part to the expectation that students have been taught successfully when they do well on standardized tests. The text books
are presumably written to further that purpose. I strongly suspect that the Ministry of Education in BC (where I teach) is unlikely to approve a textbook which is not in keeping with that goal. I
happen to be spared this dilemma in my subject area – secondary school music – because standardized testing is very seldom required and the practicality of measuring outcomes in music teaching is
very evident: the students have been successfully taught when the musical performance outcome sounds better than it did last month.
No job requiring mathematical aptitude, however, requires students to be able to pass standardized tests (unless, I suppose, it is a job as a designer of such tests). The texts are presumably
therefore designed to teach students to do something which they will never, as adults, be asked to do and which no adult who uses mathematics in his/her daily life (whether for employment,
household management, or for hobby activities) ever needs to do either. It is small wonder, then, that these texts make little if any sense to those who apply mathematics in the “real world” or,
for that matter, to those who teach it.
□ Hooray! I think this is more about people and their identities than students and their learning and understanding.
68. Is the effectiveness of the “new math” in these comments the “new math”, the new “new math” or the original “new math”? I remember the problems kids and their parents and teachers had with the
“new math” introduced in the 1970′s. Professional educators are still squirming around, hoping the next revolution resolves the last round of introduced deficiencies. Remember, all the material
introduced K-12 represents the mathematical knowledge of roughly 1760. A university degree gets you up to about 1850, with a few sparks of more recent insights.
Start math with the basic tools, and a child’s experience with success..
69. School math results are actually even worse than they appear. They do not tell you how many of their A students had outside help. When I heard that 30% of the class ahead of my daughter’s
failed Gr. 9 math (both schools in a high socioeconomic area), I enrolled her in Kumon. When she graduated high school with 100% on her calculus exam, no one put an asterisk next to the school’s
average indicating “supplemented by private tutoring”. Most elementary school teachers have been poorly taught themselves and are downright math phobic. It’s the blind leading the blind
expecting them to teach kids math. Meanwhile, the excuse for throwing out what worked, rote learning of the basics is the same idiotic justification as with phonics and Whole Language. The kids
were not bored. Teachers and excess administrators were bored. Kids are excited as they gain mastery and progress to another level. But idle administrator hands make the work of the devil.
Rote learning is the critical beginner stage and solid base for higher levels of conceptual thinking later on as Kumon progression reflects (structured by a Japanese math teacher). BTW, all
these curriculum dabblers are former teachers, every last one, so they cannot escape responsibility by shrugging their shoulders and saying “the higher ups are forcing them to give up what works
and substitute whooey”. And when have teachers ever had problems taking on authorities for what really interests them, their remuneration and hours?
□ Teachers have a lot of trouble taking on authority when it comes in the form of principals, supers, and higher ups!
Ambitious teachers are forced to “jump through hoops” to demonstrate to others “how great” the flavour of the day techniques are if they wish to have any hope of being promoted to VP or
Even questioning of the latest “social experiments” is highly frowned upon. Blind acceptance is expected. People who refuse to do this are considered “trouble-makers,” and speaking up can
put their career path in jeopardy.
Sorry to be so blunt, but this is what I have observed in the last 15 years.
70. The new new new new math reminds me of the whole language ideas. In LA we teach mini lessons, model how to be better readers and writers. From my experience, the discovery method is not all it is
cracked up to be. Pardon my old fashioned ideas.
71. Multiplication tables up of 1 to 10, or preferably 1 to 12 (my school preferred 1-13) are just so incredibly simple. They provide the tools required for every subsequent step. I think the
difficulty faced by some students and many teachers is that due to time issues, they take a class to step three before confirming that step one and two are successfuly learned.
And for commentary on subtraction, the word “borrowing” should be stricken from the teaching vocabulary. Once you have firmly established that the numbers in each place in the decimal system are
the counts for 1s, 10s, 100s etc, you should “subtract”, not “borrow” from the next bigger place and add that unit to the smaller place, then complete the subtraction. The system is identical for
any base, so you don’t have to relearn anything in more advanced concepts. It is easy to use blocks, strips of paper, tiddlywinks or whatever to demonstrate the concept.
72. How many of you have encountered a high school student working in retail, who when the total is $10.24 – you give a $20 then add, after they have punched in the numbers on the till — oh, here’s a
quarter… they stare at you blankly, because they no longer know what change to give…
73. I completely agree with the article. My son is currently in grade 8 and I can’t seem to understand the strategies he’s being taught in class. I also don’t want to confuse him by teaching him
‘old’ strategies that were taught to me long time ago. Luckily he’s able to sign onto a website for free math tutoring at night through a live chat with teachers. The program is called
HomeworkHelp. I think this program is only available in Ontario though.
74. I’m a math teacher and I can tell you that the only reason this new way of teaching math exists is because most teachers did very well in school in everything EXCEPT math. They are the ones who
changed the way math is taught thinking “if only I had been taught it this way I would have succeeded.” All the best math teachers I’ve ever met ignore these fads and teach the way math has been
taught for 1000′s of years. Most principals, etc don’t care as long as the parents are happy and the standardized test scores are up.
□ I agree. There was a completely interesting article about the “feminization” of public school education in the Globe and Mail. Perhaps this is another side effect. I was a bit of a prodigy in
math and know the method they are describing. It can be used to calculate large multi-digit products in your head, but there are a few things you need to know first. You need to know your
basic multiplication table by rote. You also need to have a spatial IQ over 150.
Trying to teach this method to children without the fundamentals is as realistic as getting a toddler to run a 100 yard dash under 10 seconds.
75. I am not here to dismiss Strategies, but to highlight the relevance of Algorithms.
eg 1. Algorithms play an intrinsic role in Computer Science. The vary nature of many computer programs are algorithmic in nature. Many of them rely heavily on mathematics to solve both everyday
and complex problems. Try creating a small computer program using Strategies instead of an Algorithm – that would be counter-intuitive since the aim of a program is to solve problems of the
general case which cannot be done with Strategies technique since they solve special cases.
eg 1. The famous mathematician who discovered Algebra, created a step by step method for solving for x for certain equations. The word Algorithm means Step by Step. He first experimented in a
similar fashion to what Educators are asking children to do as described in this article, but then why did he create an algorithm (which by the way comes from the Mathematician’s name). For whom
was this meant? It was meant for us to use. And then if we are interested in learning the intricacies of the theory, we can explore this further. We are not all going to be pure theoretical
mathematicians, but we certainly can apply a step by step method and later, once we develop a comfort level, we can dig deeper into what, how, why the algorithm works and when it may not work or
coming up with another method altogether.
76. The big problem is that many teachers who teach math are not comfortable with math concepts because they have not taken Math since grade 10. The grade 8 math we teach now is similar to the grade
10 I learned. There is no specific Math course required for elementary teachers to teach math. So some grade 7 and 8 students are learning Math from teachers who have only completed up to grade
10 level Math. I believe the Ministry of Education should mandate that all middle school teachers (Grade 6, 7, & 8) must have at least one university Math credit and an Intermediate Math course
So if you have a student in middle school, ask their Math teacher when was the last time they took a Math course.
□ Yes. Part of the problem is that there is a growing trend of forcing all homeroom teachers to teach their own math, whether they are comfortable doing so or not.
Meanwhile, teachers who have expertise in math are not allowed to teach it as a rotary subject, but instead they are limited to teaching it only to their own class.
77. I’m interested in setting up a group like WISEMath for Ontario. If you have had similar experiences to those described in the article and would like to see changes made to the Ontario curriculum
to improve the standard of math education, please contact me. I can be found on Facebook or email clive.packer (at) gmail (dot) com.
78. I agree 100% with the points addressed in this article…I’m an elementary school teacher and parent – we need to get back to the basics…the pendulum has swung too far to the left!
79. Well, that was an unbiased article…
Weren’t university math professors all up in arms over the last decade about how poor the math skills were in ‘this generation’ of students? This was well before new curricula hit the
In fact, weren’t the new curricula designed to address that issue in the first place? I think we have an obligation to speak up for the kids. Maybe all of that memorization just didn’t work
EITHER. Maybe it is NOT the curriculum but rather the lack of support for schools and teachers both financially and the view/attitude towards learning and the teaching profession as a whole.
Maybe it is the fact that we have many grade 9 and 10 students working 20 to 30 hours a week at a paid job, often on top of extracurricular activities. Maybe it is the 7 hours of non-school
based screen time the average high school kid chooses (with the help of their parents) to spend their time on.
□ I agree with you that academics don’t seem to be very important anymore. But this new math is doing harm to those students who value education.
80. This is a mass over generalization of the methods teachers are using to engage, empower, and educate students in math. It seems that these parents are faced with the dilemma of many of us who
experienced the more traditional ways of math teaching…we were taught tricks, not why and how the math actually works. That is truly the reason that people are frustrated and are unable to assist
their children. I have a mathematics degree, an elementary school math specialist, and have taught for more than ten years in Ontario and have seen huge gains in my students through these methods
hat allow children to explore and engage in math in ways that make sense to them. To see the students engaged in class and having conversations about the math, as well as expressing now much more
they now understand is the reward. It is empowering to children when they are stuck and can work their way through a problem because they have the ability to apply their knowledge to the
situation, not just a trick someone showed them. We want these children to be confident and creative thinkers…problem solvers. Providing them with situations where they can practice this safely
in the classroom, will only help them in the future. I am sorry for those who have had negative experiences with these methods, but as the article mentioned, there is pressure on teachers, and
when they themselves are not confident in the subject, perhaps because of their own math experiences, then it does prove much more difficult. When done well, the math class should be based on
what the students need. A balance of exploration, application, and practice gives students the foundation they need to carry them from year to year, the confidence that they need to take risks
and apply what they know, as well see the practical application in what they are learning.
□ I am a math major…not an “Elementary Math Specialist” and the new methods are destroying young students ability to do math. The so-called elementary math specialists that I frequently meet
with know very little about math. They do, however, know how to make fun lessons using Smarties.
☆ ̶̶l̶i̶k̶e̶ love
81. I do not want to think that Stokke is unnecessarily picking on teachers as a profession that doesn’t require the learning of basic Maths concepts, teacher education as I know it is not devoid of
general basic Mathematical curriculum. I will suggest that Stokke take a 2 years teacher education training to find out.
82. A major part of the problem parents are having comes from having memorized the math they learned, which does not lead to an understanding that would allow creative real world problem solving.
The new curricula are aimed at making math useful, at giving students with many different learning styles the problem solving tools, and basing it on understanding not rote memorization. One of
the most important methods in understanding math concepts is to use concrete objects like base-10 blocks or strips of graph paper to convey a visual and tactile understanding of the operation,
like multiplying (10-1) times 6. Six tens minus 6 ones can be visualized as an area touched with fingers and does not need to be a rote memorized fact. Once this method is understood for
finding basic number facts, it is easily transferred to algebra like (X-1) times (X - 6).
Previous teaching methods depended upon rote memorization of techniques that did not convey understanding that left high school graduates unable to use the algebra to solve real world problems.
Teachers and parents need to diagnose the way each student learns. Many students who have language difficulties will find problem solving by manipulating objects with their fingers works well
for them, because the jumble of numbers and symbols written on paper, makes no sense. Surely those of us educated 30+ years ago persevered to memorize the symbol manipulation techniques on
paper, did not truly understand the underlying concept. A good example is the old method of doing square root on paper and pencil. But finding the length of a side of a square as the square
root of a number which is the area of the square, conveys understanding, and using square tiles or graph paper cut into pieces. This jumble of words can be confusing, but just put a pile of
squares on the table and aske the person the make a square and tell them to give you the length of one side. That is easy for someone who struggles with language, such as immigrants and many
First Nations students that are also trying to learn English as a second language at school.
People who found math easy have the most difficulty understanding that they learned to memorize their number facts and then years later came to understand what they were doing with the various
mysterious algorithms. ‘Memorize now to understand later’ is not preferred to ‘understand first and use technology to provide the number facts’.
The level of math has increased in the past few years so that students in grade 10 are doing the math the retiree writing this learned in grade 12 or first year university. Probability and
statistics were not taught in high school 30 years ago. There is a lot more to learn in this modern world. Many students are trying to learn concepts when they are not ready for them–with a bit
more maturity the concepts will be easier, but every learner is different. Larger class sizes reduce the time the teacher has to analyze each student’s learning style.
□ “Previous teaching methods depended upon rote memorization of techniques that did not convey understanding that left high school graduates unable to use the algebra to solve real world
Then why is it that my previously-homeschooled daughter, who placed into grade ten math when she started public high school, found herself tutoring ninth graders who could barely add,
subtract, or multiply, and who had absolutely no understanding of fractions? I taught her through the elementary years with ”previous teaching methods” (and I’m an English major who
squeaked through grade 13 Functions and Relations–not a mathie), and yet she was the one who ended up trying to explain numbers to those taught by the wonderful new way. Something–if you’ll
pardon the expression–doesn’t add up here.
83. You should look into JUMP Math, designed by Professor John Mighton at the Fields Math Institute at the University of Toronto. The teacher manual is free online, and the books are reasonable since
it’s nonprofit. It combines both the conceptual understanding with plenty of practice on the basics. As a teacher, I have used it for years, and all my kids are confident mathematicians who do
extremely well. http://jumpmath1.org
□ Thumbs up for JUMP Math–lots of homeschoolers use it too.
84. Thank you for publishing what a lot of elementary teachers and parents (like my wife and me) are thinking but can’t say.
85. This is an interesting article that strikes very close to home. After one public and two private schools we decided to home school our son. The public school curriculum in Canada has now aimed
so low even a retarted child can no longer benefit from it. Fortunately we have the money and the time to be able to home school. Our son now belongs to the advance boys network and has completed
3 years math and science in one year. Next year looks even better. There is no chance he will see the inside of a public school ever again, and I do feel sorry for those that have no choice but
to attend.
86. I absolutely agree with Anna Stokke. I came from different country couple years ago, and wanna say that my son in Grade 4 know absolutely nothing about traditional Math method. I am in shock
what they learn in Grade 4 , they don’t teach kids in normal way. They teach some games and some visual strategies that for me absolutely wrong. I bought couple Math books from Amazon and
teaching my son by myself which is ridiculous that I have to do it instead teachers.Math is Math , kids have to learn at least 2 way or methods to understand math, and also they need practice. In
Winnipeg schools we don’t have books at all … How my kid should practice math with no books ?
Compare to Universal level Grade 4 in Canada = Grade 2 in Worlwide. What a shame !!!
87. I am really worry about my son’s math education here !
88. Ohh this is a painful subject for us as well. I don’t trust this new method of teaching at all.
Lets start from that our kids have no books at all in elementary school ??? WTH going wrong with our educational system ? Compare to Universal level our kids way way behind in Math .
89. This article has some good points about the ‘no-fail’ policy and a lack of basic math drills. I think is misses some important points though. First, while it’s true that you can’t do higher
order math without basic math concepts it is also true that memorization alone isn’t enough. The math curriculum as I see it is undergoing a similar struggle to the one faced by language duing
the 1970′s. The debate was over ‘whole language’ vs. ‘phonics’. Eventually it was realized that BOTH were needed. This is essentially the same thing. We need BOTH the memorization AND the
understanding. Unfortunately, educators are being told that the ‘kill and drill’ method is out. Why? We teach kids to read by having them memorize the alphabet and all the sound combinations
and then practice with ‘kill and drill’ until reading becomes automatic. THEN we focus on comprehension. Why can’t we do the same with math? It’s rediculous to think that if you solve a
problem once or twice and understand it that you will remember it the following year. You need to practice before the skill is really developed. Since educators are no longer allowed to do this
at the base elementary level, the kids miss out and get pushed forward to the higher grades were they no longer have the skills needed to understand what they are being taught.
Secondly, and more importantly, it misses the point that PARENTS ARE RESPONSIBLE FOR THEIR CHILDRENS WELL-BEING. This means that we spend time with them teaching them, and yes, helping them
learn. If a person lacks the time and responsibility needed to ensure that their children get the help they need then they shouldn’t have them. If your kid can’t count….HELP THEM. Don’t just
complain. Make sure they do their homework and get the extra help they need. While you’re at it……teach them to be respectful and responsible, then maybe they will pay attention in class, ask
for help and do their homework on their own. That’s what we did when I was a kid.
90. I teach math in high school, and I cringe every time I see a grade 10 student reach for their calculator
to find 5 x 6, something they should be able to do in their head in
grade 3. The problem is that students are forwarded into grade 8 with too many of them not knowing their basic skills (multiplication, division, fractions, decimals, etc.) instead of being held
back until they know them. Many kids learn that they get forwarded whether they do the work or not, so why should they work? This means teachers have grade 8 and 9 to try to get them up to the
level they were supposed to be at entering high school, otherwise they end up in the basic math stream and can say goodbye to any plans they had of a career requiring a university degree (doctor,
nurse, vet, engineer, etc.), no matter how bright they are.
91. The problem that I see, as an elementary teacher, is that we expect kids to have a deep understanding of these concepts too soon, before they are developmentally ready. I agree with the push for
understanding and critical thinking, but in my opinion it is thrust upon kids too early. I teach things to 10-year-olds that I specifically remember learning in late high school, when I was
actually ready to learn them.
92. I’m a grade 2 teacher in Ontario and I have to agree with the message of this article. I’m required to teach the new way, and i do it, but twice a week I slip in drill work. I know from
observation that it increases my students’ confidence in math and the brighter ones prefer using the “old” algorithms because they are simple and quick. Luckily my principal knows what I’m doing
and turns a blind eye.
93. As a private tutor I see the disastrous effects of new-age math instruction all the time. Kids need to be drilled–yes drilled–on multiplication tables and even basic adding and subtractng facts.
there is nothing wrong with rote memorization of times tables. Rote memorization i how we remember birthdays, phone numbers,a nd computer passwords.
94. We should master “what is” before expecting “why is”. It is an unfair expectation (and generally beyond students’ developmental cognitive abilities) to expect them to come to conclusions about
“why” math concepts are, before they really learn what the concept is. Frankly, most adults can’t do this either (with reason!). Let’s stop putting the horse before the cart and teach students
math they can do, and use. Higher-level thinking will come to some students, once they’ve mastered the basics.
95. I don’t agree in rote memorization. I was a kid that excelled in math and could answer anything if it was given to me as 12 + 59, but as soon as you put it in a context or showed it in a
different way I was totally lost. Rote memorization is how you end up with kids that go 12 + 3 and end up with an answer of 42 and don’t understand why it’s wrong.
96. While discovery learning is powerful the problem with allowing children to “do it however works best for them” in math is that not all methods will carry forward into the higher skill sets of
math, thats just the way it is. So we need to teach children the method that will work best long term mathematically speaking. It is that simple.
97. “But do you understand why the multiplication table is why it is?”
This is the very root of the problem that plagues the curriculum. Do you need to know a reason for this? Does the fulfillment of your life rest in being able to answer why 3X3=9?
Meandering drivel. Who cares. News flash: there are things you need to just learn and accept and memorize and move on.
98. “Teachers, who can go through the system with minimal math training and
arrive in class expected to inspire children to create and conceptualize
their own mathematical knowledge—and relying on a new set of parents to
fill the gap. “This is a never-ending cycle of innumeracy,” says
Stokke. “And we have an obligation to speak up for the kids.””
No doubt teachers who reply to this comment feel outrage and are indignant at the suggestion that we have minimal training.
As an elementary teacher, I completely agree with the above statement. Nowhere was I “trained” to teach math. My brief year in teachers’ college was spent explaining why I had to feel a certain
way about the injustice of residential schools.
Teachers need to accept the fact that they are simply not trained to teach math. Unless you take additional professional development courses or additional qualifications in math, or unless you
have a formal math background, you have no training. Period.
There are a certain number of math facts that frankly do not need to be explained ad nauseam. There is simply not enough room or time in the curriculum to allow for children to explain how they
got 5+3. In light of time constraints, asking a child to justify their thinking when they add 20+20 is a waste of time.
Consider the gap that is produceds between the generation we are educating now and the previous generation stll teaching math, as evidenced in the article. We focus more on strategies instead of
the algorithm because education policy wonks ensconced in their ivory towers churn out paper after paper exalting the need for children to express themselves through math. The problem is, 2-3
children can do this, and the rest fall by the sidelines because they are not cognitively ready yet to declare this. I find it a complete waste of time and feel that I am doing a child an
injustice when I assign them a low mark, simply because they were unable to use a strategy yet the algorithm is fine.
We are comfortable in our self-entitled nanny state in Ontario, clinging falsely to the notion that we have one of the best curriculums in the world. Meanwhile on other continents young children
are still taught by rote up to a certain age and they are far outpacing our students on an international level. The world will not be made up of future jobs with a need for someone to constantly
express themselves. The world will most likely to continue being a niche-driven economy and we are essentially creating a generation of children with shaky foundations who, all the while being
able to “express” themselves, are bereft of any base. If you disagree, consider the silent mathematic illiteracy epidemic sweeping high schools, along with their literacy levels. High school and
university students are simply incapable of writing papers suited for their grade levels.
99. So has anybody actually used the SuccessMaker program which cost $12,500? Was it better than the terrible “Math Makes Sense” textbooks being used in schools?
100. What I find find most disgusting about this is the blatant hypocrisy of the Duvall’s of the world.“We want to provide options for kids,”
Ya, right. But if your kid does it any other way than the 4 strategies that we show him, we’ll mark it wrong and label him as stupid.
101. There is alot of overgeneralizations being made here, “…they won’t be able to do fractions, which means they won’t be able to handle algebra, which means calculus is out, which means they can’t
I must admit that as a teacher, there’s alot of new exciting ways to understand a concept and there is a push in a curriculum for conceptual understanding. The content and skill that the students
have to learn stay the same. They gotta know their facts and they gotta know their procedural skills. The endeavour of teaching math is for students to make connections between the pictoral (or
representation by manipulatives like blocks or shapes), the graph (scatter plot, Singapore math, pie chart, etc…), the tabular (organizing ideas using some thinking map to realize patterns), and
with the numeracy (math sentences, the abstract, procedural knowledge). The key is balance. In the past, the emphasis has been on the latter. This is more prevalent in the high school level (old
school). I don’t disagree that parents should take advocate but they should be addressing their concerns as well to the Department of Education. They have a voice and a right to express their
concerns about the curriculum or the textbook/resource being used in their school division/district and they definitely have the voice to express what content should be taught in the schools and
102. By the way, people, please learn that (a+b)*(c+d)=ac+ad+bc+bd, that will be good for any multiplication now, is it not? For example, suppose you want 67 * 97, that would be (60*90+60*7+7*90+7*7=
5400+420+630+49=6499). If you do not know that, you have serious problems in math. | {"url":"http://www.macleans.ca/society/life/have-you-finished-your-homework-mom/","timestamp":"2014-04-16T22:03:46Z","content_type":null,"content_length":"647387","record_id":"<urn:uuid:9964e18d-bc98-40e7-8611-752f1e6bc7d2>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00594-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Non-real critical values possible?
October 13th 2011, 02:51 PM #1
Non-real critical values possible?
The question in question:
Find the critical numbers of the function $f(x)=2x^3+x^2+2x$.
I know that $f'(x)=6x^2+2x+2=2(3x^2+x+1)$.
Solving using the quadratic formula gives me $x=\frac{-1\pm\sqrt{11}i}{6}$
I'm only taking Calc 1, so I figured that I should just say that there aren't any critical values, but do non-real values count?
Re: Non-real critical values possible?
The question in question:
Find the critical numbers of the function $f(x)=2x^3+x^2+2x$.
I know that $f'(x)=6x^2+2x+2=2(3x^2+x+1)$.
Solving using the quadratic formula gives me $x=\frac{-1\pm\sqrt{11}i}{6}$
I'm only taking Calc 1, so I figured that I should just say that there aren't any critical values, but do non-real values count?
not if you're studying calculus of a real variable (which you most probably are)
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Motion Graphs Worksheet
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1: Motion Name Worksheet B: Interpreting Motion Graphs AP/Inquiry Physics answer questions 1 and 2 in complete sentences 1. What does the slope of a distance vs. time graph indicate about an object’s
Motion Graphs 1 M. Poarch – 2003 http://science-class.net Name_____ Motion Graphs Describing the motion of an object is occasionally hard to do with words.
Worksheet: Motion Graphs Name_____ PHYSICSFundamentals 2004, GPB 3-10 Questions 1-4 refer to the velocity-time graph of a car’s
Name _____ Date _____ Teacher _____ Period _____ Graphing Motion Worksheet 1. Answer the following questions about the car whose motion is graphed in the
Graphical Analysis of Motion Part 1: Concepts: 1. The graph below shows the position vs time for an object in motion. Give a description of what
Describing motion with graphs involves representing how a quantity such as the object's position can change with respect to the time. The key to using position-time graphs is knowing that the slope
of a
Name Date Pd Unit 2 CP Worksheet 5 - Quantitative Motion Graphs 1. Robin, roller skating down a marked sidewalk, was observed to be at the following positions at the times
Most of the questions on this worksheet were collected from a variety of internet sources, and are reproduced here in accordance with the doctrine of fair use. Motion Graphs Page 2 3. The graph on
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Worksheet: Interpreting Graphs C H A P T E R 4 : L I N E A R M O T I O N INTERPRETING GRAPHS- As I have said many ... ing graphs describing motion (displacement, speed, velocity, and acceleration).
Distance vs. Time Graphs - To the right is a
Study Guide and Review Worksheet: Graphing Motion 13 Answer the next series of questions using the following velocity-time graph. 19. During which interval(s) is the object accelerating?
©Modeling Instruction 2011 2 U2 Constant Velocity – ws2 v3.2Trinity Draw the velocity vs time graphs for an object whose motion produced the position vs time graphs
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Motion Graphs 1 M. Poarch 2003 http://science-class.net Name_____ Motion Graphs Describing the motion of an object is occasionally hard to do with words.
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Grade 9 Science – Velocity / Time Graphs Worksheet – Solutions Motion of the car
Kinematics Animated Graphs Worksheet GRAPH 4 Watch the animation (several times) and draw what you think the displacement vs time graph should look like below.
1-D Motion Worksheet 2.Create a new simple position graph (not too complex!), play the motion and then write the corresponding scenario which describes the motions of the object.
Topic 3: Worksheet D-1 – Displacement, Velocity and Acceleration (Graphical Approach) (A) Displacement 1. ... Graphical analysis of motion is illustrated by three graphs, and three graphical items
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©Modeling Instruction 2010 1 U2 Constant Velocity ws 1 v3.0 Name Date Pd Constant Velocity Particle Model Worksheet 1: Motion Maps and Position vs. Time Graphs
Example It is very important that you are able to look at a graph and explain it's motion in great detail. These graphs can be very conceptual.
Worksheet 1: Motion Maps and Position vs. Time Graphs 5. Worksheet 2: Motion Maps and Velocity vs. Time Graphs 6. Unit Review This sample is designed to provide prospective users with enough material
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Motion can be described using words, diagrams, numerical information, equations, and graphs. Describing motion with graphs involves representing how a quantity such as the object's velocity = changes
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Interpreting Distance–Time Graphs MARS Shell Center University of Nottingham & UC Berkeley Beta Version !!!!! If you ... graph means that the motion is along a straight path. steady or is it
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PMO1.2 – Graphs of Motion Page 3 of 5 June 2012 Some further examples Example 1: A car accelerates from a stationary position. The acceleration is constant
Analyzing Graphs of Motion With Numbers 2.4 Speed can be calculated from position-time graphs and distance can be calculated from speed-time graphs. Both calculations rely on the familiar speed
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Physics IA Worksheet #2 on Motion Graphing Given the motion graphs below, fill in the complementary graph. A-10 1. 2. 3. t d t d t d v t v t v t 4. 5. 6.
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Name Date Pd UNIT 2 CP Worksheet 3 - Velocity vs Time Graphs Sketch velocity vs time graphs corresponding to the following descriptions of the motion of an
4 x t v t a t x x t v t a t x t v t a t 2 2 These graphs have numbers! Find the total displacement, speed (with direction), and acceleration. Label any numbers that are relevant.
7. IN-CLASS ACTIVITY: WORKSHEET – Interpreting Motion Graphs I. 8. HOMEWORK: Interpreting Motion Graphs II. 9. HOMEWORK: Graphing Worksheet # 1.
Worksheet 85 Position, Velocity and Acceleration – Graphs 1. Particle Motion The accompanying figure shows the velocity v=f(t) of a particle moving on a coordinate line.
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Worksheet: Uniformly Accelerated Motion Problems For all of these problems, give formulas and graphs for acceleration, velocity and position.
'Modeling Workshop Project 2002 2 Unit II ws2 v2.0 Draw the velocity vs time graphs for an object whose motion produced the position vs time graphs shown below at left.
Lesson 4 Describing Motion with Position vs. Time Graphs A person walking along a straight line is being timed while their position from the starting point is ... Lesson #4 - Worksheet Distance-Time
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Numerical Analysis
Quadrature/Numerical Integration
December 17, 2009
Quadrature is the numerical evaluation of definit inegrals in the following form:
$\displaystyle F(x) = \int_{a}^{b} {f(x) dx}$
This can approximated by in this form:
$\displaystyle F(x) \approx \sum_{k} w_k f(x_k)$
There are many forms of quadratures available. Each of these methods are in the same form above. The only difference is the choice of $x_k$ called the quadrature nodes, and weights $w_k$.
Interpolatory Quadrature:
Previous subject area had been about polynomial interpolation. The integral of the polynomial intepolant $p_n(x)$ of the $f(x)$ maybe be used to derive the quadrature formula of $F(x)$. Lets this
polynomial integral be denoted as $\hat{F(x)}$ from this point on.
$\displaystyle p_n(x) \approx f(x)$ through $n+1$ points.
$\displaystyle \hat{F(x)}$=$\displaystyle \hat{F(x)}=\int_a^b {p_n(x)}dx$
Example using Lagrange form:
The polynomial interpolation in the Lagrange form is: $\displaystyle p_n(x) = \sum_{k=0}^n {L_{n,k}(x) f(x_k)}$. The integral of a given function $f(x)$ which is interpolated in Lagrange form can be
approximated by:
$\displaystyle \hat{F(x)}=\int_a^b{(\sum_{k=0}^n {L_{n,k}(x) f(x_k)}})dx$
Interchanging the order of summation and integration, we obtain:
$\hat{F(x)}$=$\displaystyle \sum_{k=0}^n(\int_a^b{L_{n,k}(x) dx})f(x_k)$
The quadrature formula is then:
$\displaystyle \hat{F(x)} = \sum_{k=0}^n{w_k f(x_k)}$
the weight is now $\displaystyle w_k = \int_a^b {L_{n,k}(x) dx}$. The most commonly used and the simplest quadrature formula are those with equally spaced nodes and its knows as the $(n+1)$
Newton-Cotes formulas. Here is an example of 2-point closed Newton-Cotes formula using the Lagrange form above:
set $x_0=a$, $x_1=b$ and $\Delta{x}=h=b-a$. The interpolatory quadrature is equivalent to $\displaystyle \hat{F_1 (x)} = w_0 f(x_0) + w_1 f(x_1)$, where
$\displaystyle w_0 =\int_{x_0}^{x_1} L_{1,0}(x)dx$ and $\displaystyle w_1 = \int_{x_0}^{x_1} L_{1,1}(x)dx$
From my previous on Lagrange Interpolation,
$\displaystyle w_0 = \int_{x_0}^{x_1} {\frac{x-x_1}{x_0-x_1}dx}$ and $\displaystyle w_1 = \int_{x_0}^{x_1} {\frac{x-x_0}{x_1-x_0}dx}$
Here is the Interpolatory quadrature that Andrew Perry and I wrote:
This form can be extended for $n+1$ point of any Polynomial base interpolation:
$\displaystyle \hat{F_n (x)}= \sum_{k=0}^n w_k f(x_k)$=$w_0 f(x_0) + w_1 f(x_1) + \cdots + w_{n-1} f(x_{n-1) + w_{n} f(x_n)}$
The above equation can be used for any of the Rienmann Sum rules.
Gauss-Chebyshev Quadrature Rule:
The Newton-Cotes formulas are based on interpolating polynomials. However, interpolation at equidistant points experiences the Runge Phenomenon even for well-behaved function. For explicitly given $f
(x)$, other choices of quadrature should be considered, such as the Gaussian quadrature.
$\displaystyle \int_{-1}^1 f(x) dx = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} g(x) dx = \sum_{i=0}^n w_i g(x_i)$
$\displaystyle g(x_i) = f(x_i) \sqrt{1-x^2}$ and the weight is $w_i=\frac{\pi}{n}$ for $i=0,1,2, \cdots n$.
Here is a Matlab code implementing Gauss-Chebyshev Quadrature:
Interpolation Using Splines
December 10, 2009
This method has advantages over the use of high degree polynomial, which have tendency to oscillate between data values. This method is a series of third degree polynomials joining the points
together. Only four unknown coefficients need to computed to construct the cubic spline.
Given $n+1$ data points, there will be $n$ cubic polynomials $s_n(x)$:
$s_0 (x) = a_0 + b_{0}x + c_0 x^2 + d_0 x^3 \\ s_1 (x) = a_1 + b_{1}x + c_1 x^2 + d_1 x^3 \\ s_2 (x) = a_2 + b_{2}x + c_2 x^2 + d_2 x^3 \\ \vdots \\ s_n (x) = a_n + b_n x + c_n x^2 + d_nx^3$
Each of these cubic polynomial must pass through the two points it joins.
At the point $(x_0,y_0)$ and $(x_1,y_y)$, $s_0(x)$ will pass through both points. At the second point,$s_1(x)$ will also pass trough.
Smoothness conditions is most important in the spline method. Therefore, continuity of the slope $f'(x)$ and the second derivative $f''(x)$, which determines the concavity of the function $f(x)$,
must also be agree between the adjacent data points. These conditions controls the oscillations that usually happens in the high order polynomials from happening in Cubic Spline.
Properties of Cubic Spline:
1. $S(x)=s_k(x)=s_{k,0} + s_{k,1}(x-x_k) + s_{k,2}(x-x_k)^2 + s_{k,3}(x-x_k)^3$ for $k=0,1,2, \cdots n$
2. $s(x_k)= f(x_k)$ for $k=0,1,2, \cdots n$. The splines passes through each data points.
3. $s_k(x_{k+1})=s_{k+1}(x_{k+1})$ for $k=0,1,2, \cdots n$. The spline forms a continuous functions on the interval.
4. $s'_k(x_{k+1})=s'_{k+1}(x_{k+1})$ for $k=0,1,2,\cdots n$. The spline forms a smooth function.
5. $s''_k(x_{k+1})=s''_{k+1}(x_{k+1})$ for $k=0,1,2,\cdots n$. The second derivative is continuous.
The properties of cubic splines makes sure that Runge phenomenon does not occur in this interpolation method.
Matlab Code:
close all; clc; clear all
%x = linspace(-1,1,15);
x=-cos(pi*(0:20)/20); % the quadrature points [-1,1]
y = 1./(1+25*x.^2);
xx = linspace(-1,1,100);
yy= 1./(1+25*xx.^2);
y = spline(x,y,xx);
hold on
The result of this code is the following:
The Chebyshev points distributions visual results is a lot better than equally spaced points (with 10 points).
Newton Interpolation Polynomial
December 10, 2009
Polynomial interpolation can be used to construct the polynomial of degree $n$ that passes through $n+1$ points $(x_k,y_k)_{k=0}^n$ just like many other polynomials . The difference is that
Newton’s Method uses divided difference method for calculating its polynomial coefficients.
Assume that
Divided Difference is a recursive division process. The are symmetric in their arguments. The order in which the date points are supplied does not affect the leading coefficient (associative
Computing Newton’s Polynomial’s leading coefficient:
$a_2=\frac{f(x_2)-f(x_0)-(x_2-x_0) a_1}{(x_2-x_0)(x_2-x_1)}$
This method in essence is adding weights to the leading coefficients, by calculating different “centers”. A huge difference between this polynomial and the many others.
Using these coefficients, newton’s polynomial is constructed to approximate
The divided differences for a function
The coefficients needed to construct Newton’s polynomial is the top diagonal of the divided-difference matrix.
Runge’s Phenomenon
December 10, 2009
Runge’s Phenomenon:
Its a problem that occurs with large data during interpolation of polynomial of higher degrees. The more data points used, the higher the polynomial degree. To investigate this phenomena, I will use
$f(x)=\frac{1}{1+25x^2}$ on $[-1,1]$.
Equally Spaced Nodes:
The error plot looks great. Anyone would be happy to get an error like that, almost machine precision(10e-16). However, the reconstruction of the function(in blue) in poor. With more points, we see
the runge’s phenomena, the oscillation toward the end:
The error plot of this interpolation shows the impact of Runge’s Phenomena:
The error toward the end point, where the oscillations occur in the interpolation, is very poor.
Chebyshev Points:
Chebyshev point is better in polynomial interpolation because it reduces the ocilation as $n$ increases. Below is a plot of $f(x)$ with just 16 points.
Newton Divided Difference:
Vandermonde Matrix & Lagrange Polynomial
December 10, 2009
Vandermonde Matrix & Lagrange Interpolation:
(I got this idea from Chuck. During our chat, he pointed out that they are really not a big difference between the two method except for how the coefficient are gathered. Thanks Chuck!)
The quadratic equation example above is in fact Vandermonde Matrix with only 3 points. However, lets consider $n+1$ points. The polynomial can be expressed in this form:
$P_n(x)=\sum_{k=0}^n{a_k x^k}$ and
$f(x_k)=P_n(x_k), k=0,1,2 \cdots ,n$
For large data points, usually the case in real life problem (compare to $P_2(x)$ example above), we will obtain a linear system of equations for the coefficients $a_0,a_1,a_2, \cdots, n$:
$\begin{bmatrix} 1 & x_0 & x_0^2 & \cdots & x_0^n \\ 1 & x_1 & x_1^2 & \cdots & x_1^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^n \end{bmatrix}$$\begin
{bmatrix} a_0 \\ a_1 \\ \vdots \\a_n \end{bmatrix}$=$\begin{bmatrix} f(x_0) \\ f(x_1) \\ \vdots \\f(x_n) \end{bmatrix}$
The coefficient matrix is known as the Vandermonde Matrix.
Here is the Matlab code written in class:
for j=1:N
x(j)=((j-1)*(2*(pi))/(N-1));% set up grid
%alternativel use x=linspace(0,2*pi,N)
f=cos(x’); %set up function values
for j=1:N
c=inv(V)*f;; % polynomial coefficients
%now make the polynomial
for j=1:N
cp(N+1-j)=c(j)% rearrange order of the coefficients
You can increase the number of points to achieve better accurate approximation. However, the numerical solution of the linear equations above for large $n$ is very expensive and give rise to
stability problems. Using matlab, you will start getting the following warning:
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 9.377592e-48.
And the approximation(in blue) vs. the function( $f(x)=cos(x)$ on $[0,2 \pi]$ in red) would look like this:
The error of this interpolation is on the right. It is plotted again the number of point. As you increase data point, the error increases tremendously. This is not good. You should’ve expected the
error to decrease. The figures above are with just 50 data points. If more points are added, you can see the error even more vividly.
For this reason, a better option for polynomial interpolation is needed. Say you’re trying to find an interpolation polynomial that agrees with the function $f(x)$ at two points, a $P_1(x)$. The
function can then be approximated by taking linear combination of two polynomial of degree 1:
$L_{1,0}=\frac{x_1 -x}{x_1 -x_0}$$L_{1,0}=\frac{x -x_0}{x_1 -x_0}$
Hence, whenever $L_{1,0}(x_0)=1$ then $L_{1,1}(x_1)=0$, and whenever $L_{1,1}(x_0)=0$ then $L_{1,1}(x_1)=1$.This way, the following equation will creates secant lines connecting to each other: $P_1
The above equation can be extended for $n+1$ points as the following:
$P_n(x)= \sum_{k=0}^{n} {L_{n,k}}(x) f(x_k)$
The index $n$ denotes the degree of the polynomial and $k$ indicates $f(x_k)$ being the coefficient in $P_n(x)$.
$L_{n,k}(x)=\prod_{j=0, j eq k}^n \frac{x-x_j}{x_k-x_j}$
The Lagrange polynomials satisfies:
$L_{n,k}(x_j) = \begin{cases} 1 & j = k\\ 0 & j eq k \end{cases}$
The same thing happens with Lagrange Interpolation that happened in Vandermonde Matrix. This problem, or misbehavoir, is known as Runge’s phenomenom.
Polynomial Interpolation: Function Approximation
December 10, 2009
The search for polynomials fits to data
One of the common polynomial interpolation that is familiar to all is Linear Interpolation, it is also the simplest form. This is when two points are given $(x_0,y_o) \& (x_1,y_1)$ and we use them to
find the equation of a line that passes through them. In algebra, we will use the slope intercept form and the slope equation to find such a line:
This model can be used to approximating any $f(x^*)$ given that $x^* \in [x_0,x_1]$. This is interpolation approximation.When two point are given, we can build a linear model. This linear model is a
polynomial of degree one. Given a set of $n+1$ point, we can build a interpolating polynomial of degree $n$.
An example for a a quadratic polynomial, $P_2(x)$, interpolation would be the following:
Given 3 points, $(1,3), (2,12),(5,63)$. It can be assumed that the interpolation polynomial takes this form:
$P_2(x)=a_0 +a_{1}x+a_{2}x^2$,
where the polynomial coefficients are $a_0,a_1,a_2$. These coefficients can then be solved for using the condition:
$P_{2}(1)=3$, $P_{2}(2)=12$, $P_{2}(5)=63$
This can be written into:
We can transform this into a matrix and have Matlab solve it for us.
$A= \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 5 & 25 \end{bmatrix}$$x=\begin{bmatrix} a_0 \\ a_1 \\ a_2 \end{bmatrix}$$b=\begin{bmatrix} 3 \\ 12 \\ 63 \end{bmatrix}$
We can solve the coefficient, the $x=\frac{A}{b}= \begin{bmatrix} -2 \\ 3 \\2 \end{bmatrix}$ Hereby constructing $P_2(x)=-2+3x+2x^2$
Weierstrass Approximation Theorem:
for a given $\epsilon >0$ there exists a polynomial $P(x)$ defined on [a,b] such that
$|f(x)-P-n(x)| \leq \epsilon$ for all $x\in [a,b]$
This theorem says that a continuous function can be approximated by an interpolation polynomial. There are many different interpolation polynomial that could be used. Taylor Polynomials, Lagrange
Polynomial, Cubic Spline Interpolation, these are just a few.
Newton’s Method
October 29, 2009
Given a function $f(x)$ and whose $f'(x)$ continuous near a root $r$, then the newton’s method can be used to produce a sequence {${p_k}$} that converges faster than Bisection method. Newton method
is also a one-point method (i.e it only requires one initial guess), whereas the bisection is a two-point method.
Newton method uses the tangent line at one point, the initial guess $x_0$, on the given function $y=f(x)$. The equation of the tangent at the initial guess is the written as the following:
Newton’s Method Theorem:
Assume that $f(x) \in C[a,b]$ & $\exists$ root $r \in [a,b]$ such that $f(r)=0$. The Newton’s Method sequence is defined by the following iteration:
$x_{k+1}=f(x)$=${x_k}-\frac{f(x_k)}{f'(x_k)}$ for $k \geq 0$
This is $x_{k+1}$ is the intersection about the x-axis of the tangent line above. For the next iteration, our initial guess is replaced by the x-intercept obtain in the first calculation. This
iteration will continue until the root $r$ is found or the user is satisfied with level of accuracy.
The Newton Method iteration, similar to the fixed point iteration, will converge to the root $r$ for any initial guess close to the root and if $f'(x) eq 0$ on $[a,b]$.
Python Code for Newton Method:
#Sidafa Conde
#Newton Method
#MTH 361
from math import*
def F(x): #Defines the function here
return f
def F1(x): #derivative function
return f
def newtonmethod():
p.insert(0,1) #initial condition
for i in range(0,10,1):
x=p[i]-(F(t)/F1(t)) #newton method formula
I will test this code to find $sqrt(2)=1.41421356237$ with 20 iterations. I will the function as $f(x)=x^2-2$. $x_0=5$. Here is the program output: | {"url":"http://scondemth361.wordpress.com/","timestamp":"2014-04-17T18:22:58Z","content_type":null,"content_length":"83578","record_id":"<urn:uuid:6916e03f-25b0-4c64-85e5-7d3952cf14f5>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00283-ip-10-147-4-33.ec2.internal.warc.gz"} |
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The Math Forum || Annie's Sketchpad Activites || Java Version
Morphing in Sketchpad
Or: How to Win an Academy Award
(shamelessly stolen from the GSP Diety himself, Nick Jackiw)
In this activity, we're going to turn an ordinary polygon into a circle. Doesn't sound very interesting when it's described like that, but what we're really going to do is "morph" a polygon onto a
circle, which means create an image that at one point is a polygon, then becomes a circle.
What we want to do is create a "mapping" that takes each point on the polygon to a point on the circle. So we will pick a point on the polygon and find the corresponding point on the circle. Each
point on the polygon has to "go" to a point on the circle.
Move the point on your polygon and make sure that the ray moves appropriately, as does the corresponding point on the circle.
This point can move between the polygon and the circle. We'd like to have the position of this point for EVERY point on the polygon and their corresponding points on the circle. So select this new
point and select the point on the polygon. Under the Construct menu, choose "Locus". You should have a blob-shaped line that's between the polygon and the circle. I've colored mine blue.
Select the point on the segment and drag it onto the polygon. Ooooohhhheee! Now drag the point out to the circle. Whhoooo doggies!
Wouldn't it be really cool to animate that? Of course it would :-) Select the segment and the point on the segment and from the Edit menu, choose Action Button and go over and select Animation... Say
"Okay",and then double click your button. The coolest thing since sliced bread! | {"url":"http://mathforum.org/~annie/gsp.handouts/morphing/index.nojava.html","timestamp":"2014-04-18T19:04:25Z","content_type":null,"content_length":"3981","record_id":"<urn:uuid:20d720a1-c662-4821-83ee-b67f8fc3845c>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00631-ip-10-147-4-33.ec2.internal.warc.gz"} |
Woven fabric consumption formula.
Learn Apparel Merchandising, Learn Fashion design.
Before calculate the woven fabric consumption, you must need the followings information as mentioned below:
1) Measurement chart with technical spec.
2) Style Description.
3) Fabric Description.
4) Fabrics width/weight.
5) Washing shrinkage if any.
This formula is as under : Formula = Length X Width / Fab width X Fab Unit
Length = length of the specific parts + allowance
Width = width of the specific parts + allowance
Fab width = Fabric width ( after considering the shrinkage
Say, fabric width is 45" & the shrinkage
allowance is 1", Then the Fabric width will
be (45"-1") = 44" in the formula.
Fab unit = Fabric calculation unit, here it will be 36 because
we are going to calculate the consumption in
Now we will calculate the consumption of a Shirt (front part)
Center front length = 32"+1" (Sewing allowance) = 33"
Width (Chest) = 24" + 1" (Sewing allowance) + 3"
Pleat.W (1.5X2) = 28"
= Length X Width / Fab width X Fab Unit
= 33" X 28" / 44" X 36"
= 924" / 1584"
= 0.5833333
= 0.59 Yards. (for front part)
39 comments:
1. This is masud. I am working in Beximco as a sr. merchandiser.I saw above and appreciate for this. It will be useful for all merchandiser professional.
1. It'll be fruitful tips for all the merchandisers. Thanks.
2. Thanks Brother!
3. Dear Sir,
Have a nice day. Thanks for your valuable kind informations. Now i need Boxer shorts (6)pocket consumptions.How can i calculate it? Pls inform me.
Thanking you,
Md. Abdul Baki
1. I'am not more expert but i can try to help u.You first take every pocket length and width and then put their into following formula......
Pocket=[{(Length+allowance+shrinkage if required)*(width+allowance+shrinkage if required)}*no of parts/fabric width*12/36+wastage%]
All measurement puts sequence on a normal calculator.I think u get accurate consumption...brother...shajol
4. Please wait.
1. Dear Sir,
I need fabric consumption for bottom (trousers, pant, etc) and yarn dyed price for all color.
Kindly help me in this regard.
Many thanks
Md.Habib Ullah (Shimul)
2. Chitiye.....same wau you have to calculate for bottoms...length -width and fabric cw width is required everywhere...
don't create strong image of fool bangladeshies.
5. Send Me the list of your all color.
6. its great helpful job. i personally appreciate for your sharing.
1. Thanks!!
7. Great Sharing.Really APpreciate it.Having said that it would be very helpful if you can help us to find the consumption of a whole garment.
8. Great sharing.Appreciate it.Would be very helpful if you help us to find the consumption of a whole garment.
9. Very useful
10. Very useful and thanks bipul vai
11. Hi Bipul,
can you pls advise the woven bottom consumption details formula. i'm confused on several way of making consumption as i knew the below like as;
Consumption example:
Inseam = 58
Back rise= 27
= 0000
(+) W/b 4x2+allowance = 12
= 0000 ÷ 2.54(convert to inch) = result 40”
Thigh = 20 × 4(part) =80 ÷ 2.54(convert to inch) = result 37”
Total: L × W= ans ÷ 36 yds ÷ 56/58//60 (fab width) × 12 dzs × extra 10% = result Fab consumption
1. Dear Bipul Vhai
Asking for you kind permission to solve Mr. Tanin confusion regarding his computed Woven Bottom item formula on your blog comments box .....
Mr Tanin wrote as under .......
Hi Bipul,
can you pls advise the woven bottom consumption details formula. i'm confused on several way of making consumption as i knew the below like as;
Consumption example:
Inseam = 58
Back rise= 27
= 0000
(+) W/b 4x2+allowance = 12
= 0000 ÷ 2.54(convert to inch) = result 40”
Thigh = 20 × 4(part) =80 ÷ 2.54(convert to inch) = result 37”
Total: L × W= ans ÷ 36 yds ÷ 56/58//60 (fab width) × 12 dzs × extra 10% = result Fab consumption
Reply of Mr.Tanin
Dear Mr. Tanin
First you asked for Woven bottom garments Details formula & we think the formula given above by Bipul bhai is the unique formula ,based on that i will explain details formula .But before that
pls find the answer of you question last part
In your Consumption example:
Inseam = 58 cm
Back rise= 27 cm (excluding west band )
length = 85 cm
(+) W/b 4 cm x2 part ( inside & out side )+allowance 4 cm= 12 cm
= 97 cm ÷ 2.54(convert to inch) = result not 40” it will be 38.19 we can consider it as, Total Length is 38 inch.
Thigh = ½ thai of one leg 20cm × 4(part.. as Two leg has 4 pc ½ Thai ) =80 cm ÷ 2.54(convert to inch) = result not 37” it will be 31.496 we can consider it 32 inch
Now you can calculate it using your below formula
Total: L × W= ans ÷ 36 yds ( it will be inch ) ÷ 56/58//60 ( consider any one of this fab width ) × 12 dzs( it will be Pcs ) ×( not multiply X , Pls add the percentage + ) extra 10% = result
Fab consumption....
Total: L = 38 inch × W = 32 inch = ans 1216 squre inch ÷ 36 inch ÷ say it is 56 inch (fab width) × 12 pcs + extra(wastage as per nature of fabric ) 10% = result Fab consumption
Total: 1216 square inch ÷ ( we can calculate it like this 36 inch X 56 inch (fab width)= total × 12 pcs + extra(wastage as per nature of fabric ) 10% = result Fab consumption
Total: 1216 square inch ÷ 2016 square inch = 0.60 yds per pc × 12 pcs = 7.24 yds /doz + extra 10% .72 yds = final result for Fab consumption is 7.96 (7.24+.72) Yds /Dz .
Hope you are clear now ..
2. Dear Mr. Tamim,
First of all I'd like to thank you very much for your reply as well for below correction....
1) yes back rise excl band.
2) thigh part to be as per your consumption 32 inch.
3) Extra percentage will not multiply s/be (+).
Can you pls advise me the actual formula for woven fabrics consumption with details example.
Thanks/ Tanin
12. wow Mr. Bipul,
Thanks for your nice formulation.
13. Hi Mr. Bipul,
thanks for your nice formula for woven GMTS
14. Dear Bipul Vhai
Seeking for you kind permission, would you pls allow me to share in details about the “Woven bottom consumption ” for those who send request regarding this issue in your blog
( For those only who need this )
In Apparel sector there are 2 types of Merchandise, One is Woven merchandise ( Such as Shirt , Pant , Blouse etc ) & the other one is Knit merchandise ( Circular Knit :- Tee shirt , Polo shirt
,Fashion knit :- Sweater Etc ) . Merchandiser is the key person of this sector, who communicates with Buyer or Buying house to get an Order. The procedure of getting an order from a buyer,
merchandiser got an inquiry directly from a Buyer or from a Buying house. In that inquiry buyer ask for price of his desired garments, it may be in Woven fabric or In Knit fabric. For quote a
price, merchandiser must calculate fabric consumption first, coz fabric price cover more than 60 % of that quoted Price.
As a Merchandiser, for any type of Garments Consumption calculation, first we have to know what type of garments we are going to make. For that, we need to know the specific parts name & the
measurement of that particular garments parts. Measurement may be in inch or in centimeter (Measurement in Inch for American order & Centimeter for European order). That’s need to calculate
consumption. For accurate measurement we need to make a “Paper pattern “ or “CAD pattern “ first . pattern is the sketch combination of different parts of a garments .
We all know every part of garments have its LENGTH & WIDTH measurement. Basically on woven garments, we calculate the consumption in Yard/ Doz (2.54 Cm= 1 inch & 36 inch = 1 Yds,). In that case
we also need to know the Width of that particular fabric which we are going to use. Now we will calculate each of the separate part based on this Length & Width. to have the total consumption,
for this we can use Separate part formula as below:-
(LENGTH X WIDTH) X Number of Parts pcs or Panels / Fabric Width (Pls consider suitable sewing allowance with each length & width for each application.)
After calculate the consumption of all t part this way, now we will add all that separate parts consumption tighter ,than we will add wastage percentage based on the fabric nature which we are
going to use for that particular garments.
We can also workout the consumption of a bottom item by using the below “Solid formula for calculate consumption"
Solid formula for calculate required Fabric consumption / Doz :-
(Total LENGTH X Total WIDTH ) X Number of Parts pcs or Panels / Fabric Width X36 inch = (Total + 5 % Wastage) X 12 Pcs= Result in ..Yds / Doz .
In here
Total Length = Inseam (Length) + Back rise (Including West band ) + Sewing Allowance (S.A )
Total Width = ½ Thai + Sewing Allowance
** we can also calculate
( A) Back rise without West band
( B) Parts pcs = there is 4 pcs of 1/2 Thigh as a bottom item has 2 Leg
(C) We multiply the total measurement with 36 inch to make the result in yds which was in inch before
( D) For wastage percentage the calculation depends on the nature of fabric shrinkage or elongation on wash & others effect
**{We calculate this wastage percentage lowest ( 5 % ) on Regular fabric but for Stretch fabric it may stand up to 25 % based on the nature of the Fabric}
To be continue
15. Dear Bipul Vhai
Seeking for you kind permission, would you pls allow me to share in details about the “Woven bottom consumption ” for those who send request regarding this issue in your blog
(Rest part of previous Discation )
For batter understanding we can follow below Example :
Say we have the bellow measurement of a "Basic Denim long pant" :-
Inseam = 34 inch,
½ Thai= 13 inch,
Back rise (Under West Band) =15 Inch
Waist band width =1.5 inch,
Back pocket length = 7 inch,
Back pocket Width = 6.5 inch
Belt Loop length =1.5 inch,
Belt Loop Width = 1/2 inch
Pocket Bag Length = 7.5 inch
Pocket Bag Width = 6.5 inch
The Formula is :- ( Length X Width ) X No of panel / ( Fabric Width X 36 inch ) + Wastage Percentage = Result X12 Pcs
Calculation: - In Here
Total Bottom Length = Inseam + Back rise + Wbw + Sewing allowance.( Wbw =West band width )
Total Bottom Width = 1/2 Thigh +2“ Sewing allowance
So, Length = 34+15 + 1.5 *2 +3 = 55 Inch.
Width = 13 +2 =15 Inch
Say the actual Fabric Width is 58 inch, but when sewing a garments as we can’t use the edge of a fabric , so at the time of calculation we must deduct 1 Inch from the fabric width . I.e. fabric
width will be 57 Inch.
Now the Applicable Formula :- ( Length X Thigh ) * 4 part /( Fabric Width X 36 inch) = Result + 5 % Wastage = Result for 1 Pc X12 pcs = Yds / Doz
Based on above information we found
Main Fabric Consumption :- (55 Inch X 15 Inch) X 4 Pcs /57 inch X 36 Inch + Wastage 5% X 12 Pcs
= 3300 square inch X 2052 square inch + 5% Wastage X12 Pcs
= 1.61 Yds +0.08(Wastage)= 1.69 Yds X12 Pcs = 20.28 Yds /Doz.
Back Pocket Consumption = (7”X 6.5 )=44.5 X 2 pc / 57 X 36 =91 /2052 =.044 +.0022=.047 X12 Pcs
= 0.56 Yds / Dz
Belt loop Consumption = (1.5”X .5 ) = .75 X 6 pc / 57 X 36 =4.5 /2052 =.0022 +.0001=.0023 X12 Pcs
= 0.028 Yds / Dz
Total Fabric =20.28 yds +0.56 Yds+0..28 Yds = 20.84 Yds / Dz
For Pocketing Fabric :- Length =7 .5 inch X 2 Ply +1 inch (SA )=( 15+1) inch = 16 inch
Width = 6.5 inch X 2 Ply + 1 inch (SA )= 14 inch
= ( 16 inch X 14 inch )2 pcs /(43 inch X 36 inch)
= 448/1548
=.29+5% wastage=.28+.0145=0.30X 12 Pcs =3.65 Yds Per
So we can say, based on above measurement we need 20.84 Yds main fabric & 3.65 Yds pocketing fabric to make 12 Pcs Basic long pant. this way we can calculate fabric consumption for any quantity
Caution :- Friends pls keep in mind that as market is so competitive now a days , so when a buyer ask for a price quotation no matter how experience you are but pls don’t take any risk. Always
figure out practical consumption by the help of your Pattern master & believe me that will be the perfect for your organization.
Thanking you on be half of Mr.Bipul
(The owner of this Blog )
Md. Tahidul islam Talukder ( Tamim)
Sr. Merchandiser
True Lotus Inc.
1. this is amazing!!!!!
2. Need proper method to work out Fabric Consumption on Outer wear, Tailored Winter Wears like Ladies/ Girls/Baby's Coats, Jacket, Rain wear. Fabric type: Melton, PVC
16. Correction on above calculation :-
Pls note that there is a typing error on.............
Main Fabric Consumption :- (55 Inch X 15 Inch) X 4 Pcs /57 inch X 36 Inch + Wastage 5% X 12 Pcs
(Type mistake in the below line )
= 3300 square inch X 2052 square inch + 5% Wastage X12 Pcs
It will be
=( 3300 square inch / 2052 square inch )+ 5% Wastage X12 Pcs
= 1.61 Yds +0.08(Wastage)= 1.69 Yds X12 Pcs = 20.28 Yds /Doz.
I am sorry for this inconvenience
17. Dear Tamim bai,
well deserve it. many many thanks for your details formula.
all are clear. by the by you calculated in consumption in inch if the measurement are in CM than should we convert into inch or do you other idea. pls advise.
yes for the correct consumption everyone must have to make the pattern mark.
Thanks you very much. as well to Bipul bai for making such a useful blog.
18. niceeeeeeeeeeeeeeeeeeeeeeee
19. Dear sir,
What is the shortcut fabric consumption formula of a basic jacket with hood.
pls inform me how can i measure it....
Thanks & Regard
20. I NEED TO KNOW WOVEN TOP FULL CONSUMPTION.
21. very much fruitful...no doubt about this
22. Dear brother,Bipu.
Good day.I want to joint as an assistant merchandiser in sweater section, not trainee.So how much and whats knowledge should I have fast to join as like.Plz inform me as if I can take preparation
to develop my career.
For your kind information,I'm a student of MBA, well skilled in English & Computer. Plz give me advise.
Your best regard
ROBIN, S@IFUR'S
23. Very thanks to you for publishing this fourmula. It is very convenient for merchandising professionals.
TRZ Group
24. Why total length is inseam + back rise..
Why not inseam + front rise????
25. Total lenght u said inseam + back rise
But why not front rise????
26. Dear Bipul Bhai
I have to know on think,
if i want do basic shirt consumption,
How many part i should calculate for yoke,
is it 1 part or two part.
Ex: Yoke length X width + S.Allownce X part ( 1 Or 2) /36/57X12=
Please help.
Best Regards
27. Dear Bipul Bhai
I want merchandising job. pls give a merchandising job.
28. hi.. i am studying fabric consumption.. when we say that the fabric width is 57/58" what does it mean.. hoping some one can help me thanks
29. hi am trying to study fabric consumption, when you say the fabric width is 57/58". what does it mean
30. if you dont know what they are talking about like what does 57/58 inch mean then you have a lot to learn Marylyn | {"url":"http://www.apparel-merchandising.com/2011/08/woven-fabric-consumption-formula.html","timestamp":"2014-04-20T06:25:05Z","content_type":null,"content_length":"217722","record_id":"<urn:uuid:70a57870-73a4-46da-a867-56e7cdc28f0e>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00345-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hi guys!a math question:
i've spent lots of hours trying to figure it out,so if it seems easy,i'm just dumb not lazy^^
i'm reading the artificial intelligence book by russel and norvig,where the 8 puzzle is described as the usual 3*3 board with 8 numbered tiles and an empty cell...
the authors claim that only 9!/2 states can be reached from a given initial state(wich means there are only 2 equivalence classes).well intuitively it seems
correct,but i can't figure out any satisfying proof.i know i just could use a few lines of code to calculate this,but the authors mentioned that so casually
that it seems like it's the result of a very simple deduction...so what am i missing?
by the way i posted that on mathoverflow,they closed it in less than 5 minutes.
Re: the 8 puzzle
My memory is that there was a simple symmetry that explained the equivalence classes. I'm wanting to think about this even less than you do...
I took a quick look at an abstract algebra book that I remember having some related problems. The 15 puzzle (4x4 board) is equivalent to A15 (the alternating group of 15 elements). Searching for
"alternating group" with A7 or A15 should find some interesting literature (hopefully not hidden in too much abstraction).
Re: the 8 puzzle
It's not about a simetry, it's about the possible states that you can reach. There are 9! possible states, but half of those states are not reachable.
Wikipedia explains this.
Re: the 8 puzzle
thanks a lot guys.i hadn't noticed the proof sketch on wikipedia.It needed more creativity than i thought(i'm talking about the definition of the parity function).
Anyhow,i still think it's a bit odd that the authors of the book easily mention that without any allusion to any proof,while they tend to prove a lot of trivial things that most often are direct
results of some definition. | {"url":"http://www.lispforum.com/viewtopic.php?f=20&t=3302","timestamp":"2014-04-19T14:33:16Z","content_type":null,"content_length":"18647","record_id":"<urn:uuid:5d9c9205-fffa-4775-b025-cba208c32d56>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00028-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOM: Truth of G
Stephen Fenner fenner at cs.sc.edu
Tue Nov 7 11:53:03 EST 2000
On Tue, 7 Nov 2000, Raatikainen Panu A K wrote:
> The structure of Goedel's proof is (very roughly) the following:
> ASSUME THAT (e.g.) PA is consistent (otherwise it proves
> EVERY sentence and is trivially complete).
> By diagonalization, one can construct a sentence G that is
> independent (neither provable nor refutable) of PA.
> So far so good, but how, then, can one conclude that G is true ?
> There are even two somewhat different ways...
> First, assuming that the provability predicate used is normal, one
> can show (and prove even inside PA) that
> G <-> Cons(PA).
> (Although one can prove neither side of the equivalence in PA).
> Therefore, the truth of the sentence is, in a sense, already
> assumed in the beginning of the proof.
> NOTE: No non-mechanical intuition, no use of the standard model,
> is involved.
Here is where I lose you. I think the original hypothesis, stated in the
metalanguage "Assume PA is consistent...." is always taken to mean,
"Assume that there is no _standard_ proof of 0=1 in PA" and so the
standard model is implicitly assumed. If PA is consistent, then
PA + ~Con(PA) is also consistent and thus has a (nonstandard) model. In
this model, G is false.
Of course, if you fix a model of PA and interpret Goedel's theorem in this
model, then you are only saying something nonvacuous if the model
satisfies Con(PA) to begin with. This makes your statement above
plausible, but I don't think most people think of Goedel's theorem this
Stephen Fenner
Computer Science and Engineering Department
University of South Carolina
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Application of the shooting method to second-order multi-point integral boundary-value problems
In this paper, we focus on the following second-order multi-point integral boundary-value problem:
where , for and are given constants. The proof is based on the shooting method. By constructing a quadratic function and a sine function as the shooting objects and combining the integral mean
value theorem with the comparison principle, we consider the existence of positive solutions to the BVP respectively under the case and the case . The method is concise and some new criteria are
MSC: 34B10, 34B15, 34B18.
shooting method; integral boundary-value problem; positive solution
1 Introduction
For the study of nonlinear second-order multi-point boundary-value problem, many results have been obtained by using all kinds of fixed point theorems related to a completely continuous map defined
in a Banach space. We refer the reader to [1-9] and the references therein. Some of the results are so classical that little work can exceed; however, most of these papers are concerned with problems
with boundary conditions of restrictions either on the slope of solutions and the solutions themselves, or on the number of boundary points [2,5-8,10].
In [8], Ma investigated the existence of positive solutions of the nonlinear second-order m-point boundary value problem
where , for , , , , and there exists a such that .
The author obtained the existence of a positive solution to (1.1)-(1.2) under the case and (super-linear case) or the case and (sub-linear case) when .
Recently, Tariboon [9] considered three-point boundary-value problem (1.1) with the integral boundary condition
Such a boundary condition might be more realistic in the mathematical models of thermal conductivity, groundwater flow, thermoelectric flexibility and plasma physics, because it describes the fluid
properties in a certain continuous medium. Under the assumption that , Tariboon and the author proved that problem (1.1)-(1.3) has at least one positive solution in the super-linear case or in the
sub-linear one.
However, the method used in the previous two papers is Krasnoselskii’s fixed point theorem in a cone, which relates to constructing a completely continuous cone map in a Banach space, and the proof
is somewhat procedural.
Constructively, Agarwal [11] explored the solution of multi-point boundary value problems by converting BVPs to equivalent IVPs, which is called shooting method. After that Man Kam Kwong [4,12] used
the shooting method to consider second-order multi-point boundary value problems. In [12], Kwong studied the existence of a positive solution to the following three-point boundary value problem:
The principle of the shooting method used in [12] is converting BVP (1.4)-(1.5) into finding suitable initial slopes such that the solution of equation (1.4) with the initial value condition
vanishes for the first time after . Denote by the solution of (1.4)-(1.6) provided it exists. Then solving the boundary value problem is equivalent to finding m such that
If we can find two solutions and of (1.4) such that
where , for , then there must exist a number m between and such that is the solution of (1.4)-(1.5). By constructing two sine functions as the shooting objects and combining with the comparison
principle, the author obtained some better results than those via fixed point techniques for the existence of positive solutions to (1.4)-(1.5).
In this paper, we try to employ the shooting method to establish the existence results of positive solutions for (1.1) with the more generalized multi-point integral boundary condition
where , for and are given constants. Following the principle of the shooting method, there are two obstacles we encounter. The first one is that the boundary condition involves integral from 0 to
( ), so we transform the integral problem into a single-point problem by using the integral mean value theorem. The other difficulty is that we cannot obtain the existence results by constructing
two sine functions as in [12] because of the particularity of in [12]. Therefore, we construct a quadratic function and a sine function as the objective ones.
The purpose of this article lies in two aspects. One is to explore the application of the shooting method in a more complicated multi-point integral boundary value problem, which demonstrates another
way in studying BVPs. The other one is to establish new criteria for the existence of positive solutions to (1.1)-(1.7) under the case and the case .
For the sake of convenience, we denote
Let be the solution of (1.1)-(1.6) and define
In this paper, we always assume:
Under the assumption, it is not difficult to prove that the initial problem (1.1)-(1.6) has at least one solution defined on . In fact, after translating second-order differential equation (1.1)
into one-order equations, one can draw the conclusion [13].
Further, we introduce the comparison results derived from [4,12], which evolved from the Sturm comparison theorem.
Theorem 1.1Let , , be the solution of the initial value problems, respectively,
and suppose thatF, G, gare nonnegative continuous functions on a certain intervalIfor and such that
If does not vanish in , then for , it yields
The paper is arranged as follows. In the next section, we put forward the basic principle of the shooting method used in this paper, and show that BVP (1.1)-(1.7) has no positive solution when . In
Section 3, the general criteria are established for the existence of positive solutions to (1.1)-(1.7) under the case . Moreover, we present the special results in the form of corollaries
corresponding to the super-linear case or the sub-linear case. Finally, we come to the conclusion and an example is presented to illustrate our results.
2 Preliminaries
Lemma 2.1If there exist two initial slopes and such that
(i) the solution of (1.1)-(1.6) remains positive in and ;
(ii) the solution of (1.1)-(1.6) satisfies for and ; then multi-point boundary value problem (1.1)-(1.7) has a positive solution with the slope between and .
Proof Since the solutions of (1.1)-(1.6) depend on the initial value continuously, then from (1.8), it implies that is continuous on m. In view of the intermediate value theorem of continuous
functions, there exists a number between and such that , that is,
Therefore, is the solution of (1.1)-(1.7).□
Lemma 2.2Let , then (1.1)-(1.7) has no positive solution.
Proof Assume that (1.1)-(1.7) has a positive solution u.
If , then , the convexity of u implies that ( ) and
which contradicts with the convexity of u.
If , then , that is, for . If there exists such that , then and , which contradicts with the convexity of u. Therefore for .
In the rest of this paper, we always assume:
3 Main results
Theorem 3.1Assume that (H[1])-(H[2]) holds. Suppose and there exists a constant such that
Then problem (1.1)-(1.7) has a positive solution.
Proof (i) Since , we can choose a positive number such that
We claim that there exists a positive number small enough such that for . The claim is based on the convexity of the function and the Sturm comparison theorem (see [12]). Hence,
From (1.8), (3.1) and combining the integral mean value theorem with Theorem 1.1, we have
The second inequality in (i) means that there exists a number M large enough such that
For this M, there exist two numbers δ and such that
and there exists another number such that for . Set
In view of (H[2]) and (3.3), it is not difficult to verify that
which implies from (3.4) that for . Thus, by the convexity of and Theorem 1.1, we have
By Lemma 2.1 and (3.2)-(3.5), there exists a number between and such that is the positive solution of (1.1)-(1.7). The proof for (i) is complete.
Now, we prove for (ii).
In view of , we can choose a number N large enough such that
For this N, there exist a number ϵ small enough and a number large enough such that and for . Therefore
Obviously, as . Thus approximately for as .
Let , . Similar to (3.2), we obtain
Since , then there exist two positive numbers and σ small enough such that
By the convexity of , for these σ and , there exists a positive number τ small enough such that
which yields
From (3.6) and (3.7), we have and for . Thus
By Lemma 2.1, the proof for (ii) is complete.□
Theorem 3.2Assume that (H[1])-(H[2]) holds. Suppose and there exists a constant such that
Then problem (1.1)-(1.7) has a positive solution under the case
Proof Note the computation of in Theorem 3.1. In (3.2), if we substitute with
then , and all the steps in the following are the same as in Theorem 3.1.□
Now, let us consider the special super-linear case or the sub-linear case. It is not difficult to verify the following corollaries.
Then problem (1.1)-(1.7) has a positive solution.
Corollary 3.2If and there exists a constant such that
Then, problem (1.1)-(1.7) has a positive solution under the case
4 Conclusion and examples
The tool which we used for the analysis in this article is the shooting method derived from [4,12]; however, we considered a more general problem which involves integral boundary-value and
multiplicity of boundary-point. The meaningful work that we have done lies in the following three aspects. The first one is that we transform the integral problem into a single-point value one by
using the integral mean value theorem. The other one is that we construct a quadratic function and a sine function as the comparison functions because it does not take effect to construct two sine
functions as in [12]. Finally, we established the new criteria for the existence of positive solutions to (1.1)-(1.7) under the case and the case . Obviously, (1.7) vanishes to (1.3) when and the
sup-linear case or the sub-linear case is sufficient for the conditions in Theorem 3.1 and Theorem 3.2, so some of our results are more general or better than those via fixed point techniques.
However, in Theorem 3.2, whether the transcendental equation has a solution is somewhat difficult to verify. It can be seen that each method has its pros and cons.
Example 4.1 Consider the BVP
It is not difficult to see that
In view of , Matlab software gives and . Hence
Therefore, the condition (ii) of Theorem 3.2 is satisfied. A numerical simulation (Figure 1) for Example 4.1 demonstrates that BVP (4.1)-(4.2) has a positive solution such that .
Figure 1. Numerical simulation for Example 4.1.
Authors’ contributions
The work was carried out in collaboration between all authors. HL practised the methods and organized this paper. ZG found the topic of this paper and suggested the methods. LG finished the Matlab
program of numerical simulation. All authors have contributed to, seen and approved the manuscript.
The authors would like to thank the editors and the anonymous referees for their valuable suggestions on the improvement of this paper. First author was partially supported by the Scientific Research
Fund of Hunan Provincial Educational Department (1200361), Project of Science and Technology Bureau of Hengyang, Hunan Province (2012KJ2). Second author was partially supported by the Doctor
Foundation of University of South China ( No. 5-XQD-2006-9), the Foundation of Science and Technology Department of Hunan Province (No. 2009RS3019), the Natural Science Foundation of Hunan Province
(No. 13JJ3074) and the Subject Lead Foundation of University of South China (No. 2007XQD13).
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The Pairing Heap Egg
You are looking at historical revision 5798 of this page. It may differ significantly from its current revision.
The Pairing Heap Egg
The pairing-heap egg provides a heap datastructure which is persistent (operations on an existing heap do not modify it, but rather return a new heap which shares some structure with the old heap).
The egg uses memoization and lazy evaluation (via force and delay) to ensure that the following operations complete within the bounds given below.
Sort a List
If the pairing-heap egg did not provide a pairing-heap-sort operation on lists, here is how you could code it:
(use srfi-1)
(use pairing-heap)
;; compare should take two objects and return -1, 0, or 1, depending
;; on whether they are <, =, >.
;; We invert the comparison relation when making the heap to avoid
;; having to reverse the output list.
(define (ph-sort compare list)
(let ((rev-compare (lambda (obj1 obj2) (fx* -1 (compare obj1 obj2)))))
(let ((h (fold pairing-heap-insert (pairing-heap-empty rev-compare) list)))
(let loop ((sorted-elts '())
(h h))
(if (pairing-heap-empty? h)
(loop (cons (pairing-heap-min h) sorted-elts)
(pairing-heap-remove-min h)))))))
;; Returns '(1 2 3 4 5 6 7 8 9 10)
(ph-sort (lambda (n1 n2) (cond ((< n1 n2) -1) ((= n1 n2) 0) (else 1)))
'(10 9 5 4 6 7 8 2 3 1))
The pairing-heap egg is released under a BSD License.
This egg requires only the base chicken system.
A Note on Compare Procedures
The pairing-heap egg is compatible with SRFI-67, though it does not require any SRFI-67 code to be loaded at runtime or compile-time. This means that comparison procedures take two arguments, OBJ1
and OBJ2, and should return
• -1 if OBJ1 < OBJ2
• 0 if OBJ1 = OBJ2
• 1 if OBJ1 > OBJ2
Thus, an appropriate fixnum-comparison procedure would be
(define (fixnum-compare n1 n2)
((fx< n1 n2) -1)
((fx= n1 n2) 0)
(else 1)))
Exported Procedures
The following procedures are provided by the pairing-heap egg. In the following, COMPARE is a comparison function as described in the last section, OBJ is a generic scheme object, H and Hi, where i
is an integer, are pairing-heaps, and n refers to the number of elements in a heap:
[procedure] (pairing-heap? OBJ)
#f unless OBJ is a pairing heap.
[procedure] (pairing-heap-empty COMPARE)
Construct an empty heap which uses COMPARE to compare elements.
[procedure] (pairing-heap-empty? H)
#f unless the pairing-heap H is empty.
[procedure] (pairing-heap-min H)
Returns the smallest element in the heap H. Completes in O(1) time.
[procedure] (pairing-heap-insert OBJ H)
Returns a new heap which contains all elements of H in addition to OBJ. Completes in O(1) time.
[procedure] (pairing-heap-merge H1 H2)
Returns a new heap which contains all elements from H1 and all elements from H2. Completes in O(1) time.
[procedure] (pairing-heap-remove-min H)
Returns a new heap which contains all elements from H except the smallest. Can complete in O(n) time, but a sequence of pairing-heap-remove-min operations complete in averaged O(log(n)) time. (i.e.
amortized log(n) bounds.)
[procedure] (pairing-heap-fold KONS KNIL H)
Applies the two-argument procedure KONS to the elements of H and an accumulation value in unspecified order. KNIL is the initial accumulation value, and the result of each KONS application is used
for the next accumulation. If H is empty, KNIL is returned. The result of pairing-heap-fold is the result of the final KONS application. Completes in O(n) time. For example, one could define
pairing-heap->list as
(define (pairing-heap->list h) (pairing-heap-fold cons '() h))
[procedure] (pairing-heap-sort COMPARE LIST-OR-VECTOR)
Returns a list or vector (depending on the type of LIST-OR-VECTOR) which contains all elements of LIST-OR-VECTOR, but in non-decreasing order. Completes in O(n*log(n)) time.
Version History
Version 1.0
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Games for Students to Learn the Equivalence of Adding & Subtracting | eHow
Games for Students to Learn the Equivalence of Adding & Subtracting
Learning how to add and subtract numbers can be daunting for a beginner, but teachers can use hands-on activities to teach their students more effectively.
Adding and subtracting isn't always easy for the beginning math student. For children entering into first or even second grade, learning how to subtract and add equal amounts to an equation can get
complicated quickly. Teachers can use simple, entertaining and hands-on games to teach their students the equivalence of adding and subtracting to a sum without solely relying on math drills and
1. Subtraction bowling
Subtraction bowling can be an interesting way of teaching addition and subtraction.
To play subtraction bowling, you'll need nine bowling pins made out of empty 2-liter soda bottles. Line them up as you would bowling pins and have students stand about 5 feet away. Allow the
students to take turns bowling, using a small ball, then have them write the equation. For example, if a student knocks down six pins, she would write "9 - 6" and try to figure out the
answer. Each player gets two turns to bowl, and must write and answer a subtraction equation based on how many pins they knock down. To show an equivalence of adding, have the student then
take the difference they found just before and add it to the original number of pins they knocked down to get the total number of pins. So the student who knocked down six pins would add "3 +
6" to get a total of nine pins.
"Getting even" addition game
Students can play a fun addition and subtraction game using playing cards.
Instead of bowling pins, use playing cards to teach students in groups of two to five how to add. Lay out all of the cards from one deck on one student's desk, face up in an overlapping line.
Look for cards next to each other that add up to an even number. Aces equal one point, jacks equal 11, queens equal 12 and kings are 13. Then remove all even sums. Continue until the student
can no longer add two remaining cards together to make an even number. Have the student count the remaining cards. If the student can remove all of the cards in subsequent rounds, he or she
is the winner. To modify this game for subtraction, look for cards next to each other that, if subtracted, would lead to a difference that is an even number. For example, if a 9 and a 7 were
next to each other, the difference would be 2, which is an even number.
Eggs-Act Sums
□ Another way to reinforce addition and subtraction uses an empty dozen egg carton, jelly beans or cotton balls and a pair of standard dice for each student. This game is best played in groups
of two to five students. The students should number each section of the egg carton in consecutive order, starting with two, until they get to 12, leaving one spot unnumbered and marked with
an asterisk (*). For each turn, the student rolls the dice and adds the numbers on the top of both dice. For that number, the student puts a jelly bean or cotton ball in the corresponding
space on the egg carton. The asterisk can be used for any number. Then the next student goes and rolls the dice for their turn. The student who fills up all spaces with a jelly bean or cotton
ball wins.
Domino Mats
Dominoes can also teach students how to add and subtract equally from a sum.
Using dominoes can also be a way for the instructor to build up a student's number sense. Take a standard sheet of computer paper and draw a line vertically down the width of the paper, in
the manner of a blank domino. Have the student select a domino randomly from a pile, and then recreate the domino on the piece of paper with colorful counter chips. Then have the student
identify what total number the domino is by assigning it a number card. For example, if the domino had two dots on one side and three dots on the other, then the student would place a number
5 card by the domino mat he created. To modify this game for subtraction, have the student subtract one side from the total amount and assign a number card to that difference. For example, if
the student took away the two dots on one side, he would assign the number card 3.
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The unsteady lift of a finite wing
Robert T. Jones
January 1939
Unsteady lift function for wings of finite aspect ratio have been calculated by approximate methods involving corrections of the aerodynamic inertia and of the angle of the infinite wing. The
starting lift of the finite wing is found to be only slightly less than that of the infinite wing; whereas the final lift may be considerably less. The calculations indicate that the distribution of
lift near the start is similar to the final distribution. Both the indicia and the oscillating lift functions are given. Approximate operational equivalents of the functions have been devised to
facilitate the calculation of lift under various conditions of motion.
An Adobe Acrobat (PDF) file of the entire report: | {"url":"http://naca.central.cranfield.ac.uk/report.php?NID=1536","timestamp":"2014-04-17T15:26:45Z","content_type":null,"content_length":"1894","record_id":"<urn:uuid:0f2ff0e8-e9af-4feb-a8fa-8fb11a97939f>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00362-ip-10-147-4-33.ec2.internal.warc.gz"} |
MySQL pagination without double-querying?
up vote 51 down vote favorite
I was wondering if there was a way to get the number of results from a MySQL query, and at the same time limit the results.
The way pagination works (as I understand it), first I do something like
query = SELECT COUNT(*) FROM `table` WHERE `some_condition`
After I get the num_rows(query), I have the number of results. But then to actually limit my results, I have to do a second query like:
query2 = SELECT COUNT(*) FROM `table` WHERE `some_condition` LIMIT 0, 10
My question: Is there anyway to both retrieve the total number of results that would be given, AND limit the results returned in a single query? Or any more efficient way of doing this. Thanks!
mysql pagination double
add comment
7 Answers
active oldest votes
No, that's how many applications that want to paginate have to do it. It's reliable and bullet-proof, albeit it makes the query twice. But you can cache the count for a few seconds
and that will help a lot.
up vote 34 down The other way is to use SQL_CALC_FOUND_ROWS clause and then call SELECT FOUND_ROWS(). apart from the fact you have to put the FOUND_ROWS() call afterwards, there is a problem with
vote accepted this: There is a bug in MySQL that this tickles that affects ORDER BY queries making it much slower on large tables than the naive approach of two queries.
Excellent, thanks for your help! – ash May 4 '09 at 2:49
1 It's not quite race-condition proof, however, unless you do the two queries within a transaction. This generally isn't a problem, though. – NickZoic May 4 '09 at 3:23
By "reliable" I meant the SQL itself is always going to return the result you want, and by "bullet-proof" I meant that there are no MySQL bugs hampering what SQL you can use.
Unlike using SQL_CALC_FOUND_ROWS with ORDER BY and LIMIT, according to the bug I mentioned. – staticsan May 4 '09 at 4:30
+1. Hadn't heard about the bug before – Manos Dilaverakis Jun 22 '10 at 17:38
On complex queries, using SQL_CALC_FOUND_ROWS to fetch the count in the same query will almost always be slower than doing two separate queries. This is because it means all rows
5 will need to be retrieved in full, regardless of the limit, then only those specified in the LIMIT clause are returned. See also my response which has links. – thomasrutter Sep 8
'11 at 5:21
show 1 more comment
I almost never do two queries.
Simply return one more row than is needed, only display 10 on the page, and if there are more than are displayed, display a "Next" button.
SELECT x, y, z FROM `table` WHERE `some_condition` LIMIT 0, 11
// iterate through and display 10 rows.
up vote 38
down vote // if there were 11 rows, display a "Next" button.
Your query should return in an order of most relevant first. Chances are, most people aren't going to care about going to page 236 out of 412.
When you do a google search, and your results aren't on the first page, you likely go to page two, not nine.
This is true, I will keep that in mind. – ash Jul 24 '10 at 18:59
13 Actually, if I don't find it on the first page of a Google query, usually I do skip to page nine. – Phil May 18 '11 at 5:42
@Phil I heard this before but why do that? – TK123 May 13 '12 at 4:56
4 COUNT is an aggregate function. How do you return the count and all the results in one query? The above query will only return 1 row, no matter what the LIMIT is set at. If you add
GROUP BY, it'll return all results but the COUNT will be inaccurate – pixelfreak Nov 29 '12 at 9:57
1 The result of count is usefull to calculate how many pages will be needed to see the total number of rows. – rvazquezglez Mar 22 '13 at 18:21
show 2 more comments
Another approach to avoiding double-querying is to fetch all the rows for the current page using a LIMIT clause first, then only do a second COUNT(*) query if the maximum number of rows
were retrieved.
In many applications, the most likely outcome will be that all of the results fit on one page, and having to do pagination is the exception rather than the norm. In these cases, the first
query will not retrieve the maximum number of results.
up vote 14
down vote For example, answers on a stackoverflow question rarely spill onto a second page. Comments on an answer rarely spill over the limit of 5 or so required to show them all.
So in these applications you can simply just do a query with a LIMIT first, and then as long as that limit is not reached, you know exactly how many rows there are without the need to do a
second COUNT(*) query - which should cover the majority of situations.
Excellent point! – Andrei Taranchenko Jun 5 '12 at 9:22
@thomasrutter I had the same approach, however discovered a flaw with it today. The final page of results will not then have the pagination data. i.e., let's say each page should have 25
results, the last page will likely not have that many, let's say it has 7... that means the count(*) will never be run, and so no pagination will be displayed to the user. – duellsy Aug
21 '12 at 6:34
No - if you are say, 200 results in, you query the next 25 and you only get 7 back, that tells you that the total number of results is 207 and therefore you don't need to do another
query with COUNT(*) because you already know what it's going to say. You have all the information you need to show pagination. If you are having a problem with pagination not showing to
the user then you have a bug somewhere else. – thomasrutter Aug 22 '12 at 2:38
add comment
In most situations it is much faster and less resource intensive to do it in two separate queries than to do it in one, even though that seems counter-intuitive.
If you use SQL_CALC_FOUND_ROWS, then for large tables it makes your query much slower, significantly slower even than executing two queries, the first with a COUNT(*) and the second with a
LIMIT. The reason for this is that SQL_CALC_FOUND_ROWS causes the LIMIT clause to be applied after fetching the rows instead of before, so it fetches the entire row for all possible
results before applying the limits. This can't be satisfied by an index because it actually fetches the data.
If you take the two queries approach, the first one only fetching COUNT(*) and not actually fetching and actual data, this can be satisfied much more quickly because it can usually use
up vote 13 indexes and doesn't have to fetch the actual row data for every row it looks at. Then, the second query only needs to look at the first $offset+$limit rows and then return.
down vote
This post from the MySQL performance blog explains this further:
For more information on optimising pagination, check this post and this post.
add comment
query = SELECT col, col2, (SELECT COUNT(*) FROM `table`) AS total FROM `table` WHERE `some_condition` LIMIT 0, 10
up vote 3 down vote
8 This query just returns the total number of record in the table; not the number of records that match the condition. – Lawrence Barsanti May 3 '10 at 0:51
add comment
My answer may be late, but you can skip the second query (with the limit) and just filter the info through your back end script. In PHP for instance, you could do something like:
if($queryResult > 0) {
$counter = 0;
foreach($queryResult AS $result) {
if($counter >= $startAt AND $counter < $numOfRows) {
//do what you want here
up vote 2 down vote $counter++;
But of course, when you have thousands of records to consider, it becomes inefficient very fast. Pre-calculated count maybe a good idea to look into.
Here's a good read on the subject: http://www.percona.com/ppc2009/PPC2009_mysql_pagination.pdf
1 thanks for sharing this link! – Vikram Aug 7 '12 at 17:01
Yes, this link is exactly has valuable information. – Nuri Akman Dec 17 '12 at 22:40
add comment
SELECT *
FROM table
WHERE some_condition
up vote -7 down vote ORDER BY RAND()
LIMIT 0, 10
add comment
Not the answer you're looking for? Browse other questions tagged mysql pagination double or ask your own question. | {"url":"http://stackoverflow.com/questions/818567/mysql-pagination-without-double-querying","timestamp":"2014-04-18T04:24:56Z","content_type":null,"content_length":"110405","record_id":"<urn:uuid:03fc521b-a3dc-4459-8ae4-260e0d56c2cf>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00412-ip-10-147-4-33.ec2.internal.warc.gz"} |
Help needed simplifying the following:
March 31st 2009, 02:39 PM #1
Mar 2009
Help needed simplifying the following:
So I'm struggling with this problem and wondering if anyone could walk me through it so I can figure out how it works.
y^8 * z^2
y^-3 * z^0.5
Help would be appreciated.
Edit: Just in case the above isn't clear, it's: (y^8 * z^2) / (y^-3 * z^0.5)
$\frac{y^8z^2}{y^{-3}z^{0.5}} = y^{8 + 3}z^{2 - 0.5}$
$= y^{11}z^{1.5}$
$= y^{11}\sqrt{z^3}$.
Thanks very much, I was almost there myself that last step just stumbled me, probably because when I was going through it I changed Z^0.5 for Sqrt(Z) then got myself confused over the rules.
For some reason I was thinking it was more complicated than that.
+rep and thanks for you.
March 31st 2009, 05:57 PM #2
March 31st 2009, 06:19 PM #3
Mar 2009 | {"url":"http://mathhelpforum.com/algebra/81702-help-needed-simplifying-following.html","timestamp":"2014-04-18T07:19:32Z","content_type":null,"content_length":"36005","record_id":"<urn:uuid:178abb2e-e395-4252-8d4e-688d36ec5a29>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00561-ip-10-147-4-33.ec2.internal.warc.gz"} |
Trains of Thought
Trains of Thought
Computing with locomotives and box cars takes a one-track mind
Solitaire Sorting
The second phase of the freight-car sorting process—putting the cars in order for delivery—is typically done in smaller, local switching yards. These are humpless "flat yards"; the cars are moved by
engines rather than gravity. The tracks can be arranged as queues, as in a hump yard, or as stacks—dead-end leads—so that cars have to be pushed in and pulled out from the same end. Suppose a train
has cars numbered 1 through n, but on arrival at the yard they are scrambled in some arbitrary order; the departing train should have the cars in ascending sequence, with car 1 just behind the engine
and car n at the end. How many classification tracks are needed to achieve this result? How many times do cars have to be pushed onto and pulled out of the tracks? These are questions of obvious
practical importance to railroaders. They are also questions that yield to mathematical and algorithmic analysis.
Robert Tarjan of Stanford University answered some of the questions in 1972. Here are a few of his results:
If a switch yard has an internal loop, allowing cars at the output to be brought back to the input for further processing, then any sequence can be sorted. Conversely, in the absence of such loops,
no finite network of stacks, queues or deques can sort all possible sequences, even if the individual storage elements are of unbounded capacity.
If the yard consists of m queues arranged in parallel, then a train can be fully sorted if and only if the longest decreasing subsequence has no more than m cars. (The cars of a decreasing
subsequence don't have to be consecutive; for example, in the sequence 1635492 the longest decreasing subsequence is 6542.) For a yard with m parallel stacks, it's the longest increasing subsequence
that governs. But these constraints are somewhat artificial. They apply only if cars must always move from the input to a stack or a queue and then directly to the output. Real rail yards are more
flexible; cars can be pulled from one stack and pushed onto another. When moves like this are allowed, it's harder to determine which sequences can be sorted.
In 2000 Elias Dahlhaus, Peter Horak, Mirka Miller and Joseph F. Ryan showed that a version of the switch-yard problem is NP-complete (which means, roughly speaking, that there's no efficient
algorithm for solving it). Specifically, they proved it is difficult to decide how many tracks are needed.
Chinese mathematicians have taken a somewhat different and more-pragmatic approach to train-sorting problems, apparently in response to a request from Chinese railroad officials. (The exact
provenance of these ideas is somewhat murky. In 1976 an American delegation to China heard a lecture on the subject by Ma Chung-fan; notes on this talk were written up by Henry O. Pollak and
published in a National Academy of Sciences report. A 1983 paper by Zhu Yongjin and Zhu Ruopeng covers similar ideas but does not mention Ma.)
Pollak's lecture notes present an example: Use an array of stacks to sort the 10-car sequence 6324135726. (Cars with the same number are going to the same destination and thus should be grouped
together.) Here I am going to consider the same example but look at it from a different point of view.
One approach to sorting the example sequence resembles a game of solitaire, building multiple stacks of cars in nondecreasing order. Working from the rear of the train toward the front, we examine
the number on each car and push the car onto a stack. Suppose the car we have just reached bears number k. If there is exactly one nonempty stack whose topmost element is greater than or equal to k,
then we put the car on that stack. If there are multiple stacks with a top element greater than or equal to k, we choose the stack with the smallest top entry. If no stack qualifies to receive car k,
then we have to start a new stack.
For the sequence 6324135726, we begin with the rightmost 6, which necessarily starts a new stack. We push 2 onto the same stack, but the 7 starts a second stack, which can also accept 5 and 3. The 1
then goes on the first stack, and the 4 inaugurates a third stack. Working through the rest of the sequence, we finally reach this configuration of four stacks:
Now the cars can be pulled out of the stacks in nondecreasing order; following the guidelines indicated by the colored blocks, this final assembly step will take seven "pulls." The sorted sequence,
of course, is 1223345667.
In his Beijing lecture, Ma gave an alternative sorting procedure; I'm going to call it the Chinese solitaire algorithm. It partitions the sequence in a way that requires just four pulls to assemble
the sorted train. Here is the final state of the four stacks:
It's easy to confirm that this configuration can be reached from the original train order, and that four pulls do indeed yield the properly sorted sequence. But by what rule were the numbers dealt
into these particular groups? Both the notes on Ma's lecture and the paper by Zhu and Zhu give a rather convoluted algorithm. In trying to explain it I can do no better than quote the lecture notes:
Start at the leftmost (in this case the only) 1, put down all 1s, all 2s to the right of the last 1, 3s to the right of the last 2 if you have covered all the 2s, etc. In this case, the first subset
defined in this way is 12.... The next subset takes the other 2 and the second 3...; it can't get to the first 3. The next subset takes the first 3, the 4, the 5, and second 6; the last subset is 67.
This procedure works, but there's an easier way to generate the same partitioning: Repeatedly scan from left to right, and on each pass extract the longest possible nondecreasing subsequence starting
with the leftmost number. In the example considered here, the first such subsequence is 67, followed by 3456, then 23 and finally 12.
Zhu and Zhu give a proof that the Chinese solitaire algorithm allows the train to be assembled with the minimal number of pulls from the classification tracks. But the proof counts only pulls. What
about "pushes"—the train movements needed to place the cars on the stacks in the first place? For the example sequence, the Chinese algorithm has the worst possible performance in this respect: Ten
separate pushes are needed to stack up the 10 cars. The non-Chinese solitaire method is somewhat better, at seven pushes. Taking the sum of pushes and pulls, the two methods score a tie at 14. I
don't know whether some other technique can do better. | {"url":"http://www.americanscientist.org/issues/pub/trains-of-thought/6","timestamp":"2014-04-16T13:05:19Z","content_type":null,"content_length":"141783","record_id":"<urn:uuid:d092951d-01ca-4e52-af3a-c3289193a9b2>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00342-ip-10-147-4-33.ec2.internal.warc.gz"} |
On the use of artificial neural networks to model sound quality.
ASA 127th Meeting M.I.T. 1994 June 6-10
2aNSb5. On the use of artificial neural networks to model sound quality.
P. Laux
P. Davies
Ray W. Herrick Labs., School of Mech. Eng., Purdue Univ., West Lafayette, IN 47907-1077
Loudness, percentile loudness, sharpness, and fluctuation strength have been shown in laboratory experiments to be correlated with annoyance. Zwicker proposed a model of unbiased annoyance (UBA)
based on time of day (d), fluctuation strength (F), sharpness (S), and N[sub 10]. In psychoacoustic tests on amplitude-modulated noise signals [P. Laux and P. Davies, NCEJ 40 (3) (May-June 1993)] UBA
has been shown to have a higher correlation to annoyance than other commonly used noise measures, although, other correlation models were found to produce an even better model of the relationship
between (S,N[sub 10],F) and the subjective ratings. The use of artificial neural networks (ANNs) to describe the relationship between (S,N[sub 10],F) and the subjective responses has been
investigated. The smaller the network, the fewer the number of weights, and the fewer the number of signals that a subject has to rate. The objective of the research was to estimate a network that
was the smallest possible model of the data. As a starting point UBA was modeled as a function of (S,N[sub 10],F); the resulting ANN model of UBA was then used as the initial conditions when
subjective data were available. | {"url":"http://www.auditory.org/asamtgs/asa94mit/2aNSb/2aNSb5.html","timestamp":"2014-04-21T02:00:09Z","content_type":null,"content_length":"1870","record_id":"<urn:uuid:ce384e47-d780-4c64-8d8b-66185712c062>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00147-ip-10-147-4-33.ec2.internal.warc.gz"} |
overload operator[] help please
This is a discussion on overload operator[] help please within the C++ Programming forums, part of the General Programming Boards category; so i have to overload this so that it returns the ith
element of a red black tree im guessing ...
so i have to overload this so that it returns the ith element of a red black tree im guessing this will be an array but im having trouble trying to start it off can give me some pointers on how to
start please
Do you know how to traverse a tree? If you do, then the only other thing you need to be able to do is count up to i and then stop. At which point, you return a reference to whatever node of your RB
tree that you've found.
If you dance barefoot on the broken glass of undefined behaviour, you've got to expect the occasional cut. If at first you don't succeed, try writing your phone number on the exam paper. I support
http://www.ukip.org/ as the first necessary step to a free Europe.
You could always do it this way: Count the number of child nodes to the left of the root, and then traverse down the left or right side according to whether your index is less than or greater than
that value. Then at subsequent nodes you only have to count the left hand side as the right hand side is parantcount-left-1. Draw a diagram to play around with, to observe how this works.
My homepage Advice: Take only as directed - If symptoms persist, please see your debugger Linus Torvalds: "But it clearly is the only right way. The fact that everybody else does it some other way
only means that they are wrong"
i do know how to traverse a tree psuedocode: node* x = new node; x->data = some value; node* temp; temp = root; //then we traverse through the tree //ill use a temp node to point to root of tree //
then move down until we find the node we are look for //if found we return the node position //if this value is not in tree return 0 while(root != NULL){ if(x->data > root->data) root = root->right;
else if(x->data <= root->data) root = root->left; else if(x->data == root->data) return value; } this is somewhat correct??? heres the prototype that my prof wrote for the assignment Code: val_type
Tree::operator[](unsigned long i) { //PROBLEM FOR THE STUDENT //Note: this should return the ith element of the tree. if i is outside //of the range of indexes, the return value is undefined. You can
just do //what I did and return the default value in the NIL node in this case. return 0; }
val_type Tree::operator[](unsigned long i) { //PROBLEM FOR THE STUDENT //Note: this should return the ith element of the tree. if i is outside //of the range of indexes, the return value is
undefined. You can just do //what I did and return the default value in the NIL node in this case. return 0; }
So what's the problem? You've described how to traverse the tree, and you have a function that is required to find a node (by traversing from root, "i" times). Do do something "i" times, a loop of
some form would be appropriate.
Right 98% of the time, and don't care about the other 3%.
Code: val_type Tree::operator[](unsigned long i) { //PROBLEM FOR THE STUDENT //Note: this should return the ith element of the tree. if i is outside //of the range of indexes, the return value is
undefined. You can just do //what I did and return the default value in the NIL node in this case. return 0; } this is the method i was given to implement i understand how to traverse the tree and
find the ith element by comparing each node its the unsigned long i thats throwing me off i have no clue how to work with it though please help!
The "i" tells you what node to find. Basically, it says find the ith node. What is the ith node? That can be difficult to define in a tree, but the most straightforward definition, I believe, is
simply that you travel the tree and count every node you find until the count is i.
For information on how to enable C++11 on your compiler, look here. よく聞くがいい！私は天才だからね！ ^_^
its the i thats really throwing me off dont really know how i need to do what i mean is this: if we had a full binary tree with 7 nodes o / \ o o / \ / \ o o o o n we wanna find lets say the 5th
element how can i do that with just unsigned i given??? if i go from the root how do i know that its the 5th element in the tree what if it wasnt a full tree o / \ o o / \ / o o o how do i get the
5th element from that???
I believe you're just confused about which direction to go in first, but I think a lot of people agree with me here: the logical thing would be to traverse in order. That is to say that if a tree has
the first ten numbers in it [0, 9], then tree[5] would be 4, for instance.
Let's say the tree is [1-10] like so: Is this the right way to label them from 1-10??? O1 /\ O2O3 /\/\ O4O5O6O7 Or should I label them according to their value ie < > Like a heap???
I mean that the value of the node is what is important when you traverse the tree in order.
sorry im just not getting it ='/ lets use a tree with actual values from 1 to 7 inserted the usual tree way 4-O /\ 2-O6-O /\/\ 1-O3-O5-O7-O lets say we want the 5th element in the tree 5th will be 6
which is the right child of the root node am i correct??? im having trouble writing the code for this sorry im been slow but please explain a bit or maybe psuedocode-ish it
Breadth-first Tree traversal - Wikipedia, the free encyclopedia If you still have problems after reading that section at least, then post, but I think you will get it anyway. | {"url":"http://cboard.cprogramming.com/cplusplus-programming/131629-overload-operator%5B%5D-help-please.html","timestamp":"2014-04-19T13:07:13Z","content_type":null,"content_length":"92597","record_id":"<urn:uuid:68436b87-144e-4fc5-bac0-30ecf4aa14c0>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00046-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: st: probit with interaction dummies (significance and marginaleffect
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Re: st: probit with interaction dummies (significance and marginaleffects)
From Alan Acock <acock@mac.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: probit with interaction dummies (significance and marginaleffects)
Date Sat, 26 Jul 2008 19:54:10 -0700
I've not followed all of this discussion, but with continuous variables,
what Tony says makes centering extremely important. Centering education at
12 years in the U.S. Makes the 0 value meaningful. Changing where you center
a variable can dramatically change interactions so the centering should be
conceptually justified. Uncentered continuous variables for which a zero
value is extremely rare, e.g., years of education, age, etc., can produce
bizarre results.
Alan Acock
On 7/25/08 8:19 AM, "Tony Lachenbruch" <Peter.Lachenbruch@oregonstate.edu>
wrote:Tony Lachenbruch
> With logistic regression, interaction is usually interpreted as effect
> modification: the effect of predictor A is different at different levels of
> predictor B. With probit regression it is likely to be similar although not
> identical.
> It is often useful to write the predicted probabilities:
> For logistic regression you have
> (1) ln(p/(1-p))=B0 when A=B=0
> (2) ln(p/(1-p))=B0+B1 when A=1, B=0
> (3) ln(p/(1-p))=B0+ B2 when A=0, B=1
> (4) ln(p/(1-p))=B0+B1+B2+B3 when A=1, B=1
> The differences (2)-(1)=B1 is the ln(OR) for A at B=0
> (3)-(1)=B2 is the ln(OR) for B at A=0
> If B2=0 the interpretation of the above is how the ln(OR)s behave. If B2 is
> not 0, then
> (4)-(2)=B2+B3 is how the ln(OR) for A is modified when B=1
> (4)-(3)=B1+B3 is how the ln(OR) for B is modified when A=1
> Tony
> Peter A. Lachenbruch
> Department of Public Health
> Oregon State University
> Corvallis, OR 97330
> Phone: 541-737-3832
> FAX: 541-737-4001
> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu
> [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Erasmo Giambona
> Sent: Friday, July 25, 2008 3:05 AM
> To: statalist@hsphsun2.harvard.edu
> Subject: Re: st: probit with interaction dummies (significance and marginal
> effects)
> Dear Statalisters,
> I have found this thread particulalrly interesting. I have found the
> questions asked by Andrea and especially the answer of Marteen very
> useful. However, despite having read a lot about it over the last
> several days, it is still hard for me to have a good intuition on how
> to intepret interaction terms in logit regressions. I have also found
> that papers in finance (my field) usually miss to provide a clear
> interpretation of interaction terms in logit regressions.
> I truly hope some other people might join the thread to provide more insights.
> Here is my major source of confusion. Consider the case of interaction
> of two continuos variables (e.g., profit and number of employees) in a
> logit model. The dependent variable is 1 if the firm's ceo is fired
> and zero otherwise. The coefficient estimate on the interaction from
> the logit output is positive (for example, +0.25) and statistically
> significant. I interpret this to mean that the odds that the ceo is
> fired are higher when both profit and number of employees are large
> (small) in absolute term (rather than changes). However, Ali et al.
> (2004) show that the marginal effect for the interaction of two
> continuos variables can be negative even if its coefficient estimate
> is positive. Assuming that the marginal effect is negative (e.g.,
> -0.2) in my example, I would interpret this to mean that the
> likelihood of firing the ceo decreases by 20% on average as the
> interaction term increases by 1%.
> Assuming that my way of interpreting coefficient and marginal effect
> of the interaction term in a logit is correct, I would still find it
> hard to reconcile the "seemingly contradictory" evidence of the above
> example.
> I hope this can stimulate further discussion on the issue.
> Best regards,
> Erasmo
> Reference
> Norton, Wang, & Ai. 2004. Computing interaction effects in logit and
> probit models. The Stata Journal 4(2):103116.
> On Fri, Jul 18, 2008 at 5:00 AM, Andrea Bennett <mac.stata@gmail.com> wrote:
>> Thank you so much!
>> May I sum up for clarification: When I am using e.g. a probit model with a
>> dependent variable Y and include an interaction term -female*wage- and I am
>> primarily interested in the interaction effect of a woman with wage then it
>> is save to use the standard regression output to interpret the direction
>> (AND the significance?) from the regression table. E.g. if the
>> beta-estimators are -female- ==0.5, -wage- == 0.34 and -female*wage- ==
>> -0.03 and all being significant then I can say that the wage effect is
>> significantly smaller when being a woman? Does this also hold when one is
>> formulating models like -female*low_education-, -female*mid_education-,
>> -female-high_education-? Or did I misinterpret you line "as long as you
>> interpret the effects in terms of the effect on the latent variable you are
>> ok in simply using the output from -probit-"?
>> When I want to know if (and for which range) the interaction of female and
>> wage has a significant effect on Y I should use -inteff-. When I want to do
>> the same for the interaction of female with the education levels, then there
>> is not yet consensus on how it shall be done. Norton et al. 2004 mention
>> -predictnl- but urge to use it with extra care. Another source would be Rich
>> Williams webpage.
>> Did I completely mess it up (I fear so!) or is it like I described?
>> Andrea
>> On Jul 17, 2008, at 6:13 PM, Maarten buis wrote:
>>> Regarding problem 1, this is just a matter of interpretation, as long
>>> as you interpret the effects in terms of the effect on the latent
>>> variable you are ok in simply using the output from -probit-, if you
>>> want to interpret the results in terms of the probability you should
>>> use -inteff-.
>>> Problem 2 is much harder to solve. Any solution would in one way or
>>> another try to controll for things that haven't been observed. It
>>> should not come as a surprise that that is hard (read: impossible). So,
>>> the fact that "the solution" hasn't been implemented yet in Stata is
>>> not so much a problem with Stata but with the state of the statistical
>>> science: we know the problem, but we just don't know the answer. Though
>>> Rich Williams discusses one solution on his website.
>>> -- Maarten
>>> --- Andrea Bennett <mac.stata@gmail.com> wrote:
>>>> Thanks for the link! Still, I wonder if there's really no Stata
>>>> command I could use to "simply" test if the interaction is
>>>> significant and what influence (direction) it has on the dependent
>>>> variable. I'd be just rather surprised if this does not exist
>>>> because it seems to me this is a very common issue in any regression
>>>> design (interaction effects).
>>> --- Maarten buis wrote:
>>>>> There are two distinct issues when interpreting interaction effects
>>>>> in a probit:
>>>>> 1) a significant positive (negative) interaction in terms of the
>>>>> latent
>>>>> variable does not mean a significant positive (negative)
>>>>> interaction effect in terms of the probability that y = 1.
>>>>> 2) The scale of the latent variable is identified by setting the
>>>>> residual variance at 1. If the residual variance differs between
>>>>> the groups than that means that the scale of the latent variable
>>>>> differs between the groups and when comparing differences in
>>>>> effects across the groups you are basically comparing apples and
>>>>> oranges.
>>> -----------------------------------------
>>> Maarten L. Buis
>>> Department of Social Research Methodology
>>> Vrije Universiteit Amsterdam
>>> Boelelaan 1081
>>> 1081 HV Amsterdam
>>> The Netherlands
>>> visiting address:
>>> Buitenveldertselaan 3 (Metropolitan), room Z434
>>> +31 20 5986715
>>> http://home.fsw.vu.nl/m.buis/
>>> -----------------------------------------
>>> __________________________________________________________
>>> Not happy with your email address?.
>>> Get the one you really want - millions of new email addresses available
>>> now at Yahoo! http://uk.docs.yahoo.com/ymail/new.html
>>> *
>>> * For searches and help try:
>>> * http://www.stata.com/help.cgi?search
>>> * http://www.stata.com/support/statalist/faq
>>> * http://www.ats.ucla.edu/stat/stata/
>> *
>> * For searches and help try:
>> * http://www.stata.com/help.cgi?search
>> * http://www.stata.com/support/statalist/faq
>> * http://www.ats.ucla.edu/stat/stata/
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2008-07/msg00970.html","timestamp":"2014-04-18T18:40:02Z","content_type":null,"content_length":"16510","record_id":"<urn:uuid:974b9fbc-39ee-43b5-ba83-2dd42d82490c>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00143-ip-10-147-4-33.ec2.internal.warc.gz"} |
Convex compact sets
April 8th 2010, 06:24 AM #1
Jan 2010
Convex compact sets
Given a Euclidean n-space, take any compact convex set $S$ in that space, with a distinguished point $p$ in its interior. Let $\lambda$ be any positive real, and let $S^{\prime}$ be the result of
scaling $S$ as centered on $p$ by the factor $\lambda$. I would have thought that $S^{\prime}$ must be compact and convex as well. However, I didn't manage prove this. Is there any easy proof for
Given a Euclidean n-space, take any compact convex set $S$ in that space, with a distinguished point $p$ in its interior. Let $\lambda$ be any positive real, and let $S^{\prime}$ be the result of
scaling $S$ as centered on $p$ by the factor $\lambda$. I would have thought that $S^{\prime}$ must be compact and convex as well. However, I didn't manage prove this. Is there any easy proof for
I don't think that it is necessary for P to be an interior point of S to prove it.
First, compactness: if S' has an open cover, then the inversely scaled cover is an open cover of S, hence has a finite subcover for S, and the scaled version of this is a finite subcover of the
original open cover of S'.
Second, convexity: the inverse scaling of two points of S', $x'_1,x'_2$ say, are two points $x_1,x_2$ of S. If S is convex, then the entire line segment between these inversely scaled points must
be in S, and therefore the scaled version of that line segment must be in S'. So to really show this in detail, you need to show that the scaled segment between $x_1$ and $x_2$ is identical to
the line segment between $x'_1$ and $x'_2$.
April 8th 2010, 06:44 AM #2 | {"url":"http://mathhelpforum.com/differential-geometry/137911-convex-compact-sets.html","timestamp":"2014-04-17T17:06:15Z","content_type":null,"content_length":"37150","record_id":"<urn:uuid:ce1d1199-afa9-4c91-8b33-5aada9dd1dfe>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00401-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bandwidth reduction of multiple matrices
up vote 1 down vote favorite
Suppose I have a symmetric matrix A, and several diagonal matrices $D_1,D_2,...$
Are there any matrix transformations, such as $P^\top A P$ so that $P^\top AP$, $P^\top D_1 P$, $P^\top D_2 P$, etc are either all tridiagonal, or all have minimimal bandwidth in some sense? If for
example, I only had $D_1$ then solving the generalized eigenvalue problem for the matrix pencil $A,D_1$ would give me a basis that simultaneously diagonalizes both $A$ and $D_1$. Obviously, one basis
will not simultaneously diagonalize more matrices in general, but a set of banded matrices would also be pretty nice.
I am aware of Tisseur and Garvey's papers on simultaneous tridiagonalization of two matrices.
linear-algebra matrix-analysis
Sorry, I should not have chosen the symbol $P$ in that case. $P$ is not a permutation matrix in my example, and $A$ is assumed to be full. As an example, suppose I have the generalized eigenvalue
problems [ A V = V D_1 \Lambda_1 ] [ A U = U D_2 \Lambda_2 ] Now I have a matrice $V$ and $U$ such that the following matrices are diagonal: $V^\top A V$, $U^\top A U$, $V^\top D_1 V$, $U^\top D_2
U$ It seems, in general, impossible to find a single matrix that diagonalized $A$, $D_1$ and $D_2$, however, if it produced a banded matrix, that would still be useful for my purposes. – Greg von
Winckel Apr 7 '12 at 13:26
In general if $A$ is symmetric and $D$ is diagonal, it is not possible to simultaneously diagonalize them because this would imply that they commute. – Chris Godsil Apr 7 '12 at 13:29
Yes. That is exactly what I said. That is precisely why I was asking if there was a way to reduce them both to banded form. – Greg von Winckel Apr 7 '12 at 20:29
Additionally, it is possible to simultaneously diagonalize two matrices, but not with a similarity transformation. The matrix of eigenvectors to a symmetric definite matrix pencil $Ax=\lambda B x$
does diagonalize both $A$ and $B$, however, it is not an orthogonal matrix (see my first comment above). – Greg von Winckel Apr 7 '12 at 20:33
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If you use the typical reorderings (like reverse Cuthill-McKee ordering), $P^T A P$ will have a smaller bandwidth (in general, not tridiagonal though). Since $P$ is a permutation
matrix, all $P^T D_i P$ will remain diagonal too.
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vote Bart
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