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Topic: DEAD MATHEMATICS : NUMBER 23789591344261 ( Prime or not) Primality
test shown by Hope research
Replies: 5 Last Post: May 7, 2012 8:07 PM
Messages: [ Previous | Next ]
Re: DEAD MATHEMATICS : NUMBER 23789591344261 ( Prime or not)
Primality test shown by Hope research
Posted: May 7, 2012 5:57 PM
On May 7, 2012 4:27 pm, YBM wrote:
> Inverse 19 mathematics a écrit :
> > Is this a prime number. This is NOT the Primality formula but a
> > initial quick Primality test. This number cannot ever be a Prime
> > number because it is a summation total of a number 6897766.
> Big deal, NUTS!
> Last week you discovered what is trivial for anyone here (summation
> formula), this week you've discoverd that n*(n+1)/2 is either a
> multiple of n or n+1.
> Are you about to do another great discovery of this kind? What about
> the sum of cubes and the square of sums I asked you last week?
> > Smart Alecs from the dead mathematics find out from your Pappas how
> > to mathematicaly tell if the number is a summatation of a summate.
> What the hell is a summatation of a summate?
> > Now leave us alone, Sour grapes. We already have got what you do not
> > tthe Primality formula, as well as a full alogarithm of Predictive
> > Primality of prime numbers
> What the hell is a alogarithm?
> > CRAP RESEARCH ," By the grace of our self-deluded idiocy" V . CaMORON/
> > Theo DeNUTter
> But remember:
> Give us TWELVE HUNDREDS MILLIONS DOLLARDS and we will hide your shame
> to the world, otherwise we will send a letter to all citizens of
> Athens Wisconsin with a complete guide about how to ridicule kooks,
> cranks, bigots and morons - with materials such as cream pies, tar
> and barrels.
> Toto Von Lojee (MD), Jobvious Proof (GMBH), Nutty Drunk (ACME),
> in the grace of our Lord the Flying Spaghetti Monster
> (http://www.venganza.org/about/)
Give up and forget them. There is nothing else to do. You're wasting your time. | {"url":"http://mathforum.org/kb/thread.jspa?threadID=2377333&messageID=7814119","timestamp":"2014-04-19T09:30:03Z","content_type":null,"content_length":"24537","record_id":"<urn:uuid:7a3cbee5-fa08-4fb9-8e96-ff63e6700bd9>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00238-ip-10-147-4-33.ec2.internal.warc.gz"} |
Advanced Algebra Functions Video Tutorials
Functions in algebra.
Functions in pre-calculus.
Functions in advanced algebra.
Evaluating Functions in advanced algebra.
Periodic Functions in advanced algebra.
Exponential Functions in advanced algebra.
Polynomial Functions in advanced algebra.
Function Rule in advanced algebra.
Composite Functions in advanced algebra.
Absolute Value Functions in advanced algebra.
Inverse Functions in advanced algebra.
Radical Functions in advanced algebra.
Domain of a Function in advanced algebra.
Range of a Function in advanced algebra.
Zeros of a Function in advanced algebra.
Logarithmic Functions in advanced algebra.
Trigonometric Functions in advanced algebra.
Slope of a Function in advanced algebra.
Advanced Algebra Functions Video Tutorials
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2 minutes 31 seconds
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This math video tutorial gives a step by step explanation to "Slopes Of Parallel Lines 2". The video tutorial is recommended for Advanced Algebra students.
Slopes Of Perpendicular Lines Video Clip Length:
3 minutes 58 seconds
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This math video tutorial gives a step by step explanation to "Slopes Of Perpendicular Lines". The video tutorial is recommended for Advanced Algebra students.
Tangent Function Video Clip Length:
3 minutes 13 seconds
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This math video tutorial gives a step by step explanation to "Tangent Function". The video tutorial is recommended for Advanced Algebra students.
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4 minutes 38 seconds
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This math video tutorial gives a step by step explanation to "Testing Whether Two Functions Are Inverses 2". The video tutorial is recommended for Advanced Algebra students.
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Zero Slope Video Clip Length:
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This math video tutorial gives a step by step explanation to "Zero Slope". The video tutorial is recommended for Advanced Algebra students. | {"url":"http://www.tulyn.com/advanced-algebra/functions/videotutorials","timestamp":"2014-04-21T04:33:15Z","content_type":null,"content_length":"120817","record_id":"<urn:uuid:14efb226-faf8-4d07-b544-ca8dc82f4641>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00404-ip-10-147-4-33.ec2.internal.warc.gz"} |
How do i get LaTex on my computer
February 28th 2011, 09:13 AM #1
Senior Member
Dec 2008
How do i get LaTex on my computer
Sorry if this is a stupid question but how to i get Latex on my computer. I would like to be able to create mathematics on my computer (like on this forum) and save them as p.d.f's (or whatever)
but I have no idea how to go about doing this.
Thanks for any help
What is your operating system? Windows? (Which version?) Mac? Linux? Other?
Thanks but is there something more like whats on this forum? I use windows 7 or linux (ubuntu)?
thanks again for any help
With Linux, you can typically get everything you need in your Linux distribution. For Windows, one way to get everything you need is to install the following software, in this order:
A Text Editor, preferably one that does LaTeX syntax highlighting. I like TextPad.
There are some nice customizations possible like inverse search, line numbers, etc., that can save you some effort once you get going.
Last edited by Ackbeet; February 28th 2011 at 12:15 PM. Reason: GS before GView.
In Ubuntu >= 10.04, you need texlive package.
I used Ubuntu and downloaded Kile 4.2, it was quite simple from there. Thanks for all the help
You're welcome for my input.
February 28th 2011, 09:16 AM #2
February 28th 2011, 09:25 AM #3
February 28th 2011, 09:48 AM #4
Senior Member
Dec 2008
February 28th 2011, 12:02 PM #5
February 28th 2011, 12:14 PM #6
MHF Contributor
Oct 2009
March 7th 2011, 01:17 PM #7
Senior Member
Dec 2008
March 7th 2011, 01:21 PM #8 | {"url":"http://mathhelpforum.com/math-software/172932-how-do-i-get-latex-my-computer.html","timestamp":"2014-04-20T17:48:36Z","content_type":null,"content_length":"51914","record_id":"<urn:uuid:c4270706-8d5a-41c9-b455-df56dff1965f>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00415-ip-10-147-4-33.ec2.internal.warc.gz"} |
A framework for drawing planar graphs with curves and polylines
Results 1 - 10 of 14
- GRAPH DRAWING (PROC. GD ’03), VOLUME 2912 OF LECTURE NOTES COMPUT. SCI , 2003
"... We introduce a new approach for drawing diagrams. Our approach is to use a technique we call confluent drawing for visualizing non-planar graphs in a planar way. This approach allows us to draw,
in a crossing-free manner, graphs—such as software interaction diagrams—that would normally have many cro ..."
Cited by 28 (7 self)
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We introduce a new approach for drawing diagrams. Our approach is to use a technique we call confluent drawing for visualizing non-planar graphs in a planar way. This approach allows us to draw, in a
crossing-free manner, graphs—such as software interaction diagrams—that would normally have many crossings. The main idea of this approach is quite simple: we allow groups of edges to be merged
together and drawn as “tracks” (similar to train tracks). Producing such confluent drawings automatically from a graph with many crossings is quite challenging, however, we offer a heuristic
algorithm (one version for undirected graphs and one version for directed ones) to test if a non-planar graph can be drawn efficiently in a confluent way. In addition, we identify several large
classes of graphs that can be completely categorized as being either confluently drawable or confluently non-drawable.
- PROC. 26TH EUR. WORKSH. COMP. GEOMETRY (EUROCG’10) , 2010
"... The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay
triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alter ..."
Cited by 27 (2 self)
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The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay
triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alternative that offers flexibility for additionally enlarging small angles. We show that angle
optimization and related questions lead to linear programming problems, and we define unique flips in arc triangulations. Moreover, applications of certain classes of arc triangulations in the areas
of finite element methods and graph drawing are sketched.
- Proc. 12th Int. Symposium on Graph Drawing, 2004, Springer LNCS 3383
"... Abstract. We present a method for modifying a force-directed graph drawing algorithm into an algorithm for drawing graphs with curved lines. Our method is based on embedding control points as
dummy vertices so that edges can be drawn as splines. Our experiments show that our method yields aesthetica ..."
Cited by 16 (0 self)
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Abstract. We present a method for modifying a force-directed graph drawing algorithm into an algorithm for drawing graphs with curved lines. Our method is based on embedding control points as dummy
vertices so that edges can be drawn as splines. Our experiments show that our method yields aesthetically pleasing curvilinear drawing with improved angular resolution. Applying our method to the GEM
algorithm on the test suite of the “Rome Graphs ” resulted in an average improvement of 46 % in angular resolution and of almost 6 % in edge crossings. 1
, 2000
"... In visualizations of large-scale transportation and communications networks, node coordinates are usually fixed to preserve the underlying geography, while links are represented as geodesics for
simplicity. This often leads ..."
Cited by 13 (3 self)
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In visualizations of large-scale transportation and communications networks, node coordinates are usually fixed to preserve the underlying geography, while links are represented as geodesics for
simplicity. This often leads
"... Abstract. In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution. However, not every graph has a Lombardi
drawing, and not every planar graph has a planar Lombardi drawing. We introduce k-Lombardi drawings, in which ..."
Cited by 4 (4 self)
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Abstract. In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution. However, not every graph has a Lombardi drawing,
and not every planar graph has a planar Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi
drawing. We show that every planar graph has a smooth planar 3-Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings. 1
- In: Proc. 19th Int. Symp. on Graph Drawing , 2011
"... Abstract. A Lombardi drawing of a graph is defined as one in which vertices are represented as points, edges are represented as circular arcs between their endpoints, and every vertex has
perfect angular resolution (angles between consecutive edges, as measured by the tangents to the circular arcs a ..."
Cited by 4 (2 self)
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Abstract. A Lombardi drawing of a graph is defined as one in which vertices are represented as points, edges are represented as circular arcs between their endpoints, and every vertex has perfect
angular resolution (angles between consecutive edges, as measured by the tangents to the circular arcs at the vertex, all have the same degree). We describe two algorithms that create
“Lombardi-style” drawings (which we also call near-Lombardi drawings), in which all edges are still circular arcs, but some vertices may not have perfect angular resolution. Both of these algorithms
take a force-directed, spring-embedding approach, with one using forces at edge tangents to produce curved edges and the other using dummy vertices on edges for this purpose. As we show, these
approaches both produce near-Lombardi drawings, with one being slightly better at achieving near-perfect angular resolution and the other being slightly better at balancing vertex placements. 1
- Proceedings of Graph Drawing 2000, Lecture Notes in Computer Science , 2000
"... Timetable graphs are used to analyze transportation networks. In their visualization, vertex coordinates are xed to preserve the underlying geography, but due to small angles and overlaps, not
all edges should be represented by geodesics (straight lines or great circles). A previously introduced ..."
Cited by 3 (1 self)
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Timetable graphs are used to analyze transportation networks. In their visualization, vertex coordinates are xed to preserve the underlying geography, but due to small angles and overlaps, not all
edges should be represented by geodesics (straight lines or great circles). A previously introduced algorithm represents a subset of the edges by Bezier curves, and places control points of these
curves using a forcedirected approach [5]. While the results are of very good quality, the running times make the approach impractical for interactive systems. In this paper, we present a fast layout
algorithm using an entirely different approach to edge routing, based on directions of control segments rather than positions of control points. We reveal an interesting theoretical connection with
Tutte's barycentric layout method [18], and our computational studies show that this new approach yields satisfactory layouts even for huge timetable graphs within seconds. 1
- In Proc. 9th Symposium on Graph Drawing , 2003
"... We present a novel way to draw planar graphs with good angular resolution. We introduce the polar coordinate representation and describe a family of algorithms for constructing it. The main
advantage of the polar representation is that it allows independent control over grid size and bend positions. ..."
Cited by 3 (2 self)
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We present a novel way to draw planar graphs with good angular resolution. We introduce the polar coordinate representation and describe a family of algorithms for constructing it. The main advantage
of the polar representation is that it allows independent control over grid size and bend positions. We first describe a standard (Cartesian) representation algorithm, CRA, which we then modify to
obtain a polar representation algorithm, PRA. In both algorithms we are concerned with the following drawing criteria: angular resolution, bends per edge, vertex resolution, bend-point resolution,
edge separation, and drawing area. The CRA algorithm achieves...
- Proc. 8th Workshop on Algorithms and Data Structures , 2003
"... Abstract. We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and
information traveling through the network. We present an efficient linear-time algorithm which draws edges a ..."
Cited by 2 (1 self)
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Abstract. We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and
information traveling through the network. We present an efficient linear-time algorithm which draws edges and vertices of varying 2-dimensional areas to represent the amount of information flowing
through them. The algorithm avoids all occlusions of nodes and edges, while still drawing the graph on a compact integer grid. 1
"... Abstract. We study the problem of creating smooth orthogonal layouts for planar graphs. While in traditional orthogonal layouts every edge is made of a sequence of axis-aligned line segments, in
smooth orthogonal layouts every edge is made of axis-aligned segments and circular arcs with common tange ..."
Cited by 1 (1 self)
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Abstract. We study the problem of creating smooth orthogonal layouts for planar graphs. While in traditional orthogonal layouts every edge is made of a sequence of axis-aligned line segments, in
smooth orthogonal layouts every edge is made of axis-aligned segments and circular arcs with common tangents. Our goal is to create such layouts with low edge complexity, measured by the number of
line and circular arc segments. We show that every biconnected 4-planar graph has a smooth orthogonal layout with edge complexity 3. If the input graph has a complexity-2 traditional orthogonal
layout we can transform it into a smooth complexity-2 layout. Using the Kandinsky model for removing the degree restriction, we show that any planar graph has a smooth complexity-2 layout. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1299602","timestamp":"2014-04-16T22:20:14Z","content_type":null,"content_length":"38069","record_id":"<urn:uuid:fe7e629e-0506-4627-ba58-322a53eba90d>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00072-ip-10-147-4-33.ec2.internal.warc.gz"} |
A highly accurate heuristic algorithm for the haplotype assembly problem
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BMC Genomics. 2013; 14(Suppl 2): S2.
A highly accurate heuristic algorithm for the haplotype assembly problem
Single nucleotide polymorphisms (SNPs) are the most common form of genetic variation in human DNA. The sequence of SNPs in each of the two copies of a given chromosome in a diploid organism is
referred to as a haplotype. Haplotype information has many applications such as gene disease diagnoses, drug design, etc. The haplotype assembly problem is defined as follows: Given a set of
fragments sequenced from the two copies of a chromosome of a single individual, and their locations in the chromosome, which can be pre-determined by aligning the fragments to a reference DNA
sequence, the goal here is to reconstruct two haplotypes (h[1], h[2]) from the input fragments. Existing algorithms do not work well when the error rate of fragments is high. Here we design an
algorithm that can give accurate solutions, even if the error rate of fragments is high.
We first give a dynamic programming algorithm that can give exact solutions to the haplotype assembly problem. The time complexity of the algorithm is O(n × 2^t × t), where n is the number of SNPs,
and t is the maximum coverage of a SNP site. The algorithm is slow when t is large. To solve the problem when t is large, we further propose a heuristic algorithm on the basis of the dynamic
programming algorithm. Experiments show that our heuristic algorithm can give very accurate solutions.
We have tested our algorithm on a set of benchmark datasets. Experiments show that our algorithm can give very accurate solutions. It outperforms most of the existing programs when the error rate of
the input fragments is high.
The recognition of genetic variations is an important topic in bioinformatics. Single nucleotide polymorphisms (SNPs) are the most common form of genetic variation in human DNA. Humans are diploid
organisms. There are two copies of each chromosome (except the sex chromosomes), one from each parent. The sequence of SNPs in a given chromosome copy is referred to as a haplotype. Haplotype
information is useful in many applications, such as gene disease diagnoses [1,2], drug design, etc. Due to their essential importance in many biological analysis, haplotypes have been attracting
great attention in recent years [3-7]. Since experimental methods for direct sequencing of haplotypes are both expensive and time consuming, computational methods are usually much more promising.
Currently, computational methods for computing haplotypes often fall into two categories: population haplotyping [8-11] and haplotype assembly (also known as single individual haplotyping) [12-15].
The former tries to compute haplotypes based on the genotype data from a sample of individuals in a population. Many software packages have been published in this field, e.g., PHASE [10]. An obvious
drawback of population haplotyping lies in its weakness in recognizing rare and novel SNPs [16]. Contrary to population haplotyping, haplotype assembly is more efficient and has received more
attention in recent years. The input to the haplotype assembly problem is a set of fragments sequenced from the two copies of a chromosome of a single individual, and their locations in the
chromosome, which can be pre-determined by aligning the fragments to a reference DNA sequence. The task here is to reconstruct two haplotypes from the input fragments. In this paper, we focus on the
haplotype assembly problem.
The haplotype assembly problem was first introduced by Lancia et al. [17]. In [17], the authors proposed three optimization criteria for solving this problem, i.e. minimum fragment removal (MFR),
minimum SNP removal (MSR) and longest haplotype reconstruction (LHR). Some polynomial time algorithms have been designed to solve some versions of such optimization problems [18,19]. Lippert et al. [
19] summarized the models in [17] and proposed some new models. Among these models, the most difficult and realistic one is minimum error correction (MEC), where we want to minimize the total number
of conflicts (errors) between the fragments and the constructed haplotypes (h[1], h[2]). The haplotype assembly problem with MEC is NP-hard [5,19] even for gapless fragments.
Levy et al. [20] designed a greedy heuristic algorithm that concatenates the fragments with minimum conflicts. The greedy heuristic algorithm is very fast but not very accurate when the error rate of
fragments is high. Later, Bansal and Bafna [21] developed a software package HapCUT and the algorithm is based on the idea of building a graph from the sequenced fragments, in which each SNP site
corresponds to a node in the graph and two nodes are connected by an edge if there exists a fragment that covers both SNP sites (which correspond to the two nodes). It then tries to minimize the MEC
cost of the reconstructed haplotypes by iteratively finding max-cuts in the associated graph. Bansal et al. [22] designed a Markov Chain Monte Carlo (MCMC) algorithm, HASH. Both HASH and HapCUT have
better performance than the greedy heuristic algorithm proposed in [20].
Recently, He et al. [16] gave a dynamic programming algorithm that can give the optimal solution to the haplotype assembly problem with MEC. The time complexity of the algorithm is O(m × 2^k × n),
where m is the number of fragments, n is the number of SNP sites, and k is the length of the longest fragments. This algorithm works well for k ≤ 15. However, it becomes impractical when k is large.
In this paper, we propose a heuristic algorithm for the haplotype assembly problem with MEC. It is worth mentioning that in HapCUT [21] and the dynamic programming algorithm proposed in [16], the
authors assumed that the two constructed haplotypes are complementary with each other, i.e. there are only 2 choices at a SNP site in the reconstructed haplotypes. We drop this assumption in our
heuristic algorithm. As a result, there are 4 choices at a SNP site in the reconstructed haplotypes. We have tested our algorithm on a set of benchmark datasets and compare it with several
state-of-the-art algorithms. Experiments show that our algorithm is highly accurate. It outperforms most of the existing programs when the error rate of input fragments is high.
The input to the haplotype assembly problem is a set of fragments sequenced from the two copies of a chromosome of a single individual. Each fragment covers some SNP sites. We assume that all the
fragments have been pre-aligned to a reference DNA sequence. As a result, we can organize the input fragments as an m × n matrix M (called fragment matrix), where m is the number of fragments and n
is the number of SNP sites. Each row of M corresponds to a fragment that can be represented as a string on the alphabet ∑ = {a, c, g, t, -}, where '-' indicates a space when the SNP site is not
covered by the fragment or the SNP value cannot be determined with enough confidence. The start (respectively, end) position of a fragment is defined as the first (respectively, last) position in the
corresponding row that is not a '-'. In the middle of a fragment, '-'s are allowed due to data missing or paired-end fragments. Throughout the remainder of this paper, we will use the two notations,
i.e. fragments and rows of M interchangeably when there is no ambiguity. Moreover, we will use columns and SNP sites interchangeably when there is no ambiguity.
It is accepted that there are at most two distinct nucleotides at a SNP site. We assume that a column with more than two distinct nucleotides in M must contain errors. In this case, we keep the two
distinct nucleotides that appear the most at this column and replace the rest of them with a '-'. After removing errors, M can be converted into $M′$, in which each entry is encoded by a character
from the alphabet $∑′={0,1-}$. Figure 1(a) gives an example of an original input matrix M containing errors in some columns, Figure 1(b) is the matrix after error correction. $M′$ is given in Figure
Illustration of the preprocessing on the input fragment matrix. (a) The original fragment matrix M. (b) The matrix obtained from (a) by removing possible errors. (c) The obtained matrix $M′$.
We say that row i covers column j in M if M[i][,j ]is not a '-' or there are two integers p and q with p < j < q such that M[i][,p ]≠ - and M[i][,q ]= -. The number of rows covering column i in M is
referred to as the coverage of column i.
Two rows p and q in M are in conflict if there exists a column j such that M[p][,j ]≠ M[q][,j], M[p][,j ]≠ - and M[q][,j ]≠ -. Obviously, for error-free data, two rows from the same copy of a
chromosome should not conflict with each other, and two rows which conflict with each other must come from different copies of a chromosome. The distance between two rows i and j, denoted by D(i, j),
is defined as the generalized hamming distance as follows:
$D(i,j)= ∑k=1nd(Mi,k,Mj,k)$
Minimum error correction (MEC) is a commonly used model for the haplotype assembly problem. For the haplotype assembly problem with MEC, the input is a fragment matrix M, the task is to partition the
fragments in M into two groups and construct two haplotypes (h[1], h[2]), one from each group, such that the total number of conflicts (errors) between the fragments and the constructed haplotypes (h
[1], h[2]) is minimized.
In this section, we will describe the algorithms used to solve the problem. We first design a dynamic programming algorithm that gives an exact solution and runs in O(n × 2^t × t) time, where n is
the number of columns in M, and t is the maximum coverage of a column in M. The dynamic programming algorithm will be very slow when t is large. We then design a heuristic algorithm that first
computes an initial pair of haplotypes by using the dynamic programming algorithm on only a subset of M. This initial pair of haplotypes can be viewed as an approximation to the optimal solution. To
obtain a better solution, we further introduce some techniques to refine the initial solution.
A dynamic programming algorithm
Recall that the goal of the haplotype assembly problem is to partition the rows of the input fragment matrix M into two groups, each of which determining a haplotype. To obtain an optimal partition,
a naive approach is to enumerate all possible partitions on the rows of M, among which we then choose the one minimizing MEC. For an instance with m rows, there are 2^m total partitions, and thus the
approach does not work in practice. Here we introduce a dynamic programming algorithm for the haplotype assembly problem with MEC that runs in O(n × 2^t × t) time, where n is the number of columns in
M, and t is the maximum coverage of a column in M.
Before we give the details of the dynamic programming algorithm, we first define some basic notations that will be used later:
•R[i]: the set of rows covering column i in M.
•P[j](i): the j-th partition on R[i].
•Q[j](i): the j-th partition on R[i ]∩ R[i][+1].
•P[j](i)|[Ri][∩Ri+1]: the partition on R[i ]∩ R[i][+1 ]obtained from P[j](i) by restriction on the rows in R[i ]∩ R[i][+1].
•QQ[j](i): the set of partitions P[k](i) such that P[k](i)|[Ri][∩Ri+1 ]= Q[j](i).
•C(P[j](i)): the minimum number of corrections to be made in column i of M when the partition on R[i ]is indicated by P[j](i).
•MEC(i, P[j](i)): the optimal cost for the first i columns in M such that column i has a partition P[j](i).
In order to compute MEC(i + 1, P[j](i + 1)) efficiently, we define
Let P[j](i + 1) be the j-th partition on R[i][+1], Q[k](i) = P[j](i + 1)|[Ri][∩Ri+1]. The recursion formula of the dynamic programming algorithm is illustrated as follows:
Based on P[j](i + 1), we can get Q[k](i) in O(t) time. Furthermore, we know the majority value (0 or 1) at column (i + 1) in each group. To compute C(P[j](i + 1)), we can simply count the number of
minorities in each group (at column (i + 1)) separately, and then add them up. Thus, it takes O(t) time to compute C(P[j](i + 1)).
The optimal MEC cost for partitioning all the rows of M is the smallest MEC(n, P[j](n)) over all possible P[j](n), where n is the number of columns in M. A standard backtracking process can be used
to obtain the optimal solution.
Let us look at the time complexity of the dynamic programming algorithm. To compute each MEC(i + 1, P[j](i + 1)) in Equation (4), it requires O(t) time to compute C(P[j](i + 1)). Thus, it takes O(n ×
2^t × t) time to compute all C(P[j](i + 1))s for all the n columns in M. Now, let us look at the way to compute M E(i, Q[k](i))s. For each partition P[j](i) on R[i], we can get Q[k](i) = P[j](i)|[Ri]
[∩Ri+1 ]in O(t) time. We then update ME(i, Q[k](i)) if the current value of ME(i, Q[k](i)) is greater than MEC(i, P[j](i)). There are at most 2^t P[j](i)s on R[i]. Thus, it takes O(t × 2^t) time to
compute all ME(i, Q[k](i))s on R[i]. Since there are n columns in M, the total time required for computing all ME(i, Q[k](i))s is O(n × 2^t × t).
Theorem 1 Given a fragment matrix M, there is an O(n × 2^t × t) time algorithm to compute an optimal solution for the haplotype assembly problem with MEC, where n is the number of columns in M, and t
is the maximum coverage of a column in M.
Obtaining an initial solution via randomized sampling
The dynamic programming algorithm works well when t is relatively small. However, it will be very slow when t is large. To solve the problem when t is large, we look at each column i at a time,
randomly select a fixed number of rows, say, boundOfCoverage, from the set of rows covering it and delete the characters in the remaining rows at all the columns after i - 1. After that, the coverage
of each column in the newly obtained submatrix is at most boundOfCoverage. We then run the dynamic programming algorithm on the submatrix. The resulting pair of haplotypes, which is referred to as
the initial solution, can be viewed as an approximation to the optimal solution.
The detailed procedure for obtaining a submatrix from M via the randomized sampling approach is as follows:
1. Compute the coverage c[i ]for each column i in M.
2. For i = 1 to n, perform the following steps.
3. If c[i ]≤ boundOfCoverage, do nothing and goto the next column. Otherwise, goto step 4.
4. Randomly choose boundOfCoverage rows from the set of rows covering column i. Let $S ¯$ be the set of rows covering column i but are not chosen during this process.
5. For each row $r∈S ¯$, cut r from column i such that it no longer covers any column larger than i (including i). Accordingly, we need to reduce c[j ]by 1 for each i ≤ j ≤ k, where k is the end
position of r before being cut.
By employing this randomized sampling strategy, we can always make sure that the maximum coverage is bounded by the threshold boundOfCoverage in the selected submatrix. How to choose a proper value
for boundOfCoverage? Actually, there is a tradeoff between the running time and the quality of the initial solution output by the dynamic programming algorithm. On one hand, reducing boundOfCoverage
can reduce the running time of the algorithm. However, on the other hand, increasing boundOfCoverage can maintain more information from M. As a result, the initial solution output by the dynamic
programming algorithm has a higher chance to be close to the optimal solution. In practice, boundOfCoverage is generally no larger than 15, which is feasible in terms of running time and is large
enough to sample sufficient information from M. See Section Experiments for a detailed discussion on how the size of boundOfCoverage affects the initial solution.
Refining the initial solution with all fragments
In the newly obtained submatrix, it is possible that (1) some columns are not covered by any rows, thus leaving the haplotype values at these SNP sites undetermined in the initial solution, (2) the
haplotype values at some SNP sites in the initial solution are wrongly determined due to the lack of sufficient information sampled from M during the randomized sampling process. In this subsection,
we try to refine the initial solution with all input fragments, aiming to fill haplotype values that are left undetermined and correct those that are wrongly determined.
The refining procedure contains several iterations. In each iteration, we take two haplotypes as its input and output a new pair of haplotypes. Initially, the two haplotypes in the initial solution
are used as the input to the first iteration. The haplotypes output in an iteration are then used as the input to the subsequent iteration. In each iteration, we try to reassign the rows of M into
two groups based on the two input haplotypes. More specifically, for each row r of M, we first compute the generalized hamming distance between r and the two haplotypes. Then, we assign r to the
group associated with the haplotype that has the smaller (generalized hamming) distance with r. After reassigning all rows of M into two groups, we can compute a haplotype from each of the two groups
by majority rule. At the same time, we can also obtain the corresponding MEC cost.
The refining procedure stops when, at the end of some iteration, the obtained haplotypes no longer change, or when a certain number of iterations have been finished. The two haplotypes output in the
last iteration are the output of the refining procedure.
Voting procedure
To further reduce the effect of randomness caused by the randomized sampling process, we try to obtain several different submatrices from M by repeating the randomized sampling process several times.
Accordingly, we can obtain several initial solutions, one derived from each submatrix. Furthermore, we can refine these initial solutions with all fragments. Given a set of solutions, each of which
containing a pair of haplotypes, the goal here is to compute a single pair of haplotypes by adopting a voting procedure.
In the voting procedure, the two haplotypes are computed separately. We next see how to compute one of the two haplotypes. The other case is similar. Let S be the set of solutions used for voting.
First, we find a set of haplotypes (denoted by S[1]), one from each solution in S, such that the haplotypes in S[1 ]all correspond to the same copy of a chromosome. With S[1], we can then compute a
haplotype by majority rule. Simply speaking, at each SNP site, we count the number of 0s and 1s at the given SNP site over the haplotypes in S[1]. If we have more 0s, the resulting haplotype takes 0
at the SNP site, otherwise, it takes 1.
How to find S[1]? First, we need to clarify that the two haplotypes in each solution in S are unordered. That is, given a solution H = (h[1], h[2]), we do not know which chromosome copy h[1 ](or h
[2]) corresponds to. So, we should first find the correspondence between the haplotypes in different solutions. Let $H1=(h11,h21),...,Hy=(h1y,h2y)$ be the set of solutions in S. Without loss of
generality, assume that the MEC cost associated with H[1 ]is the smallest among all the y solutions. We use H[1 ]as our reference and try to find the correspondence between haplotypes in H[1 ]and
other solutions. For each i (1 < i ≤ y), we first compute two generalized hamming distances $D(h11,h1i)$ and $D(h11,h2i)$. If $D(h11,h1i)<D(h11,h2i)$, we claim that $h1i$ corresponds to the same
chromosome copy as $h11$. Otherwise, $h2i$ corresponds to the same chromosome copy as $h11$. As a result, the set of haplotypes in S that correspond to the same chromosome copy as $h11$ is the S[1 ]
we want to find.
Assume that at the beginning of this procedure, we obtain x solutions by repeating the randomized sampling process along with the refining procedure x times. It is worth mentioning that in the voting
procedure, we only use part of the solutions, say, the first y (y ≤ x) solutions with the highest quality. Given two solutions A and B, we say that A has higher quality than B if the MEC cost
associated with A is smaller than that of B. In this case, we assume that A is much closer to the optimal solution and contains less noises than B. To reduce the sideeffect of noises and improve the
quality of the solution output by the voting procedure, it is helpful to use only solutions with high quality in the voting procedure.
Summarization of the algorithm
Generally speaking, given an input fragment matrix M, our heuristic algorithm can be summarized as the following four steps.
Step 1: We first perform a preprocessing on M to detect possible errors in it. After removing errors from M, we further convert it into $M′$ in which each entry is encoded by a character from the
alphabet $∑′={0,1,-}$. See Section Preliminaries for more details. $M′$ is used as the input to the following steps.
Step 2: We compute an initial solution by running the dynamic programming algorithm on a subset of $M′$. The submatrix is computed by using the randomized sampling approach.
Step 3: Refine the initial solution with all the fragments in $M′$, instead of the submatrix that is used to generate the initial solution in Step 2.
Step 4: To further reduce the effect of randomness caused by the randomized sampling process, we repeat Step 2 and Step 3 several times. Each repeat ends with a solution, from which we then compute a
single pair of haplotypes by adopting the voting procedure. The resulting pair of haplotypes is the output of our algorithm.
We have tested our algorithm on a set of benchmark datasets and compare its performance with several other algorithms. The main purpose here is to evaluate how accurately our algorithm can
reconstruct haplotypes from input fragments. All the tests have been done on a Windows-XP (32 bits) desktop PC with 3.16 GHz CPU and 4GB RAM.
The benchmark we use was created by Geraci in [23]. It was generated by using real human haplotype data from the HapMap project [4]. There are three parameters associated with the benchmark, i.e
haplotype length, error rate and coverage rate, denoted by l, e, c, respectively. Each parameter has several different values, l = 100, 350, and 700, e = 0.0, 0.1, 0.2 and 0.3, c = 3, 5, 8 and 10.
Note that unlike the "coverage" defined in Section Preliminaries, the coverage rate c defined in this benchmark refers to the number of times each of the two haplotypes replicates when generating the
dataset. In other words, given an instance in the benchmark, i.e. a fragment matrix, there are up to 2c rows which take non '-' value at each column in the matrix. For each combination of the three
parameters, there are 100 instances in the benchmark. As for the details on how to generate the benchmark, the reader is referred to [23].
Throughout our experiments, we measure the performance of our algorithm by the reconstruction rate, a frequently used criterion in the haplotype assembly problem. Given a problem instance in the
benchmark, the reconstruction rate is defined as follows:
where H = (h[1], h[2]) is the pair of correct haplotypes that is used to generate the problem instance, and is thus known a prior, $Ĥ=(ĥ1,ĥ2)$ is the pair of haplotypes output by the algorithm, n is
the length of the haplotypes, and $D′$ is the hamming distance between two haplotypes. More specifically, $D′$ is defined as follows:
$D′(hi,ĥj)= ∑k=1nd′(hi[k],ĥj[k])$
Intuitively speaking, the reconstruction rate measures the ability of an algorithm to reconstruct the correct haplotypes.
Recall that in Step 2 of our algorithm, we try to compute an initial solution by using only a subset of the input matrix. The initial solution forms the basis for the following steps of our algorithm
and is closely related to the parameter boundOfCoverage. Briefly speaking, boundOfCoverage is the maximum coverage of a column in the submatrix selected during this step. For a formal description of
boundOfCoverage, we refer you to Section Methods. In this experiment, we first evaluate how the size of boundOfCoverage affects the initial solution. As aforementioned earlier, boundOfCoverage is
generally no larger than 15. Here we consider three different sizes of boundOfCoverage, i.e. 10, 12 and 15. Given a problem instance, we can obtain three initial solutions by using the three
different sizes of boundOfCoverage, respectively. As an example, we choose the set of benchmark datasets with l = 350 and e = 0.2. For each combination of the coverage rate c and boundOfCoverage,
there are 100 instances and we compute the average of the reconstruction rates over the 100 instances. The results are listed in Table Table11.
Evaluation of how the size of boundOfCoverage affects the initial solution.
From Table Table1,1, we can see that for a fixed coverage rate c, when increasing the size of boundOfCoverage, the reconstruction rate of the obtained initial solution gets higher, and the running
time increases accordingly. Since the reconstruction rate in the case where boundOfCoverage = 12 is relatively high, and the running time is feasible, we will set boundOfCoverage to be 12 in the
following experiments.
Next, to evaluate the performance of our algorithm, we have tested it on the set of benchmark datasets. The parameters we use are as follows: boundOfCoverage = 12, x = 100, and y = 11, where x is the
number of initial solutions obtained in Step 4, i.e. the number of times we repeat Step 2 and Step 3, and y is the number of solutions used for voting in Step 4. The results for l = 100, 350 and 700
are given in the last column in Table Table22 Table Table33 and Table Table4,4, respectively. In [23], Geraci compared the reconstruction rates of seven state-of-the-art algorithms on the same
benchmark datasets. These seven algorithms are SpeedHap [24], Fast hare [25], 2d-mec [26], HapCUT [21], MLF [27], SHR-three [28] and DGS, the greedy heuristic proposed in [20]. For a full review of
these seven algorithms, the reader is referred to [23]. For the sake of comparison, we list the reconstruction rates of the seven algorithms, see Columns 3 - 9 in Table Table2,2, Table Table33 and
Table Table4.4. Note that the results for the seven algorithms are directly taken from [23]. Each reconstruction rate shown in the three tables is the average over 100 instances under the same
parameter setting.
Comparisons of the algorithms when l = 100.
Comparisons of the algorithms when l = 350.
Comparisons of the algorithms when l = 700.
Take a close look at the three tables, we can see that (1) each of the seven algorithms studied in [23] only works well in some cases, e.g., SpeedHap works well when the error rate is low (≤ 0.1),
while MLF works well when the error rate is high (≥ 0.2); (2) the reconstruction rates of all the seven algorithms are relatively low when the error rate of input fragments is high. For example, in
the case where l = 700, e = 0.3 and c = 10, the best reconstruction rate of the seven other algorithm is 0.645. Compared with its competitors, our algorithm can give solutions with high
reconstruction rate. It outperforms its competitors in almost all cases, especially in cases in which the error rate of fragments is high (≥ 0.2). We also notice that when the error rate is 0, our
algorithm may introduce some errors in the output solution, e.g., in the case where l = 700, c = 3, e = 0.0. However, even in this case, the reconstruction rate can still reach up to 0.997.
In the first step of our algorithm, we perform a preprocessing on the input fragment matrix. This allows us to detect errors in the input. For example, for the benchmark datasets with l = 350 and e =
0.3, Step 1 of our algorithm can identify about 47%, 55%, 59% and 61% of the total errors for the cases c = 3, 5, 8 and 10, respectively. Thus, Step 1 has significant importance to the following
steps of our algorithm.
Next, we further investigate how the voting procedure in Step 4 affects the performance of our algorithm. In Step 4, we first obtain x solutions, from which we then choose the first y (y ≤ x)
solutions with the smallest MEC cost. The y solutions are then used in the voting procedure to compute the final solution of our algorithm. To demonstrate the effect of the voting procedure, we
compare the final version of our algorithm with the one without the voting procedure. For the version without the voting procedure, we simply outputs the solution with the smallest MEC cost among all
the x solutions in Step 4. As an example, we have tested both versions on the set of benchmark datasets with l = 350. The parameters are as follows: boundOfCoverage = 12, x = 100 and y = 11. Figure 2
(a) (respectively, 2(b)) shows the results for the two versions in the case where e = 0.2 (respectively, e = 0.3). The results for e = 0.0 and 0.1 are similar as that of e = 0.2, and we omit it here.
Illustration of the effect of the voting procedure. The reconstruction rates for the final version of our algorithm and the one without the voting procedure are depicted by black and gray bar,
respectively. The error rate for the benchmark used in (a)(respectively, ...
From Figure 2(a), we can see that the two versions of our algorithm have almost the same reconstruction rate. However, when e = 0.3, the final version of our algorithm has higher reconstruction rate
than the one without the voting procedure, see Figure 2(b). Thus, the voting procedure is essential for our algorithm.
To see how the size of the parameter x affects the reconstruction rate of our algorithm, we have tested our algorithm with three different sizes of x, i.e. 25, 50, 100. The values of boundOfCoverage
and y are fixed to be 12 and 11, respectively. The tests are done on the set of benchmark datasets with l = 350. The results for e = 0.2 and 0.3 are shown in Figure 3(a) and 3(b), respectively. For e
= 0.0 and 0.1, the reconstruction rates are almost the same in all the three cases, and we do not list it here. As can be seen from Figure 3(a), the reconstruction rate increases with the increasing
of x in the cases where c = 3 and 5. For c = 8 and 10, the cases where x = 50 and 100 have almost the same reconstruction rate which is higher than that in the case where x = 25. As for Figure 3(b),
it is much more obvious that the reconstruction rate increases as x gets larger.
Evaluation of how the size of x affects the performance of our algorithm. The reconstruction rates for x = 25, 50 and 100 are depicted by white, gray and black bar, respectively. The error rate for
the benchmark used in (a)(respectively, (b)) is 0.2 (respectively, ...
In this paper, we propose a heuristic algorithm for the haplotype assembly problem. Experiments show that our algorithm is highly accurate. It outperforms most of the existing programs when the error
rate of input fragments is high.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
FD participated in the design of the study, performed the experiments and drafted the manuscript. WC participated in the design of the study and helped to draft the manuscript. LW conceived the
study, participated in its design and helped to draft the manuscript. All authors read and approved the final manuscript.
The publication costs for this article were funded by the corresponding author's institution.
The authors would like to thank Filippo Geraci for kindly providing us with the set of benchmark datasets. This work is fully supported by a grant from City University of Hong Kong (Project No.
This article has been published as part of BMC Genomics Volume 14 Supplement 2, 2013: Selected articles from ISCB-Asia 2012. The full contents of the supplement are available online at http://
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The birthday paradox - Pie Cubed
Take a group of 23 people. What is the probability that two of them share the same birthday?
Or let’s take a different approach. How many people do you need to make sure that the probability that two of them share the same birthday is close to 100%?
On first glance, 367 seems like a probable answer (as there as 366 days during a year including February 29th). But while you would need as many to get to exactly 100%, you need much less to reach
99% probability.
The fact is, we humans tend to be rather egocentric. The problem is not: what is the probability of someone sharing a birthday with you, but that any two people share the same birthday. The two
people are not chosen in advance.
In the case of the probability of someone sharing the same birthday as you, the result is much more intuitive and less surprising.
Let’s go back to our original problem. You have 23 people. You are looking if anyone’s birthday matches with anyone else’s. What matters is the possibilities each individual has to share a birthday
with anyone else in the group.
For simplicity’s sake, let’s assume that there are 365 possible birthdays and that they are all equally likely. Therefore, the probability of two people having the same birthday is 1/365 or 0.274%.
But there are 23 people, not just two. The first person can test this with 22 other people and has therefore 22 chances of matching a birthday (each chance has a 0.274%). But, the second person has
another 21 people to try to match his birthday with (one less because the possibility with the first person has already been tested). And the third person has 20 other people with whom to check. The
fourth has 19, etc.
Hence, the actual chances are 22+21+20+19+18+17….+1=253.
As said before, the probability of two people sharing a birthday is only 0.274%. However, in a group of 23 people there are 253 chances to test this probability. For each of these chances the
probability is 0.274%, but if we take them all together the probability that any two people share the same birthday is actually 50% (in a group of 23). (For a more detailed mathematical analysis see
this wiki page).
This result is quite surprising to most people. We tend to think only of ourselves and we automatically assume that there are only 22 chances to share a matching birthday which is eleven times less
than the real chances, when considering everyone.
For a real life example, there are 22 players in a football match, which means that almost every other match there are two players that share a birthday.
Also,to have 99% probability of sharing a birthday you don’t need 367 people, 57 is enough.
More Maths
How to take the perfect penalty kick
Distribution of first digits: Benford’s Law
Further reading | {"url":"http://piecubed.co.uk/birthday-paradox/","timestamp":"2014-04-19T11:57:36Z","content_type":null,"content_length":"46354","record_id":"<urn:uuid:ff117d93-b206-41d5-b046-68d4408aed69>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00553-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wave function at high symmetry point
A clean discussion involves the assumption and discussion of time reversal symmetry. If there are no spin orbit coupling effects, time reversal will be represented by complex conjugation and the
single particle wavefunctions in a periodic potential can always be chosen real as then E(k)=E(-k) so that instead of the solutions [itex]\psi_k(x)=u_k(x)\exp(ikx)[/itex] and [itex]\psi_{-k}=(\psi_k
(x))^*[/itex] real valued combinations can be chosen. For k=0, only one real function will be obtained.
If spin orbit coupling is taken into account, time reversal is no longer just complex conjugation so that it does not always guarantee real valuedness. This is known as Kramers degeneracy. | {"url":"http://www.physicsforums.com/showthread.php?p=3897051","timestamp":"2014-04-20T18:39:53Z","content_type":null,"content_length":"39957","record_id":"<urn:uuid:b8b426ef-6ee0-4bca-83b7-e7c7e8c241cb>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00385-ip-10-147-4-33.ec2.internal.warc.gz"} |
Cartesian oval
A curve that actually consists of two ovals, one inside the other. It is the locus of a point whose distances s and t from two fixed points S and T satisfy the equation s + mt = a. When c is the
distance between S and T then the curve can be expressed in the form:
((1 - m^2)(x^2 + y^2) + 2m^2cx + a^2 - m^2c^2)^2 = 4a^2(x^2 + y^2)
The curves were first studied by René Descartes in 1637 and are sometimes called the ovals of Descartes. They were also investigated by Isaac Newton in his classification of cubic curves.
If m = +1 then the Cartesian oval is a central conic. If m = a/c then it becomes a limacon of Pascal, in which case the inside oval touches the outside one. Cartesian ovals are anallagmatic curves.
Related category
PLANE CURVES | {"url":"http://www.daviddarling.info/encyclopedia/C/Cartesian_oval.html","timestamp":"2014-04-18T10:39:08Z","content_type":null,"content_length":"6536","record_id":"<urn:uuid:6e8362d7-ad86-4795-80d6-ad8c03046069>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00511-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum: Alejandre & Lanius: NCTM 2006
The Math Forum @ Drexel: Technology Enhances
Problem Solving and Assessment
presented by Suzanne Alejandre & Cynthia Lanius
NCTM 2006 Annual Meeting and Exposition
Asking Questions - Generating Solutions
Session 670 in St. Louis, Missouri
Friday, April 28, 2006, from 1:00 pm to 2:30 pm
America's Center - 262 (capacity: 50)
What is a tPoW?
Let's do math!
Balloon Booths - Scale the size of a hot air balloon so you can make it fit through a narrow passage and hit a nail to pop it.
How might technology enhance or detract from students' mathematical understanding?
Traffic Jam Activity - Find the fewest number of moves for the ten people to end up on the opposite side from where they started. Kinesthetic experience (people), manipulative (plastic people),
technology (Java applet)
What is the advantage of using one method over another?
Explore tPoW Selection
What are the largest number of cubes to make the house? What's the smallest number of cubes?
Find the fewest number of weighings that will identify which of the nine coins is counterfeit.
Using a vending machine (Java applet) determine the relative values of the coins used on the planet Orange.
Given the front view and side view of some cubes, students are asked what the larget number of cubes can be used to make the arrangement and what the smallest number of cubes would be.
Students use a spreadsheet to show when they will have collected their million pennies.
Use a spreadsheet to compare salaries of the star basketball player and the rookie.
Use the Sam the Chameleon applet to learn more about the coordinate plane.
How many stones did Kathy need to buy to pave an area in her yard?
Use an interactive applet to discover some of the relationships among the pieces in a set of Tangrams.
Use the given information to find the temperature.
How fast is the train moving to allow Herman and Sheila to each escape the tunnel?
Students use a GSP sketch or an interactive web page to drag and manipulate four triangles. They identify one of each type: scalene, isosceles, equilateral, and right.
Let's Submit
Temperature Change submission
Hints and Answer Checks
Student Sample Solutions
Questions for Discussion:
How engaging are the problems? Are they at the right level of rigor for your students? Will they challenge students? What learning goals will they help achieve?
How might communication (reading, writing, talking, and explaining) around the problem enhance or detract from students' mathematical understanding?
What kinds of support will you and your students need to solve the problems and use the technologies effectively? How can we best deliver that support?
Resources to Browse with Internet Access:
List of Technology Problems of the Week
Notes of Appreciation
Thank you to Joel Duffin of the National Library of Virtual Manipulatives (NLVM), Utah State University for his assistance and permission to use the Space Blocks, Coin Problem, Tangrams, Pattern
Blocks, and Color Chips Java applets.
Thank you to the Freudenthal Institute, Utrecht, The Netherlands, for their assistance and permission to use the Building houses with side views Java applet.
Thank you to George Reese and Pavel Safronov of MSTE, a division of the College of Education at the University of Illinois at Urbana-Champaign, for their assistance and permission to use the
Chameleon Java applet. | {"url":"http://mathforum.org/workshops/nctm2006/session670.html","timestamp":"2014-04-21T10:32:33Z","content_type":null,"content_length":"8945","record_id":"<urn:uuid:f5bc8236-0ba9-4e28-b485-198918b1aacf>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00127-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Mann Whitney U Test
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Comparing medians: the Man Whitney U-test The Mann Whitney U-test is a fairly complicated statistical test to understand, though it is quite easy to apply to a set of data.…
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1. Comparing medians: the Man Whitney U-test The Mann Whitney U-test is a fairly complicated statistical test to understand, though it is quite easy to apply to a set of data. So, while the
calculation is relatively easy, knowing when to apply it, and what the calculation actually means, is a little more difficult. It is also important not to be put off by the formula.
□ What is the Mann Whitney U-test?
□ The Mann Whitney U-test is a nonparametric test which is used to analyse the difference between the medians of two data sets. By using the critical values tables, it is possible to assess the
degree to which any observed difference is a result of chance or fluke.
□ If the answer to each of the following questions is ‘yes’ then you may use the Mann Whitney U-test.
□ Are you investigating the difference between two samples of data?
□ Is the data nonparametric?
□ Are there more than five pieces of data in each sample?
□ Are there 20 or fewer pieces of data in each sample (recommended)?
KEY TERMS Nonparametric test : statistical test that assumes that the data is not normally distributed. Ordinal data : data that can be ranked, i.e. put into order from highest to lowest
3. Though this is a nonparametric statistical test, both samples should have a similar distribution. You can plot the data for each set on a simple graph to check this. Like many of the other
statistical tests, you have to start with a null hypothesis (H o). However, unlike some of the other tests, the null hypothesis (H o) is always the same: There is no significant difference
between the two samples.
4. Applying the Mann Whitney U-test Comparing two traffic flows in a town centre A student was interested in finding out if a new retail development had an impact upon traffic (and therefore
congestion) in the local area near to the development. There were two parts to the primary data collection. The first part was conducted before the construction of the planned development (sample
x). Methodology for primary data collection She recorded the time of day and date. She counted traffic (in both directions) on 10 streets around the development selected randomly). She counted
for 10 minutes. She used a stopwatch for timing and a simple tally chart for recording the data. She completed the tally at different times of the day. TIP Is it a one-tailed or a two-tailed
test? This relates to the difference between the data sets. If you assume one specified data set will be larger than the other, you are investigating a one-tailed distribution. If you assume
differences can operate in both directions, i.e. up or down, you are investigating a two-tailed distribution. This is important when you interpret you findings using the critical values. In this
example, the student is assuming traffic can go up or down in study 2 for all sites, despite the fact that more customers are likely to be attracted to the development. In this case, this makes
it a two-tailed test
5. For the second study (sample y), she waited until 2 months after the development had been completed. She went to another 10 sites (selected randomly) and repeated the test. She then devised the
following null hypothesis (H ?): ‘ There is no significant difference in traffic flows before and after the development.’ Now let’s take a look at the formula: U ? = N?.N ? + N ?(N? + 1) 2 - ? r?
U ? is the Mann Whitney calculation for sample x n is the number in the sample ? r? is the sum of ranks for sample x (‘sum of’ just means added together)
6. The best way to proceed is to incorporate the findings into a table that also allows you to calculate the result. When you get two or more equal values, use the mean rank. Here are the student’s
findings: Complete the table by ranking all the data from highest to lowest. Total traffic flow in 10 minutes ( ? ) Rank r ? Site Number Total traffic flow in 10 minutes ( ? ) Rank r ? 126 11 1
194 148 7 2 128 85 15.5 3 69 61 19 4 135 179 4 5 171 93 12.5 6 149 45 20 7 89 189 3 8 248 1 85 15.5 9 79 93 12.5 10 137
7. ? r? = 120 ?r ? = Ranking puts values in order from highest to lowest . Next, she substituted the data into the formula : U ? = 10x10 + U? = 100 + U? = 100 + 55 – 120 U? = 35
□ Mann Whitney U-test can be used to compare any two data sets that are not normally distributed . As long as the data is capable of being ranked, then the test can be applied.
□ Investigating differences in questionnaire responses relating to a new development.
□ Investigating differences in species diversity near to footpaths.
□ Investigating differences in vegetation cover between two different slopes
U ? = N?.N ? + N ?(N? + 1) 2 - ? r? 10(10 + 1) 2 - 120 110 2 - 120 TIP There is a useful way of checking the accuracy of your calculations. U ? + U ? should be the same number as N?.N ? (which in
this case is 10x10 = 100). If it is not, you have made a mistake somewhere
8. TASKS
□ You now have the figure for U ?. Use the same method of calculation to work out the figure for U ? . Here is the formula
□ You now need to select the smaller of the two figures and use the critical values table to decide on the statistical significance of your result. For this test, if your result equal to or
smaller than the critical value as the 0.05 level of significance, then you can reject the null hypothesis (H?)
□ For 10 figures in each sample, the critical value at the 0.05 level of significance is 23.
□ Do you accept or reject the null hypothesis (H?)? Give reasons for your answer.
□ What do these findings suggest about the difference in traffic flow before and after the retail development in this study?
U ? = N?.N ? + N ?(N ? + 1) 2 - ? r ?
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Math Forum Discussions - Re: Matheology § 191
Date: Jan 13, 2013 9:16 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 191
On 13 Jan., 13:15, Zuhair <zaljo...@gmail.com> wrote:
> What mean nothing more than saying that we have Countably many FINITE
> paths
Yes, and it is not intuitive nor needs it any formalization to
recognize that everything that happens in Cantor-lists happens withing
finite paths (or sequences of digits). It is absoluteley impossible
that something happens elsewhere! And if a list contains all possible
finite paths (which is possible as they are countable) then Cantor's
"proof" proves the uncountability of a countable set.
Note again: everything that happens in a Cantor-list happens withing
finite paths or finite initial segments of the anti-diagonal.
And please do me a favour and stop parroting of uncountable sets
unless you can explain how something can happen *after* all finite
initial segments.
Regards, WM | {"url":"http://mathforum.org/kb/plaintext.jspa?messageID=8065250","timestamp":"2014-04-16T22:00:14Z","content_type":null,"content_length":"1880","record_id":"<urn:uuid:18f70a1a-424f-4395-b63c-db99aecee1c9>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00524-ip-10-147-4-33.ec2.internal.warc.gz"} |
Algebra Unit 4 Graphing Lines
From OpenContent Curriculum
Algebra Unit 4 Graphing Lines
Unit Goal Description Here!
Focus Questions:
Focus Questions Listed Here!
Learning Objectives / Standards:
MA.9.13 Graph lines using intercepts, slope-intercept form, and point-slope form. Verify if lines are parallel or perpendicular using slope and write equations of lines parallel or perpendicular to a
given line. (A.REI.10,12; F.IF.4,7a; G.GPE.5)
Key Vocabulary: Resources/Materials:
Key Vocabulary Terms Listed Here! Resources for Unit Listed and Linked Here!
Assessment (formative):
Assessment Alternatives Listed and Described Here!
Closure or Culminating Activities Described Here!
Teacher notes /refinements:
Notes for Teachers About Unit Can Go Here! | {"url":"http://wiki.bssd.org/index.php/Algebra_Unit_4_Graphing_Lines","timestamp":"2014-04-17T21:31:41Z","content_type":null,"content_length":"22027","record_id":"<urn:uuid:52cee370-21f0-458f-9544-2d0e52c44912>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00518-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pi Day Challenge And Happy Pi Day
There is an awesome challenge online. Since yesterday was pi day (March 14) they have something really cool.
3.14159265358979... Ok.
It's a series of levels where you have to answer lots of math questions.
Here are some of the questions below:
1. Name the first 5 decimal digits of pi.
3. (This is a trick question) What does the license plate say?
6A. The ___________ Theorem states that for any right triangle, the sum of the lengths equals the square of the length of the hypotenuse.
What is x?
9. What are the roots of the function
11. What is Archimedes' ratio approximation of pi?
I'm completely stuck on #23:
You run downwards at a speed of 3 steps per second.
It takes you 180 steps to get to the bottom.
You walk upwards at a speed of 1 step per second.
It takes you 20 steps to get to the top.
How many steps do you need if the escalator stands still?
Last edited by n872yt3r (2013-03-17 22:02:19) | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=290346","timestamp":"2014-04-20T21:18:56Z","content_type":null,"content_length":"43474","record_id":"<urn:uuid:c442738d-03b5-4cfc-a253-d85daf65d158>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00321-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Excel CHITEST Function
The Excel CHITEST Function
Related Function:
CHIDIST Function
The chi-square test uses the chi-square distribution of one or more sets of data, to test whether there is a significant difference between observed frequencies and expected frequencies.
The chi-square distribution is given by the formula :
A[ij] = actual frequency in the i'th row & j'th column
E[ij] = expected frequency in the i'th row & j'th column
r = number of rows
c = number of columns
The chi-square test can then be used to determine whether the value of this function is likely to have occurred by chance alone, in independent sets of data.
CHITEST and CHISQ.TEST
In Excel 2010, the Chitest function has been replaced by the Chisq.Test function, which has improved accuracy.
Although it has been replaced, the Chitest function is still available in Excel 2010 (stored in the list of compatibility functions), to allow compatibility with earlier versions of Excel.
Basic Description
The Excel CHITEST function uses the chi-square test to calculate the probability that the differences between two supplied data sets (of observed and expected frequencies), are likely to be simply
due to sampling error, or if they are likely to be real.
The syntax of the function is :
CHITEST( actual_range, expected_range )
Where the function arguments are:
actual_range - An array of observed frequencies
expected_range - An array of expected frequencies (must have the same dimension as the actual_range array)
You should bear in mind that the chi-square test is not reliable when the expected values are too small. As a guideline, if any of the expected values are less than 5, or if the total of the expected
values is less than 50, you should not rely on the result of the chi-square test.
Chitest Function Example
Cells B3-C5 and F3-G5 of the spreadsheet below show the observed and expected frequencies of responses from men and women to a simple question.
The chi-square test for independence, for the above data sets, is calculated using the Excel Chitest function as follows:
=CHITEST( B3:C5, F3:G5 )
This gives the result 0.000699103.
Generally, a probability of 0.05 or less is considered to be significant. Therefore, the returned value of 0.000699103 in this example, indicates that there is a significant difference between the
observed frequencies and the expected frequencies, which is unlikely to be simply due to sampling error.
Further examples of the Excel Chitest function can be found on the Microsoft Office website.
Chitest Function Errors
If you get an error from the Excel Chitest function this is likely to be one of the following :
Common Errors
Occurs if either:
#N/A - - the two supplied data arrays have different dimensions
- the supplied data arrays have width and height 1 (i.e. contain just one value)
#DIV/0! - Occurs if any of the supplied expected_values are zero
#NUM! - Occurs if any of the supplied expected_values are negative | {"url":"http://www.excelfunctions.net/Excel-Chitest-Function.html","timestamp":"2014-04-19T11:57:12Z","content_type":null,"content_length":"17989","record_id":"<urn:uuid:9585e4b2-0a65-45e9-9470-b687e3066278>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00624-ip-10-147-4-33.ec2.internal.warc.gz"} |
MATLAB - Graphics
This chapter will continue exploring the plotting and graphics capabilities of MATLAB. We will discuss:
• Drawing bar charts
• Drawing contours
• Three dimensional plots
Drawing Bar Charts
The bar command draws a two dimensional bar chart. Let us take up an example to demonstrate the idea.
Let us have an imaginary classroom with 10 students. We know the percent of marks obtained by these students are 75, 58, 90, 87, 50, 85, 92, 75, 60 and 95. We will draw the bar chart for this data.
Create a script file and type the following code:
x = [1:10];
y = [75, 58, 90, 87, 50, 85, 92, 75, 60, 95];
bar(x,y), xlabel('Student'),ylabel('Score'),
title('First Sem:')
print -deps graph.eps
When you run the file, MATLAB displays the following bar chart:
Drawing Contours
A contour line of a function of two variables is a curve along which the function has a constant value. Contour lines are used for creating contour maps by joining points of equal elevation above a
given level, such as mean sea level.
MATLAB provides a contour function for drawing contour maps.
Let us generate a contour map that shows the contour lines for a given function g = f(x, y). This function has two variables. So, we will have to generate two independent variables, i.e., two data
sets x and y. This is done by calling the meshgrid command.
The meshgrid command is used for generating a matrix of elements that give the range over x and y along with the specification of increment in each case.
Let us plot our function g = f(x, y), where −5 ≤ x ≤ 5, −3 ≤ y ≤ 3. Let us take an increment of 0.1 for both the values. The variables are set as:
[x,y] = meshgrid(–5:0.1:5, –3:0.1:3);
Lastly, we need to assign the function. Let our function be: x^2 + y^2
Create a script file and type the following code:
[x,y] = meshgrid(-5:0.1:5,-3:0.1:3); %independent variables
g = x.^2 + y.^2; % our function
contour(x,y,g) % call the contour function
print -deps graph.eps
When you run the file, MATLAB displays the following contour map:
Let us modify the code a little to spruce up the map:
[x,y] = meshgrid(-5:0.1:5,-3:0.1:3); %independent variables
g = x.^2 + y.^2; % our function
[C, h] = contour(x,y,g); % call the contour function
print -deps graph.eps
When you run the file, MATLAB displays the following contour map:
Three Dimensional Plots
Three-dimensional plots basically display a surface defined by a function in two variables, g = f (x,y).
As before, to define g, we first create a set of (x,y) points over the domain of the function using the meshgrid command. Next, we assign the function itself. Finally, we use the surf command to
create a surface plot.
The following example demonstrates the concept:
Let us create a 3D surface map for the function g = xe^-(x^2 + y^2)
Create a script file and type the following code:
[x,y] = meshgrid(-2:.2:2);
g = x .* exp(-x.^2 - y.^2);
surf(x, y, g)
print -deps graph.eps
When you run the file, MATLAB displays the following 3-D map:
You can also use the mesh command to generate a three-dimensional surface. However, the surf command displays both the connecting lines and the faces of the surface in color, whereas, the mesh
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Simpson's Paradox, Lord's Paradox, and Suppression Effects are the same phenomenon – the reversal paradox
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Emerg Themes Epidemiol. 2008; 5: 2.
Simpson's Paradox, Lord's Paradox, and Suppression Effects are the same phenomenon – the reversal paradox
This article discusses three statistical paradoxes that pervade epidemiological research: Simpson's paradox, Lord's paradox, and suppression. These paradoxes have important implications for the
interpretation of evidence from observational studies. This article uses hypothetical scenarios to illustrate how the three paradoxes are different manifestations of one phenomenon – the reversal
paradox – depending on whether the outcome and explanatory variables are categorical, continuous or a combination of both; this renders the issues and remedies for any one to be similar for all
three. Although the three statistical paradoxes occur in different types of variables, they share the same characteristic: the association between two variables can be reversed, diminished, or
enhanced when another variable is statistically controlled for. Understanding the concepts and theory behind these paradoxes provides insights into some controversial or contradictory research
findings. These paradoxes show that prior knowledge and underlying causal theory play an important role in the statistical modelling of epidemiological data, where incorrect use of statistical models
might produce consistent, replicable, yet erroneous results.
This article discusses three statistical paradoxes that pervade epidemiological research: Simpson's paradox, Lord's paradox, and suppression. These paradoxes are not just tantalising puzzles of
purely academic interest; potentially, they have serious implications for the interpretation of evidence from observational studies. Scenarios which are associated with and can be explained by these
paradoxes are discussed. A concise explanation of these paradoxes and an historical overview is also provided. Simulated data based upon the foetal origins of adult diseases hypothesis [1,2] are used
to illustrate how the three paradoxes are different manifestations of one phenomenon – the reversal paradox – depending on whether the outcome and explanatory variables are categorical, continuous or
a combination of both; this renders the issues and remedies for any one to be similar for all three. All statistical analyses were performed within SPSS 15.0 (SPSS Inc, Chicago, USA).
Foetal origins hypothesis
The 'foetal origins of adult disease' hypothesis (FOAD), which has evolved into the 'developmental origins of health and disease' (DOHaD) hypothesis [1,2], was proposed to explain the associations
observed between low birth weight and a range of diseases in later life. These associations have been interpreted as evidence that growth retardation in utero has adverse long-term effects on the
development of vital organ systems which predispose the individuals to a range of metabolic and related disorders in later life. Nevertheless, although an inverse association between birth weight and
disease in later life was found in some studies, this relationship was only established in many studies after the current body size variables such as body mass index (BMI), body weight and/or body
height were adjusted for in the regression analysis. As body sizes may be in the causal pathway from birth weight to health outcomes in later life, the justification of this adjustment of current
body sizes has been questioned recently [3-8].
Using the inverse relationship between birth weight and systolic blood pressure in later life as an example, Figure Figure11 shows the directed acyclic graphs [9-11] for the possible relationships
between the three observed variables: birth weight, current body weight and systolic blood pressure. In Figure Figure1a,1a, current body weight is on the causal pathway from birth weight to systolic
blood pressure, so current body weight is not a genuine confounder and should not be adjusted for. In Figure Figure1b,1b, there is no relationship between birth weight and current body weight, and
therefore the latter is not a confounder for the relationship between birth weight and blood pressure either. However, this model cannot explain the observed positive correlations between birth
weight and current body weight in many epidemiological studies. In Figure Figure1c,1c, current body weight is a confounder because it is ancestor to both birth weight and blood pressure in the
directed acyclic graph [9-11]. Obviously, this scenario is implausible in reality because current body weight cannot affect birth weight. In Figure Figure1d,1d, the observed positive correlation
between birth weight and current body weight is due to an unobserved confounder, UC, which affects both birth weight and current body weight. Also, there is no path from birth weight and current body
weight [7], i.e. if UC could be identified and measured, birth weight and current body weight would be independent, conditional on UC [12]. More complex causal diagrams for the three variables are
possible by incorporating more unobserved variables in the model. However, the four scenarios in Figure Figure11 are sufficient for our discussion in this study, so we do not pursue them further.
Causal models expressed as directed acyclic graphs for possible relationships between the three observed variables: birth weight (BW), current body weight (CW) and systolic blood pressure (BP). UC is
an unobserved variable that affects both BW and CW ...
Figure 1a,c and and1d1d all explain the observed correlation structure amongst birth weight, current body weight and blood pressure equally well, and it is not possible to judge which one is true
based upon the observed data. For example, researchers may argue current body weight is a genuine confounder in Figure Figure1d1d and therefore should be adjusted for [7]. This can only be confirmed
when the unobserved confounder (be parental, genetic, or environmental factors) is identified and the conditional independence between birth weight and current weight is satisfied.
Nevertheless, the adjustment of current body weight in the statistical analysis will change the estimated relationship between birth weight and blood pressure, as the adjusted relationship is a
conditional relationship. Differences between the unadjusted and adjusted (i.e. unconditional and conditional) relationships frequently cause confusion in the interpretations of statistical analyses
and they also give rise to three statistical paradoxes, which we shall explain in the next section.
Simpson's Paradox
Simpson's paradox [13], or Yule's paradox [14], is a well known statistical phenomenon. It is observed when the relationship between two categorical variables is reversed after a third variable is
introduced to the analysis of their association, or alternatively where the relationship between two variables differs within subgroups compared to that observed for the aggregated data. Although
first discussed by Karl Pearson in 1899 [15], it is George Udny Yule, once Pearson's assistant, who provides a detailed assessment of this problem in 1903 [14].
A numerical example
Table Table11 provides a summary of a hypothetical survey of 1000 adult males in England based on data simulated using values derived from the literature [16] and surveys conducted by the UK
Department of Health [17]. Data are simulated such that the three variables systolic blood pressure (BP), birth weight (BW), and current body weight (CW) are positively correlated: the correlation
between BP and birth weight (r[BW-BP]) is weak (0.11); whereas the correlations between birth weight and current weight (r[BW-CW]) and between current weight and BP (r[CW-BP]) are reasonably strong
(0.52 and 0.50, respectively).
Summary of the analysis of simulated systolic blood pressure, birth weight and current body weight data for 1000 adult males
Suppose the research question is to investigate whether or not there is an association between low birth weight and high blood pressure in later life. In this hypothetical study, low birth weight is
defined as birth weight lower than the population mean (i.e. < 3.5 Kg), and high blood pressure is defined as systolic BP greater than the mean value (i.e. > 135 mmHg). The results are summarized in
Table Table2.2. It is noted that the probability of developing high blood pressure is 0.272 for subjects with low birth weight and 0.362 for subjects with high birth weight. This indicates that low
birth weight has a protective effect of developing high blood pressure. However, when these subjects are stratified according to their current weight (> 90 Kg vs. < = 90 Kg), the risk of developing
high blood pressure is consistently higher amongst subjects with low birth weight compared to those with high birth weight. It seems to be quite counter-intuitive that low birth weight has an adverse
effect on blood pressure for both subgroups of current weight, yet a protective effect on the groups as a whole.
Numbers and Percentages of subjects with high blood pressure (> 135 mmHg) according to their birth weight and current body weight
In this scenario, there are substantial differences in the numbers of subjects with low birth weight between the two subgroups of current weight, because lower birth weight babies on average are
smaller in adulthood. Therefore, the overall relation between low birth weight and high blood pressure is a sum of weighted relations between the two variables in each subgroup. A graphical
representation of this paradox, first proposed by Paik [18], is given in Figure Figure2.2. Due to a greater influence of the lower risk of developing high blood pressure in the subjects with low
birth weight and lower current weight, the adverse relation is reversed in the whole-group analysis (solid line in Figure Figure2).2). Note that, in the following two scenarios, the adjustment for
current weight will not change the relationship between birth weight and BP [12], if: (a) there is no difference in the percentages of subjects with high current weight between the two subgroups of
birth weight (i.e. no correlation between birth weight and current weight); or (b) there is no association between CWb and BP in the subgroups stratified by BWb (i.e. the association between BP and
current weight is entirely caused by the association between birth weight and BP). The problem is whether the relation between low birth weigh and high blood pressure in the whole group provides an
answer to the intended research question, or whether the relation in the two subgroups does this. In other words, should CWb be considered a confounder and hence adjusted for in the statistical
Graphical representation of Simpson's paradox. The two circles on the top of the panel represent subjects with lower (0) and higher birth weight (1), respectively, in the subgroup with current weight
> 90 Kg. The two circles on the bottom of the ...
In statistical language, adjustment for current body weight represents a conditional relationship; the relationship between birth weight and blood pressure is conditional on current body weight.
Although there are substantial differences in the numbers of subjects with low birth weight between the two subgroups of current weight, the adjustment for CWb indicates that if all subjects had the
same level of current body weight, subjects with low birth weight would have a greater risk of developing high blood pressure, i.e. the adjustment of CWb erases the greater influence of subjects with
low birth weight and lower current weight on the association between birth weight and blood pressure, as people born smaller in general grow into a smaller adults.
Simpson's paradox has broad implications for epidemiological research since it indicates that making causal inference from any non-randomised study (e.g. cohort studies, case-control studies) can be
difficult, because, whilst it is possible to control for the differences between cases and controls, there will always be the possibility that an unobserved and therefore unadjusted confounder might
attenuate the association (or even reverse its direction) between exposure and outcome, due to the difference in the mean values or the distribution of confounders between the case or control group.
Nevertheless, whether or not there is any unobserved (and therefore unadjusted) confounder may not always be an issue of debate, because in most epidemiologic studies, the important confounders are
generally known. The controversy in making causal inference arises in situations where the adjusted variable may not be a genuine confounder [6,7,19,20]. Within epidemiology, Simpson's paradox is
closely linked to the concepts of confounding [9] and incollapsibility [10].
Lord's Paradox
Lord's paradox was named after two short articles in the psychology literature by Frederick M Lord regarding the use of analysis of covariance (ANCOVA) within non-experimental studies [21,22]. In
contrast to Simpson's paradox, little discussion of Lord's paradox can be found in the statistical and epidemiological literature [23], though social scientists have shown a great interest in this
phenomenon [24-28]. Lord's paradox refers to the relationship between a continuous outcome and a categorical exposure being reversed when an additional continuous covariate is introduced to the
analysis. One specific example is that the additional covariate is a measure made at baseline within a longitudinal study, where the outcome is the same variable measured some time later (e.g.
following an intervention). Therefore, the aim is to measure change in the outcome by adjusting for the baseline measurements, and the categorical covariate might be the exposure/control groups –
this is the familiar design for ANCOVA. This controversy was first discussed in 1910 between Karl Pearson and Arthur C Pigou when they debated the role of parental alcoholism and its impact on the
performance of children [29].
A numerical example
Considering the previous numerical example for Simpson's paradox, we examine current body weight (CW) and blood pressure (BP) as continuous variables, retaining birth weight as a binary (BWb). The
two-sample t-test shows that, on average, the blood pressure of subjects with higher birth weight is 2.49 mmHg (95%CI: 1.12, 3.87) greater than those with lower birth weight. However, using ANCOVA
(i.e. linear regression with a (categorical) group-allocating variable and with the adjustment of a continuous confounding variable), adjusting for current weight as a covariate, the blood pressure
of subjects with higher birth weight becomes 2.94 mmHg (95%CI: 1.12, 3.87) lower than those with lower birth weight.
Differences in the results of the two analyses are due to adjustment in the second analysis for current body weight (CW). As current weight is positively associated with both BP and BWb, it is
expected that the relation between BP and BWb will change when current weight is adjusted for. In randomised controlled trials, mean values of the adjusted baseline covariate are expected to be
approximately equal across treatment and control groups since, assuming randomisation has been achieved, baseline variation should be within groups rather than between groups), i.e. there is no
correlation between the group variable and adjusted covariate (i.e. in our numerical example, no correlation between BWb and current weight). In such circumstances it is well known that using ANCOVA
achieves the same estimated treatment difference across groups as found by the t-test, though the former will generally have greater power [30,31]. Recall our previous discussion of two scenarios in
the section on Simpson's paradox, where the adjustment for CWb will not change the relationship between BWb and BP. Randomised controlled trials may thus be seen as a special case of scenario (a)
where there is no difference in the mean current weight between the two sub-groups of birth weight.
Figure Figure33 is a three-dimensional representation of the associations amongst the three variables. Although the solid black line shows that subjects with higher birth weight (coded as 1) have on
average a greater blood pressure than those with lower birth weight (coded as 0), the various horizontal red lines with a negative slope indicate that at each level of current weight, subjects with
higher birth weight have a lower mean blood pressure than those with lower birth weight.
A 3-dimensional scatter plot for the numerical example in Lord's paradox. The solid line shows that the mean blood pressure of subjects with higher birth weight (BWb = 1) is greater than those with
lower birth weight (BWb = 0). However, at each level ...
In statistical language, results from the regression analyses are conditional on both birth weight groups having equal mean current weight in later life, and if true there would be a benefit from low
birth weight in terms of blood pressure. However, since the two groups have a different mean current weight in later life, results from the regression analysis need to be interpreted with caution. In
Simpson's paradox, the discussion surrounds the differences in results between unconditional and conditional risk/probability, and in Lord's paradox, discussion is around the differences in results
between unconditional and conditional means.
Of the three paradoxes, suppression effects within multiple regression are probably the least recognised amongst clinical and epidemiological researchers, though the suppression phenomenon has been
extensively discussed by statisticians [32-34] and methodologists from the social sciences [35,36]. The classical definition of suppression is that a potential covariate that is unrelated to the
outcome variable (i.e. has a bivariate correlation of zero) increases the overall model fit within regression (as assessed by R^2, for instance) when this covariate is added to the model. This seems
counter-intuitive and needs some explanation.
Suppose y is the outcome variable, and x[1]and x[2 ]are two covariates (i.e. 'explanatory' variables). Denote the bivariate Pearson correlation between y and x[1 ]as r[y1]; the correlation between y
and x[2 ]as r[y2]; and the correlation between x[1 ]and x[2 ]as r[12]. Within multiple regression, where y = b[0 ]+ b[1 ]x[1 ]+ b[2 ]x[2], the standardized partial regression coefficients of b[1 ]and
b[2 ]for x[1 ](β [1]) and x[2 ](β [2]), respectively, are given by [37]:
Now suppose that y is adult blood pressure (BP), x[1 ]birth weight (BW), and x[2 ]adult current weight (CW). Many studies have shown the bivariate correlation (r[y1]) between BP (y) and birth weight
(x[1]) to be negative though weak [38,39], whilst others show this to be positive [40]; for illustrative purposes only, assume that r[y1 ]is zero. Many studies show the bivariate correlation (r[y2])
between BP (y) and current weight (x[2]) to be positive [41]. When BP is regressed on birth weight and current weight, the model fit assessed by R^2 becomes [37]:
R^2 = r[y1 ]* β [1 ]+ r[y2 ]* β [2].
Since r[y1 ]is equal to zero, equation (2) becomes:
Since $1−r122$ will always be smaller than 1, $R2=ry221−r122$ will always be greater than $ry22$. By including x[1 ]in the regression model, more variance of y is 'explained', i.e. the predictability
of the model increases. However, this seems counterintuitive, since the zero bivariate correlation between y and x[1 ](r[y1 ]= 0) indicates that no more variance in y can be explained by x[1]. So
where does the additional 'explained variance' in y come from when x[1 ]is entered in the regression model? The answer is that the additional explained variance in y comes from x[2].
Although x[1 ]is not correlated with y, it is positively correlated with x[2], which in turn is positively correlated with y. When x[1 ]is entered in the model, it 'suppresses' the part of x[2 ]that
is uncorrelated with y, thereby increasing overall predictability. In other words, the role of x[1 ]in the model is to suppress (reduce) the noise (the uncorrelated component of x[2]) within the
correlation between y and x[2], as though any uncertainty in x[2 ]'predicting' y is 'explained' by x[1].
It is not only R^2 that is increased; the coefficient for x[2], $β2=ry21−r122$, becomes greater than r[y2]. Furthermore, although r[y1 ]is equal to zero, β[1 ]is not zero and becomes negative: $β1=
−r12ry21−r122$. In general, the greater the positive correlation between x[1 ]and x[2], the greater the absolute value of β[1 ]and β[2]. However, having r[y1 ]equal zero (or being negative) is not
necessary to observe suppression; r[y1 ]may be positive and x[1 ]may still be a suppressor [35].
It was Paul Horst, in 1941, who first explored this curious phenomenon within educational research [42], and in the last few decades, many statisticians have been interested in this topic [33-35].
There are still very few discussions within the clinical and epidemiological literature regarding the impact of suppression (i.e. the impact on the changes in the regression coefficients and R^2) on
the interpretation of non-randomised studies whilst making statistical adjustment for covariates within regression [12,43].
A numerical example
Considering the previous numerical examples for Simpson's paradox and Lord's paradox, all three variables are now treated as continuous. Simple regression shows a positive association between BP and
birth weight: the regression coefficient for birth weight is 1.861 mmHg/Kg (95% CI: 0.770, 2.953). Simple regression also reveals a positive association between BP and current weight: the regression
coefficient for current weight is 0.382 (95% CI = 0.341, 0.423) mmHg/Kg. Following the practice of many previous studies, BP is regressed on birth weight and current weight simultaneously and the
partial regression coefficients for birth weight and current weight are -3.708 (95% CI = -4.794, -2.622) and 0.465 (95% CI = 0.418, 0.512) mmHg/Kg respectively, and both are highly statistically
significant (Table (Table3).3). Thus, after adjusting for current weight, birth weight has a significant inverse association with BP, suggesting that hypertension is associated with lower birth
Simple and multiple regression models for simulated hypothetical data on birth weight (BW), blood pressure (BP), and current body weight (CW); the dependent variable in all three models is BP.
It is noteworthy that not only the association of birth weight with BP is reversed (coefficients change from 1.861 to -3.708 mmHg/Kg), but that the impact of current weight also increases from 0.382
to 0.465 mmHg/Kg. The R^2 for multiple regression is 0.283, which is greater than the sum of the squared correlations for birth weight ((0.105)^2 = 0.011) and current weight ((0.501)^2 = 0.251), i.e.
0.262. Therefore, the explained variance of BP is greater than the sum of the explained variances for the two simple regression models.
Figure Figure44 is a three-dimensional representation of the associations amongst the three continuous variables. Although the solid black line shows that birth weight has a positive association
with blood pressure, the various horizontal red lines with a negative slope indicate that at each level of current weight, birth weight has an inverse relationship with blood pressure.
A 3-dimensional scatter plot for the numerical example of suppression. The solid black line shows that the marginal relation between BP and birth weight is positive. However, the conditional relation
between BP and birth weight (conditional on current ...
In the hypothetical foetal origins example, the strength of association between BP and birth weight differs considerably between simple regression and multiple regression. Which model genuinely
reflects their true causal relationship depends on whether or not current weight should be adjusted for; whether or not current weight is a confounder for the relationship between BP and birth
weight, which depends upon biological and clinical knowledge, not ad hoc statistical analyses and changes in the estimated effects [11]. The question is whether or not it is also biologically and
clinically feasible to isolate the independent effect of birth weight on BP by removing the impact of current weight on BP [3,5-7,44]. In other words, changes in the regression coefficient for birth
weight caused by current weight being adjusted for in multiple regression is irrelevant to whether or not current weight is viewed to be a confounder. The definition of confounding depends upon the a
priori causal model assumed by the investigator [8,11], which then dictates which statistical model is adopted.
In statistical language, results from adjustment for current weight are conditional on all babies growing to the same size in adulthood. In Simpson's paradox, the 'paradox' is due to differences in
the results between unconditional and conditional risk/probability, and in Lord's paradox, it is due to differences in the results between unconditional and conditional means. In suppression, the
paradox is due to differences in the results between the marginal (i.e. unconditional) BP-birth weight relation and the BP-birth weight relation conditional on current weight.
The reversal paradox is often used as the generic name for Simpson's paradox, Lord's Paradox, and suppression (see Table Table4).4). Whilst the original definition and naming of the reversal paradox
was derived from the notion that the direction of a relationship between two variables might be reversed after a third variable is introduced, this nevertheless may generalise to scenarios where the
relationship between two variables is enhanced, not reduced or reversed, after the third variable is introduced (as with many studies on the foetal origins hypothesis).
Comparison of Simpson's paradox, Lord's Paradox, and suppression
In non-randomised studies, the reversal paradox can often occur due to 'controlling' for what is typically termed a confounder, even though a clear definition of what is meant by 'confounder' is
rarely provided (contingent on understanding its role in the biological/clinical process being modelled). Differences in the strength or even direction of any association between outcome and exposure
might give rise to contradictory interpretations regarding potential causal relationships. Furthermore, it is very difficult, if not impossible, to compare results across studies where many varied
attempts are made to control for different confounders, especially in the absence of any consistent reasoning given for the choice of confounders. In some situations, statistical adjustment might
introduce bias rather than eliminate it [45].
It might be suggested that the adjustment of current weight in our foetal origins example can be viewed as estimations of direct and indirect effects, such as those in path analysis or structural
equation modelling. Recall Figure Figure1a,1a, the path from birth weight to BP is to estimate the direct effect of birth weight → BP, and then the path from birth weight → current weight → BP is to
estimate the indirect effect. For instance, in the model 3 of Table Table3,3, the regression coefficient for birth weight, -3.708, is the direct effect, and the indirect effect is derived from 0.465
(the regression coefficient for current weight in model 3) multiplied by 11.976 (the simple regression coefficient for birth weight when current weight is regressed on birth weight) = 5.569. The
total effect is therefore -3.708 + 5.569 = 1.861, which is the simple regression coefficient for birth weight in the model 1 of Table Table3.3. Our reservation with interpreting the results from
model 3 as the partition of the total effect into direct and indirect effect is that many variables, such as current height and current BMI, can be put in between birth weight and BP, and it can be
claimed that there is more than one indirect effect. Furthermore, any body size measured after birth, for example, body weight at year one, year two etc, can be adjusted for in the model and
presumably used to estimate the indirect effects and direct effect. Whilst the total effect of birth weight on BP is not affected by the numbers of intermediate body size variables in the model, the
estimation of 'direct' effect differs when different intermediate variables are adjusted for. Unless there is experimental evidence to support the notion that there are indeed different paths of
direct and indirect effects from birth weight to BP, we are cautious of using such terminology to label the results from multiple regression, as with model 3. In other words, to determine whether the
unconditional or conditional relationship reflects the true physiological relationship between birth weight and blood pressure, experiments in which birth weight and current weight can be manipulated
are required in order to estimate the impact of birth weight on blood pressure.
Although the three statistical paradoxes occur in different types of variables, they share the same characteristic: the association between two variables can be reversed, diminished, or enhanced when
another variable is statistically controlled for. Understanding the concepts and theory behind these paradoxes will provide insights into some of the controversial or contradictory results from
previous research. Prior knowledge and theory play an important role in the statistical modelling of non-randomised data. Incorrect use of statistical models might produce consistent, replicable, yet
erroneous results.
Competing interests
The author(s) declare that they have no competing interests.
We are very grateful for the constructive comments of two reviewers. One reviewer brought to our attention of the excellent paper by Cox and Wermuth [9]. YKT conceived the ideas of this study and
wrote the first draft. DG and MSG contributed to the discussion of these ideas and writing of the final draft.
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Proportional Fairness
(See also references collected here).
• Recent papers:
• Theoretical papers:
□ Charging and rate control for elastic traffic
European Transactions on Telecommunications, volume 8 (1997) pages 33-37.
□ Rate control in communication networks: shadow prices, proportional fairness and stability
F. P. Kelly, A.K. Maulloo and D.K.H. Tan.
Journal of the Operational Research Society 49 (1998), 237-252.
□ Mathematical models of rate control for communication networks
D.K.H. Tan.
Ph.D. thesis, University of Cambridge, 1999.
□ Resource pricing and the evolution of congestion control
R.J. Gibbens and F.P. Kelly
Automatica 35 (1999), 1969-1985.
□ Differential QoS and pricing in networks: where flow-control meets game theory
Peter Key and Derek McAuley.
IEE Proceedings Software 146 (1999), 39-43.
□ Bandwidth sharing: objectives and algorithms
L. Massoulie and J. Roberts
INFOCOM (1999).
□ Bandwidth sharing and admission control for elastic traffic
J. Roberts and L. Massoulie
□ Some observations on fairness of bandwidth sharing
Dah Ming Chiu.
□ A note on the fairness of additive increase and multiplicative decrease
P. Hurley, J. Y. Le Boudec, P. Thiran.
ITC16, Edinburgh, 1999.
□ Fair end-to-end window-based congestion control
Jeonghoon Mo and Jean Walrand.
□ Charge-sensitive TCP and rate control in the Internet
Richard J. La and Venkat Anantharam. INFOCOM (2000)
See also Richard La's dissertation Resource Allocation in the Internet
□ Global fairness of additive-increase and multiplicative-decrease with heterogeneous round-trip times
M. Vojnovic, J. Y. Le Boudec and C. Boutremans.
INFOCOM (2000)
See also J. Y. Le Boudec's Rate adaptation, congestion control and fairness: a tutorial
□ Proportionally fair pricing: dynamics, stability and pathology.
J. E. Carroll and P.A. Kirkby.
IEE Proceedings Commun. Vol 147 (2000), 23-31.
□ Impact of fairness on Internet performance
Thomas Bonald and Laurent Massoulie.
• Internet:
• Wireless:
• ATM networks:
• Important related approaches:
• Background references:
• Talks
Frank Kelly
Statistical Laboratory,
University of Cambridge | {"url":"http://www.statslab.cam.ac.uk/~frank/pf/","timestamp":"2014-04-19T12:07:33Z","content_type":null,"content_length":"10964","record_id":"<urn:uuid:a362d4d5-d8ac-4ed6-985f-f6df947e8693>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00090-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Fundamentals of Transportation/Trip Generation/Problem
Planners have estimated the following models for the AM Peak Hour
$T_i = 1.5*H_i \,\!$
$T_j = (1.5*E_{off,j}) + (1*E_{oth,j}) + (0.5*E_{ret,j}) \,\!$
$T_i\,\!$ = Person Trips Originating in Zone $i\,\!$
$T_j\,\!$ = Person Trips Destined for Zone $j\,\!$
$H_i\,\!$ = Number of Households in Zone $i\,\!$
You are also given the following data
Variable Dakotopolis New Fargo
$H$ 10000 15000
$E_{off}$ 8000 10000
$E_{oth}$ 3000 5000
$E_{ret}$ 2000 1500
A. What are the number of person trips originating in and destined for each city?
B. Normalize the number of person trips so that the number of person trip origins = the number of person trip destinations. Assume the model for person trip origins is more accurate.
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subdivision of each tile type is into tiles that can be identified with tile types, one can recursively subdivide any complex made up out of tile types. My coworkers and I have studied them for years
as part of our research program in low-dimensional topology and geometry.
Click here for more information and to see more graphic images of finite subdivision rules. | {"url":"http://www.math.vt.edu/","timestamp":"2014-04-16T19:02:52Z","content_type":null,"content_length":"8845","record_id":"<urn:uuid:643b3458-c06c-4fed-afec-a82039853910>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00035-ip-10-147-4-33.ec2.internal.warc.gz"} |
CGTalk - Help with expression? Please?
07-25-2006, 08:38 PM
I tried this post in the Character Anim thread, but have no response so since this is more of an expression question figured I would try here instead? Any tips? Pointers? A slap in the face? :D
OK... so I am running a test to get expressions right, but because I suck at programming, I can't seem to get it right. Specificly with this scene, I have a sphere that moves in the direction when
another object - a cylinder - moves (or translates, have yet to even figure out rotation). The expression I have so far works well, untill it reaches a certain point. When moving on the X axis it
works fine, but once you reach a point and then move down the Z axis the ball totally moves in the opposite direction. I know it's something that can be fixed, but I got this expression off the net
and just modified it a tad, but I have no idea WHERE to fix this.
ball.rotateZ = (box.translateX / (2 * 3.14 * 1)) * -360;
ball.rotateX = (box.translateZ / (2 * 3.14 * 1)) * -360;
I can upload the test scene (66kb) if anyone wants to see what it is I am talking about. Otherwise if you are an expression wiz I will be totally in your debt. | {"url":"http://forums.cgsociety.org/archive/index.php/t-385628.html","timestamp":"2014-04-19T04:21:57Z","content_type":null,"content_length":"8303","record_id":"<urn:uuid:5eab7c3c-d7fa-4072-9716-70380ba21391>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00028-ip-10-147-4-33.ec2.internal.warc.gz"} |
West Covina Statistics Tutor
Find a West Covina Statistics Tutor
...I have cleaned raw data from my clientele and have properly named, assigned codes for missing values and coded each variables for grouping factors, and planned contrasts. My approach to data
analysis, typically begins with a diagnostic test for normality, homogeneity of variance test, examining ...
3 Subjects: including statistics, SPSS, ecology
...I will use my experience in this area to teach strategies that will help you succeed in this subject and prepare you for future areas of math. From polynomials, to functions to graphs. I will
help make the subject of Algebra make sense and even make it a little fun!
46 Subjects: including statistics, calculus, physics, geometry
...In college I tutored high school math and science. Before graduate school, I spent 3 years tutoring and teaching at a private home school. I taught children from preschool through middle school
in all subjects.
31 Subjects: including statistics, English, reading, literature
...We ran about 50 servers and 500 workstations. I have worked with Windows: XP, Vista, & 7; Mac OSX: 10.4, 10.5, 10.6; Linux: Ubuntu & Debian. Also Server editions for most of these OS's.
32 Subjects: including statistics, English, calculus, physics
...It’s crucial for word problems - and the SAT has lots of word problems. Without math-translation, the SAT may as well be written in Greek. Once the student translates and understands what the
problem is about, I focus on the approach.
24 Subjects: including statistics, Spanish, physics, writing | {"url":"http://www.purplemath.com/West_Covina_statistics_tutors.php","timestamp":"2014-04-17T21:28:15Z","content_type":null,"content_length":"23768","record_id":"<urn:uuid:8f2c152f-7ddc-419a-8a58-6883597ec518>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00398-ip-10-147-4-33.ec2.internal.warc.gz"} |
10 Out - math card game
Today I told my dd to write down addition facts where the sum is 10 - "Which numbers make ten?" are the words I use with her. While she was writing them and using abacus, this math card came 'popped'
to my mind...
I can't claim it as my own, because I have this 'feeling' that I've read about it somewhere, sometime, but since I can't remember when or where, I can't give credit to where credit might be due.
I will just name it "10 Out" - a math card game.
Anyway, this is how it goes:
Take away the 'picture' cards and joker from normal playing cards. Then deal 10 cards to each player; put the rest of the deck in the middle. The goal is to get rid of all cards in your hand.
Find all pairs of cards in your hand that add up to a ten (or single 10's) and discard them. Then you may ask the player left to you for one card, and if she has it, she has to give it to you.
For example, say you have 2, 3, 8, and 1 left in your hand. You ask the person next to you if he has 9 (because 9 and 1 would make ten). He has, and gives it to you. Then you form another ten and
discard your 1 and the 9.
Then the next person asks the player to their left for one card, and on around.
You could modify it so that each players just blindly hands one card to the player on their left.
Then, once nobody can discard any more cards, every player takes one card from the deck in the middle, and checks if they can form a sum of ten and discard cards. Again each player will be allowed to
ask for one card from their left neighbor, if they want to (or everyone hands one card blindly to the player on their left). Continue until someone wins.
I feel this can easily make kids memorize addition facts with sum 10.
Then, of course, next math lesson it will be time for "9 Out" and "8 Out" and "11 Out" and others!
Tags: math | {"url":"http://homeschoolmath.blogspot.com/2006/06/10-out-math-card-game.html","timestamp":"2014-04-16T04:16:13Z","content_type":null,"content_length":"109113","record_id":"<urn:uuid:b4ccfe07-3ea7-40a0-bbfd-96e9094285c7>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00646-ip-10-147-4-33.ec2.internal.warc.gz"} |
Leaky Rayleigh wave scattering from elastic media with
ASA 128th Meeting - Austin, Texas - 1994 Nov 28 .. Dec 02
5aPA10. Leaky Rayleigh wave scattering from elastic media with microstructure.
Yuan Zhang
Richard L. Weaver
Dept. of Theor. and Appl. Mech., Univ. of Illinois, 216 Talbot Lab., 104 S. Wright St., Urbana, IL 61801
Scattering of leaky Rayleigh wave from a flat fluid--solid interface is studied. The fluid half-space is taken to be ideal and homogeneous while the solid half-space has randomly inhomogeneous
anisotropic elastic constants due to the microstructure of the material. For plane waves incident from the fluid onto the interface at the critical Rayleigh angle, the singly scattered incoherent
field is obtained by utilizing a first Born approximation. When the solid is a crystal aggregate and when the correlation function is of exponential form, the mean-square scattered signal level is
found to be inversely proportional to the dimensionless frequency in the high-frequency limit but proportional to the third power of frequency in the low-frequency limit. Numerical results are given
for water--aluminum interface scattering. [Work supported by NSF.] | {"url":"http://www.auditory.org/asamtgs/asa94aus/5aPA/5aPA10.html","timestamp":"2014-04-18T10:35:23Z","content_type":null,"content_length":"1566","record_id":"<urn:uuid:73400fa7-1fa4-40a0-9f33-852243c9fa91>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00466-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts by
Posts by babs
Total # Posts: 41
Hi I need a formula for the following word problem. The cost of a cellphone has increased by 400% over 10 years. Does this correspond to a decrease of 4 times its cost?
Improving Your Writing
Background: Ten years ago, you started working as a clerk for DMD Medical Supplies. Six months ago, Liz Jakowski, the human resources director, promoted you to office manager. You manage two
employees: Jack Snyder and Ruth Disselkoen. Your office provides secretarial support f...
Improving Your Writing
Revised my rough draft,could you check and see if it's better.Thank you. For the past three months, I have noticed problems in the work flow within our company, DMD Medical Supplies. The problems are
with my two employees, Jack Snyder and Ruth Disselkoen. Jack supports th...
Improving Your Writing
Thank you for all your advice.It is a rough draft,just wanted to know if I was in right direction.
Improving Your Writing
Background: Ten years ago, you started working as a clerk for DMD Medical Supplies. Six months ago, Liz Jakowski, the human resources director, promoted you to office manager. You manage two
employees: Jack Snyder and Ruth Disselkoen. Your office provides secretarial support f...
writing sentences and paragrahs
Thanks so much, really appreciate all your help..
writing sentences and paragrahs
Your favorite cousin has moved to your town and is looking for a job. Her previous experiences are working as a cashier and sales clerk at two department stores. You know she plans to apply at
similar stores in your town. But you also know she is a perfect match for a job open...
writing sentences and paragrahs
Thanks,really appreciate you info on this...screwed up. How's this? I am extremely excited about beginning my career as a computer support technician with you at Microdyne Computers. I look forward
with great anticipation to the challenges of today s ever changing com...
writing sentences and paragrahs
Background You ve applied for a specific job in your field of study. The Human Resources Department arranges an interview and tells you to bring with you a polished piece of writing for them to
evaluate your writing skills. The paragraph must describe one particular exper...
algebra 116
The year was 2003
Why can't reversing entries be used on cash accounts?
Subtract (4a^3+3b^3)-(2a^3+4a^2b-_5ab^2+3b^3) is the answer 2a^3+4a^2b-5ab^2
please tell me if these are right. Soybean meal is 12% protein, cornmeal is 6% protein. How many pounds of each should be mixed together in order to get 240-lb. mixture that is 11% protein? 80 pounds
soybean and 160 pounds of cornmeal Trains A and B are traveling in the sme di...
Solve using the multiplication principle. 3/4x=9/10. Would the answer be 2/5?
is the solution of -2x>1/9 -2x<1/9
would the answer to 8x-(4x+4)=20 would that be 6 then? I posted it wrong before thank you for helping
Solve 8x-(4x+4)=20 Ms. Sue would the answer be 3/2?
Please check these too. thanks Simplify 6[86-(94-86)]= my answer is 468 2. On three consecutive passes, a football team gains 6 yards. loses 17 yards and gainbs 31 yards. Wht number represents the
total net yards. The total net years is 20 yards. 3. Solve 8x-(4x+4)=2 The solut...
Will someone check my work here and let me know if I am right. Thanks. Evaluate x+y/7 for x=44 and y=12 My answer is x+y/7=8 2. Translate to an algebraic expression. The product of 42% and some
number. The translation is .42x=y 3. Use < or> to make the statment true. my ...
Please help i cant get right.Bayside Insurance offers two health plans. Under plan A, Gisielle would pay the first $110 of her medical bills plus 35% of the rest. Under plan B, Gisielle would pay the
first $170, but only 25% of the rest. For what amount of medical bills will p...
Describe what the graph of interval[-4,10] looks like. Please answer
Snookers Lumber can convert logs into either lumber ofr lywood. In a given day, the mill turns out three times as many units of plywood as lumber. It makes a profit of $30 on a unit of lumber and $40
on a unit of plywood. How many of each unit must be produced and sold in orde...
would someone write out the whole process of the following question, including solving for y so that i can see what i am doing wrong. Thank you Solve by the substitution method. write as an ordered
pair. 8x-3y=-94 2x+22=y
solve by substitution 2x+6y=30 -9x+y=61
I am tring to work these out and i have been in this for hours it is not clicking thanks for all the help. the question is: Snookers lumber can convert logs into either lumber or plywood. In a given
day, the mill turns out twice as many units of plywood as lumber. It makes a p...
Soybean meal is 12% protein; Cornmeal is 6% protein. How many pounds of each should be mixed together in order to get 240lb mixture that is 8% protein?
solve by elimination x+6y=12 -x+7y=1
solve by elimination x+6y-12 -x+7y=1
A disc jockey must play 14 commercial spots during one hour of radio show.Each commercial is either 30 seconds or 60 seconds long. If the total commercial time during 1 hour is 11 min, how many 30
second commercials were played that hour? How many 60-second commercials? and th...
could someone please help solve by substitution 8x-3y=-88 4x+48=y
Don't you hate it when your answer is a link? THAT IS REAL HELP!!!!
Comparez -les Example ;Michele est paresseuse. Agnes est travailleuse. Michele......[travaille]..>Michele travaille moins que Agnes. 1]Alain est bavard .Paul est timide--Alain.....[parler]. 2]Marie
est triste. Jacqueline est gaie ->Marie......[s'amuser] 3]Pierre est ...
Caracteriser les actions les adverbes a] elle travaille bien ,mal , vite. b] adverbes formes avec l' adjectif 1] adjectif au feminin +ment; correcte -correctement-joyeux-joyeusement 2]adjectif en
i,ai, e, u, au masculin+ment; joli- joliment-gai -gaiment 3]adjectif en, ant,...
Le souvenir -l'oubli 1] la memoire -un souvenir avoir une bonne \mauvaise memoire retenir une lecon apprendre\ savoir une lecon par coeur 2]se souvenir de quelquechose se rappeler quelque chose 3]
oublier quelque chose de faire quelque chose un oubli etre distrait [e] -etou...
La Verite Le Mensonge 1]c'est vrai -c'est juste -c'est exact\c'est faux dire la verite \mentir un mensonge un menteur 2] cacher la verite decouvrir la verite [une decouverte] reveler-faire une
revelation deviner -une devinetteatruth lies
PORTRAITS Le Francais moyen Lui ;taille;1,72m,Poids;72kg. Elle;taille;1,60m,Poids;60kg. Leurs yeux ;noirs ou marron a 55% bleus ou gris a 45% Leurs cheveux ;plutot bruns [les blonds sont en
diminution] Decrivez ces personnes Existe -t-il pour vous un type physique francais? CA...
Vocabulaire 1]Donnez le contraire des phrases suivantes Il dit la verite.--La population a augmente.--Elle a critique le film .---Je me souviens de cette histoire. ---Cette region est restee la meme.
2]Completez ces phrases Dans le desert du Sahel, l'eau ................so...
Comparaison et caracterisation comparez la consommation de poisson et de lait dans les differents pays. Consommation annuelle par pays Poisson [en kg par habitant] Grande Bretagne:127 Norvege : 50
Japon: 36 France: 9 Lait [en litre par habitant] Islande: 200 Espagne: 110 Franc...
Remplacez les mots soulignes par un adverbe en ' ment' Example ;Il marche dans la rue se depecher. Il marche lentement1] Il a critique le film avec ironie. 2]Nous avons fete la nouvelle annee dans la
joie. 3]Il a explique la situation sans s'enerver. 4]Elle a repon...
2Expression du futur Mettrez les verbes entre parentheses au temps qui convient Demain je [ revoir]Mireille.Elle m'[attendre]a 9 heures au Cafe des Arts. Nous [aller] dans maison de campagne des
Dubos. Ils nous [emmener] visiter la region. Nous [passer] toute la journee av...
perimeter of right triangle equals 1400 and hypotenuse equals 600 how do you find out what legs measure | {"url":"http://www.jiskha.com/members/profile/posts.cgi?name=babs","timestamp":"2014-04-16T23:07:57Z","content_type":null,"content_length":"16247","record_id":"<urn:uuid:7d23e1f7-3d53-4c4b-b280-9e6e2fc8c2b4>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00125-ip-10-147-4-33.ec2.internal.warc.gz"} |
Unit Conversion Question
I didn't check the math, but it looks like you are on the right track.
Remember what I said about carrying along units in your equations, though. Even if you don't show the units while you type here on the forum, hopefully you are now showing units for each of the
quantites in the equations that have them, and cancelling them out when they appear on top and bottom like cm/cm = 1, so cross them both out. That will save you many, many errors over the years, and
also help you figure out what goes on the top and bottom of a fraction when you aren't sure.
Like, what is the relationship between wavelength, speed and frequency? Well, if wavelength's units are m, speed is m/s, and frequency is 1/s (=Hz), then you can figure out the equation just based on
keeping the units consistent, right? | {"url":"http://www.physicsforums.com/showpost.php?p=1354539&postcount=4","timestamp":"2014-04-16T04:30:28Z","content_type":null,"content_length":"8140","record_id":"<urn:uuid:7c1d398f-f013-4f84-8de3-b5dcd85c1d7a>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00560-ip-10-147-4-33.ec2.internal.warc.gz"} |
namespace std {
template<class T, class A>
class slist;
// TEMPLATE FUNCTIONS
template<class T, class A>
bool operator==(
const slist<T, A>& lhs,
const slist<T, A>& rhs);
template<class T, class A>
bool operator!=(
const slist<T, A>& lhs,
const slist<T, A>& rhs);
template<class T, class A>
bool operator<(
const slist<T, A>& lhs,
const slist<T, A>& rhs);
template<class T, class A>
bool operator>(
const slist<T, A>& lhs,
const slist<T, A>& rhs);
template<class T, class A>
bool operator<=(
const slist<T, A>& lhs,
const slist<T, A>& rhs);
template<class T, class A>
bool operator>=(
const slist<T, A>& lhs,
const slist<T, A>& rhs);
template<class T, class A>
void swap(
slist<T, A>& lhs,
slist<T, A>& rhs);
} // namespace std
Include the STL standard header <slist> to define the container template class slist and several supporting templates.
template<class T, class A>
bool operator!=(
const slist <T, A>& lhs,
const slist <T, A>& rhs);
The template function returns !(lhs == rhs).
template<class T, class A>
bool operator==(
const slist <T, A>& lhs,
const slist <T, A>& rhs);
The template function overloads operator== to compare two objects of template class slist. The function returns lhs.size() == rhs.size() && equal(lhs. begin(), lhs. end(), rhs.begin()).
template<class T, class A>
bool operator<(
const slist <T, A>& lhs,
const slist <T, A>& rhs);
The template function overloads operator< to compare two objects of template class slist. The function returns lexicographical_compare(lhs. begin(), lhs. end(), rhs.begin(), rhs.end()).
template<class T, class A>
bool operator<=(
const slist <T, A>& lhs,
const slist <T, A>& rhs);
The template function returns !(rhs < lhs).
template<class T, class A>
bool operator>(
const slist <T, A>& lhs,
const slist <T, A>& rhs);
The template function returns rhs < lhs.
template<class T, class A>
bool operator>=(
const slist <T, A>& lhs,
const slist <T, A>& rhs);
The template function returns !(lhs < rhs).
allocator_type · assign · back · begin · clear · const_iterator · const_pointer · const_reference · difference_type · empty · end · erase · front · get_allocator · insert · iterator · slist ·
max_size · merge · pointer · pop_back · pop_front · previous · push_back · push_front · reference · remove · remove_if · resize · reverse · size · size_type · sort · splice · swap · unique ·
template<class T, class A = allocator<T> >
class slist {
typedef A allocator_type;
typedef typename A::pointer pointer;
typedef typename A::const_pointer
typedef typename A::reference reference;
typedef typename A::const_reference const_reference;
typedef typename A::value_type value_type;
typedef T0 iterator;
typedef T1 const_iterator;
typedef T2 size_type;
typedef T3 difference_type;
explicit slist(const A& al);
explicit slist(size_type n);
slist(size_type n, const T& v);
slist(size_type n, const T& v, const A& al);
slist(const slist& x);
template<class InIt>
slist(InIt first, InIt last);
template<class InIt>
slist(InIt first, InIt last, const A& al);
iterator begin();
const_iterator begin() const;
iterator end();
const_iterator end() const;
iterator previous(const_iterator it);
const_iterator previous(const_iterator it) const;
void resize(size_type n);
void resize(size_type n, T x);
size_type size() const;
size_type max_size() const;
bool empty() const;
A get_allocator() const;
reference front();
const_reference front() const;
reference back();
const_reference back() const;
void push_front(const T& x);
void pop_front();
void push_back(const T& x);
void pop_back();
template<class InIt>
void assign(InIt first, InIt last);
void assign(size_type n, const T& x);
iterator insert(iterator it, const T& x);
void insert(iterator it, size_type n, const T& x);
template<class InIt>
void insert(iterator it, InIt first, InIt last);
iterator erase(iterator it);
iterator erase(iterator first, iterator last);
void clear();
void swap(slist& x);
void splice(iterator it, slist& x);
void splice(iterator it, slist& x, iterator first);
void splice(iterator it, slist& x, iterator first,
iterator last);
void remove(const T& x);
templace<class Pred>
void remove_if(Pred pr);
void unique();
template<class Pred>
void unique(Pred pr);
void merge(slist& x);
template<class Pred>
void merge(slist& x, Pred pr);
void sort();
template<class Pred>
void sort(Pred pr);
void reverse();
The template class describes an object that controls a varying-length sequence of elements of type T. The sequence is stored as a singly linked list of elements, each containing a member of type T.
The object allocates and frees storage for the sequence it controls through a stored allocator object of class A. Such an allocator object must have the same external interface as an object of
template class allocator. Note that the stored allocator object is not copied when the container object is assigned.
List reallocation occurs when a member function must insert, erase or splice elements of the controlled sequence. In all such cases, only the following iterators or references become invalid:
• iterators that designated a position immediately beyond an inserted element
• iterators that designate an erased element or a position immediately beyond an erased element
• iterators that designate a spliced element or a position immediately beyond a spliced element
All additions to the controlled sequence occur as if by calls to insert, which is the only member function that calls the constructor T(const T&). If such an expression throws an exception, the
container object inserts no new elements and rethrows the exception. Thus, an object of template class slist is left in a known state when such exceptions occur.
typedef A allocator_type;
The type is a synonym for the template parameter A.
template<class InIt>
void assign(InIt first, InIt last);
void assign(size_type n, const T& x);
If InIt is an integer type, the first member function behaves the same as assign((size_type)first, (T)last). Otherwise, the first member function replaces the sequence controlled by *this with the
sequence [first, last), which must not overlap the initial controlled sequence. The second member function replaces the sequence controlled by *this with a repetition of n elements of value x.
reference back();
const_reference back() const;
The member function returns a reference to the last element of the controlled sequence, which must be non-empty.
const_iterator begin() const;
iterator begin();
The member function returns a forward iterator that points at the first element of the sequence (or just beyond the end of an empty sequence).
void clear();
The member function calls erase( begin(), end()).
typedef T1 const_iterator;
The type describes an object that can serve as a constant forward iterator for the controlled sequence. It is described here as a synonym for the implementation-defined type T1.
typedef typename A::const_pointer
The type describes an object that can serve as a constant pointer to an element of the controlled sequence.
typedef typename A::const_reference const_reference;
The type describes an object that can serve as a constant reference to an element of the controlled sequence.
typedef T3 difference_type;
The signed integer type describes an object that can represent the difference between the addresses of any two elements in the controlled sequence. It is described here as a synonym for the
implementation-defined type T3.
bool empty() const;
The member function returns true for an empty controlled sequence.
const_iterator end() const;
iterator end();
The member function returns a forward iterator that points just beyond the end of the sequence.
iterator erase(iterator it);
iterator erase(iterator first, iterator last);
The first member function removes the element of the controlled sequence pointed to by it. The second member function removes the elements of the controlled sequence in the range [first, last). Both
return an iterator that designates the first element remaining beyond any elements removed, or end() if no such element exists.
Erasing N elements causes N destructor calls. Reallocation occurs, so iterators and references become invalid for the erased elements and iterators become invalid for any remaining element
immediately beyond an erased element.
The member functions never throw an exception.
reference front();
const_reference front() const;
The member function returns a reference to the first element of the controlled sequence, which must be non-empty.
A get_allocator() const;
The member function returns the stored allocator object.
iterator insert(iterator it, const T& x);
void insert(iterator it, size_type n, const T& x);
template<class InIt>
void insert(iterator it, InIt first, InIt last);
Each of the member functions inserts, before the element pointed to by it in the controlled sequence, a sequence specified by the remaining operands. The first member function inserts a single
element with value x and returns an iterator that designates the newly inserted element. The second member function inserts a repetition of n elements of value x.
If InIt is an integer type, the last member function behaves the same as insert(it, (size_type)first, (T)last). Otherwise, the last member function inserts the sequence [first, last), which must not
overlap the initial controlled sequence.
Inserting N elements causes N constructor calls. Reallocation occurs, so iterators become invalid for any element that was immediately beyond it.
If an exception is thrown during the insertion of one or more elements, the container is left unaltered and the exception is rethrown.
typedef T0 iterator;
The type describes an object that can serve as a forward iterator for the controlled sequence. It is described here as a synonym for the implementation-defined type T0.
size_type max_size() const;
The member function returns the length of the longest sequence that the object can control.
void merge(slist& x);
template<class Pred>
void merge(slist& x, Pred pr);
Both member functions remove all elements from the sequence controlled by x and insert them in the controlled sequence. Both sequences must be ordered by the same predicate, described below. The
resulting sequence is also ordered by that predicate.
For the iterators Pi and Pj designating elements at positions i and j, the first member function imposes the order !(*Pj < *Pi) whenever i < j. (The elements are sorted in ascending order.) The
second member function imposes the order !pr(*Pj, *Pi) whenever i < j.
No pairs of elements in the original controlled sequence are reversed in the resulting controlled sequence. If a pair of elements in the resulting controlled sequence compares equal (!(*Pi < *Pj) &&
!(*Pj < *Pi)), an element from the original controlled sequence appears before an element from the sequence controlled by x.
An exception occurs only if pr throws an exception. In that case, the controlled sequence is left in unspecified order and the exception is rethrown.
typedef typename A::pointer pointer;
The type describes an object that can serve as a pointer to an element of the controlled sequence.
void pop_back();
The member function removes the last element of the controlled sequence, which must be non-empty. This operation takes time proportional to the number of elements in the controlled sequence (linear
time complexity).
The member function never throws an exception.
void pop_front();
The member function removes the first element of the controlled sequence, which must be non-empty.
The member function never throws an exception.
iterator previous(const_iterator it);
const_iterator previous(const_iterator it) const;
The member function returns an iterator that designates the element immediately preceding it, if possible; otherwise it returns end(). This operation takes time proportional to the number of elements
in the controlled sequence (linear time complexity).
void push_back(const T& x);
The member function inserts an element with value x at the end of the controlled sequence.
If an exception is thrown, the container is left unaltered and the exception is rethrown.
void push_front(const T& x);
The member function inserts an element with value x at the beginning of the controlled sequence.
If an exception is thrown, the container is left unaltered and the exception is rethrown.
typedef typename A::reference reference;
The type describes an object that can serve as a reference to an element of the controlled sequence.
void remove(const T& x);
The member function removes from the controlled sequence all elements, designated by the iterator P, for which *P == x.
The member function never throws an exception.
templace<class Pred>
void remove_if(Pred pr);
The member function removes from the controlled sequence all elements, designated by the iterator P, for which pr(*P) is true.
An exception occurs only if pr throws an exception. In that case, the controlled sequence is left in an unspecified state and the exception is rethrown.
void resize(size_type n);
void resize(size_type n, T x);
The member functions both ensure that size() henceforth returns n. If it must make the controlled sequence longer, the first member function appends elements with value T(), while the second member
function appends elements with value x. To make the controlled sequence shorter, both member functions call erase(begin() + n, end()).
void reverse();
The member function reverses the order in which elements appear in the controlled sequence.
size_type size() const;
The member function returns the length of the controlled sequence.
typedef T2 size_type;
The unsigned integer type describes an object that can represent the length of any controlled sequence. It is described here as a synonym for the implementation-defined type T2.
explicit slist(const A& al);
explicit slist(size_type n);
slist(size_type n, const T& v);
slist(size_type n, const T& v,
const A& al);
slist(const slist& x);
template<class InIt>
slist(InIt first, InIt last);
template<class InIt>
slist(InIt first, InIt last, const A& al);
All constructors store an allocator object and initialize the controlled sequence. The allocator object is the argument al, if present. For the copy constructor, it is x.get_allocator(). Otherwise,
it is A().
The first two constructors specify an empty initial controlled sequence. The third constructor specifies a repetition of n elements of value T(). The fourth and fifth constructors specify a
repetition of n elements of value x. The sixth constructor specifies a copy of the sequence controlled by x. If InIt is an integer type, the last two constructors specify a repetition of (size_type)
first elements of value (T)last. Otherwise, the last two constructors specify the sequence [first, last).
void sort();
template<class Pred>
void sort(Pred pr);
Both member functions order the elements in the controlled sequence by a predicate, described below.
For the iterators Pi and Pj designating elements at positions i and j, the first member function imposes the order !(*Pj < *Pi) whenever i < j. (The elements are sorted in ascending order.) The
member template function imposes the order !pr(*Pj, *Pi) whenever i < j. No ordered pairs of elements in the original controlled sequence are reversed in the resulting controlled sequence. (The sort
is stable.)
An exception occurs only if pr throws an exception. In that case, the controlled sequence is left in unspecified order and the exception is rethrown.
void splice(iterator it, slist& x);
void splice(iterator it, slist& x, iterator first);
void splice(iterator it, slist& x, iterator first,
iterator last);
The first member function inserts the sequence controlled by x before the element in the controlled sequence pointed to by it. It also removes all elements from x. (&x must not equal this.)
The second member function removes the element pointed to by first in the sequence controlled by x and inserts it before the element in the controlled sequence pointed to by it. (If it == first || it
== ++first, no change occurs.)
The third member function inserts the subrange designated by [first, last) from the sequence controlled by x before the element in the controlled sequence pointed to by it. It also removes the
original subrange from the sequence controlled by x. (If &x == this, the range [first, last) must not include the element pointed to by it.)
If the third member function inserts N elements, and &x != this, an object of class iterator is incremented N times. For all splice member functions, If get_allocator() == str.get_allocator(), no
exception occurs. Otherwise, a copy and a destructor call also occur for each inserted element.
Iterators or references that designate spliced elements, or that designate the first element beyond a sequence of spliced elements, become invalid.
void swap(slist& x);
The member function swaps the controlled sequences between *this and x. If get_allocator() == x.get_allocator(), it does so in constant time, it throws no exceptions, and it invalidates no
references, pointers, or iterators that designate elements in the two controlled sequences. Otherwise, it performs a number of element assignments and constructor calls proportional to the number of
elements in the two controlled sequences.
void unique();
template<class Pred>
void unique(Pred pr);
The first member function removes from the controlled sequence every element that compares equal to its preceding element. For the iterators Pi and Pj designating elements at positions i and j, the
second member function removes every element for which i + 1 == j && pr(*Pi, *Pj).
For a controlled sequence of length N (> 0), the predicate pr(*Pi, *Pj) is evaluated N - 1 times.
An exception occurs only if pr throws an exception. In that case, the controlled sequence is left in an unspecified state and the exception is rethrown.
typedef typename A::value_type value_type;
The type is a synonym for the template parameter T.
template<class T, class A>
void swap(
slist <T, A>& lhs,
slist <T, A>& rhs);
The template function executes lhs.swap(rhs).
See also the Table of Contents and the Index.
Copyright © 1992-2006 by P.J. Plauger. All rights reserved.
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To do any metric conversion there has an invented formula by the mathematician. From the ancient age people uses this formula and day by day this formula being develops and gives an accurate result
for any conversion. 158 Cm in Feet and Inches is one of the conversions.
Here are three different metric formula related term cm (centimeter), feet or foot, and inches.
These three types of different term mention us three different measures in calculation purpose.
Now we will discuss about the formula at first:
A. One foot/foot equals to 30.48 centimeter and 12 inches
B. One inches equals to 0.08333 feet and 2.54 centimeter
So this is the basic formula if any one can remember this formula than he can easily calculate any measurement.
Here we have to show that what will be the result of 158 centimeter in feet and inches:
At first 158 centimeters to feet
30.48 = 1 feet
So 158 centimeter equals will be 5.166 feet
And 158 centimeter equals will be 62 inches.
We got this calculation by using online calculating software.
There has lots of online metric calculating software in the internet. Anyone wants to do such conversion can go to the internet for help.
Such types of website are like iceryder.net/measureconvert, and you make type in google search by typing metric calculator.
158 Cm in Feet and Inches is basically a calculating factor.
The process of this calculation will start if anyone knows the formula of this. Than, you can calculate very easily even by hand.
But if you don't know the formula of this than this is not a problem for you to make conversion. If anyone have internet connection he may also can convert and calculate this factor.
For the development of modern science it gives us lots of computer software by which any one can calculate and solve any critical solve.
But to do this conversion you may choose redimate software. Or you may download software to do this conversion.
To buy software can be costly for you. So it will be batter for you if you can use the demo version of this software which is provided in the various website for free.
Finally we can say that this the modern age here any types of critical conversion are possible by software. We need to know about the internet and being always with computer related issue. If you
want to convert measurements you can always search the internet for conversions.
This entry was posted in
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Wooders, Myrna (2008): Market games and clubs. Published in: Springer Encyclopeida of Complexity and Systems Science (2009): pp. 5359-5377.
Download (253Kb) | Preview
The equivalence of markets and games concerns the relationship between two sorts of structures that appear fundamentally different -- markets and games. Shapley and Shubik (1969) demonstrates that:
(1) games derived from markets with concave utility functions generate totally balanced games where the players in the game are the participants in the economy and (2) every totally balanced game
generates a market with concave utility functions. A particular form of such a market is one where the commodities are the participants themselves, a labor market for example. But markets are very
special structures, more so when it is required that utility functions be concave. Participants may also get utility from belonging to groups, such as marriages, or clubs, or productive coalitions.
It may be that participants in an economy even derive utility (or disutility) from engaging in processes that lead to the eventual exchange of commodities. The question is when are such economic
structures equivalent to markets with concave utility functions. This paper summarizes research showing that a broad class of large economies generate balanced market games. The economies include,
for example, economies with clubs where individuals may have memberships in multiple clubs, with indivisibile commodities, with nonconvexities and with non-monotonicities. The main assumption are:
(1) that an option open to any group of players is to break into smaller groups and realize the sum of the worths of these groups, that is, essential superadditivity is satisfied and :(2) relatively
small groups of participants can realize almost all gains to coalition formation. The equivalence of games with many players and markets with many participants indicates that relationships obtained
for markets with concave utility functions and many participants will also hold for diverse social and economic situations with many players. These relationships include: (a) equivalence of the core
and the set of competitive outcomes; (b) the Shapley value is contained in the core or approximate cores; (c) the equal treatment property holds -- that is, both market equilibrium and the core treat
similar players similarly. These results can be applied to diverse economic models to obtain the equivalence of cooperative outcomes and competitive, price taking outcomes in economies with many
participants and indicate that such results hold in yet more generality.
Item Type: MPRA Paper
Original Market games and clubs
Language: English
Markets; games; market games; clubs; core; market-game equivalence; Shapley value; price taking equilibrium; small group effectiveness; inessentiality of large groups; per capita
Keywords: boundedness; competitive equilibrium; games with side payments; balanced games; totally balanced games; local public goods, core convergence; equal treatment property; equal treatment
core; approximate core; strong epsilon core; weak epsilon core; cooperative game; asymptotic negligibility
D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice; Clubs; Committees; Associations
Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games
H - Public Economics > H4 - Publicly Provided Goods > H41 - Public Goods
C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C70 - General
Item ID: 33968
Depositing Myrna Wooders
Date 09. Oct 2011 17:51
Last 18. Feb 2013 12:25
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(ed), Lincoln Institute of Land Policy, Cambridge, Mass.
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Rationality and Equilibrium: A Symposium in Honour of Marcel K. Richter, Studies in Economic Theory Series 26, Springer: Springer Verlag, 141-169.
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Shubik, M. and M. Wooders (1982) "Near markets and market games," Cowles Foundation Discussion Paper No. 657, on line at http://www.myrnawooders.com/
Shubik, M., Wooders, M., 1982. Clubs, markets, and near-market games. In: Wooders, M. (Ed.), Topics in Game Theory and Mathematical Economics: Essays in Honor of Robert J. Aumann. Field
Institute Communication Volume, American Mathematical Society, originally Near Markets and Market Games, Cowles Foundation, Discussion Paper No. 657.
Shubik, M. and Wooders, M. (1983) "Approximate cores of replica games and economies: Part I. Replica games, externalities, and approximate cores," Mathematical Social Sciences 6, 27-48.
Shubik, M. and Wooders, M. (1983) "Approximate cores of replica games and economies: Part II. Set-up costs and firm formation in coalition production economies." Mathematical Social
Sciences 6, 285-306.
Shubik, M. and Wooders, M. (1986) "Near-markets and market-games," Economic Studies Quarterly 37, 289-299.
Sondermann, D. (1974) "Economics of scale and equilibria in coalition production economies," Journal of Economic Theory 8, 259--291.
Sun, N., W. Trockel, and Z. Yang (2008) "Competitive outcomes and endogenous coalition formation in an n-person game," Journal of Mathematical Economics 44, 853-860.
Tauman, Y. (1987) "The Aumann-Shapley prices: A survey", in The Shapley Value: Essays in Honor of Lloyd S. Shapley, ed. A. Roth, Cambridge University, Cambridge, MA.
Tauman, Y., A. Urbano and J. Watanabe (1997) "A model of multiproduct price competition," Journal of Economic Theory 77, 377-401.
Tiebout, C. (1956) "A pure theory of local expenditures," Journal of Political Economy 64, 416-424.
Weber, S. (1979) On ε-cores of balanced games, International Journal of Game Theory 8, 241-250.
Weber, S. (1981) Some results on the weak core of a non-sidepayment game with infinitely many players, Journal of Mathematical Economics 8:101-111.
Winter, E. and M. Wooders (1990) "On large games with bounded essential coalition sizes," University of Bonn Sondeforschungsbereich 303 Discussion Paper B-149, on-line at http://
www.myrnawooders.com/ International Journal of Economic Theory (2008) 4, 191 - 206.
Wooders, M. (1977) "Properties of quasi-cores and quasi-equilibria in coalition economies," SUNY-Stony Brook Department of Economics Working Paper No. 184, revised (1979) as "A
characterization of approximate equilibria and cores in a class of coalition economies", State University of New York Stony Brook Economics Department, on-line at http://
www.myrnawooders.com/ (on-line at www.myrnawooders.com)
Wooders M. (1978) "Equilibria, the core, and jurisdiction structures in economies with a local public good," Journal of Economic Theory 18, 328-348.
Wooders, M. (1983) "The epsilon core of a large replica game," Journal of Mathematical Economics 11, 277-300, on-line at http://www.myrnawooders.com/
Wooders, M. (1988) "Large games are market games 1. Large finite games," C.O.R.E. Discussion Paper No. 8842 http://www.myrnawooders.com/
Wooders, M.H. (1989) `A Tiebout Theorem', Mathematical Social Sciences 18, 33-55.
Wooders, M. (1991a) "On large games and competitive markets 1: Theory," University of Bonn Sonderforschungsbereich 303 Discussion Paper No. (B-195, Revised August 1992), on line at http:
Wooders, M. (1991b) "The efficaciousness of small groups and the approximate core property in games without side payments," University of Bonn Sonderforschingsbereich 303 Discussion
Paper No. B-179on-line at http://www.myrnawooders.com/
Wooders, M. (1992a) "Inessentiality of large groups and the approximate core property; An equivalence theorem," Economic Theory 2, 129-147.
Wooders, M. (1992b) "Large games and economies with effective small groups," University of Bonn Sonderforschingsbereich 303 Discussion Paper No. B-215.(revised) in Game-Theoretic Methods
in General Equilibrum Analysis, eds. J-F. Mertens and S. Sorin, Kluwer Academic Publishers Dordrecht/Boston/London, on-line at http://www.myrnawooders.com/
Wooders, M. (1993) "The attribute core, core convergence, and small group effectiveness; The effects of property rights assignments on the attribute core," University of Toronto Working
Paper No. 9304
Wooders, M.(1994) "Equivalence of games and markets," Econometrica 62, 1141-1160, on-line at www.myrnawooders.com/ Wooders, M. (1997) "Equivalence of Lindahl equilibria with
participation prices and the core," Economic Theory 9,113-127. Wooders, M. (2007) " Cores of many-player games; Nonemptiness and equal treatment," Review of Economic Design (to appear).
Wooders, M. (2008a) "Small group effectiveness, per capita boundedness and nonemptiness of approximate cores," Journal of Mathematical Economics 44, 888-906.
Wooders, M. (2009) "Market Games and Clubs," Springer Encyclopedia of Complexity and Systems Science 2009: 5359-5377. Wooders, M. and Zame, W.R. (1987) "Large games; Fair and stable
outcomes', Journal of Economic Theory 42, 59-93. Zajac, E. (1972) "Some preliminary thoughts on subsidization," presented at the Conference on Telecommunications Research, Washington
URI: http://mpra.ub.uni-muenchen.de/id/eprint/33968 | {"url":"http://mpra.ub.uni-muenchen.de/33968/","timestamp":"2014-04-18T23:21:36Z","content_type":null,"content_length":"56162","record_id":"<urn:uuid:828cbf36-17e5-46b3-abff-daf7a85e5427>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00438-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(a,b]$ and infinitesimals to singletons
up vote 7 down vote favorite
Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that
1. assigns $b-a$ to $[a,b)$ and to $(a,b]$ for all $a\lt b,$ and
2. assigns an infinitesimal--ideally, the same one--to each singleton?
It's (1) that's a problem. The Bernstein-Wattenberg construction yields a finitely-additive measure that gives (1) up to infinitesimals. But it would be nice to have (1) exactly.
pr.probability nonstandard-analysis
3 We get the "same one" property of 2 from 1. – Gerald Edgar Sep 26 '12 at 18:45
I don't see it. Could you elaborate? – Alexander Pruss Sep 26 '12 at 21:02
Actually, now that I think about it, we don't get the "same one" property from 1, unless it does so trivially because there is no measure satisfying 1. For suppose that $\mu$ satisfies 1 and 2.
Let $\nu(A) = \mu(A) - \mathrm{st} \mu(A)$ be the infinitesimal part of $\mu$. Let $\rho(A) = \mu(A) + \nu(A \cap [0,1/2)) + 2\nu(A \cap [1/2,1))$. Then $\rho(A)$ satisfies 1 but not 2. –
Alexander Pruss Sep 26 '12 at 21:07
1 @Alexander: In the title, it should be $(a, b]$ not $(b, a]$. – Michael Albanese Sep 26 '12 at 22:36
1 I can throw some buzzwords around, but I'm out of my depth here. If you take a nonstandard extension that is an enlargement (or polysaturated) then there is a hyperfinite set $b\subseteq *[0,1]$
with $[0,1] \subseteq b$. This feels relevant, but I'm not sure how exactly. – Kevin O'Bryant Sep 27 '12 at 1:41
show 3 more comments
2 Answers
active oldest votes
Yes, by compactness.
Let $R$ denote your favorite hyperreal ordered field and let $\delta\in R$ be a positive infinitesimal. Let $\mathcal{E}$ denote the set of all (standard) finite Boolean subalgebras of
$\mathcal{P}([0,1))$. For every $A\in\mathcal{E}$, let $\lambda_A(I)$ be the (exact) length of $I$ for all half-open intervals $I\in A$; for all open or closed intervals $I\in A$,
respectively subtract or add $\delta$ to the length of $I$ to define $\lambda_A(I)$; let $\lambda_A(S)=\delta$ for all singletons $S\in A$. Extend $\lambda_A$ to a probability measure $
\mu_A$ on $A$.
up vote 6
down vote (Specifically, for each minimal finite union of intervals $F\in A$, let the connected components of $F$ be $[a_0,b_0),\ldots,[a_k,b_k)$ with $p$ elements of $\{a_i,b_i:i\leq k\}$ added
accepted and $q$ removed. Partition $F$ into its atomic subsets $H_0,\ldots,H_n$. Choose a positive $\mu_A(H_i)\in R$ for each $i$, such that $\sum_{i\leq n}\mu_A(H_i)=(p-q)\delta+\sum_{i\leq k}
(b_i-a_i)$. Now extend $\mu_A$ from the atoms to all of $A$.)
Let $U$ be a fine ultrafilter on $\mathcal{E}$. ("Fine" means that $\{B\in\mathcal{E}:A\subseteq B\}\in U$ for all $A\in\mathcal{E}$.) The ultraproduct measure $\mu_U$ is $R^U$-valued
and has the two properties you seek.
+1. Very nice! – Joel David Hamkins Sep 27 '12 at 16:09
I just minorly corrected the answer: the parenthetical paragraph now correctly handles cases such as $[0,1/3)\cup[2/3,1)\in A$ but $[0,1/3)\not\in A$. – David Milovich Sep 27 '12 at
But probably we want more than the OP asks for. In addition to assigning lengths to intervals and $\delta$ to points, maybe we want to pick a certain infinitesimal $\delta_s$ for each
$0<s<1$ and try to get $\delta_s$ times the $s$-dimensional Hausdorff measure when we have a set of dimension $s$ (up to smaller-size infinitesimals). And, for that matter, when we do
Hausdorff measures there is no reason to use only constant gauge functions. Maybe we would want to do this with come other nonarchimedean extension of the reals, rather than NSA. –
Gerald Edgar Sep 27 '12 at 17:57
@Gerald Edgar: Indeed, we can do much much more with the same technique, which boils down to replacing a single ultraproduct with an iterated ultraproduct to get more control. (In
this case, a two-step iteration was sufficient.) To my mind, the most salient barriers are finite obstructions like the Banach-Tarski obstruction to congruency invariance in three
dimensions. I don't see a finite obstruction to what you propose regarding Hausdorff measure. – David Milovich Sep 27 '12 at 20:27
add comment
I think this is a very interesting question.
In response to your comment, let me argue that if 1 holds and the measure is additive, then the singleton values are all the same. This is the sense in which the strong form of 2
follows from the weak form of 2.
up vote 4 down To see this, following Sean's comment, observe that $\mu (\{a\})+\mu((a,b])=\mu([a,b])=\mu([a,b))+\mu(\{b\})$, and so $\mu(\{a\})=\mu(\{b\})$. So all singletons must have the same
vote measure, and so the strong form of 2 follows from the weak form of 2.
In particular, the proposed function $\rho$ in your comment to the question does not exhibit the desired properties, in light of the decomposition $[0,\frac{1}{2}]=\{0\}\cup(0,\frac12]
Perhaps more simply, if $a<b$ then $$\mu(\lbrace a\rbrace) + (b-a) = \mu(\lbrace a\rbrace\cup(a,b]) = \mu([a,b]) = \mu([a,b)\cup\lbrace b\rbrace) = (b-a) + \mu(\lbrace b\rbrace).$$ –
Sean Eberhard Sep 26 '12 at 22:18
Sean, I agree, and I have edited. – Joel David Hamkins Sep 26 '12 at 22:46
I stand corrected about my comment above. I kind of forgot that I also required $\mu((a,b])=b-a$. (My initial thinking about the problem only required $\mu([a,b))=b-a$. I still don't
know the answer to that one, either.) – Alexander Pruss Sep 27 '12 at 12:58
add comment
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UM-StL Physics 341
Thermal and Statistical Physics (3)
UM-StL Physics 341 - Fall '98
What's New?
• 8:50am 14 Dec 1998: A key to Exam 4 was posted across from the physics office, and the graded exam papers were put into the physics office for distribution. The average grade was around 50 +/- 17
out of 80 possible points./pf
This is an exciting time in statistical physics because of developments this half-century in numerous fields (among them information theory and complex systems), and in the discovery of new paths to
"simpler, deeper, and more widely applicable" understanding that such paradigm changes offer. Some implications for the introductory physics curriculum are mentioned briefly in the web-outline for a
talk I gave, primarily on curriculum content improvements, this summer at the American Association of Physics Teachers meeting in Nebraska. I have also been asked to work on optional sections for the
new edition of a major calculus-based physics text, so that an opportunity to "get your two bits" in on that process may be available as well!
This semester, we will be working from an authorized copy (provided to you by the department) of a promising new book by Dan Schroeder (Weber University in Ogden Utah, to be published by
Addison-Wesley sometime in 1999) called "An Introduction to Thermal Physics".
The book by Garrod (half of which is worked problems) listed below and in stock at the bookstore will be used as a reference.
This course also now has a university-hosted web-wizard page and discussion server. The login ID and password for the latter are both [Physcs308.E01], at least for the time being...
Is it possible to describe recent insights underlying statistical physics simply? Send your thoughts on one attempt at this to pfraundorf@umsl.edu.
Questions this course might help you answer...
• How many "ways to wiggle" per molecule does water evidence at room temperature?
• Where might one observe the "herd-behavior" of bosons sharing a single ground state?
• In what way do plants act as heat engines, and how efficient are they in practice?
• What is the equation of state for a gas containing only one atom?
• What keeps small white dwarves from shrinking past a certain point?
• Which expansion requires more energy: adiabatic, isothermal, or isobaric?
• How can spin-temperatures approach absolute zero from the negative direction?
• How to get mass action, equipartition, & the equation of state from a single function!
• Why and how might one convert between Kelvins, and eV/nat of uncertainty?
• How an invention might increase the winter heat from a natural gas flame 8-fold?
• What's the heat capacity of iron, in bits of uncertainty per two-fold increase in energy?
• How can Lagrange multipliers help you consider whatever you know and don't know?
• Why do the "several electron-volt" electrons in metals really not burn your fingers?!
• As an information engine, what is the upper limit on your productivity in bytes/day?
• How might statistical inference (and entropy) apply elsewhere, to images for example?
Other resources of possible interest:
• Notes for Dan on solving problems with MathCad.
• University of California at Irvine, thermal physics applets!
• U of I Physics Heat Engines Lecture.
• Tom Schneider's page on Information Theory and Molecular Machines.
• Browser-interactive solver for constant acceleration problems.
• A question involving relativistic acceleration which contains what you need to solve it.
• Try focussing a high-res electron microscope image on-line!
• Does making a hotdog require 50 nanoseconds or more of life's power stream?
• Is statistical physics a dead subject, or is there another paradigm change afoot?
• In preparation: assignment list, example tests, course calendar, homework/exam solutions...
• What other resources might help you? E-mail suggestions to philf@newton.umsl.edu.
• At UM-StLouis see also: a1toc, cme, i-fzx, phys&astr, programs, stei-lab, & wuzzlers.
• Some current and previous courses: p111, p112, p231, p341, p400.
• Cite/Link: http://newton.umsl.edu/~philf/p341f95s.html
• This release dated 15 Sep 1996 (Copyright by Phil Fraundorf 1988-1996)
Assumed Background:
• Math 180: Analytic Geometry and Calculus III (5)
• Physics 231: Introduction to Modern Physics (3)
Prof: Phil Fraundorf 516-5933; Benton Hall 421 (office)
Office Hours: after class and by appointment
Text: An Introduction to Thermal Physics by Schroeder (Addison-Wesley, 1999, provided by department) and Statistical Mechanics & Thermodynamics by Garrod (Oxford, 1995) for reference
Lectures: Ref#:33345, 06:55PM-08:45pm MW, B443 Section:E01
Approximate Distribution for Grade:
• (1) Collected HomeWork / Quizzes - 20%
• (3) Three 1-Hour Exams - 50%
• (3) Comprehensive Final Exam - 30%
Some Suggested Supplementary Reading
on subjects considered in this course...
• Keith Stowe, Intro to Statistical Mechanics and Thermodynamics (Wiley, 1984).
• Kittel & Kroemer, Thermal Physics (WH Freeman, 1980).
• George Arfken, Mathematical Methods for Physicists (Academic Press, 1970 & later)
on stuff of more general interest...
• Galileo Galilei - Dialog Concerning the Two Chief World Systems (1632, translated by Stillman Drake, UC Press, 1962)
• Thomas Kuhn, The Structure of Scientific Revolutions, 2nd edition (U. of Chicago Press, Chicago IL, 1970)
• Jearl Walker - The Flying Circus of Physics (Wiley 1977)
• Joel A. Barker, The Business of Paradigms (ILI Press, Lake Elmo MN, 1985)
• R. P. Feynman - "Surely You're Joking, Mr. Feynman!" (Bantam 1986)
• K. Eric Drexler, Engines of Creation (Anchor Doubleday, New York NY, 1986)
• Stephen W. Hawking - A Brief History of Time | {"url":"http://newton.umsl.edu/philf/p341/index.html","timestamp":"2014-04-17T10:10:23Z","content_type":null,"content_length":"10568","record_id":"<urn:uuid:93dad1c5-2e3a-4ff6-9185-a827a755476a>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00326-ip-10-147-4-33.ec2.internal.warc.gz"} |
TADP 640 Transmission Line Design Advanced
Course Information: 3 credits. Advanced structures, foundation testing and design, thermal conductor ratings, sag and tension calculation, survey methods. This course further develops the strategies
that were learned in the Introduction course and introduces advanced concepts for designing transmission lines.
Course Description/Objectives:
• Week Zero: Introduction to Blackboard tools
• Week 1: Guyed Structures
• Week 2: Lattice Towers
• Week 3: Steel Poles
• Week 4: Foundations (soil properties, foundations under compression)
• Week 5: Foundations (under lateral load, under uplift)
• Week 6: Advanced Sag and Tension
• Week 7: Special Problems in Sag and Tension
• Week 8: LiDAR Survey Technology, Thermal Ratings
As exercises students will be given data on a line segment and will be required to compute the conductor temperature, do a clearance assessment, and be given a task of upgrading the line to a greater
MVA rating using the several techniques presented.
Module Descriptions:
Module 1: Sag & Tension & LiDAR Survey Technology (2 weeks)
Week 1: Special problems in Sag and Tension
● Derivation of the Catenary Hyperbolic equation
● Calculation of conductor tension with survey points.
● Adding / changing a structure.
● Adding / removing conductor.
● Moving structures.
Week 2: Advanced Sag and Tension, Thermal Ratings, LiDAR Survey
● Actual sag and tension behavior.
● Thermal Ampacity Ratings of Transmission Lines
● Upgrading of Transmission Lines
● LiDAR Survey Technology
Module 2: Structures (3 weeks)
Weeks 3 & 4: Lattice Towers
● Introduction to lattice towers, anatomy of tower geometry
● Lattice tower families
● Analysis of truss members, two dimensional and three dimensional, Matrix methods of analysis, Tower analysis details with Software, Illustrated Examples
● Buckling capacity of compression members, Euler's Theory, Buckling modes
● Compression capacity of angle members, Global and local buckling, K factors, ASCE 10-97 code, Illustrated examples
● Tension capacity of angle members, ASCE 10-97 code, Illustrated examples
● Connection capacity under shear and bearing, ASCE 10-97 code, Illustrated examples
● Other design considerations, ASCE 10-97 code, Tower weight calculations, Tower testing, Analysis of existing lattice towers
Weeks 4 &5: Poles
● Analysis and Design of wood poles, NESC code requirements, Illustrated examples
● Design of steel poles_polygonal tubular members and round members, ASCE 48-05, Illustrated examples
● Design consideration of anchor bolts, Wood equivalent steel poles: Wood vs. steel poles: economic considerations, life-cycle costs
Week 5: Guyed Structures
● Configurations of guyed structures: Single poles (Steel poles, Wood poles), Stub poles, Multi-pole structures,
● Analysis and deign of guyed structures, Compression capacity, Illustrated examples
Module 3: Foundations (3 weeks)
Week 6: Geotechnical Properties
● Methods of determining soil properties: SPT test, Pressuremeter test and other tests, Empirical correlations, Soil boring log interpretation
& Foundations under Compression
● Bearing capacity theories, guidelines for bearing capacity of soils
● Applications: Footings, Grillages, Worked-out examples, software and codes
Week 7: Foundations under Lateral load
● Lateral capacity theories: Broms and Brinch-Hansen theories, Elastic analysis
● Direct Embedment foundations, worked-out examples
● Concrete Pier foundations, worked-out examples
& Foundations under Uplift
● Uplift capacity theories
● Applications: H-frames, Grillages, Footings, Screw foundations, worked-out examples, software and codes
Week 8: Conductors and Conductor Motion
● Conductor Stress-Strain-Creep Chart.
● Conductor Galloping & control.
● Conductor Vibration & control | {"url":"http://www.gonzaga.edu/Academics/Colleges-and-Schools/School-of-Engineering-and-Applied-Science/Majors-Programs/Transmission-Distribution/Courses/TADP640TransmissionLineDesignAdvanced-print.asp","timestamp":"2014-04-17T18:33:28Z","content_type":null,"content_length":"12144","record_id":"<urn:uuid:02a62d77-fc6c-4aa4-b9db-a4b8bdd0dca4>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00655-ip-10-147-4-33.ec2.internal.warc.gz"} |
EPA On-line Tools for Site Assessment Calculation
Estimated Henry's Law Constants
Background: Henry's Law Constants characterize the equilibrium distribution of dilute concentrations of volatile, soluble chemicals between gas and liquid. For this calculator, the liquid is water.
Temperature-dependence is calculated by two methods: one developed by the EPA Office of Solid Waste and Emergency Response and the other published in the journal Ground Water and written by John
Washington in 1996. Background information on each method is given on a separate page.
Special background information on methyl tert-butyl ether (MTBE) is available
1) Chemicals are only included if there is data for the temperature-dependence calculation. Henry's Constants for many petroleum hydrocarbons and oxygenated additives are available from a data set of
estimated properties. For other chemicals see the chemical properties page.
2) The unit choices for Henry's constants include atm-m^3/mol, atm, and two separate dimensionless values. These are listed below.
3) Previously the calculator contained some single-temperature values. These have been eliminated from the calculation but are available for reference.
Unit Choices for the Henry's Law Constant
H [cc] = Concentration/Concentration (dimensionless--volumetric basis) ^1
H [yx] = Mole Fraction Y / Mole Fraction X (dimensionless)
H [px] = Partial Pressure / Mole Fraction X (atmospheres)
H [pc] = Partial Pressure / Solubility (atm m ^3 /mol)
^1The dimensionless form based on concentrations (volumetric basic) is the most commonly used of the dimensionless values. See Staudinger and Roberts, 1996, A Critical Review of Henry's Law Constants
for Environmental Applications, in Critical Reviews in Environmental Science and Technology, 26(3):205-297 for more information on various units (specifically page 292).
Home | Glossary | Notation | Links | References | Calculators | {"url":"http://www.epa.gov/athens/learn2model/part-two/onsite/esthenry.html","timestamp":"2014-04-19T22:52:55Z","content_type":null,"content_length":"25375","record_id":"<urn:uuid:7a4f7fff-996e-4e7c-8b86-3d6c2e192fa5>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00646-ip-10-147-4-33.ec2.internal.warc.gz"} |
CLAWPACK Bibliography
Bibliography of papers using CLAWPACK
or related wave-propagation methods
J. R. Appel. Sensitivity calculations for conservation laws with application to discontinuous fluid flow. PhD thesis, Virginia Polytechnic Institute, 1997.
J. D. Au, D. Reitebuch, M. Torrilhon, and W. Weiss. The Riemann-problem in extended thermodynamics. In H.~{Freist\"uhler} and G.~Warnecke, editors, Proc. 8'th Intl. Conf. on Hyperbolic Problems,
pages 79-88. {Birkh\"auser}, 2000.
L. Andersson, H. van Elst, E. C. Lim, and C. Uggla. Asymptotic silence of generic cosmological singularities. arXiv:gr-qc/0402051, 2005.
L. Andersson, H. van Elst, and C. Uggla. Gowdy phenominology in scale-invariant variables. Class. Quantum Grav., 21:S29-S57, 2004.
D. S. Bale. Wave propagation algorithms on curved manifolds with applications to relativistic hydrodynamics. PhD thesis, University of Washington, 2002. {\\ \verb+http://faculty.washington.edu/
J. M. Bardeen and L. T. Buchman. Numerical tests of evolution systems, gauge conditions, and boundary conditions for 1D colliding gravitational plane waves. Phys. Rev. D, 65:064037-1, 2002.
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C. Zhang and R. J. LeVeque. Immersed interface methods for wave equations with discontinuous coefficients. Wave Motion, 25:237-263, 1997. | {"url":"http://depts.washington.edu/clawpack/clawpack-4.3/bib.html","timestamp":"2014-04-25T09:21:49Z","content_type":null,"content_length":"28629","record_id":"<urn:uuid:c9a94c4d-46c9-40fb-9285-306151524e48>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00632-ip-10-147-4-33.ec2.internal.warc.gz"} |
Classes for Bit manipulation
Go to the previous, next section.
libg++ provides several different classes supporting the use and manipulation of collections of bits in different ways.
• Class Integer provides "integer" semantics. It supports manipulation of bits in ways that are often useful when treating bit arrays as numerical (integer) quantities. This class is described
• Class BitSet provides "set" semantics. It supports operations useful when treating collections of bits as representing potentially infinite sets of integers.
• Class BitSet32 supports fixed-length BitSets holding exactly 32 bits.
• Class BitSet256 supports fixed-length BitSets holding exactly 256 bits.
• Class BitString provides "string" (or "vector") semantics. It supports operations useful when treating collections of bits as strings of zeros and ones.
These classes also differ in the following ways:
• BitSets are logically infinite. Their space is dynamically altered to adjust to the smallest number of consecutive bits actually required to represent the sets. Integers also have this property.
BitStrings are logically finite, but their sizes are internally dynamically managed to maintain proper length. This means that, for example, BitStrings are concatenatable while BitSets and
Integers are not.
• BitSet32 and BitSet256 have precisely the same properties as BitSets, except that they use constant fixed length bit vectors.
• While all classes support basic unary and binary operations ~, &, |, ^, -, the semantics differ. BitSets perform bit operations that precisely mirror those for infinite sets. For example,
complementing an empty BitSet returns one representing an infinite number of set bits. Operations on BitStrings and Integers operate only on those bits actually present in the representation. For
BitStrings and Integers, the the & operation returns a BitString with a length equal to the minimum length of the operands, and |, ^ return one with length of the maximum.
• Only BitStrings support substring extraction and bit pattern matching.
BitSets are objects that contain logically infinite sets of nonnegative integers. Representational details are discussed in the Representation chapter. Because they are logically infinite, all
BitSets possess a trailing, infinitely replicated 0 or 1 bit, called the "virtual bit", and indicated via 0* or 1*.
BitSet32 and BitSet256 have they same properties, except they are of fixed length, and thus have no virtual bit.
BitSets may be constructed as follows:
BitSet a;
declares an empty BitSet.
BitSet a = atoBitSet("001000");
sets a to the BitSet 0010*, reading left-to-right. The "0*" indicates that the set ends with an infinite number of zero (clear) bits.
BitSet a = atoBitSet("00101*");
sets a to the BitSet 00101*, where "1*" means that the set ends with an infinite number of one (set) bits.
BitSet a = longtoBitSet((long)23);
sets a to the BitSet 111010*, the binary representation of decimal 23.
BitSet a = utoBitSet((unsigned)23);
sets a to the BitSet 111010*, the binary representation of decimal 23.
The following functions and operators are provided (Assume the declaration of BitSets a = 0011010*, b = 101101*, throughout, as examples).
~a returns the complement of a, or 1100101* in this case.
sets a to ~a.
a & b; a &= b;
returns a intersected with b, or 0011010*.
a | b; a |= b;
returns a unioned with b, or 1011111*.
a - b; a -= b;
returns the set difference of a and b, or 000010*.
a ^ b; a ^= b;
returns the symmetric difference of a and b, or 1000101*.
returns true if a is an empty set.
a == b;
returns true if a and b contain the same set.
a <= b;
returns true if a is a subset of b.
a < b;
returns true if a is a proper subset of b;
a != b; a >= b; a > b;
are the converses of the above.
sets the 7th (counting from 0) bit of a, setting a to 001111010*
clears the 2nd bit bit of a, setting a to 00011110*
clears all bits of a;
sets all bits of a;
complements the 0th bit of a, setting a to 10011110*
sets the 0th through 1st bits of a, setting a to 110111110* The two-argument versions of clear and invert are similar.
returns true if the 3rd bit of a is set.
a.test(3, 5)
returns true if any of bits 3 through 5 are set.
int i = a[3]; a[3] = 0;
The subscript operator allows bits to be inspected and changed via standard subscript semantics, using a friend class BitSetBit. The use of the subscript operator a[i] rather than a.test(i)
requires somewhat greater overhead.
a.first(1) or a.first()
returns the index of the first set bit of a (2 in this case), or -1 if no bits are set.
returns the index of the first clear bit of a (0 in this case), or -1 if no bits are clear.
a.next(2, 1) or a.next(2)
returns the index of the next bit after position 2 that is set (3 in this case) or -1. first and next may be used as iterators, as in for (int i = a.first(); i >= 0; i = a.next(i))....
returns the index of the rightmost set bit, or -1 if there or no set bits or all set bits.
a.prev(3, 0)
returns the index of the previous clear bit before position 3.
returns the number of set bits in a, or -1 if there are an infinite number.
returns the trailing (infinitely replicated) bit of a.
a = atoBitSet("ababX", 'a', 'b', 'X');
converts the char* string into a bitset, with 'a' denoting false, 'b' denoting true, and 'X' denoting infinite replication.
a.printon(cout, '-', '.', 0)
prints a to cout represented with '-' for falses, '.' for trues, and no replication marker.
cout << a
prints a to cout (representing lases by 'f', trues by 't', and using '*' as the replication marker).
diff(x, y, z)
A faster way to say z = x - y.
and(x, y, z)
A faster way to say z = x & y.
or(x, y, z)
A faster way to say z = x | y.
xor(x, y, z)
A faster way to say z = x ^ y.
complement(x, z)
A faster way to say z = ~x.
BitStrings are objects that contain arbitrary-length strings of zeroes and ones. BitStrings possess some features that make them behave like sets, and others that behave as strings. They are useful
in applications (such as signature-based algorithms) where both capabilities are needed. Representational details are discussed in the Representation chapter. Most capabilities are exact analogs of
those supported in the BitSet and String classes. A BitSubString is used with substring operations along the same lines as the String SubString class. A BitPattern class is used for masked bit
pattern searching.
Only a default constructor is supported. The declaration BitString a; initializes a to be an empty BitString. BitStrings may often be initialized via atoBitString and longtoBitString.
Set operations ( ~, complement, &, &=, |, |=, -, ^, ^=) behave just as the BitSet versions, except that there is no "virtual bit": complementing complements only those bits in the BitString, and all
binary operations across unequal length BitStrings assume a virtual bit of zero. The & operation returns a BitString with a length equal to the minimum length of the operands, and |, ^ return one
with length of the maximum.
Set-based relational operations (==, !=, <=, <, >=, >) follow the same rules. A string-like lexicographic comparison function, lcompare, tests the lexicographic relation between two BitStrings. For
example, lcompare(1100, 0101) returns 1, since the first BitString starts with 1 and the second with 0.
Individual bit setting, testing, and iterator operations (set, clear, invert, test, first, next, last, prev) are also like those for BitSets. BitStrings are automatically expanded when setting bits
at positions greater than their current length.
The string-based capabilities are just as those for class String. BitStrings may be concatenated (+, +=), searched (index, contains, matches), and extracted into BitSubStrings (before, at, after)
which may be assigned and otherwise manipulated. Other string-based utility functions (reverse, common_prefix, common_suffix) are also provided. These have the same capabilities and descriptions as
those for Strings.
String-oriented operations can also be performed with a mask via class BitPattern. BitPatterns consist of two BitStrings, a pattern and a mask. On searching and matching, bits in the pattern that
correspond to 0 bits in the mask are ignored. (The mask may be shorter than the pattern, in which case trailing mask bits are assumed to be 0). The pattern and mask are both public variables, and may
be individually subjected to other bit operations.
Converting to char* and printing ((atoBitString, atoBitPattern, printon, ostream <<)) are also as in BitSets, except that no virtual bit is used, and an 'X' in a BitPattern means that the pattern bit
is masked out.
The following features are unique to BitStrings.
Assume declarations of BitString a = atoBitString("01010110") and b = atoBitSTring("1101").
a = b + c;
Sets a to the concatenation of b and c;
a = b + 0; a = b + 1;
sets a to b, appended with a zero (one).
a += b;
appends b to a;
a += 0; a += 1;
appends a zero (one) to a.
a << 2; a <<= 2
return a with 2 zeros prepended, setting a to 0001010110. (Note the necessary confusion of << and >> operators. For consistency with the integer versions, << shifts low bits to high, even though
they are printed low bits first.)
a >> 3; a >>= 3
return a with the first 3 bits deleted, setting a to 10110.
deletes all 0 bits on the left of a, setting a to 1010110.
deletes all trailing 0 bits of a, setting a to 0101011.
cat(x, y, z)
A faster way to say z = x + y.
diff(x, y, z)
A faster way to say z = x - y.
and(x, y, z)
A faster way to say z = x & y.
or(x, y, z)
A faster way to say z = x | y.
xor(x, y, z)
A faster way to say z = x ^ y.
lshift(x, y, z)
A faster way to say z = x << y.
rshift(x, y, z)
A faster way to say z = x >> y.
complement(x, z)
A faster way to say z = ~x.
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word problem (HELP!!) 'a field is to be enclosed by a fence' [Archive] - Free Math Help Forum
hi this is a special question my teacher gave me see if you can help
a rectangular field is to be enclosed by a fence and then divided into 2 smaller plots by a fence parallel to one of the sides. find the dimensions of the field of greatest possible area if 1800 m of
fence is available. state the total area
if you figure it out i will be very thankful | {"url":"http://www.freemathhelp.com/forum/archive/index.php/t-44862.html","timestamp":"2014-04-18T03:09:49Z","content_type":null,"content_length":"5079","record_id":"<urn:uuid:19297985-d3ec-48e2-92e2-22020f5d1167>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00475-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Math Help
October 9th 2006, 10:21 AM #1
Junior Member
Jul 2006
Let (xn) and (yn) be two sequences. Let (zn) be the shuffled sequence:
z1=x1, z2=y1, z3=x2, z4=y2.....z(2n-1)=xn, z(2n)=yn...
Prove that (zn) converges, which implies that both (xn) and (yn) converge.
Can someone help?
x_n = n
y_n = n^2
None of the sequences x, y, z converge.
Do you mean the question to say that "if {z_n} converges then {x_n}, {y_n} converge"?
I did not try to solve this problem but I am thinking:
If the sequence {z_n} converges then the odd and even subsequences converge that is,
z_1,z_3,z_5,... -----> Converges
z_2,z_4,z_6,... -----> Converges
But the odd-even subsequences are the sequences x_n and y_n. Thus, {x_n} and {y_n} both converges.
Is that not true?
October 9th 2006, 12:43 PM #2
October 9th 2006, 01:11 PM #3
Global Moderator
Nov 2005
New York City
October 10th 2006, 01:42 AM #4
Grand Panjandrum
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The Reciprocal of a Number
Year 7 Interactive Maths - Second Edition
The Reciprocal of a Number
One number is the reciprocal of another if their product is 1.
In general:
The reciprocal of a fraction is obtained by interchanging the numerator and the denominator, i.e. by inverting the fraction.
Example 7
Find the reciprocal of 20.
Example 8
Example 9
To find the reciprocal of a mixed number, change it into an improper fraction and then invert it.
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Section: Melting user-guide (1) Updated: 2009 April 01 Local index Up
melting - nearest-neighbor computation of nucleic acid hybridation
melting [options]
Melting computes, for a nucleic acid duplex, the enthalpy and the entropy of the helix-coil transition, and then its melting temperature. Three types of hybridisation are possible: DNA/DNA, DNA/RNA,
and RNA/RNA. The program uses the method of nearest-neighbors. The set of thermodynamic parameters can be easely changed, for instance following an experimental breakthrough. Melting is a free
program in both sense of the term. It comes with no cost and it is open-source. In addition it is coded in ISO C and can be compiled on any operating system. Some perl scripts are provided to show
how melting can be used as a block to construct more ambitious programs.
The options are treated sequentially. If there is a conflict between the value of two options, the latter normally erases the former.
Informs the program to use file.nn as an alternative set of nearest-neighbor parameters, rather than the default for the specified hybridisation type. The standard distribution of melting
provides some files ready-to-use: all97a.nn (Allawi et al 1997), bre86a.nn (Breslauer et al 1986), san96a.nn (SantaLucia et al 1996), sug96a.nn (Sugimoto et al 1996) san04a.nn (Santalucia et al
2004) (DNA/DNA), fre86a.nn (Freier et al 1986), xia98a.nn (Xia et al 1998), (RNA/RNA), and sug95a.nn (Sugimoto et al 1995), (DNA/RNA).
The program will look for the file in a directory specified during the installation. However, if an environment variable NN_PATH is defined, melting will search in this one first. Be careful, the
option -A changes the default parameter set defined by the option -H.
Enters the complementary sequence, from 3' to 5'. This option is mandatory if there are mismatches between the two strands. If it is not used, the program will compute it as the complement of the
sequence entered with the option -S.
Informs the program to use the file dnadnade.nn to compute the contribution of dangling ends to the thermodynamic of helix-coil transition. The dangling ends are not taken into account by the
approximative mode.
This is the a correction factor used to modulate the effect of the nucleic acid concentration in the computation of the melting temperature. See section ALGORITHM for details.
Magnesium concentration (No maximum concentration for the moment). The effect
of ions on thermodynamic stability of nucleic acid duplexes is complex,
and the correcting functions are at best rough approximations.The published
Tm correction formula for divalent Mg2+ ions of Owczarzy et al(2008) can
take in account the competitive binding of monovalent and divalent ions on DNA.
However this formula is only for DNA duplexes.
-h Displays a short help and quit with EXIT_SUCCESS.
Specifies the hybridisation type. This will set the nearest-neighbor set to use if no alternative set is provided by the option -A (remember the options are read sequentially). Moreover this
parameter determines the equation to use if the sequence length exceeds the limit of application of the nearest-neighbor approach (arbitrarily set up by the author). Possible values are dnadna,
dnarna and rnadna (synonymous), and rnarna. For reasons of compatibility the values of the previous versions of melting A,B,C,F,R,S,T,U,W are still available although strongly deprecated. Use the
option -A to require an alternative set of thermodynamic parameters. IMPORTANT: If the duplex is a DNA/RNA heteroduplex, the sequence of the DNA strand has to be entered with the option -S.
Provides the name of an input file containing the parameters of the run. The input has to contain one parameter per line, formatted as in the command line. The order is not important, as well as
blank lines. example:
Informs the program to use file.nn as an alternative set of inosine pair
parameters, rather than the default for the specified hybridisation type.
The standard distribution of melting provides some files ready-to-use: san05a.nn
(Santalucia et al 2005) for deoxyinosine in DNA duplexes, bre07a.nn (Brent M Znosko
et al 2007)for inosine in RNA duplexes. Note that not all the inosine mismatched
wobble's pairs have been investigated. Therefore it could be impossible to compute
the Tm of a duplex with inosine pairs. Moreover, those inosine pairs are not taken
into account by the approximative mode.
Permits to chose another correction for the concentration in sodium. Currently, one can chose between wet91a, san96a, san98a. See section ALGORITHM. TP. BI. "-k" "x.xxe-xx"
Potassium concentration (No maximum concentration for the moment). The effect of ions
on thermodynamic stability of nucleic acid duplexes is complex, and the correcting
functions are at best rough approximations.The published Tm correction formula for
sodium ions of Owczarzy et al (2008)is therefore also applicable to buffers containing Tris or
KCl. Monovalent K+, Na+, Tris+ ions stabilize DNA duplexes
with similar potency, and their effects on duplex stability are additive. However this formula
is only for DNA duplexes.
-L Prints the legal informations and quit with EXIT_SUCCESS.
Informs the program to use the file dnadnamm.nn to compute the contribution of mismatches to the thermodynamic of helix-coil transition. Note that not all the mismatched Crick's pairs have been
investigated. Therefore it could be impossible to compute the Tm of a mismatched duplex. Moreover, those mismatches are not taken into account by the approximative mode.
Sodium concentration (between 0 and 10 M). The effect of ions on thermodynamic
stability of nucleic acid duplexes is complex, and the correcting functions
are at best rough approximations. Moreover, they are generally reliable only
for [Na+] belonging to [0.1,10M]. If there are no other ions in
solution, we can use only the sodium correction. In the other case, we use the Owczarzy's
The output is directed to this file instead of the standard output. The name of the file can be omitted. An automatic name is then generated, of the form meltingYYYYMMMDD_HHhMMm.out (of course,
on POSIX compliant systems, you can emulate this with the redirection of stdout to a file constructed with the program date).
Concentration of the nucleic acid strand in excess (between 0 and 0.1 M).
-p Return the directory supposed to contain the sets of calorimetric parameters and quit with EXIT_SUCCESS. If the environment variable NN_PATH is set, it is returned. Otherwise, the value defined
by default during the compilation is returned.
-q Turn off the interactive correction of wrongly entered parameter. Useful for run through a server, or a batch script. Default is OFF (i.e. interactive on). The switch works in both sens.
Therefore if -q has been set in an input file, another -q on the command line will switch the quiet mode OFF (same thing if two -q are set on the same command line).
Sequence of one strand of the nucleic acid duplex, entered 5' to 3'. IMPORTANT: If it is a DNA/RNA heteroduplex, the sequence of the DNA strand has to be entered. Uridine and thymidine are
considered as identical. The bases can be upper or lowercase.
Size threshold before approximative computation. The nearest-neighbour approach will be used only if the length of the sequence is inferior to this threshold.
Tris buffer concentration (No maximum concentration for the moment).
The effect of ions on thermodynamic stability of nucleic acid
duplexes is complex, and the correcting functions are at best
rough approximations.The published Tm correction formula for sodium ions of
Owczarzy et al(2008)is therefore also applicable to buffers containing Tris or
KCl. Monovalent K+, Na+, Tris+ ions stabilize DNA duplexes with similar potency, and
their effects on duplex stability are additive. However this formula is only for DNA
duplexes. Be careful, the Tris+ ion concentration is about half of the total tris buffer
-v Control the verbose mode, issuing a lot more information about the current run (try it once to see if you can get something interesting). Default is OFF. The switch works in both sens. Therefore
if -v has been set in an input file, another -v on the command line will switch the verbose mode OFF (same thing if two -v are set on the same command line).
-V Displays the version number and quit with EXIT_SUCCESS.
-x Force the program to compute an approximative tm, based on G+C content. This option has to be used with caution. Note that such a calcul is increasingly incorrect when the length of the duplex
decreases. Moreover, it does not take into account nucleic acid concentration, which is a strong mistake.
Thermodynamics of helix-coil transition of nucleic acid
The nearest-neighbor approach is based on the fact that the helix-coil transition works as a zipper. After an initial attachment, the hybridisation propagates laterally. Therefore, the process
depends on the adjacent nucleotides on each strand (the Crick's pairs). Two duplexes with the same base pairs could have different stabilities, and on the contrary, two duplexes with different
sequences but identical sets of Crick's pairs will have the same thermodynamics properties (see Sugimoto et al. 1994). This program first computes the hybridisation enthalpy and entropy from the
elementary parameters of each Crick's pair.
DeltaH = deltaH(initiation) + SUM(deltaH(Crick's pair))
DeltaS = deltaS(initiation) + SUM(deltaS(Crick's pair))
See Wetmur J.G. (1991) and SantaLucia (1998) for deep reviews on the nucleic acid hybridisation and on the different set of nearest-neighbor parameters.
Effect of mismatches and dangling ends
The mismatching pairs are also taken into account. However the thermodynamic parameters are still not available for every possible cases (notably when both positions are mismatched). In such a case,
the program, unable to compute any relevant result, will quit with a warning.
The two first and positions cannot be mismatched. in such a case, the result is unpredictable, and all cases are possible. for instance (see Allawi and SanLucia 1997), the duplex
A T
T A
is more stable than
The dangling ends, that is the umatched terminal nucleotides, can be taken into account.
) = DeltaH(AG/-C)+DeltaH(A-/TT)
+Delta(AT/TG mismatch) +DeltaG(TC/GG mismatch)
(The same computation is performed for DeltaS)
The melting temperature
Then the melting temperature is computed by the following formula:
Tm = DeltaH / (DeltaS + Rx ln ([nucleic acid]/F))
Tm in K (for [Na+] = 1 M )
+ f([Na+]) - 273.15
correction for the salt concentration (if there are only sodium cations in the solution)and to get the temperature in degree Celsius. (In fact some corrections are directly included in the DeltaS see
that of SanLucia 1998)
Correction for the concentration of nucleic acid
If the concentration of the two strands are similar, F is 1 in case of self-complementary oligonucleotides, 4 otherwise. If one strand is in excess (for instance in PCR experiment), F is 2 (Actually
the formula would have to use the difference of concentrations rather than the total concentration, but if the excess is sufficient, the total concentration can be assumed to be identical to the
concentration of the strand in excess).
Note however, MELTING makes the assumption of no self-assembly, i.e. the computation does not take any entropic term to correct for self-complementarity.
Correction for the concentration of salt
If there are only sodium ions in the solution, we can use the following corrections:
The correction can be chosen between wet91a, presented in Wetmur 1991 i.e.
16.6 x log([Na+] / (1 + 0.7 x [Na+])) + 3.85
san96a presented in SantaLucia et al. 1996 i.e.
12.5 x log[Na+]
and san98a presented in SantaLucia 1998 i.e. a correction of the entropic term without modification of enthalpy
DeltaS = DeltaS([Na+]=1M) + 0.368 x (N-1) x ln[Na+]
Where N is the length of the duplex (SantaLucia 1998 actually used 'N' the number of non-terminal phosphates, that is effectively equal to our N-1). CAUTION, this correction is meant to correct
entropy values expressed in cal.mol-1.K-1!!!
Correction for the concentration of ions when other monovalent ions such as Tris+ and K+ or divalent Mg2+ ions are added
If there are only Na+ ions, we can use the correction for the concentration of salt(see above). In the opposite case , we will use the magnesium and monovalent ions correction from Owczarzy et al
(2008). (only for DNA duplexes)
[Mon+] = [Na+] + [K+] + [Tris+]
Where [Tris+] = [Tris buffer]/2. (in the option -t, it is the Tris buffer concentration which is entered).
If [Mon+] = 0, the divalent ions are the only ions present
and the melting temperature is :
1/Tm(Mg2+) = 1/Tm(1M Na+) + a - b x ln([Mg2+]) + Fgc x (c + d x ln([Mg2+]) + 1/(2 x (Nbp - 1)) x (- e +f x ln([Mg2+]) + g x ln([Mg2+]) x ln([Mg2+]))
where : a = 3.92/100000. b = 9.11/1000000. c = 6.26/100000. d = 1.42/100000. e = 4.82/10000. f = 5.25/10000. g = 8.31/100000. Fgc is the fraction of GC base pairs in the sequence and Nbp is the
length of the sequence (Number of base pairs).
If [Mon+] > 0, there are several cases because we can have a competitive DNA binding between monovalent and divalent cations :
If the ratio [Mg2+]^(0.5)/[Mon+] is inferior to 0.22, monovalent ion influence is dominant, divalent cations can be disregarded and the melting temperature is :
1/Tm(Mg2+) = 1/Tm(1M Na+) + (4.29 x Fgc - 3.95) x 1/100000 x ln([mon+]) + 9.40 x 1/1000000 x ln([Mon+]) x ln([Mon+])
where : Fgc is the fraction of GC base pairs in the sequence.
If the ratio [Mg2+]^(0.5)/[Mon+] is included in [0.22, 6[, we must take in account both Mg2+ and monovalent cations concentrations. The melting temperature is :
1/Tm(Mg2+) = 1/Tm(1M Na+) + a - b x ln([Mg2+]) + Fgc x (c + d x ln([Mg2+]) + 1/(2 x (Nbp - 1)) x (- e + f x ln([Mg2+]) + g x ln([Mg2+]) x ln([Mg2+]))
where : a = 3.92/100000 x (0.843 - 0.352 x [Mon+]0.5 x ln([Mon+])).
b = 9.11/1000000. c = 6.26/100000.
d = 1.42/100000 x (1.279 - 4.03/1000 x ln([mon+]) - 8.03/1000 x ln([mon+] x ln([mon+]).
e = 4.82/10000.
f = 5.25/10000.
g = 8.31/100000 x (0.486 - 0.258 x ln([mon+]) + 5.25/1000 x ln([mon+] x ln([mon+] x ln([mon+]).
Fgc is the fraction of GC base pairs in the sequence and Nbp is the length of the sequence (Number of base pairs).
Finally, if the ratio [Mg2+]^(0.5)/[Mon+] is superior to 6, divalent ion influence is dominant, monovalent cations can be disregarded and the melting temperature is :
1/Tm(Mg2+) = 1/Tm(1M Na+) + a - b x ln([Mg2+]) + Fgc x (c + d x ln([Mg2+]) + 1/(2 x (Nbp - 1)) x (- e + f x ln([Mg2+]) + g x ln([Mg2+]) x ln([Mg2+]))
where : a = 3.92/100000. b = 9.11/1000000. c = 6.26/100000. d = 1.42/100000. e = 4.82/10000. f = 5.25/10000. g = 8.31/100000.
Fgc is the fraction of GC base pairs in the sequence and Nbp is the length of the sequence (Number of base pairs).
Long sequences
It is important to realise that the nearest-neighbor approach has been established on small oligonucleotides. Therefore the use of melting in the non-approximative mode is really accurate only for
relatively short sequences (Although if the sequences are two short, let's say < 6 bp, the influence of extremities becomes too important and the reliability decreases a lot). For long sequences an
approximative mode has been designed. This mode is launched if the sequence length is higher than the value given by the option -T (the default threshold is 60 bp).
The melting temperature is computed by the following formulas:
Tm = 81.5+16.6*log10([Na+]/(1+0.7[Na+]))+0.41%GC-500/size
Tm = 67+16.6*log10([Na+]/(1.0+0.7[Na+]))+0.8%GC-500/size
Tm = 78+16.6*log10([Na+]/(1.0+0.7[Na+]))+0.7%GC-500/size
This mode is nevertheless strongly disencouraged.
Miscellaneous comments
Melting is currently accurate only when the hybridisation is performed at pH 71.
The computation is valid only for the hybridisations performed in aqueous medium. Therefore the use of denaturing agents such as formamide completely invalidates the results.
Allawi H.T., SantaLucia J. (1997). Thermodynamics and NMR of internal G.T mismatches in DNA. Biochemistry 36: 10581-10594
Allawi H.T., SantaLucia J. (1998). Nearest Neighbor thermodynamics parameters for internal G.A mismatches in DNA. Biochemistry 37: 2170-2179
Allawi H.T., SantaLucia J. (1998). Thermodynamics of internal C.T mismatches in DNA. Nucleic Acids Res 26: 2694-2701.
Allawi H.T., SantaLucia J. (1998). Nearest Neighbor thermodynamics of internal A.C mismatches in DNA: sequence dependence and pH effects. Biochemistry 37: 9435-9444.
Bommarito S., Peyret N., SantaLucia J. (2000). Thermodynamic parameters for DNA sequences with dangling ends. Nucleic Acids Res 28: 1929-1934
Breslauer K.J., Frank R., Bl�ker H., Marky L.A. (1986). Predicting DNA duplex stability from the base sequence. Proc Natl Acad Sci USA 83: 3746-3750
Freier S.M., Kierzek R., Jaeger J.A., Sugimoto N., Caruthers M.H., Neilson T., Turner D.H. (1986). Improved free-energy parameters for predictions of RNA duplex stability. Biochemistry 83:9373-9377
Owczarzy R., Moreira B.G., You Y., Behlke M.B., Walder J.A. (2008) Predicting stability of DNA duplexes in solutions containing Magnesium and Monovalent Cations. Biochemistry 47: 5336-5353.
Peyret N., Seneviratne P.A., Allawi H.T., SantaLucia J. (1999). Nearest Neighbor thermodynamics and NMR of DNA sequences with internal A.A, C.C, G.G and T.T mismatches. dependence and pH effects.
Biochemistry 38: 3468-3477
SantaLucia J. Jr, Allawi H.T., Seneviratne P.A. (1996). Improved nearest-neighbor parameters for predicting DNA duplex stability. Biochemistry 35: 3555-3562
Sugimoto N., Katoh M., Nakano S., Ohmichi T., Sasaki M. (1994). RNA/DNA hybrid duplexes with identical nearest-neighbor base-pairs hve identical stability. FEBS Letters 354: 74-78
Sugimoto N., Nakano S., Katoh M., Matsumura A., Nakamuta H., Ohmichi T., Yoneyama M., Sasaki M. (1995). Thermodynamic parameters to predict stability of RNA/DNA hybrid duplexes. Biochemistry 34:
Sugimoto N., Nakano S., Yoneyama M., Honda K. (1996). Improved thermodynamic parameters and helix initiation factor to predict stability of DNA duplexes. Nuc Acids Res 24: 4501-4505
Watkins N.E., Santalucia J. Jr. (2005). Nearest-neighbor t- hermodynamics of deoxyinosine pairs in DNA duplexes. Nucleic Acids Research 33: 6258-6267
Wright D.J., Rice J.L., Yanker D.M., Znosko B.M. (2007). Nearest neighbor parameters for inosine-uridine pairs in RNA duplexes. Biochemistry 46: 4625-4634
Xia T., SantaLucia J., Burkard M.E., Kierzek R., Schroeder S.J., Jiao X., Cox C., Turner D.H. (1998). Thermodynamics parameters for an expanded nearest-neighbor model for formation of RNA duplexes
with Watson-Crick base pairs. Biochemistry 37: 14719-14735
For review see:
SantaLucia J. (1998) A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics. Proc Natl Acad Sci USA 95: 1460-1465
SantaLucia J., Hicks Donald (2004) The Thermodynamics of DNA structural motifs. Annu. Rev. Biophys. Struct. 33: 415 -440
Wetmur J.G. (1991) DNA probes: applications of the principles of nucleic acid hybridization. Crit Rev Biochem Mol Biol 26: 227-259
Files containing the nearest-neighbor parameters, enthalpy and entropy, for each Crick's pair. They have to be placed in a directory defined during the compilation or targeted by the environment
variable NN_PATH.
A Graphical User Interface written in Perl/Tk is available for those who prefer the 'button and menu' approach.
Scripts are available to use MELTING iteratively. For instance, the script multi.pl permits to predict the Tm of several duplexes in one shot. The script profil.pl allow an interactive
computation along a sequence, by sliding a window of specified width.
New versions and related material can be found at http://www.pasteur.fr/recherche/unites/neubiomol/meltinghome.html and at at https://sourceforge.net/projects/melting/
You can use MELTING through a web server at http://bioweb.pasteur.fr/seqanal/interfaces/melting.html
The infiles have to be ended by a blank line because otherwise the last line is not decoded.
If an infile is called, containing the address of another input file, it does not care of this latter. If it is its own address, the program quit (is it a bug or a feature?).
In interactive mode, a sequence can be entered on several lines with a backslash
If by mistake it is entered as
The backslash will be considered as an illegal character. Here again, I do not think it is actually a bug (even if it is unlikely, there is a small probability that the backslash could actually be a
mistyped base).
Melting is copyright (C) 1997, 2009 by Nicolas Le Novère and Marine Dumousseau
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.
You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
Nicolas Joly is an efficient and kind debugger and advisor. Catherine Letondal wrote the HTML interface to melting. Thanks to Nirav Merchant, Taejoon Kwon, Leo Schalkwyk, Mauro Petrillo, Andrew
Thompson, Wong Chee Hong, Ivano Zara for their bug fixes and comments. Thanks to Richard Owczarzy for his magnesium correction. Thanks to Charles Plessy for the graphical interface files. Finally
thanks to the usenet helpers, particularly Olivier Dehon and Nicolas Chuche.
Nicolas Le Novère and Marine Dumousseau, EMBL-EBI, Wellcome-Trust Genome Campus Hinxton Cambridge, CB10 1SD, UK lenov@ebi.ac.uk
See the file ChangeLog for the changes of the versions 4 and more recent.
This document was created by man2html, using the manual pages.
Time: 21:23:49 GMT, April 16, 2011 | {"url":"http://www.makelinux.net/man/1/M/melting","timestamp":"2014-04-17T12:36:08Z","content_type":null,"content_length":"38076","record_id":"<urn:uuid:26bc7618-8ba6-4007-8cc6-b9c85814fddb>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00331-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rowan University
Department of Mathematics
Course Proposal
Operations Research II
Math 03.512
I. Details
a) Course Title: Operations Research II
b) Sponsor: Dr. Chris Lacke, Department of Mathematics
c) Credit Hours: 3 credit hours
d) Course Level: Graduate
e) Curricular Effect: Bank B elective for the Master of Arts in Mathematics,
elective for the Master of Science in Engineering and
Computer Science.
f) Prerequisites: An undergraduate course in probability
1) Math 03.511 Operations Research I,
2) an undergraduate course in linear algebra
an undergraduate course in multivariate calculus
3) permission of the instructor.
g) Suggested Time, Implementation: This course will be offered once every other spring.
h) Resources: Faculty, equipment and library resources are adequate.
II. Rationale
The development of mathematical applications in business, government and the military during the 20th century has been phenomenal. It is no longer sufficient to find a solution to a problem, but
rather to develop a methodology to provide the optimal solution to a problem. In its early days, operations research was used by the British military to find the best way to move all of its forces
and equipment across the English Channel. Modern applications include portfolio theory, environmental and communication systems development, hospital staffing, and production planning. Operations
researchers use their strong mathematical background to find a best solution to a problem, given a specific set of limitations, or constraints. A vast mathematical background is required, as
operations researchers are required to build mathematical models that can involve virtually any type of mathematical function. Moreover, operations researchers need to be able to convert everyday
phenomena to mathematical expressions. The extensive growth of the Institute for Operations Research and the Management Sciences (INFORMS) during the 1980’s and 1990’s has been, in large part, to the
demand for qualified students in this field.
While there are a number of graduate programs in Operations Research, many Masters degree programs in Mathematics now include a pair of courses in Operations Research. At Rowan University, many of
our graduate students are teachers at the secondary level. Many of the concepts and tools in Operations Research are needed at some level on the High School Proficiency Assessment (HSPA), so the
class provides skills that the students can bring back to their classrooms. Moreover, the addition of Operations Research courses will increase industrial interest in our program, as those in
industry will be able to use the knowledge gained in these courses on a day in, day out basis.
This is the second member of a two-course sequence designed to provide a strong introduction in Operations Research, regardless of the student’s industrial or academic future pursuits. An
undergraduate version of this course, 1703.4XX, Stochastic Models in Operations Research, will be offered concurrently. As is often necessary in the work place, graduate students will be expected to
cover additional relevant topics outside of the class lecture. Furthermore, graduate students will be expected to produce a project that displays strong comprehension of a portion of the course’s
subject matter. Further details of the extra expectations placed on the graduate students are described in the following sections.
III. Essence of the Course
a) Objectives in Relation to Student Outcomes
Students in this course will become familiar with the process of Operations Research: learning how to create and validate a mathematical model, as well as the processes and optimization/
sub-optimization. They will be able to create and solve Markovian and general queuing models. They will also learn how to use decision analysis to make optimal decisions in the face of uncertainty.
They will learn how to determine optimal inventory policies under the assumption of variable demand. Lastly, they will learn how to model and solve a variety of problems by using computer simulation.
They will complete this process for a variety of model types; however, all of the types of modeling covered in this course will be stochastic, that is, including uncertainty. Reliance on the tools in
the Calculus, Linear Algebra and Probability will be substantial, but we will also examine the reasons why these tools provide us with an optimal solution in each scenario. In addition, we will
examine how multiple modeling procedures can be used to arrive at the same result, as well as the benefits and pitfalls of the different techniques. Furthermore, students will learn a procedure
called sensitivity analysis, which is used to determine what types of changes are necessary for our optimal solution to become sub-optimal. Use of some of the leading software in the field, which is
included in the text, will be required.
b) Topical Outline (Additional graduate topics denoted by *)
1. Markov Chains
Stochastic Processes
Discrete Time Markov Chains
Chapman-Kolmogorov Equations
Transition Matrices
Steady-State Behavior
Passage Times
Absorbing and Transient States
Continuous Time Markov Chains (Markov Processes)
2. Queuing Theory
Exponential Distribution
Birth-Death Processes
Single Server Queues
Finite, Multiple Server Queues
Infinite Server Queues*
Little’s Law
Finite and Infinite Capacity Queues
Non-Exponential Service Disciplines*
Throughput Analysis*
3. Decision Analysis
Decision Trees
Utility Theory*
Expected Utility Maximization*
4. Stochastic Inventory Theory
Continuous Review Models
Periodic Review Models
Models Involving Perishables
5. Stochastic Dynamic Programming and Markov Decision Processes
6. Reliability Theory
Parallel Systems
Series Systems
Mixed Systems
7. Simulation*
Random Number Generation
Transformation of Uniform Random Variates to another Probability Distribution
Spreadsheet Tools
Stopping Criteria
Statistical Analysis of Output
c) Evaluation and Grading
Students will be evaluated by traditional methods of homework, which will include analytic and computer-based problems, and written exams. Students will also be required to devise and complete a
substantial project. Possible projects can come from applied problems in the student’s major, an application from the individual’s place of employment, applications in relevant journals, theoretical
derivations of solutions, research on a topic not covered in the course, or in the form of annotated bibliographies. A presentation on the project will be required.
d) Course Evaluation
The course will be evaluated through customary student evaluations as well as regular departmental review.
IV) Consultation
The content and nature of this course was discussed with:
1. Dr. T. R. Chandrupatla , Department of Mechanical Engineering
2. Dr. Ralph Dusseau, Department of Civil and Environmental Engineering
3. Dr. Jennifer S. Kay, Department of Computer Science
4. Dr. Jooh Lee, Department of Management and M.I.S.
V) This proposal has been reviewed by the Department of Mathematics’ Curriculum Committee.
VI) Catalogue Description
1703.412 Operations Research II
(Prerequisites: An undergraduate course in probability and either 1703.5xx Operations Research I, an undergraduate course in linear algebra and an undergraduate course in multivariate calculus, or
permission of the instructor.)
This course is an introduction to mathematical modeling, analysis, and solution procedures applicable to decision-making problems in an uncertain (stochastic) environment. Methodologies covered
include dynamic programming, simulation, Markov chains, queuing theory, decision analysis, dynamic programming, system reliability and inventory theory. Solutions will be obtained using theoretical
methods and software packages. | {"url":"http://www.rowan.edu/colleges/csm/departments/math/syllabi/OperationsresearchII.html","timestamp":"2014-04-17T11:13:40Z","content_type":null,"content_length":"21284","record_id":"<urn:uuid:6a2e086a-e200-4705-98c9-1f30f457be5e>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00345-ip-10-147-4-33.ec2.internal.warc.gz"} |
: Survival Analysis in R
Although survival is the most used package in survival analysis, there are useful tools in other packages. R has a wide list of resources for performing survival analysis, including some specific
packages in the CRAN website: bayesSurv, cmprsk, dblcens, eha, emplink, frailtypack Icens, intcox, kinship, Kmsurv, msm, musa, relsurv, smoothSurv, subrayes, survival, survnnet, survrec and
zicount, and the general package Hmisc. There are also other functions that work with certain models in survival models, such as addreg for the Aalen's additive model. The purpose of this course
is to present many useful tools to perform a Survival Analysis, including some methodological aspects and examples. | {"url":"http://www.r-project.org/useR-2006/Tutorials/Borges.html","timestamp":"2014-04-20T13:24:10Z","content_type":null,"content_length":"3021","record_id":"<urn:uuid:5ae9fd53-8d13-460e-a35b-b2c70c21a2f5>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00531-ip-10-147-4-33.ec2.internal.warc.gz"} |
Decimal Bianry Converter - Certain Expectation for assignment
March 18th, 2012, 08:16 PM
Decimal Bianry Converter - Certain Expectation for assignment
Hello, I am stuck on this assignment:
Write a DBConverter class that has a single public static void printInBin (int n) method that implements the following algorithm: (4)
1. Let S be an empty stack of the type Integer.
2. While n is greater than 0
a. Rem = n % 2
b. Push Rem into stack S
c. n = n / 2
3. While S is not empty
a. digit = S.pop()
b. Print digit
Write a DBConverterTester that creates this array:
Integer[] numbers = { new Integer(23), new Integer(47), new Integer(257),
new Integer(1023), new Integer(0), new Integer(82), new Integer(512),
new Integer(100)};
and prints each number and then calls the printInBin method for it..
MY SOLUTION (WHAT IS WRONG):
package stack;
public class DBConverter{
public static void printInBin(int n)
ArrayStack<Integer> s = new ArrayStack<Integer>(n);
while( n > 0)
int rem = n % 2;
n = n / 2;
while ( n <= 0)
int digit = s.pop();
public static void main(String[] args)
Integer[] numbers = { new Integer(23), new Integer(47), new Integer(257),
new Integer(1023), new Integer(0), new Integer(82), new Integer(512),
new Integer(100)};
for(int i = 0; i < numbers.length; i++)
System.out.println(numbers.printInBin(i) );
March 18th, 2012, 11:33 PM
Re: Decimal Bianry Converter - Certain Expectation for assignment
WHAT IS WRONG
Perhaps *you* could say what's wrong. Specifically, does that code compile? If not, and you can't understand the compiler's message, post it and say which lines of your code it's referring to.
I'm sure someone can explain what it means.
March 19th, 2012, 05:56 AM
Re: Decimal Bianry Converter - Certain Expectation for assignment
Hello biddum1!
Code :
ArrayStack<Integer> s = new ArrayStack<Integer>(n);
Do you have an ArrayStack class? If yes, can you post it?
Code :
for(int i = 0; i < numbers.length; i++)
System.out.println(numbers.printInBin(i) );
The appropriate way to call a static method is ClassName.methodName | {"url":"http://www.javaprogrammingforums.com/%20whats-wrong-my-code/14668-decimal-bianry-converter-certain-expectation-assignment-printingthethread.html","timestamp":"2014-04-19T14:55:01Z","content_type":null,"content_length":"7293","record_id":"<urn:uuid:b4bb1dcd-5658-4ac2-aa4e-85847b6b13bf>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00467-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculation of the Effect of Machines
Coverpage of Gustave Coriolis' 1829 textbook Calculation of the Effect of Machines.
famous publications
Calculation of the Effect of Machines, or Considerations on the Use of Engines and their Evaluation
is an 1829 textbook by French physicist
Gustave Coriolis
that introduced a number of fundamental modern concepts, such as the
principle of the transmission of work
, i.e. the standard definition of
times distance, the
as a unit of work, and
kinetic energy
as one-half the mass times the squared velocity of an object. [1]
To a good extent, Coriolis' 1829 textbook seems to be the main reference to German physicist
Rudolf Clausius
’ derivation of the
energy U
of a
internal energy
in the modern sense in the
mathematical introduction
to his 1875 textbook
The Mechanical Theory of Heat
, although, to note, he does not explicitly mention Coriolis
1. (a) Coriolis, Gustave. (1829).
Calculation of the Effect of Machines, or Considerations on the Use of Engines and their Evaluation
Du Calcul de l'effet des Machines, ou Considérations sur l'emploi des Moteurs et sur Leur Evaluation
). Paris: Carilian-Goeury, Libraire.
(b) Coriolis, Gustave. (1844).
Treatise on the Mechanics of Solid Bodies and Calculation of the Effect on Machines
Traité de la Mécanique des Corps Solides et du Calcul de l'effet des Machines
) (section:
Principle of the Transmission of Work in the Movement of a Material Point
, pgs. 35-40). 2nd. Ed. Paris.
Latest page update: made by Sadi-Carnot , Aug 18 2009, 8:04 PM EDT (about this update About This Update Edited by Sadi-Carnot
9 words added
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Topic: Quick Solution to Puzzle?
Replies: 12 Last Post: Aug 13, 2013 6:11 AM
Messages: [ Previous | Next ]
Quick Solution to Puzzle?
Posted: Aug 7, 2013 8:12 AM
Hello, all. I came across this puzzle in an old computer reference book
in a chapter dealing with efficient programming techniques:
Find a 7-digit phone number (I told you the book was old) such that when
the first three digits are subtracted from the last four and the result
squared you obtain the original 7-digit number. The answer is 1201216.
I'm uncertain if there are any other integer solutions. The problem
can be reduced to a quadratic in two unknowns and numbers substituted
until the answer is obtained. Does anyone know of an alternative,
relatively fast (pencil & paper) solution? Thanks for your time and
comment. Sincerely,
J. B. Wood e-mail: arl_123234@hotmail.com
Date Subject Author
8/7/13 Quick Solution to Puzzle? J.B. Wood
8/7/13 Re: Quick Solution to Puzzle? dan73
8/7/13 Re: Quick Solution to Puzzle? dan73
8/7/13 Re: Quick Solution to Puzzle? Jim Burns
8/8/13 Re: Quick Solution to Puzzle? James Waldby
8/9/13 Re: Quick Solution to Puzzle? Jim Burns
8/8/13 Re: Quick Solution to Puzzle? Dr J R Stockton
8/9/13 Re: Quick Solution to Puzzle? Jim Burns
8/9/13 Re: Quick Solution to Puzzle? Peter Percival
8/12/13 Re: Quick Solution to Puzzle? J.B. Wood
8/12/13 Re: Quick Solution to Puzzle? Jim Burns
8/12/13 Re: Quick Solution to Puzzle? antani
8/13/13 Re: Quick Solution to Puzzle? J.B. Wood | {"url":"http://mathforum.org/kb/thread.jspa?threadID=2585237&messageID=9188302","timestamp":"2014-04-19T12:45:35Z","content_type":null,"content_length":"30574","record_id":"<urn:uuid:5214e450-96c3-4f9c-afa3-98b9f8af2ef1>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00634-ip-10-147-4-33.ec2.internal.warc.gz"} |
Algebraic Correspondences 'Expressible' as Vector Bundles
up vote 0 down vote favorite
For algebraic curves $C$ over a closed field, a correspondence on $C$ is a the same thing as a divisor, and so, a line bundle on $C \times C$. Can I assume that this simplification does not extend to
the case of general varieties? Does there exist a nice characterization of those correspondences 'expressible' as vector bundles?
In the case of finite fields, is there a bundle formation of the Frobenius correspondence $(v,$Fr$(v)) \in V \times V$ in terms of bundles?
I am particularly interested in the case of projective and Grassmannian spaces.
ag.algebraic-geometry vector-bundles
2 the Chern character induces an isomorphism $K^0(X)_{\mathbb{Q}} \simeq CH^*(X)_{\mathbb{Q}}$ so every $\mathbb{Q}$-correspondance is equivalent to a $\mathbb{Q}$-polynomial in Chern classes of
vector bundles. – YBL Jun 12 '10 at 12:32
I'm not too sure I understand. Are you suggesting that the group of correspondences is the same as $K^0(X)_{\mathbb Q}$? – Jean Delinez Jun 12 '10 at 19:03
add comment
1 Answer
active oldest votes
I am not sure what is meant by "expressible as a vector bundle". Let me just make a few elementary observations. Fulton's "Intersection theory" has a great deal of related material.
1. An algebraic correspondence is an algebraic cycle $Z$ on $X\times Y$, but there is no reason why you should get a vector bundle out of it. An algebraic cycle of codimension 1 is a Weil
divisor and a line bundle is associated with a Cartier divisor (this is a typical situation when $X$ and $Y$ are curves). These two notions generalize in different ways when codimension
is larger. In fact, even in the case of curves, Weil divisors and Cartier divisors on $X\times Y$ are different notions if $X$ or $Y$ (and hence $X\times Y$) is singular.
up vote 2
down vote 2. There is a way to associate codimension $k$ cycles (modulo some equivalence relation) to a rank $k$ vector bundle $E$. If $E$ splits into a direct sum of $k$ line bundles then we get a
scheme-theoretic intersection of $k$ codim=1 cycles corresponding to the individual summands. It is an interesting and nontrivial problem to characterize complete intersections, even
for a simple case of codim=2 cycles on the projective plane $\mathbb{P}^2$. (I realize that this isn't very relevant to your question about correspondences, but it gives you an idea of
what happens for higher rank vector bundles.)
add comment
Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry vector-bundles or ask your own question. | {"url":"http://mathoverflow.net/questions/27846/algebraic-correspondences-expressible-as-vector-bundles","timestamp":"2014-04-20T18:31:38Z","content_type":null,"content_length":"52820","record_id":"<urn:uuid:d5091a27-2ce7-4aca-aa57-938c893766f8>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00647-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Find the real positif number
Find the real positif number a such that : $<br /> \int_{0}^{a}\frac{tan(\frac{\pi }{4}+\frac{x}{2})}{sec^{2}(x)}dx=\frac{1}{16}<br />$
Hello dhiabThere are many positive solutions. Here is the first one. But it doesn't work out very neatly, so check my working! $\tan\left(\frac{\pi }{4}+\frac{x}{2}\right)=\frac{1+\tan\frac x2}{1-\
tan\frac x2}$ $=\frac{\cos\frac x2+\sin\frac x2}{\cos\frac x2-\sin\frac x2}$ $=\frac{(\cos\frac x2+\sin\frac x2)^2}{\cos^2\frac x2-\sin^2\frac x2}$ $=\frac{1+2\sin\frac x2\cos\frac x2}{\cos x}$ $=\
frac{1+\sin x}{\cos x}$ $\Rightarrow \int_0^a\frac{\tan\left(\frac{\pi }{4}+\frac{x}{2}\right)}{\sec^2x}\;dx = \int_0^a(\cos x + \cos x\sin x)\;dx$ $=\int_0^a(\cos x +\tfrac12 \sin2x)\;dx$ $=\Big[\
sin x -\tfrac14\cos2x\Big]_0^a$ $=\sin a - \tfrac14\cos2a +\tfrac14$ $=\tfrac1{16}$ $\Rightarrow 16\sin a -4(1-2\sin^2a)+4=1$ $\Rightarrow 8\sin^2a+16\sin a -1=0$ $\Rightarrow \sin a = \frac{-16+\
sqrt{256+32}}{16}$, taking the positive root $=\frac{-4+\sqrt{18}}{4}$So the first positive solution is $a = \arcsin\left(\frac{-4+\sqrt{18}}{4}\right)$ Grandad | {"url":"http://mathhelpforum.com/pre-calculus/148045-find-real-positif-number-print.html","timestamp":"2014-04-17T04:11:37Z","content_type":null,"content_length":"9168","record_id":"<urn:uuid:42ac88aa-0dfa-4244-9e20-41e440a3333d>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00169-ip-10-147-4-33.ec2.internal.warc.gz"} |
Why is the Common Core Such a Big Change?
Articulating why the Common Core State Standards are such a big change for mathematics can be challenging, especially if your audience is unfamiliar with them, like parents and other non-educators.
After all, isn’t it just changing the standards? Fellow educators also feel the stress, but may have trouble stating exactly why that is. How do you describe what is so different?
The major changes seem to fall into five categories which form the acronym SPARCK. Looking at each category, you get a better sense of why this is such a big deal.
The most obvious change is that 45 of 50 states and 5 of 6 districts or territories have adopted the Common Core State Standards in lieu of their previous standards. If you’ve ever been to a grocery
store that decided to reorganize the location of its products, then you’ll know what it feels like. At first you wonder if you are even in the right store. Looking around, you notice that some of
the items you regularly buy are in the same place you expect them. Other products have moved to locations unknown and you either have to ask for help or find yourself wandering around looking all
over for them. You even encounter new products you’ve never heard of and wonder what they’re for. This alone is a notable change and it will take teachers time to acclimate to these new
standards. Real problems include dealing with students who have content knowledge gaps as a result of the transition and learning how the grade level standards fit together to form a cohesive
Teachers are not only expected to change what they teach but how they teach it. Students must have conceptual understanding and know how to apply the mathematics in addition to having procedural
skill and fluency. Children will be expected to articulate their mathematical understandings orally and in writing. Some teachers have already been doing these things with their students for years
and will see it as a validation of all their hard work. For other teachers it will be another huge change. It is notable that many teachers were not taught this way when they were students nor was
this style of teaching emphasized when they were getting their credential. Accordingly, it is not as simple as telling someone what to do. Educators will require significant amounts of professional
development to improve.
This is the biggest change in educational assessment in decades. Previous assessments were rather straightforward: paper and pencil multiple choice tests. The new assessments from Smarter Balanced
Assessment Consortium (SBAC) and Partnership for Assessment of Readiness for College and Careers (PARCC) will be computer-based, have several types of questions including those that require students
to explain themselves in writing, and are computer adaptive (meaning that they will change in difficulty, getting harder if students get questions right and easier if they get them wrong). Most
teachers and students have never done anything like this and it will take practice to get used to it.
Some people may compare adopting the Common Core State Standards to adopting new resources like textbooks. Educators may be accustomed to using activities, lessons, or problems that were aligned to
their state’s outgoing standards, so integrating new resources will take time. This alone would be a significant change. However, normally when you adopt new resources, the standards, pedagogy,
assessments, and content knowledge needed to teach them do not also change. It is a different magnitude of adjustment.
Content Knowledge
Perhaps the biggest elephant in the room is that some educators will be expected to teach lessons on mathematics that they currently do not understand themselves. It isn’t their fault either, as
many of the new standards were not part of the curriculum when they were in school. I am certainly one of them. I was a mathematics major at the University of California, Los Angeles (UCLA).
However, in retrospect, I realize I was a math robot. I could do the mathematics well but I didn’t really understand why it worked. For example, I’ve known for years how to find the area of a
circle using the formula πr^^2. However it wasn’t until after graduating from UCLA that I had any clue where the formula came from or why it worked. How can teachers be expected to help students
fill in these missing gaps when they haven’t had the professional development to fill their own? This too will take time.
To be clear, I am a strong supporter of the Common Core State Standards. I believe they represent a large step forward for mathematics education in the United States. If educators are feeling
stressed out over this change, it is rightfully so. However, with time and support, this will get easier for all.
Tags: Common Core State Standards, Educational Policy, Smarter Balanced
1. Robert, you make many good points, but I think you miss many important ones. First, there’s little that’s new under the sun here in terms of the Practice Standards (what you are really talking
about under pedagogy). Those ideas are distilled for the most part from NCTM standards volumes dating back at least to 1989. The problem now is: 1) people who don’t have any understanding or
knowledge of the history of reform efforts in US math education think that the CCSS-M Practice Standards have arrived from some NEW thinking on the part of whomever put together these standards
and that is simply untrue; 2) ignorance, rejection, distortion, and misunderstanding of those Practice Standards (or Process Standards, as NCTM used to call them) is hardly new. It dates back to
the early 1990s and resulted in vehement anti-progressive math groups like Mathematically Correct and NYC-HOLD. Their willful misreadings of those standards informed their ceaseless
propagandizing against any meaningful reform in K-12 math teaching, fed the growth of smaller local groups throughout the country, and can still be seen, sometimes word for word, in the current
attacks on the math standards the Common Core has produced; 3) there really are problems with the new Content Standards, not the least of which is the insane and often arbitrary push-down of
topics into the primary grades, at the expense of what most developmental psychologists and lower-elementary teachers tell us about children. Of course, caring about children’s emotional health
is not on the radar screen of many current corporate deformers, sad to say. Their focus has very little to do with kids, with teachers, with parents, with schools, or anything but $ $ $ $. And
it’s damned difficult to separate in the mind of many people the truly heinous things about the overall Obama-Duncan (and Bush, and Clinton, and Bush the First and Reagan) educational policies
from specific pedagogical or content changes.
So while you’re right that we need lots of professional development to improve math education, it’s not ONLY because teachers might have to teach some math content with which they are less
familiar, though that’s a possible concern. It’s because many teachers in this country are well below where they need to be in their own mathematics knowledge to begin with, coupled with their
staunch resistance to teaching methods that require that not only kids deepen their thinking but that TEACHERS leave their procedural comfort zone and start doing a lot more thinking about math
themselves. That’s why they fought the NCTM Standards in many instances, and it’s why only a madman could believe that things are different now than they were a quarter century ago so that many
teachers will warmly embrace the Practice Standards now.
On my view, even if I could support the overall CCSSI, which unlike you I don’t, I would have to assert gross idiocy or incompetence on the part of those in charge in their failure to adequately
try to change teachers’ thinking about math and to prepare them for the requisite shifts in pedagogy that should have taken place over the last 25 years and did not. There is too much greed
coupled with too much blindness to the reality of K-12 math classrooms pushing the standards juggernaut. It is a recipe for disaster that, coupled with anti-Obama right-wing lunacy, is going to
set US math education back even further.
□ I agree with you that the Practice Standards are not new to mathematics education and that they come from the NCTM process standards as well as the National Research Council’s Adding It Up
report. However, as these standards were not adopted at the state level, they were generally not required to be implemented in the classroom. Previous state mathematics standards, for example
in California, primarily covered content and not pedagogy. Now the pedagogy is explicitly a part of what teachers will be required to do.
The other critical change is that this pedagogy will also be assessed. Previous standards may have stated that students needed rigorous mathematical understandings, but didn’t assess it on
the standardized tests so it wasn’t implemented by teachers with fidelity. I also agree with you about pedagogy training being very important as I stated under “Pedagogy” that “educators will
require significant amounts of professional development to improve.”
Thank you Michael for your thoughtful reply and for pushing the conversation further.
☆ Nice (but discouraging) breakdown of the challenges facing CCSS. Also discouraging is why it is supposed to work anyway:
“However, as these standards were not adopted at the state level, they were generally not required to be implemented in the classroom. … Now the pedagogy is explicitly a part of what
teachers will be required to do…The other critical change is that this pedagogy will also be assessed.”
“Required”, “required”, and “assessed”. What plan do we have when the elephantine teacher community sits down and just looks at us? Fire them all? Then whom do you hire? Uh-oh, CCSS just
fell on its face after wasting everyone’s energy for years when that energy could have gone into much more interesting experiments in leveraging technology and new models such as
learner-centric study.
Teachers can no longer threaten students with bad grades (thank goodness). Instead, they have to figure out conditions under which students will choose to learn. Great. Likewise CCSS
enforcers need to step up their game: when has top-down mandatory global curriculum change ever worked? Math education has tried that several times since Sputnik and accomplished nothing.
The CCSS premise is that better teaching can be squeezed out of teachers by doing nothing more than setting a new bar and standing back while teachers figure out how to jump it, which
presupposes that teachers are the problem. Lip service to teacher training is precisely that.
While CCSS is huffing and puffing its thinly veiled threats, educators on the ground are exploring ways to engage and energize students with, again technology and new models of education,
a thousand experiments in a thousand classrooms.
If CCSS thinks it is so hot, just do a convincing pilot somewhere like the rest of us explorers. Win on the merits, not by coercion.
○ Thank you for your thoughts Kenneth.
2. Appreciate the reply, Robert. My sense of where things are heading is that the generally sound ideas in the Practice Standards will be thrown under the bus by a combination of the right-wing
lunatic reactions to Common Core (e.g., “The Common Core Standards were written by Bill Ayers” – which must be a shock to Bill, who has been consistently critical of the Obama-Duncan education
policy package) and broader negative reaction to the high stakes tests and their consequences. In NY State last year, for example, we got a chance to sample what public reaction is going to look
like in June 2015. Instead of focusing on the absolutely crucial professional development, the folks in the USDOE will continue to play political/rhetorical games and fail to take useful action.
I’ve written more than once that for this set of standards to work (to the extent that it should work), a year-by-year roll-out coupled with serious training of cadre after cadre of teachers who
actually get the Practice Standards, and a moratorium on the high-stakes testing would be absolutely necessary. Didn’t happen and won’t. The horse long ago left the barn, and 10-20 years from
now, the next “Big Fix” will likely make most of the same dumb-ass mistakes this one already has. A lot of well-paid people should be giving their salaries back to the American people. And that
won’t happen, either. ;^)
□ Personally, I stay out of the politics as much as possible. I focus on ideas that everyone can rally around such as:
- “We all want students to make sense of problems”
- “We all want students to be able to articulate their understandings”
I don’t see how anyone can be against those ideas so I spend my time working on ways to make them happen.
3. But in fact, there is a lot of anger directed at both of those ideas, particularly the latter. Not only students resent having to write or explain in math class. That is one of the many buttons
groups like Mathematically Correct love to press. To them, coming up with the correct answer suffices and anything else is “fuzzy.” I don’t think it’s possible, in the long run, to avoid the
politics, even if it isn’t manifesting itself as partisan party politics. After all, some of the most vocal members of Mathematically Correct are self-proclaimed Socialists, though the majority
are libertarian, conservative, reactionary and/or Republican.
□ It has been my experience as both a classroom teacher and district math teacher specialist that any resistance students have to explaining themselves is temporary and more often stems from a
change in the status quo. In fact, I also see students who resent NOT being asked to explain themselves when they go to a math class where that is not valued.
What I find, in general, is that students love explaining the creative ways they approached a problem. They are excited by seeing how the different approaches connect and build upon each
Perhaps it isn’t possible to avoid politics in the long run, but for now I am enjoying being able to focus on helping kids make sense of mathematics and becoming more articulate members of
our society.
Thanks again for helping me broaden my thinking, Michael.
4. I am impressed with how efficiently distilled your SPARCK list is Robert. I believe it can be a very useful tool for my work with nearby districts in communicating with teachers, parents,
principles, etc. what the CCSS-M “change” means. For many, it is a large hurdle to consider an integrated rather than segmented approach to implementation. The more difficult challenges are not
yet understood.
I also greatly appreciate Michael’s comments. My first take on them is that they live in a discussion space not as immediate to classroom teachers. But, that district math leaders and
administration need to fully understand–both the radical cultural change called for, the historic difficulties of making such changes, and the backlash that will come with high levels of
I, like Michael, don’t think the “law” will make a difference. I live and work in CA where since 1997 all students were supposed to take the equivalent of Algebra in 8th grade, and that never
happened. Also in CA, the CCSS-M were adopted in 2010 and 3.5 years later I don’t believe there has been much pedagogical change at all–my focus is the HS level.
And finally, I am less certain that the CCSS-M prescribe or define a pedagogy, nor will it be assessed. I think I understand your notion that it emphasizes understanding, which to do will require
a different pedagogy. But I don’t quite see any “standard” for that pedagogy defined, or a manner in which it will be assessed. I see pedagogy as teacher actions.
However, the 1991 NCTM Professional Standards for Teaching Mathematics “present[s] a vision of what teaching should entail to support the changes in curriculum set out in the Curriculum and
Evaluation Standards. This document spells out what teachers need to know to teach toward new goals for mathematics education and how teaching should be evaluated for the purpose of improvement”
(p. vii). This to me has been one of NCTM’s finest documents, and quite sadly largely ignored.
□ Thank you Brian. You mentioned that you “don’t quite see any ‘standard’ for that pedagogy defined, or a manner in which it will be assessed.” The CCSS for Mathematics require that educators
“pursue with equal intensity” conceptual understanding, procedural skill and fluency, and applications. The standards that define what that pursuit will look like are the Standards for
Mathematical Practice (SMP). In terms of how the SMP will be assessed, in California we are part of the Smarter Balanced Assessment Consortium. They have listed four claims that each problem
they include must be connected back to. Those four claims are:
Claim #1 – Concepts & Procedures
“Students can explain and apply mathematical concepts and interpret and carry out
mathematical procedures with precision and fluency.”
Claim #2 – Problem Solving
“Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.”
Claim #3 – Communicating Reasoning
“Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.”
Claim #4 – Modeling and Data Analysis
“Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.”
These claims match up nicely with the SMP. For example, Claim 3 is nicely connected to MP3 and MP6.
So, while I do not believe that all teachers will be proactive in adjusting their pedagogy, I do believe some will and others will follow as they see that their students are not performing as
well as they hoped.
5. “3) there really are problems with the new Content Standards, not the least of which is the insane and often arbitrary push-down of topics into the primary grades, at the expense of what most
developmental psychologists and lower-elementary teachers tell us about children. Of course, caring about children’s emotional health is not on the radar screen of many current corporate
deformers, sad to say. Their focus has very little to do with kids, with teachers, with parents, with schools, or anything but $ $ $ $. And it’s damned difficult to separate in the mind of many
people the truly heinous things about the overall Obama-Duncan (and Bush, and Clinton, and Bush the First and Reagan) educational policies from specific pedagogical or content changes.”
I will agree here with Michael. From my perspective as an elementary school math specialist working to help teachers and struggling students, the point he makes above is most relevant. Take a
look at the difference between the grade 2 standards related to fractions (they’re not so easy to find) and the grade 3 fraction standards.
I also agree with Brian that this conversation is taking place in a discussion space “not as immediate to classroom teachers.” But the sad fact is that it is classroom teachers who have to deal
with the fall-out from the emotional and psychological trauma that the testing-industrial complex inflicts on their children. Perhaps if the writers of the common core standards (math and ELA
both) were not themselves part of this complex, I might view them with a different lens. But I can’t and I don’t.
□ Thanks Joe. I appreciate your comments. Yes, there is going to be a lot of fall out from the transition to the Common Core State Standards. Clearly there is no perfect solution, but I believe
that they are a step in the right direction.
Also, thank you for the kind words. I just checked out your post on the highway sign lesson. Very cool. Do you mind if I link to it from the lesson? Have you seen the new butter fraction
number line lesson?
☆ Go right ahead and link away!
I have not seen the butter fraction number line lesson yet, but will check it out.
Thanks again.
6. And Robert, I’d also like to let you know how important your work has been to mine. I’ve used the highway sign project and the movie theater project, which have been the subjects of detailed
posts at my blog: exit10.blogspot.com
I continue to mine your site for meaningful activities and thought-provoking content. Thanks!
7. Typo: that’s exit10a.blogspot.com | {"url":"http://robertkaplinsky.com/why-is-the-common-core-such-a-big-change/","timestamp":"2014-04-20T16:08:55Z","content_type":null,"content_length":"74713","record_id":"<urn:uuid:a9904ac7-fead-4ada-aea2-34ca93751c8e>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00521-ip-10-147-4-33.ec2.internal.warc.gz"} |
You limited your search to:
Degree Discipline: Mathematics
Date: August 2012
Creator: Akter, Hasina
Description: Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a
parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the
super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their
Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To
prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and
hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed
Markov systems with infinite number of edges in ...
Contributing Partner: UNT Libraries
Date: August 2011
Creator: Backs, Karl
Description: A disintegration of measure is a common tool used in ergodic theory, probability, and descriptive set theory. The primary interest in this paper is in disintegrating σ-finite measures on
standard Borel spaces into families of σ-finite measures. In 1984, Dorothy Maharam asked whether every such disintegration is uniformly σ-finite meaning that there exists a countable collection of
Borel sets which simultaneously witnesses that every measure in the disintegration is σ-finite. Assuming Gödel’s axiom of constructability I provide answer Maharam's question by constructing a
specific disintegration which is not uniformly σ-finite.
Contributing Partner: UNT Libraries
Date: August 2011
Creator: Bass, Jeremiah Joseph
Description: Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose
coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that
minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure
generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.
Contributing Partner: UNT Libraries
Date: May 2013
Creator: Cohen, Michael Patrick
Description: In this thesis we study descriptive-set-theoretic and measure-theoretic properties of Polish groups, with a thematic emphasis on the contrast between groups which are locally compact and
those which are not. The work is divided into three major sections. In the first, working jointly with Robert Kallman, we resolve a conjecture of Gleason regarding the Polish topologization of
abstract groups of homeomorphisms. We show that Gleason's conjecture is false, and its conclusion is only true when the hypotheses are considerably strengthened. Along the way we discover a new
automatic continuity result for a class of functions which behave like but are distinct from functions of Baire class 1. In the second section we consider the descriptive complexity of those subsets
of the permutation group S? which arise naturally from the classical Levy-Steinitz series rearrangement theorem. We show that for any conditionally convergent series of vectors in Euclidean space,
the sets of permutations which make the series diverge, and diverge properly, are ?03-complete. In the last section we study the phenomenon of Haar null sets a la Christensen, and the closely related
notion of openly Haar null sets. We identify and correct a minor error in the proof of Mycielski that a ...
Contributing Partner: UNT Libraries
Date: August 2012
Creator: Das, Tushar
Description: The theory of discrete groups acting on finite dimensional Euclidean open balls by hyperbolic isometries was borne around the end of 19th century within the works of Fuchs, Klein and
Poincaré. We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite dimensional separable Hilbert space. We present our investigations on the
geometry of limit sets at the sphere at infinity with an attempt to highlight the differences between the finite and infinite dimensional theories. We discuss the existence of fixed points of
isometries and the classification of isometries. Various notions of discreteness that were equivalent in finite dimensions, no longer turn out to be in our setting. In this regard, the robust notion
of strong discreteness is introduced and we study limit sets for properly discontinuous actions. We go on to prove a generalization of the Bishop-Jones formula for strongly discrete groups, equating
the Hausdorff dimension of the radial limit set with the Poincaré exponent of the group. We end with a short discussion on conformal measures and their relation with Hausdorff and packing measures on
the limit set.
Contributing Partner: UNT Libraries
Date: August 2011
Creator: Farmer, Matthew Ray
Description: In the study of Banach spaces, the development of some key properties require studying topologies on the collection of closed convex subsets of the space. The subcollection of closed
linear subspaces is studied under the relative slice topology, as well as a class of topologies similar thereto. It is shown that the collection of closed linear subspaces under the slice topology is
homeomorphic to the collection of their respective intersections with the closed unit ball, under the natural mapping. It is further shown that this collection under any topology in the
aforementioned class of similar topologies is a strong Choquet space. Finally, a collection of category results are developed since strong Choquet spaces are also Baire spaces.
Contributing Partner: UNT Libraries
Date: August 2012
Creator: Foster-Greenwood, Briana A.
Description: A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry,
combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given
a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semi-direct product of the group with a polynomial ring on the vector space.
Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as
deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible
representation. We consider in-depth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques
valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex
reflection groups. We give algorithms using character theory to compute ...
Contributing Partner: UNT Libraries
Date: December 2012
Creator: Herath, Dushanthi N.
Description: Receiver operating characteristic (ROC) analysis is one of the most widely used methods in evaluating the accuracy of a classification method. It is used in many areas of decision making
such as radiology, cardiology, machine learning as well as many other areas of medical sciences. The dissertation proposes a novel nonparametric estimation method of the ROC surface for the
three-class classification problem via Bernstein polynomials. The proposed ROC surface estimator is shown to be uniformly consistent for estimating the true ROC surface. In addition, it is shown that
the map from which the proposed estimator is constructed is Hadamard differentiable. The proposed ROC surface estimator is also demonstrated to lead to the explicit expression for the estimated
volume under the ROC surface . Moreover, the exact mean squared error of the volume estimator is derived and some related results for the mean integrated squared error are also obtained. To assess
the performance and accuracy of the proposed ROC and volume estimators, Monte-Carlo simulations are conducted. Finally, the method is applied to the analysis of two real data sets.
Contributing Partner: UNT Libraries
Date: May 2011
Creator: Jasim, We'am Muhammad
Description: Let G be a Polish group. We say that G is an algebraically determined Polish group if given any Polish group L and any algebraic isomorphism from L to G, then the algebraic isomorphism
is a topological isomorphism. We will prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This will smooth
the way for the proofs for some special groups. For example, let H be a separable Hilbert space and let G be a subset of the unitary group U(H) acting transitively on the unit sphere. Assume that -I
in G and G is a Polish topological group in some topology such that H x G to H, (x,U) to U(x) is continuous, then H x G is a Polish topological group. Hence H x G is an algebraically determined
Polish group. In addition, we apply the above the above result on the unitary group U(A) of a separable irreducible C*-algebra A with identity acting transitively on the unit sphere in a separable
Hilbert space H and proved that the natural semidirect product H x U(A) is an algebraically determined Polish group. A similar theorem is true ...
Contributing Partner: UNT Libraries
Date: May 2010
Creator: Kieftenbeld, Vincent
Description: This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order
topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is
completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete
metrizability, generalizing results of Sierpiński, Vain tein, and Ostrovsky. Third, an equivalence relation on a Polish space has the Laczkovich-Komjáth property if the following holds: for
every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection
of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the Laczkovich-Komjáth property, extending a theorem
of Balcerzak and Głąb.
Contributing Partner: UNT Libraries | {"url":"http://digital.library.unt.edu/explore/partners/UNT/browse/?fq=untl_decade%3A2010-2019&fq=str_degree_discipline%3AMathematics&sort=creator","timestamp":"2014-04-17T23:13:12Z","content_type":null,"content_length":"45449","record_id":"<urn:uuid:0c3f3333-9451-4a5c-b7d5-6d226231fff6>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00270-ip-10-147-4-33.ec2.internal.warc.gz"} |
ocal ring
Algebraic theories
Algebras and modules
Higher algebras
Model category presentations
Geometry on formal duals of algebras
Local rings
A local ring is a ring (with unit, usually also assumed commutative) such that:
• $0 e 1$; and
• whenever $a + b = 1$, $a$ or $b$ is invertible.
Here are a few equivalent ways to phrase the combined condition:
• Whenever a (finite) sum equals $1$, at least one of the summands is invertible.
• Whenever a sum is invertible, at least one of the summands is invertible.
• Whenever a sum of products is invertible, for at least one of the summands, all of its multiplicands are invertible.
• The non-invertible elements form an ideal. (Unlike the previous clauses, this requires excluded middle to be equivalent.)
The ideal of non-invertible elements is in fact a maximal ideal, so the quotient ring is a field. (This quotient can also be taken constructively, where one mods out by an anti-ideal.)
In geometry
In algebraic geometry or synthetic differential geometry and commutative algebra, the most commonly used definition of a local commutative ring is a commutative ring $R$ with a unique maximal ideal.
Hence the Spec of such an $R$ has a unique closed point. Intuitively it can be thought of as some kind of “infinitesimal neighborhood” of a closed point.
The topos theory formulation of this is a local topos.
An important example of a local ring in algebraic geometry is $R = k[\epsilon]/\epsilon^2$. This ring is known as the ring of dual numbers. Intuitively, we can think of its spectrum as consisting of
a closed point and a tangent vector. Indeed this is justified, as morphisms from $\operatorname{Spec} R$ to a scheme $X$ correspond exactly to pairs $(x,v)$, where $x \in X$ and $v$ is a (Zariski)
tangent vector at $x$.
Local rings are also important in deformation theory. One might define an infinitesimal deformation of a scheme $X_0$ to be a deformation of $X_0$ over $\operatorname{Spec} R$ where $R$ is a local
In weak foundations
Local rings are often more useful than fields when doing mathematics internally. For one thing, the definition make sense in any coherent category. But unlike the definition of discrete field, which
is also coherent, it is satisfied by rings such as the ring of (located Dedekind) real numbers. Rather than mod out by the ideal of non-invertible elements, you take care to use only properties that
are invariant under multiplication by an invertible element.
In constructive mathematics, one could do the same thing, but it's more common to use the notion of Heyting field. This is closely related, however; the quotients of local rings are precisely the
Heyting fields (which are themselves local rings). In fact, one can define an apartness relation (like that on a Heyting field) in any local ring: $x \# y$ iff $x - y$ is invertible. Then the local
ring is a Heyting field if and only if this apartness relation is tight. | {"url":"http://ncatlab.org/nlab/show/local+ring","timestamp":"2014-04-16T04:47:53Z","content_type":null,"content_length":"28965","record_id":"<urn:uuid:35217a37-e77b-4b8b-8db7-72fb2584cfc0>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00367-ip-10-147-4-33.ec2.internal.warc.gz"} |
Arizona Math Standards - 3rd Grade
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Arizona Math Standards - 3rd Grade
MathScore aligns to the Arizona Math Standards for 3rd Grade. The standards appear below along with the MathScore topics that match. If you click on a topic name, you will see sample problems at
varying degrees of difficulty that MathScore generated. When students use our program, the difficulty of the problems will automatically adapt based on individual performance, resulting in not only
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View the Arizona Math Standards at other levels.
Number Sense and Operations
C1 Number Sense
1. Read whole numbers in contextual situations (through six-digit numbers).
2. Identify six-digit whole numbers in or out of order.
3. Write whole numbers through six-digits in or out of order.
4. State whole numbers, through six-digits, with correct place value, by using models, illustrations, symbols, or expanded notation (e.g., 53,941 = 50,000 + 3,000 + 900 + 40 +1).
5. Construct models to represent place value concepts for the one's, ten's, and hundred's places.
6. Apply expanded notation to model place value through 9,999 (e.g., 5,378 = 5,000 + 300 + 70 + 8). (Place Value )
7. Sort whole numbers into sets containing only odd numbers or only even numbers. (Odd or Even )
8. Compare two whole numbers, through six-digits. (Number Comparison )
9. Order three or more whole numbers through six-digit numbers (least to greatest, or greatest to least). (Order Numbers )
10. Make models that represent proper fractions (halves, thirds, fourths, eighths, and tenths).
11. Identify symbols, words, or models that represent proper fractions (halves, thirds, fourths, eighths and tenths). (Fraction Pictures )
12. Use proper fractions in contextual situations.
13. Compare two proper fractions with like denominators. (Fraction Comparison )
14. Order three or more proper fractions with like denominators (halves, thirds, fourths, eighths, and tenths).
15. Count amounts of money through $20.00 using pictures or actual bills and coins. (Making Change 2 , Counting Money )
16. Use decimals through hundredths in contextual situations. (Counting Money )
17. Compare two decimals, through hundredths, using models, illustrations, or symbols. (Compare Decimals )
18. Order three or more decimals, through hundredths, using models, illustrations, or symbols. (Decimal Place Value , Order Decimals )
19. Determine the equivalency among decimals, fractions, and percents (e.g., half-dollar = 50¢ = 50% and 1/4 = 0.25 = 25%). (Fractions to Decimals , Decimals To Fractions , Compare Mixed Values ,
Positive Number Line , Percentages )
20. Identify whole-number factors and/or pairs of factors for a given whole number through 24. (Factoring )
21. Determine multiples of a given whole number with products through 24 (skip counting). (Skip Counting )
C2 Numerical Operations
1. Demonstrate the process of subtraction using manipulatives through three-digit whole numbers.
2. Add two three-digit whole numbers. (Long Addition to 1000 , Long Addition )
3. Subtract two three-digit whole numbers. (Long Subtraction , Long Subtraction to 1000 )
4. Add a column of numbers. (Addition Grouping )
5. Select the grade-level appropriate operation to solve word problems.
6. Solve word problems using grade-level appropriate operations and numbers. (Arithmetic Word Problems , Basic Word Problems 2 )
7. Demonstrate the process of multiplication as repeatedly adding the same number, counting by multiples, combining equal sets, and making arrays. (Understanding Multiplication )
8. Demonstrate the process of division with one-digit divisors (separating elements of a set into smaller equal sets, sharing equally, or repeatedly subtracting the same number). (Understanding
Division )
9. Demonstrate families of equations for multiplication and division through 9s. (Inverse Equations 2 )
10. State multiplication and division facts through 9s. (Fast Multiplication , Fast Multiplication Reverse , Multiplication Facts Strategies , Fast Division )
11. Demonstrate the commutative and identity properties of multiplication. (Commutative Property 2 )
12. Identify multiplication and division as inverse operations. (Inverse Equations 2 )
13. Apply grade-level appropriate properties to assist in computation.
14. Apply the symbols: ×, ÷, /, *, %, and the grouping symbols ( ) and ",".
15. Use grade-level appropriate mathematical terminology.
16. Add or subtract fractions with like denominators (halves, thirds, fourths, eighths, and tenths) appropriate to grade level. (Basic Fraction Addition , Basic Fraction Subtraction )
17. Apply addition and subtraction in contextual situations, through $20.00. (Money Addition , Money Subtraction , Making Change )
C3 Estimation
1. Solve grade-level appropriate problems using estimation.
2. Estimate length and weight using U.S. customary units.
3. Record estimated and actual linear measurements for real-life objects (e.g., length of fingernail; height of desk). (Requires outside materials )
4. Compare estimations of appropriate measures to the actual measures. (Estimated Addition , Estimated Subtraction )
5. Evaluate the reasonableness of estimated measures.
Data Analysis, Probability, and Discrete Mathematics
C1 Data Analysis (Statistics)
1. Formulate questions to collect data in contextual situations.
2. Construct a horizontal bar, vertical bar, pictograph, or tally chart with appropriate labels and title from organized data.
3. Interpret data found in line plots, pictographs, and single-bar graphs (horizontal and vertical). (Tally and Pictographs , Bar Graphs , Line Graphs )
4. Answer questions based on data found in line plots, pictographs, and single-bar graphs (horizontal and vertical). (Tally and Pictographs , Bar Graphs , Line Graphs )
5. Formulate questions based on graphs, charts, and tables to solve problems.
6. Solve problems using graphs, charts and tables. (Tally and Pictographs , Bar Graphs , Line Graphs )
C2 Probability
1. Name the possible outcomes for a probability experiment.
2. Make predictions about the probability of events being more likely, less likely, equally likely or unlikely.
3. Predict the outcome of a grade-level appropriate probability experiment.
4. Record the data from performing a grade-level appropriate probability experiment.
5. Compare the outcome of an experiment to predictions made prior to performing the experiment.
6. Compare the results of two repetitions of the same grade-level appropriate probability experiment.
C3 Discrete Mathematics - Systematic Listing and Counting
1. Make a diagram to represent the number of combinations available when 1 item is selected from each of 3 sets of 2 items (e.g., 2 different shirts, 2 different hats, 2 different belts).
C4 Vertex-Edge Graphs
1. Color maps with the least number of colors so that no common edges share the same color (increased complexity throughout grade levels).
Patterns, Algebra, and Functions
C1 Patterns
1. Communicate a grade-level appropriate iterative pattern, using symbols or numbers. (Patterns: Shapes )
2. Extend a grade-level appropriate repetitive pattern (e.g., 5, 10, 15, 20, . . . rule: add five or count by five's). (Patterns: Numbers )
3. Solve grade-level appropriate pattern problems.
C2 Functions and Relationships
1. Describe the rule used in a simple grade-level appropriate function (e.g., T-chart, input/output model, and frames and arrows). (Function Tables , Function Tables 2 )
C3 Algebraic Representations
1. Use variables in contextual situations.
2. Solve equations with one variable using missing addends to sums of 18 (e.g., ? + 9 = 18, 9 + ? = 18); and using minuend through 18 (e.g., 18 - ? = 9, 18 - 9 = ?). (Missing Term )
C4 Analysis of Change
1. Identify the change in a variable over time (e.g., an object gets taller, colder, heavier). (Line Graphs )
2. Make simple predictions based on a variable (e.g., increases in allowance as you get older). (Line Graphs )
Geometry and Measurement
C1 Geometric Properties
1. Build geometric figures with other common shapes (e.g., tangrams, pattern blocks, geoboards).
2. Name concrete objects and pictures of 3-dimensional solids (cones, spheres, and cubes).
3. Describe relationships between 2-dimensional and 3-dimensional objects (squares/cubes, circles/spheres, triangles/cones).
4. Recognize similar shapes. (Congruent And Similar Triangles )
5. Identify a line of symmetry in a 2-dimensional shape.
C2 Transformation of Shapes
1. Recognize same shape in different positions (turn/rotation).
C3 Coordinate Geometry
1. Identify points in the first quadrant of a grid using ordered pairs.
C4 Measurement - Units of Measure - Geometric Objects
1. Select the appropriate measure of accuracy:
• length - cetimeters, meters, kilometers,
• capacity/volume - liters, and
• mass/weight - grams.
2. Tell time with one-minute precision (analog). (Telling Time )
3. Determine the passage of time across months (units of days, weeks, months) using a calendar.
4. Measure a given object using the appropriate unit of measure:
• length - centimeters, millimeters, meters, kilometers,
• capacity/volume - liters, and
• mass/weight - grams.
5. Record temperatures to the nearest degree in degrees Fahrenheit and degrees Celsius as shown on a thermometer.
6. Compare units of measure to determine more or less relationships for:
• length - inches to feet; centimeters to meters
• time - minutes to hours; hours to days; days to weeks; months to years, and
• money - pennies, nickels, dimes, quarters, and dollars.
7. Determine relationships for:
• volume - cups and gallons
• weight - ounces and pounds
• money - extend to amounts greater than one dollar.
8. Compare the length of two objects using U.S. customary or metric units.
9. Determine the perimeter using a rectangular array.
10. Represent area using a rectangular array.
Structure and Logic
C1 Algorithms and Algorithmic Thinking
1. Discriminate necessary information from unnecessary information in a given grade-level appropriate word problem.
C2 Logic, Reasoning, Arguments, and Mathematical Proof
1. Draw conclusions based on existing information (e.g., All students in Ms. Dean's 1st grade class are less than 7 years old. Rafael is in Ms. Dean's class. Conclusion: Rafael is less than 7 years
Learn more about our online math practice software. | {"url":"https://www.mathscore.com/math/standards/Arizona/3rd%20Grade/","timestamp":"2014-04-20T08:30:21Z","content_type":null,"content_length":"20828","record_id":"<urn:uuid:07ddaa08-0b0a-40b8-981f-1a319c31b20b>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00588-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts about Cédric Villani on Xi'an's Og
On Tuesday, there was a series of talks (in French) celebrating Statistics, with an introduction by Cédric Villani. (The talks are reproduced on the French Statistical Society (SFDS) webpage.)
Rather unpredictably (!), Villani starts from an early 20th Century physics experiment leading to the estimation of the Avogadro constant from a series of integers. (Repeating an earlier confusion of
his, he substitutes the probability of observing a rare event under the null with the probability of the alternative on the Higgs boson to be true!) A special mention to/of Francis Galton’s “supreme
law of unreason”. And of surveys, pointing out the wide variability of a result for standard survey populations. But missing the averaging and more statistical effect of accumulating surveys, a
principle at the core of Nate Silver‘s predictions. A few words again about the Séralini et al. experiments on Monsanto genetically modified maize NK603, attacked for their lack of statistical
foundations. And then, hear hear!, much more than a mere mention of phylogenetic inference, with explanations about inverse inference, Markov Chain Monte Carlo algorithms on trees, convergence of
Metropolis algorithms by Persi Diaconis, and Bayesian computations! Of course, this could be seen more as numerical probability than as truly statistics, but it is still pleasant to hear.
The last part of the talk more predictably links Villani’s own field of optimal transportation (which I would translate as a copula problem…) and statistics, mostly understood as empirical
distributions. I find it somewhat funny that Sanov’s theorem is deemed therein to be a (or even the) Statistics theorem! I wonder how many statisticians could state this theorem… The same remark
applies for the Donsker-Varadhan theory of large deviations. Still, the very final inequality linking the three types of information concepts is just… beautiful! You may spot in the last minute a
micro confusion in repeating twice the definition for Fisher’s information rather than deducing that the information associated with a location family is constant. (And a no-so-necessary mention of
the Cramer-Rao bound on unbiased estimators. Which could have been quoted as the Fréchet-Darmois-Cramer-Rao bound in such historical grounds ) A pleasant moment, all in all! (There are five other
talks on that page, including one by Emmanuel Candés.) | {"url":"http://xianblog.wordpress.com/tag/cedric-villani/","timestamp":"2014-04-19T04:21:01Z","content_type":null,"content_length":"83129","record_id":"<urn:uuid:c171234d-4df9-4843-bf9a-1f2016a8be40>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00228-ip-10-147-4-33.ec2.internal.warc.gz"} |
Trigonometry. QuestionA ladder leaning against a wall reaches a height of 10m up the wall and makes an angle of 75... - Homework Help - eNotes.com
Trigonometry. Question
A ladder leaning against a wall reaches a height of 10m up the wall and makes an angle of 75 degrees with the ground. What is the length of the ladder?
Let us say the point touching the ground in ladder is A, the point of ladder that touch the wall is B and the point of bottom of wall as C.
According to the data given;
`BC = 10m`
`angleBAC = 75 deg`
Length of ladder `= AB`
By the geometry;
`sinA = BC/AB`
`sin75 = 10/AB`
`AB = 10/sin75`
`AB = 10.353`
So the length of ladder is 10.353m.
Join to answer this question
Join a community of thousands of dedicated teachers and students.
Join eNotes | {"url":"http://www.enotes.com/homework-help/trigonometry-question-371237","timestamp":"2014-04-21T05:36:39Z","content_type":null,"content_length":"25055","record_id":"<urn:uuid:e3cd999b-d425-473f-af21-623666e169cb>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00219-ip-10-147-4-33.ec2.internal.warc.gz"} |
Geometry Tutors
Camarillo, CA 93012
Tutoring with a smile :)
I am a credentialed teacher in the State of CA (K-12 Science-Biology, K-8 Math - Algebra/
) with over 20 years of experience including consultation and development of IEP's/504's. I am brave enough to start conversations that matter. Professional experiences...
Offering 10+ subjects including geometry | {"url":"http://www.wyzant.com/Ventura_CA_Geometry_tutors.aspx","timestamp":"2014-04-21T16:14:30Z","content_type":null,"content_length":"60506","record_id":"<urn:uuid:819120e9-a1f4-44a0-9e81-4e29c01a1fc5>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00322-ip-10-147-4-33.ec2.internal.warc.gz"} |
Population growth equation
JackF90000 jackf90000 at aol.com
Sun Sep 17 19:01:18 EST 1995
The "basic" exponential growth equation is the solution to the
differential equation:
dx/dt = k x(t)
which says the rate of change of x(t) is proportional to x(t), where x(t)
is taken to be the population size at time t.
The solution is:
x(t) = k1 exp(-k2 t)
the constants k1 & k2 depend on K an the initial population size at
time t=0.
For more on population, get the book "[I can't remember the title,
something like"Introduction to Population Genetics"] by Crow & Kimura.
For the binary fission problem, Suppose that at the population size
doubles every generation:
N(1) = 2* No
N(2) = 2* N(1) = 2*2*No
N(k) = 2**k * No
Does this help?
More information about the Bioforum mailing list | {"url":"http://www.bio.net/bionet/mm/bioforum/1995-September/016192.html","timestamp":"2014-04-16T23:37:25Z","content_type":null,"content_length":"2884","record_id":"<urn:uuid:a6fb7902-fc99-4bef-832b-6ad88af220ed>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00182-ip-10-147-4-33.ec2.internal.warc.gz"} |
Nilpotent Matrices
Primary tabs
Submitted by finles on Thu, 04/10/2003 - 12:41
Im just new to linear algebra, and am having trouble proving that a nilpotent matrix must be singular. Can anyone give me a hand in the right direction slash provide the proof?
Let A be a nilpotent matrix. Then by definition for some natural n, A^n = 0.
Let d = |A| (the determinant). |0| = 0 = |A^n| = |A|^n = d^n. Hence, d = 0 and A is singular.
Another way to think about this is to suppose instead that we had an invertible matrix A such that A^n=0. A product of invertible matrices is invertible, so A^n=0 is invertible, but 0 is not
Yet another approach. Consider N nilpotent (but non-zero) with n the smallest integer (>1) such that N^n = 0. Assume N is non-singular with inverse M, then N^(n-1) = M N^n = M 0 = 0, but n was
suppose to be the smallest integer...proof by contradiction. | {"url":"http://planetmath.org/node/1538","timestamp":"2014-04-16T22:13:46Z","content_type":null,"content_length":"28202","record_id":"<urn:uuid:522a766d-8cfa-4985-a87b-1b846022a53c>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00078-ip-10-147-4-33.ec2.internal.warc.gz"} |
Weekly Problem 7 - 2011
Copyright © University of Cambridge. All rights reserved.
'Weekly Problem 7 - 2011' printed from http://nrich.maths.org/
Any three positive integers that multiply to make $2009$ would create viable cuboids.
The prime factors of $2009$ are $7\times 7\times 41$, so the options are:
$1 \times1 \times 2009$
$1\times 7\times 287$
$1\times 41\times 49$
$7\times 7 \times 41$
The first three cuboids all have two faces which each require $2009$ stickers ($1\times2009$, $7\times287$ and $41\times49$ respectively) so Roo cannot cover them.
The last cuboid has surface area: $2\times( 7\times7+7\times41 + 41\times 7) = 1246$
This leaves $2009-1246=763$ stickers left over.
This problem is taken from the UKMT Mathematical Challenges.
View the previous week's solutionView the current weekly problem | {"url":"http://nrich.maths.org/7154/solution?nomenu=1","timestamp":"2014-04-18T21:06:26Z","content_type":null,"content_length":"4001","record_id":"<urn:uuid:e056b425-c95d-4c26-976f-b0ca1c9a985c>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00583-ip-10-147-4-33.ec2.internal.warc.gz"} |
abcteach is your one-stop resource for teaching math, with printable worksheets, interactives, and fun activities, from addition to algebra.
Never run out of math drills with our Math Worksheet Generators, part of abctools®. Create your own random problem sets from our templates, featuring dozens of problem types and customization
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• Use this form to record people's favorite characters from "Trumpet of the Swan". Ten names are suggested and blank spaces are also included. Good tallying practice.
Looking for children's books featuring math? Here's a list of fiction and non-fiction books (for ages 4-14) to get you started!
• [member-created with abctools] From "bar graph" to "vertical". These vocabulary building word strips are great for word walls.
Use a line graph to learn how many students at Lawson Middle School have been on the honor roll for the last five years.
"Which player made the most touchdowns?" Use the chart to answer the five questions.
Use a pie chart to find out which foreign languages are studied at this middle school.
• [member-created with abctools] From "2x0" to "2x12". These multiplication skills strips are great for word walls.
Eight colorful math mats and cards for practice in addition within 100. Printable manipulatives are included or you can use real items. CC: Math: 2.OA.1-4
Not really magic-- just math fun! Numbers should be inserted so that rows, columns, and diagonals all add up to the same sum. Solve the two squares 3 different ways.
Find the math rule and follow it to correctly produce a completed matrix. Three sets; six matrices in each set.
• Three pages of complex math problems draw on critical thinking skills and logic in order to find the correct solution.
Write math problems on the fish cards. Students "fish for" problems and compete to have the highest "catch."
[member-created with abctools] Use this sign when you are testing. Post on your door.
A line graph that illustrates the number of tournament wins over a five year period of a video game club, with a set of ten questions on the same page and a separate answer sheet.
Students determine the rule for input/output tables, and complete the table using the rule.
• "____ did a great job in MATH today!" Three math award certificates for achievement in math skills
A line graph that illustrates the number of pounds of paper recycled by grades 1-5 with a set of ten questions on the same page, plus an answer sheet.
Use the line graph to find out how many library books the students read, by grade.
Related Links | {"url":"http://www.abcteach.com/directory/subjects-math-19-7-1","timestamp":"2014-04-21T02:18:59Z","content_type":null,"content_length":"150852","record_id":"<urn:uuid:2887783a-f6bb-4595-99ad-56444bf11457>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00280-ip-10-147-4-33.ec2.internal.warc.gz"} |
#include <boost/math/special_functions/zeta.hpp>
namespace boost{ namespace math{
template <class T>
calculated-result-type zeta(T z);
template <class T, class Policy>
calculated-result-type zeta(T z, const Policy&);
}} // namespaces
The return type of these functions is computed using the result type calculation rules: the return type is double if T is an integer type, and T otherwise.
The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more
template <class T>
calculated-result-type zeta(T z);
template <class T, class Policy>
calculated-result-type zeta(T z, const Policy&);
Returns the zeta function of z:
The following table shows the peak errors (in units of epsilon) found on various platforms with various floating point types, along with comparisons to the GSL-1.9 and Cephes libraries. Unless
otherwise specified any floating point type that is narrower than the one shown will have effectively zero error.
Table 47. Errors In the Function zeta(z)
Significand Size Platform and Compiler z > 0 z < 0
Peak=0.99 Mean=0.1 Peak=7.1 Mean=3.0
53 Win32, Visual C++ 8 GSL Peak=8.7 Mean=1.0 GSL Peak=137 Mean=14
Cephes Peak=2.1 Mean=1.1 Cephes Peak=5084 Mean=470
64 RedHat Linux IA_EM64, gcc-4.1 Peak=0.99 Mean=0.5 Peak=570 Mean=60
64 Redhat Linux IA64, gcc-4.1 Peak=0.99 Mean=0.5 Peak=559 Mean=56
113 HPUX IA64, aCC A.06.06 Peak=1.0 Mean=0.4 Peak=1018 Mean=79
The tests for these functions come in two parts: basic sanity checks use spot values calculated using Mathworld's online evaluator, while accuracy checks use high-precision test values calculated at
1000-bit precision with NTL::RR and this implementation. Note that the generic and type-specific versions of these functions use differing implementations internally, so this gives us reasonably
independent test data. Using our test data to test other "known good" implementations also provides an additional sanity check.
All versions of these functions first use the usual reflection formulas to make their arguments positive:
The generic versions of these functions are implemented using the series:
When the significand (mantissa) size is recognised (currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double) then a series of rational approximations
devised by JM are used.
For 0 < z < 1 the approximating form is:
For a rational approximation R(1-z) and a constant C.
For 1 < z < 4 the approximating form is:
For a rational approximation R(n-z) and a constant C and integer n.
For z > 4 the approximating form is:
ζ(z) = 1 + e^R(z - n)
For a rational approximation R(z-n) and integer n, note that the accuracy required for R(z-n) is not full machine precision, but an absolute error of: ε/R(0). This saves us quite a few digits when
dealing with large z, especially when ε is small. | {"url":"http://www.boost.org/doc/libs/1_49_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/zetas/zeta.html","timestamp":"2014-04-16T17:30:49Z","content_type":null,"content_length":"16754","record_id":"<urn:uuid:60831f3e-1332-45de-becd-55177dd7ebd9>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00658-ip-10-147-4-33.ec2.internal.warc.gz"} |
My First Quarter Grades (PLEASE READ!!!!)
Number of results: 134,832
My First Quarter Grades (PLEASE READ!!!!)
When I means what is my grades I mean my average (srry about that)
Friday, November 18, 2011 at 5:36pm by Laruen
My First Quarter Grades (PLEASE READ!!!!)
We are also very happy for you! We know you work hard for your grades! Bravo! Sra
Friday, November 18, 2011 at 5:36pm by SraJMcGin
My First Quarter Grades (PLEASE READ!!!!)
Friday, November 18, 2011 at 5:36pm by Laruen
My First Quarter Grades (PLEASE READ!!!!)
I don't know.
Friday, November 18, 2011 at 5:36pm by Ms. Sue
My First Quarter Grades (PLEASE READ!!!!)
What is my grades for the first quarter? I didn't get my report card because my parents didn't go to the parents teacher conference so yeah but today my guides counselor show it to me (my grades for
the first quarter) they didn't calculated my average. Here are my grades: ...
Friday, November 18, 2011 at 5:36pm by Laruen
My First Quarter Grades (PLEASE READ!!!!)
so if they did count it I'll make it to the pricipal list?
Friday, November 18, 2011 at 5:36pm by Laruen
My First Quarter Grades (PLEASE READ!!!!)
Yay! Oops -- sorry, I missed the 99 in English. Your average is now 93.5 = 94% You've almost made the principal's list. If the school counts gym, you may have a 95% average.
Friday, November 18, 2011 at 5:36pm by Ms. Sue
My Grades (this is different)
What is my grades for the first quarter? I didn't get my report card because my parents didn't go to the parents teacher conference so yeah but today my guides counselor show it to me (my grades for
the first quarter) they didn't calculated my average. Here are my grades: ...
Saturday, November 19, 2011 at 5:17pm by Laruen
Translate the following into an inequality. Do not solve. Profits in the second quarter for a local retailer were all 112% of that in the first quarter. Profits in the third quarter were 87% of that
in the first quarter. The profits for the three quarterscombined exceeded $...
Tuesday, June 14, 2011 at 8:08pm by cody
My First Quarter Grades (PLEASE READ!!!!)
91 + 91 + 80 + 91 + 100 + 100 = 553 553/6 = 92.2 Your average is 92% Note that I didn't count gym or content support. Even if we count gym as 100, it doesn't raise your average much. Congratulations!
You're on the honor roll. :-)
Friday, November 18, 2011 at 5:36pm by Ms. Sue
Oh ok because I thought it was a program so something because I saw someone wrote that she or he got accepted to this program snd that they showed there grades in 9th grade (first quarter).
Tuesday, January 3, 2012 at 7:34pm by Laruen
one taxi charges 75 cents for the first quarter-mile and 15 cents for each additional quarter-mile. the second company charges $1.00 for the first quarter-mile and 10 cents for each additional
quarter mile. What distance would produce the same fare for the two companies?
Friday, February 18, 2011 at 9:54pm by Anonymous
At 0.75 percent, I don't think you'd more than double your money in just 2.5 years. I got a different answer. I used 0.0075 as the multiplier and figured it the long way -- quarter by quarter. In the
first quarter you'd earn $3.00. Add that to the original 400 and again ...
Monday, February 16, 2009 at 8:06pm by Ms. Sue
My Grades
I'm really glad I did well in the 1st quarter.
Wednesday, November 23, 2011 at 4:53pm by Laruen
Please..I need help asap If a firm pays its bill with a 30 day delay, what fraction of its purchases will be in paid in the current quarter? Please show me how you get this answer Since a quarter is
three months long, then 2/3 of the firm's purchases are paid in the quarter in...
Wednesday, June 13, 2007 at 2:55pm by je
A company made $4 million in the second quarter. This is 1/3 more than it made in the first quarter and 4/5 of what it made in the third quarter. How much did the company make in the three quarters
Monday, September 19, 2011 at 4:51pm by eric
With reference to an earlier question:- The liquid is to be scooped at a quarter of the depth of the container it is in. Hence the first sentence is correct? Or is there a better way of phrasing? The
liquid should be scooped at a quarter depth of the container? Posted by ann ...
Tuesday, October 26, 2010 at 6:12pm by ann
I think it depends on what this means. I think it'll be the first sentence, but it's not really clear. a quarter of the depth of the measuring cup? a quarter of the depth of the container the liquid
is in? the depth of the size of a US quarter ($.25)? What?
Tuesday, October 26, 2010 at 5:31pm by Writeacher
extra credit
what does this puzzle mean QUARTER QUARTER QUARTER QUARTER
Friday, April 10, 2009 at 1:15pm by Delaney
Well first you add up the currently known grades: 84, 65, and 76. (known grades) Next multiply 80 by 4 to get the total amount for his test scores. (wanted average) Finally subtract the total for his
known grades from his wanted average.
Wednesday, October 24, 2012 at 7:57pm by Ravin
math 10
1250 * 0.06 = $75 yearly interest 75 / 4 = $18.75 first quarter interest 1250 + 18.75 = $1,268.75 balance at the end of the first quarter Repeat the above steps to find the balance after the second
Tuesday, January 5, 2010 at 12:18pm by Ms. Sue
Managerial accounting
The marketing department of Jessi Corporation has submitted the following sales forecast for the upcoming fiscal year (all sales are on account): 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter Units
to be produced 12,000 14,000 13,000 11,000 The selling price of the company's...
Monday, September 5, 2011 at 12:18am by kim
In the first quarter of a football game ,a quarterback threw for 25 yds and was sacked for -36 yds.What was the quarterback's total yardage for the first quarter?
Wednesday, April 10, 2013 at 10:01am by Praphul
verbal linear equations
a taxi costs $2.75 for the first quarter of a mile and $0.85 for each subsequent quarter mile.A ride that cost $23.15 is how many miles longer than a ride that costs $17.20?(please i don't want
example show me how u work it out )
Sunday, February 5, 2012 at 5:44pm by tanya
Antoine estimates he can read one and a quarter chapter of a novel in one hour. Jenna estimates she can read three chapters of the same novel in 2 5/6 hours. Antoine read for 8 hours. Jenna read for
6 hours. Who read more chapters? Show work. Please help!!!!
Monday, November 12, 2012 at 2:07pm by Cherie
Cost Accounting
Katherine Company's Sales Budget Has The Following Unit Sales Projection For Each Quarter Of The Calendar Year 2009. January -March 540,000 April-June 680,000 July-September 490,000 October-December
550,000 Total 2,260,000 Sales for the first quarter of 2010 are expected to be...
Saturday, November 7, 2009 at 11:19am by Jessi
Cost Accounting
PLEASE, PLEASE, can you give me the step to calculate these: 1.Prepare the beginning inventory for the first quarter 2.Prepare the budgeted beginning inventory for the second - fourth quarters
3.Prepare the budgeted production for each quarter 4.Prepare the budgeted production...
Thursday, November 5, 2009 at 9:29pm by Jessi
Sorry, this is not true. "If you studied hard, then your grades are good" only guarantees good grades if you studied hard, which means that if you don't get good grades, you didn't study hard. If you
get good grades, it does not mean that you studied hard! It also means that ...
Tuesday, July 21, 2009 at 9:40pm by MathMate
human relations
one taxi cab company charges 75 cents for the first quarter mile and 15 cents for each additional quarter mile. the competing taxi company charges 1.00 for the first quarter mile and 10 cents for
each additional mile. what distance would produce the same fare for the two taxi ...
Tuesday, February 15, 2011 at 11:28pm by eva
English 7 - Journal Entry Check
English Homework: Journal Entry Due Tomorrow Topic - Dicuss a time when you made a decison to do something (Determination), and perservered as a result. Please check my journal entry to see if it's
good. Determination When I first came in the 7th grade at (school name), I was ...
Thursday, February 2, 2012 at 5:38pm by Laruen
On the last school day of the first grading period Michael learns his final grade for each class. If order is not important and we know he has earned at least 1 of each of the 5 letter grades, what
is the total number of combinations of 7 final grades (one for each class) that...
Tuesday, November 16, 2010 at 11:19pm by Anonymous
A building is 4 feet higher than twice the height of a tree. Define variable expressions for the height of the building and the height of the tree? A companys profit for the third quarter is 22,300
more than the profit for the first quarter. Define variable expressions for the...
Tuesday, July 3, 2012 at 1:08pm by Ariel
1. The mean and standard deviation of the grades of a group of students who took an economic exam were 69 and 7, respectively. The grades have a mound-shaped distribution. According to the Empirical
Rule, a) Approximately 68% of the grades were between which grade? b) ...
Friday, September 9, 2011 at 9:10pm by Larry English
8th grade
On the last school day of the first grading period Michael learns his final grade for each class. If order is not important and we know he has earned at least 1 of each of the 5 letter grades, what
is the total number of combinations of 7 final grades (one for each class) that...
Monday, November 15, 2010 at 10:41pm by Anonymous
jewerly marking
I tried to find a site that would allow you to read all about silver grades but I found good sites only for grades from 90% silver to 99.9% silver. Sterling silver is 92.5% silver. 825 silver is
82.5% silver.
Tuesday, December 30, 2008 at 11:18pm by DrBob222
Math Word problem
Can someone please help guide me with this one? I am not good at word problems I think I may be dislexic because sometimes I may add or misread a number incorrectly... Statistics. Beach Channel High
School's football team scored one field goal (3pts) and gave up a touchdown (...
Tuesday, September 1, 2009 at 8:12pm by Anonymous
Math Word problem
Can someone please help guide me with this one? I am not good at word problems I think I may be dislexic because sometimes I may add or misread a number incorrectly... Statistics. Beach Channel High
School's football team scored one field goal (3pts) and gave up a touchdown (...
Tuesday, September 1, 2009 at 8:12pm by Anonymous
Teacher aide
I'm stuck on this question. Could someone help me please. Which of the following would you find in the new English and language arts curriculum guidelines? A An oral language program only at the
primary grades B A phonics program at all grades C A writing program at all grades...
Friday, April 22, 2011 at 9:25am by Bree
Nafari Company's sales budget has the following unit sales projection for each quarter of the calendar year 2011. January -March 1,080,000 April-June 1,360,000 July-September 980,000 October-December
1,100,000 Total 4,520,000 Sales for the first quarter of 2012 are expected to...
Wednesday, October 13, 2010 at 10:43pm by sylvia
Nafari Company's sales budget has the following unit sales projection for each quarter of the calendar year 2011. January -March 1,080,000 April-June 1,360,000 July-September 980,000 October-December
1,100,000 Total 4,520,000 Sales for the first quarter of 2012 are expected to...
Wednesday, October 13, 2010 at 10:44pm by sylvia
Percentage grades in a large physics class follow a normal distribution with mean 58. It is known that 3% of students receive grades less than 39.2. What is the standard deviation of grades in the
Monday, March 4, 2013 at 1:39pm by Anonymous
Plan Book
for my binder I'm putting my grades for grade 7th-12th and stuff I did in those grades. I don't care if Yale is not looking at my middle school grades, no matter what grade I'm in I ALWAYS WORK HARD.
Saturday, December 10, 2011 at 12:59pm by Laruen
Yes. Most colleges take your overall GPA, although they may weight your junior and senior grades more heavily. Besides if you goof off and get poor grades when you're a freshman, you may not have the
knowledge and skills to achieve top grades later.
Saturday, October 19, 2013 at 6:37pm by Ms. Sue
Suppose a young couple deposits $700 at the end of each quarter in an account that earns 7.1%, compounded quarterly, for a period of 6 years. After the 6 years, they start a family and find they can
contribute only $200 per quarter. If they leave the money from the first 6 ...
Saturday, April 30, 2011 at 7:35pm by cant figure this out :(
a quarter sits at the bottom of a wishing well where the gauge pressure reads 294 kPa. If the quarter has a diameter of 24mm, what is the total force acting on the top surface of the quarter? Note:
Atmospheric pressure = 101.3
Monday, August 20, 2012 at 10:23am by Drew
1st quarter: $8,000 * 0.06 = $480. $480 / 4 = $120 Interest for 1st quarter 2nd quarter: $8,000 + $120 = $8,120 $8,120 * 0.06 = $487.20 $487.20 / 4 = $121.80 Interest for 2nd quarter 3rd quarter:
$8,120 + $121.80 = $8,241.80 I think you can finish the 3rd quarter and do the ...
Saturday, May 10, 2008 at 8:25pm by Ms. Sue
Assume it takes 180 quarter credits to get a baccalaureate degree. If 1 quarter credit counts for one classroom hour of lecture each week of the quarter and you study 2.5 hours for each hour in
class,how many hours must you invest to get a degree? (You may assume each quarter ...
Friday, October 2, 2009 at 7:09pm by Stella
The word "of" means multiply. Please give me a quarter of a dollar. 1/4 * 1 = 1/4 He read half of the 300 pages. 1/2 * 300 = 300/2 = 150
Thursday, October 11, 2012 at 4:38pm by Ms. Sue
Question (this is important)
Ok my friend told me about John Hopkins Talent Search (for gr 7th and 8th) and like you take sat's and stuff. She said that I can't do it because this is for honor students only which is unfair. I
don't believe her so I'm asking you guys is she is right. I did check the site ...
Friday, December 2, 2011 at 6:50pm by Laruen
If a firm pays its bills with a 30-day delay, what fraction of its purchases will be paid for in the current quarter? In the following quarter? What if its payment delay is 60 days? It will pay 1/4
of its yearly bills each quarter -- no matter what its payment delay is. With a...
Sunday, February 18, 2007 at 8:15pm by mike
A box of 6 coins (penny, nickel, dime or quarter) worth $0.67 is shaken. What is the probability that a nickel is drawn first and then a quarter? Assume no replacement and that all coins are equally
Tuesday, September 28, 2010 at 12:47am by taylor
Office Management
One taxicab charges 75 cents for the first quarter-mile and 15 cents for each additional mile. The competing taxi company charges $1.00 for the first quarter-mile and 10 cents for each additional
mile. 1. what distance would produce the same fare for the two taxi companies? 2...
Thursday, February 17, 2011 at 10:20am by sandra
Suppose that, for a certain exam, a teacher grades on a curve. It is known that the grades follow a normal distribution with a mean of 70 and a standard deviation of 7. There are 45 students in the
class. How many students should receive an A? Please, round your answer to the ...
Wednesday, December 22, 2010 at 10:31pm by Anonymous
Joseph has just retired and is trying to decide between two retirement income option as to where he would place his life savings. Fund A will pay him quarterly payments for 25years starting at $1000
at the end of the first quarter. Fund A will increase his payments each ...
Sunday, September 25, 2011 at 6:19pm by Anonymous
First of all, read each of these aloud -- or better yet, ask someone to read them aloud to you. You'll hear (and be able to fix) several very obvious errors. Please post again once you've done your
first-round clean-up.
Monday, January 25, 2010 at 8:32am by Writeacher
Exam Grades. A statistics professor is used to having a variance in his class grades of no more than 100. He feels that his current group of students is different, and so he examines a random sample
of midterm grades (listed below.) At á = 0.05, can it be concluded that the ...
Wednesday, April 17, 2013 at 2:37pm by Gabriela
"Scale" means what you multiply something by to get the proper dimensions. So, 1/4 inch on the drawing is 1 foot in real life. How to do this one is to first figure out how many quarter inches are in
4 5/8 inches. Do this by dividing 4 5/8 by .25. The number of quarter inches ...
Wednesday, February 13, 2008 at 6:24pm by Sarah
Your teacher has given you very clear directions. Have you followed them? Take them one step at a time: First, On the first read-through, read mainly for a sense of plot. What is happening here?
After you’ve done the first read-through, answer question 1 below.
Monday, September 24, 2012 at 6:45pm by Writeacher
thinking outside of the box
one taxi company charges 75 cents for the first quarter mile and 15 cents for each additional mile the competing taxi company charges 1.00 for the first quarter mile and 10 each additional mile what
distance would produce the same fare for the two taxi companies?
Thursday, February 17, 2011 at 12:10am by eva
Think of it this way: there are 4 quarters in a dollar. one quarter is 25% two quarter is 50% etc.. four quarter is 100% so 25% of 100 is 25.
Monday, September 24, 2007 at 12:18pm by John
I really need help in understand what I read. Everyday we have to summarize a chapter in a book. Well I do read the chapter but I always find myself writing 2 sentences out of the chapter for my
summary. My grades are not good in reading and I want to know if you can help ...
Sunday, January 25, 2009 at 8:04am by Nehemiah
could you please translate in english to make sure you can read it correctly thank you so much GOD BLESS YOU
Sunday, January 18, 2009 at 8:12pm by Alex B.
Two students are talking after school: What is the average of your grades in Math for September? 4.6 That is impossible since the year just started. You could not possiblly have that many grades.
What was the doubting student thinking of of the possible grades teachers give ...
Thursday, October 15, 2009 at 6:09pm by Julie
Math EASY
Suppose your 83% is based on 25 grades with a total possible of 125 points. You earned about 104 points up til the last three days. Now you added 13 more points, so your total now is 117 points out
of a total of 140 points. 117/150 = 84% Please remember that three days' grades...
Friday, January 28, 2011 at 9:57pm by Ms. Sue
1. If your friend has bad grades, what would you say to help her get better grades. 2. If your friend had bad grades, what would you say to help her get better grades. ------------------ Which one is
grammatical? Do we have to use 'has' or 'had'? Are both grammatical? What is ...
Thursday, December 12, 2013 at 10:04pm by rfvv
If a quarter circle is placed on a square target and the radius of the quarter circle is the same as the length of a side of the square, what is the probability that a dart that hits the square does
not hit the quarter circle
Wednesday, October 26, 2011 at 10:29pm by Nikki
1. Achieving goals you set is greatly helped by: writing down your long-term goals. your learning style. managing your time efficiently. getting good grades. is it A 2. To actively read means to:
read aloud. concentrate on understanding. read fast. minimize note-taking. is it ...
Monday, August 19, 2013 at 9:31am by anny
Ricky, the 5.8 kg snowboarding raccoon, is on his snowboard atop a frictionless ice-covered quarter-pipe with radius 9 meters. If he starts from rest at the top of the quarter-pipe, what is his speed
at the bottom of the quarter-pipe? Use SI units in your answer. v_{Ricky} =
Friday, March 7, 2014 at 11:19pm by Anonymous
If both grades are weighted the same, the average to a C.
Thursday, September 26, 2013 at 5:45pm by Ms. Sue
algebra (i need help calculating my grade)
i have exams tomorrow and i would like to figure out what i need to make on my exam to pass, daily grades are 30% major grades are 70% i averaged my daily grades to 84.5 my major grades so far are
77,85,91 what do i do next to figure out what i need to make on my exam to ...
Thursday, October 22, 2009 at 8:31pm by Belle
How many grades will you have for your research project? Notes its like points or grades
Tuesday, September 7, 2010 at 9:08pm by Tamy
Use statistics to explain how the times for 6 grades compare to the timesfor 8 grades
Friday, July 16, 2010 at 4:28pm by Luisdavid
Add up all the grades that aren't B's. Then put that over the total amount of grades.
Sunday, December 5, 2010 at 10:02am by Jen
Hello. I've already asked a similar question but will you please clarify whether the articles are used correctly: 1)a quarter of the Russian people support this reform (I mean "the" with people) 2) a
quarter of Russians support ...(no article with "Russians", right?) Thank you...
Sunday, May 15, 2011 at 5:45am by Ilma
teachers aide curriculum comtent
literary themes understood by all people regardless of their background or the time they live in are (a)universal (b)skill-based(my answer) (c)motivational (d)literature -based what does the
curriculum web show? (a)how one theme unites different subjects (my answer) (b)how one...
Saturday, August 27, 2011 at 10:19am by susue
What grades did you earn in each of these classes? You should contact your school to find your official GPA.
Monday, June 22, 2009 at 8:01pm by Ms. Sue
A Peculiar Question
I told you before. Those grades are above average so they are good grades.
Wednesday, March 19, 2014 at 9:29pm by Ms. Sue
About Paris
Artists live many places in Paris. I've always known the 5th Arrondissement, also known as the Latin Quarter, as the traditional artists' quarter. The famed Sorbonne University is located there.
However, a Google search brought up Montmartre also as an artists' quarter.
Tuesday, March 29, 2011 at 7:28pm by Ms. Sue
Finance Question
700 x 1.17 x .3 The question was wrong, it is not greater 17% in each quarter, it is greater 17% from the FIRST quarter.
Monday, October 11, 2010 at 1:26pm by Anonymous
Suppose you start a business that has a soft opening and sells half of the expected product in the first quarter. You notice towards the end of the quarter that sales pick up near what was expected.
How much working capital might you need to budget ahead of time to overcome ...
Friday, May 3, 2013 at 3:38pm by at
math (find my grade)
1%= TEST 2%=IN CLASS WORK 3%=HOMEWORK 4%=QUIZ AND THE GRADES UNDER EACH ONE ARE MY GRADES!!!!!!!!!!
Sunday, April 18, 2010 at 2:57pm by rosie
lf you added a fifth of one and a quarter to one and a quarter, what would the answer be ? ..... The answers is 1 1/2 have tries 0.2 + 0.25 + 0.75 = 1.20 or 1 1/5 ???? 0.2 + 0.75 + 1 + 0.25 = 2.20
??? Please help!
Friday, November 1, 2013 at 1:08am by LARA
extra credit
If you are doing fractions, a "quarter" is represented by 1/4 = or quarter" = 4 times. 1/4 + 1/4 + 1/4 +1/4 = 4/4 (or a whole) Sra
Friday, April 10, 2009 at 1:15pm by SraJMcGin
No, (read)
No. My friend got a 99 for th 1st quarter and her science teacher recommend her.
Friday, December 2, 2011 at 6:50pm by Laruen
That's a good start. :-) What are you doing to make sure you get good grades? You could go on and talk about why you want to please your mother. What dreams do you have for your adult life? Before
you post more of your essay, please proof read it carefully. This spell check ...
Monday, September 15, 2008 at 5:48pm by Ms. Sue
MATH 156
Cutting a pie into equal slices so you can add or subtract them. EXAMPLE: 1/4 + 1/2 = 3/4. To get the half slice to equal the size of the quarter slice, you need to cut it in half. Then the half
slice becomes two quarter slices. When you add all of the quarter slices together...
Tuesday, June 23, 2009 at 6:03pm by Anonymous
School (Plz read, really important)
I want YOUR OPINION base on my grades and stuff and since your a teacher
Sunday, April 28, 2013 at 12:40pm by LARUEN
1. It is a quarter past seven. 2. It is a quarter after seven. 3. It is a quart to seven. 4. It is a quarter before seven. (#1 is the same as #2, right? #3 is the same as #4? Why is 'to' used here in
#3 instead of 'before'?)
Friday, June 18, 2010 at 9:10pm by rfvv
I live in The United Kingdom. And we do not do grades we do Year. I am in Year 8 in United Kingdom. What is that in the United States? No it is an essay. I have been completing it because it is due
for tomorrow. If you can please read it and make corrections that will be Fab. ...
Thursday, March 4, 2010 at 3:11pm by Ruth
I live in The United Kingdom. And we do not do grades we do Year. I am in Year 8 in United Kingdom. What is that in the United States? No it is an essay. I have been completing it because it is due
for tomorrow. If you can please read it and make corrections that will be Fab. ...
Thursday, March 4, 2010 at 3:23pm by Ruth
Think it through... If a quarter has approximately 90 days and purchases in 30 of those days, the payment will be in the following quarter, and for purchase in 60 of those 90 days, the payment will
be made in the current quarter.... means?
Sunday, November 18, 2007 at 9:15pm by economyst
college English
No, sorry. We don't do students' work for them. Please read several/many of the articles linked in those search results. Don't take notes; just read. Read and read. (Lots of reading will improve your
writing.) Once you get your ONE SENTENCE (thesis statement) written up, ...
Tuesday, July 5, 2011 at 5:24pm by Writeacher
How would you interpret the messages in the poem Anne received from her father for her birthday? I read them, but don't understand at all.. of course it makes sense I don't understand.. I am a 10
year old doing a 13 year old's work because I skipped a few grades. But anyway, I...
Tuesday, June 28, 2011 at 1:19pm by Carlie
AE ...
Teachers use databases to enter their grades. Grades are posted on the school's website. Many schools supply students with laptop computers.
Friday, June 19, 2009 at 8:37pm by Ms. Sue
a 'School court'. -- Why the quotation marks and capital S? who will be judges to their fellow students who who have -- "judges to ?? "who who" -- ?? And those are just in the first 4 lines. Please
read over your writing very carefully -- or better yet, have someone else read ...
Wednesday, March 28, 2012 at 5:13am by Writeacher
Essay; show all work. Bob’s Barber Shop estimates their gross revenue for the second quarter to be given by the polynomial 2x3 – 3x2 + 7x – 1. The shop estimates their costs for that quarter to be
given by x2 + 4x + 2. For the second quarter, find and simplify a polynomial ...
Tuesday, May 31, 2011 at 2:44pm by joel
9. Essay; show all work. Bob’s Barber Shop estimates their gross revenue for the second quarter to be given by the polynomial 2x3 + 5x2 – x – 1. The shop estimates their costs for that quarter to be
given by 4x2 – 9x – 8. For the second quarter, find and simplify a polynomial ...
Saturday, July 16, 2011 at 2:03am by Ms.Ellis
Please help with this question that has been bothering me! I take bio college class and we have graduate/student assistant for labs & when taking an exam. The professor is never there during those
sections.Due to my low exam results,I feel I'm not being graded on fairly ...
Tuesday, March 19, 2013 at 8:36pm by Leah
If the grades on a police entrance exam are normally distributed with a mean of 100 and a standard deviation of 15, what percentage of students received grades of less than 115?
Sunday, April 1, 2012 at 2:04pm by Laura
The Torrey Pine Corporation's purchases from suppliers in a quarter are equal to 75 percent of the next quarter's forecast sales. The payables period is 60 days. Wages, taxes, and other expenses are
18 percent of sales, and interest and dividends are $60 per quarter. No ...
Sunday, October 10, 2010 at 4:28pm by Mike
Pages: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Next>> | {"url":"http://www.jiskha.com/search/index.cgi?query=My+First+Quarter+Grades+(PLEASE+READ!!!!)","timestamp":"2014-04-19T02:40:45Z","content_type":null,"content_length":"42849","record_id":"<urn:uuid:d638ae31-0857-4065-ae3c-f60a3283316f>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00094-ip-10-147-4-33.ec2.internal.warc.gz"} |
Visual Gauss
Visual Gauss --- Introduction ---
Visual Gauss is an exercise on the method of Gauss elimination to diagonalise a linear system or to find the inverse of a matrix. This method consists of transforming a given matrix to the identity
matrix, by successive operations of modifications of rows.
Other exercises on: Gauss linear_systems matrix field linear_algebra
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
Description: step by step Gauss elimination (for matrix or system). This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters,
mathematical recreation and games
Keywords: wims, mathematics, mathematical, math, maths, interactive mathematics, interactive math, interactive maths, mathematic, online, calculator, graphing, exercise, exercice, puzzle, calculus,
K-12, algebra, mathématique, interactive, interactive mathematics, interactive mathematical, interactive math, interactive maths, mathematical education, enseignement mathématique, mathematics
teaching, teaching mathematics, algebra, geometry, calculus, function, curve, surface, graphing, virtual class, virtual classes, virtual classroom, virtual classrooms, interactive documents,
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[FOM] Proof "from the book"
Torkel Franzen torkel at sm.luth.se
Tue Aug 31 04:45:29 EDT 2004
Arnon Avron says:
>That "if S is consistent then G is true" is provable not only in S (which
>might prove false sentences), but also in PA (which proves only true
>sentences). Hence I do say that if S is a (formal) consistent extension of Q
>then Godel's proof shows G to be true.
I don't understand your "hence". Since we can prove Godel's theorem for
consistent extensions S of Q such that we have no idea whether or not
the Godel sentence G of S is true, what do you mean by saying that
Godel's proof shows G to be true in such a case?
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/2004-August/008434.html","timestamp":"2014-04-18T10:39:19Z","content_type":null,"content_length":"3027","record_id":"<urn:uuid:de89a114-4a9e-41de-8edc-b4b94892f622>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00041-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: st: Why is Mata much slower than MATLAB at matrix inversion?
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Re: st: Why is Mata much slower than MATLAB at matrix inversion?
From Richard Herron <richard.c.herron@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: Why is Mata much slower than MATLAB at matrix inversion?
Date Fri, 20 Jul 2012 21:19:37 -0400
Snap. Yes, your m from -runiform()- will certainly be invertible.
Richard Herron
On Fri, Jul 20, 2012 at 7:14 PM, Patrick Roland
<patrick.rolande@gmail.com> wrote:
> To be clear, my point was that all Mata matrix inverse functions are
> slower than MATLAB. It does seem though that this is not true for
> small matrices (e.g. 100x100), but the difference is easily an order
> of magnitude when it comes to larger matrices (2000x2000).
> The fact that I compared cholinv() and a general inverse function
> should be to Mata's favor, since cholinv should presumably be faster
> if it exploits the special structure of the matrix.
> X'X is positive definite if X is invertible (as in my example),
> because a'X'Xa = (Xa)'(Xa) > 0.
> On Fri, Jul 20, 2012 at 2:48 PM, David M. Drukker <ddrukker@stata.com> wrote:
>> Patrick Roland <patrick.rolande@gmail.com> posted that the Mata function
>> -cholinv()- is slower than a Matlab function for large matrices.
>> Others have discussed some issues with Patrick's example. Despite these
>> issues, we took Patrick's post seriously, looked at the code, and found
>> something that could be sped up.
>> We will release a faster version of -cholinv()- in an upcoming executable
>> update.
>> Note that any speed difference related to -cholinv()- is only noticeable for
>> large matrices. For small matrices, such as variance-covariance matrices
>> for models with 100 or fewer parameters, the difference is much harder to
>> find. For example, the computation takes about .001 seconds on my machine.
>> Best,
>> David
>> ddrukker@stata.com
>> *
>> * 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/2012-07/msg00758.html","timestamp":"2014-04-19T15:11:33Z","content_type":null,"content_length":"10943","record_id":"<urn:uuid:8d20ec22-6a71-4998-a858-a5d513643bc3>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00078-ip-10-147-4-33.ec2.internal.warc.gz"} |
A review of the curve-fitting method of least squares as applied to petroleum engineering
U.S. Bureau of Mines; [for sale by the Supt. of Docs., U.S. Govt. Print. Off.
, 1970 -
Least squares
27 pages
We haven't found any reviews in the usual places.
Abstract 1
program for least squares fit 10
Bibliographic information | {"url":"http://books.google.com/books?id=Kwi7CPqbS34C&source=gbs_similarbooks_r&cad=3","timestamp":"2014-04-17T12:39:12Z","content_type":null,"content_length":"93122","record_id":"<urn:uuid:707ab5d4-2efb-4249-beda-4506548e2087>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00377-ip-10-147-4-33.ec2.internal.warc.gz"} |
Next Article
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Other Issues
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Minimal fixing systems for convex bodies
V.G. Boltyanski and E. Morales Amaya
V. G. Boltyanski E. Morales Amaya
Steklov Mathematical Institute Centro de Investigationes
of Russian Academy of Sciences en Mathematicas, A.P. 402
Vavilov-str. 42, Moscow 117966 36000 Guanajuato, GTO
Russia Mexico
Abstract: L. Fejes Tóth [1] introduced the notion of {\it fixing system} for a compact, convex body $\, M\subset R^n.\, $ Such a system $\, F\subset \bd \, M\, $ stabilizes $\, M\, $ with respect to
translations. In particular, every {\it minimal} fixing system $\, F\, $ is {\it primitive}, i.e., no proper subset of $\, F\, $ is a fixing system. In [2] lower and upper bounds for cardinalities of
mimimal fixing systems are indicated. Here we give an improved lower bound and show by examples, now both the bounds are exact. Finally, we formulate a {\it Fejes Tóth Problem.}
Keywords: Convex body, fixing system, illumination, minimal dependency, functional $md$, indecomposable bodies, combinational geometry
Classification (MSC2000): 52A20, 52A37, 52B05
Full text of the article:
Electronic fulltext finalized on: 28 May 2002. This page was last modified: 21 Dec 2002.
© 2002 Heldermann Verlag
© 2002 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition | {"url":"http://www.emis.de/journals/JAA/vol1i1/1.html","timestamp":"2014-04-20T21:50:37Z","content_type":null,"content_length":"4128","record_id":"<urn:uuid:67f192b5-295e-44d9-a9d5-80f5596f388c>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00371-ip-10-147-4-33.ec2.internal.warc.gz"} |
Class Number / Structure
May 25th 2010, 03:47 AM
Class Number / Structure
I've got quite a few practice problems for finding class numbers and structures of imaginary quadratic number fields. For instance, I've just been trying to find the class structure of $\mathbb
{Q}(\sqrt{-34})$ and I've got an answer of $C_4$, the cyclic group of order 4. Is there a reference online that gives the class structure for "small" imaginary quadratic number fields (so that I
can check my answers)? I've found that the class numbers are given at, for instance, Tables of small class numbers of imaginary quadratic fields but obviously this doesn't give you all the
information. For instance it says that the class number of $\mathbb{Q}(\sqrt{-34})$ is 4 but the class structure could then be either $C_2 \times C_2$ or $C_4$.
May 25th 2010, 04:13 AM
I've got quite a few practice problems for finding class numbers and structures of imaginary quadratic number fields. For instance, I've just been trying to find the class structure of $\mathbb
{Q}(\sqrt{-34})$ and I've got an answer of $C_4$, the cyclic group of order 4. Is there a reference online that gives the class structure for "small" imaginary quadratic number fields (so that I
can check my answers)? I've found that the class numbers are given at, for instance, Tables of small class numbers of imaginary quadratic fields but obviously this doesn't give you all the
information. For instance it says that the class number of $\mathbb{Q}(\sqrt{-34})$ is 4 but the class structure could then be either $C_2 \times C_2$ or $C_4$.
Well, yes: to find exactly what option it will be you'll have to actually do some work with some ideal generators of ideal group. For this you''ll need Minkowski's theorem and its sequels to
bound up the norm for ideals and then take some of them and "play" with them.
I remember in a final exam in graduate schoolw I was given $\mathbb{Q}(\sqrt{105})$ , and while playing with the generators I found they all were of order 2, so the class group is $C_2\times C_2\
times C_2$ ...Perhaps in your case it won't be that hard, either
May 25th 2010, 04:36 AM
Perhaps I didn't make myself clear: I know how to go about finding the class structure, but I want to check my answers. I was just wondering if there was something online that tells you the class
group structure of all $\mathbb{Q}(\sqrt{-d})$ for $1 \leq d \leq 100$, say; as in if anyone knows of a website or book or other reference that just lists (without proof) lots of class groups so
that I can check the answers I've ended up with. | {"url":"http://mathhelpforum.com/number-theory/146338-class-number-structure-print.html","timestamp":"2014-04-20T18:00:25Z","content_type":null,"content_length":"9012","record_id":"<urn:uuid:96436c32-4505-4d8a-ab4d-9725b32ea997>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00280-ip-10-147-4-33.ec2.internal.warc.gz"} |
Another linear dependent/independent question
March 26th 2009, 11:56 PM
Another linear dependent/independent question
For this augmented matrix, the answer is that it they are linearly dependent. We are asked to find whether the coefficients are linearly independent or dependent?
For some reason I though considering there is no solution that they would be independent.
Could someone please show me why. Thanks
$<br /> \left( {\begin{array}{*{20}c}<br /> 1 & 0 & 0 & 2 \\<br /> 0 & 1 & 0 & 1 \\<br /> 0 & 0 & 0 & 3 \\<br /> \end{array}} \right)<br />$
March 27th 2009, 04:43 AM
WHAT are linearly independent? What does "they" refer to?
[qute] We are asked to find whether the coefficients are linearly independent or dependent?[/quote]
The coefficients of WHAT?
For some reason I though considering there is no solution that they would be independent.
Could someone please show me why. Thanks
$<br /> \left( {\begin{array}{*{20}c}<br /> 1 & 0 & 0 & 2 \\<br /> 0 & 1 & 0 & 1 \\<br /> 0 & 0 & 0 & 3 \\<br /> \end{array}} \right)<br />$
March 27th 2009, 05:16 AM
It's a augmented matrix, so the co-efficients are the first 3 columns.
I don't know how else to explain, that is what the question asks.
March 27th 2009, 02:30 PM
[unsolved] Another linear dependent/independent question
Still needing help with this, please. Sorry re-read question this morning, question is to find whether the columns of the coefficient part of the matrix are independent or dependent
March 27th 2009, 06:00 PM
they are linearly dependent. one reason is that the reduced form of a square matrix with linearly independent columns is always the identity matrix. but in your matrix (ignore the last column)
$3 \times 3$ matrix is not the identity matrix (the last column is 0). another reason is that if the columns of a quare matrix are linearly independent, then the augmented matrix will always show
a unique
solution but in your case there's no solution. | {"url":"http://mathhelpforum.com/advanced-algebra/80882-another-linear-dependent-independent-question-print.html","timestamp":"2014-04-18T09:39:32Z","content_type":null,"content_length":"8225","record_id":"<urn:uuid:3557b388-7bd9-4275-8537-860db71a0efe>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00004-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability of compound events?
I am having difficulty with these two questions
1. The probability of at least 4 tails facing up when 6 coins are dropped on the floor.
2. Find probability using standard deck of cards
P(both king or aces) if two cards are drawn without replacement.
I know there are 8 aces and kings in 52 card deck but I am confused on how to set problem up I think it mutually inclusive. | {"url":"http://mathhelpforum.com/statistics/183090-probability-compound-events.html","timestamp":"2014-04-16T07:20:44Z","content_type":null,"content_length":"48145","record_id":"<urn:uuid:52fc050f-2293-407f-a1cc-22d0d03ed2a9>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00537-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: May 1995 [00181]
[Date Index] [Thread Index] [Author Index]
Re: Question from Blind User
• To: mathgroup at christensen.cybernetics.net
• Subject: [mg1186] Re: Question from Blind User
• From: IANC (Ian Collier)
• Date: Mon, 22 May 1995 01:22:44 -0400
• Organization: Wolfram Research, Inc.
In article <3pjl8a$7h0 at news0.cybernetics.net>, mentat at telerama.lm.com
(Godshatter) wrote:
> Just installed the DOS version of Mathematica - the very last copy
> available, so they told me - and have run up against a problem in the way
> it displays answers. If you take the derivative of x^3, it gives the
> answer as:
> 2
> 3 x
> If I use a voice synthesyser to read the answer, it will come out as
> 2 3x. Using a braille display isn't much better, since I have to
> remember to look on more than one line for the answer.
> Is there any way to get the program to display answers in a more
> Fortran-like method?
> 3x^2
> I don't know how useful the program will be for me, if I have to look at
> multiple lines for the results. It is feasible to read equations spaced
> out on paper, but except for some very early arithmetic books, braille
> math books write equations and expressions one symbol after another.
> Besides, refreshable braille devices can display only one line at a time,
> and reading answers that way would involve a lot of extra keystrokes over
> the course of just a few problems.
> I would be grateful for any help. I would think that a program as
> powerful as Mathematica would allow for this kind of adjustment. Since
> the manual is rather large, I have not read it all yet. If the answer is
> in the documentation, just point me to it and I'll do the rest.
> Thanks in advance.
> Evan Reese
> mentat at telerama.lm.com
> "People are born with legs, not roots."
> R. Buckminster Fuller
You can do what you need using $Post, a global variable, which
is applied to every output expression if it is set.
Here is an example of a calculation carried out before and
after setting $Post to InputForm.
Integrate[ 1/(x^4 + 1),x]
-Sqrt[2] + 2 x Sqrt[2] + 2 x
ArcTan[--------------] ArcTan[-------------]
Sqrt[2] Sqrt[2]
---------------------- + --------------------- -
2 Sqrt[2] 2 Sqrt[2]
Log[1 - Sqrt[2] x + x ] Log[1 + Sqrt[2] x + x ]
----------------------- + -----------------------
4 Sqrt[2] 4 Sqrt[2]
$Post is a global variable whose value, if set, is applied
to every output expression.
$Post = InputForm
Integrate[ 1/(x^4 + 1),x]
ArcTan[(-2^(1/2) + 2*x)/2^(1/2)]/(2*2^(1/2)) +
ArcTan[(2^(1/2) + 2*x)/2^(1/2)]/(2*2^(1/2)) -
Log[1 - 2^(1/2)*x + x^2]/(4*2^(1/2)) +
Log[1 + 2^(1/2)*x + x^2]/(4*2^(1/2))
You could apply any other formating that you wished using
I hope this helps.
Ian Collier
Technical Sales Support
Wolfram Research, Inc.
tel:(217)-398-0700 fax:(217)-398-0747 ianc at wri.com
Wolfram Research Home Page: http://www.wri.com/ | {"url":"http://forums.wolfram.com/mathgroup/archive/1995/May/msg00181.html","timestamp":"2014-04-17T18:49:49Z","content_type":null,"content_length":"37382","record_id":"<urn:uuid:1509eea8-09f3-4f43-877a-321aaa1ce961>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00235-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Mathematics Education Resources on the World Wide Web.
ED402157 Sep 96 Mathematics Education Resources on the World Wide Web. ERIC Digest.
Authors: Haury, David L.; Milbourne, Linda A.
ERIC Clearinghouse for Science, Mathematics, and Environmental Education, Columbus, Ohio.
"The best aspect of the Internet has nothing to do with technology. It's us." Steven Levy
If 1995 was The "Year of the Internet" (Newsweek, Jan. 1, 1996), for educators it is quickly becoming the era of information-on-demand and collaboration-at-a-distance. Among the Internet's many
resources is the World Wide Web, a global network of information "servers" provided by individuals, organizations, businesses, and federal agencies who are offering documents, data, images, and
interactive sessions. For teachers, students, and parents, this means access to information not in textbooks or the local library, fast-breaking news, ideas for lessons and activities, and, best of
all, collaboration with others on projects of mutual interest.
Here we provide an annotated listing of Web resources relating to mathematics education. Though not an exhaustive list of what is available, these sites represent the range of resources, and they are
excellent places to begin your own journey through the web of interconnected sites.
-The Math Forum
Offers a large searchable database and Ask Dr. Math, a question-answering service. For teachers, students, parents, researchers.
-Mathematics Archives
Good place to start a search; includes educational software, laboratory notebooks, problem sets, lecture notes, and an extensive hotlist of K-12 teaching materials. Useful to mathematicians,
educators, students, and researchers.
-The Geometry Center
Offers multimedia documents; a downloadable software graphics archive; course materials; workshops, seminars, and courses; and news about events. Useful to mathematicians, students, and educators.
Provides over 100,000 pages of Mathematica programs, documents, examples, etc. Browse or search the archive by author, title, keyword, or item number. Useful to mathematicians, software users,
students, educators, and engineers.
Offers a searchable database of teaching materials and a hotlist of math-related Websites, from "Math Art" to "Word Problems for Kids."
-The Schools of California Online Resource for Educators: Mathematics
Extensive, searchable database of lesson plans, activities, projects, etc. by grade level. Other databases of assessment tools and organizations. Submissions are invited.
-American Mathematical Society
Offers an extensive array of services and guides to the literature in mathematics. A resource for mathematicians and mathematics educators.
-MAA Online
Site of the Mathematical Association of America. See the online version of "Quantitative Reasoning for College Graduates: A Complement to the Standards." It addresses the goal of quantitative
literacy for college students. Useful to college mathematics educators and others.
-Eisenhower National Clearinghouse for Mathematics and Science Education
Supported by the U.S. Department of Education, this site offers curriculum resources in support of reform efforts in math and science. Provides an online copy of the NCTM standards for K-12
mathematics. For K-12 educators and students.
-ERIC Clearinghouse for Science, Mathematics, and Environmental Education
Supported by the U.S. Department of Education, this site offers documents, links to other sites in the ERIC system, and access to the world's largest database of education-related materials. Useful
to anyone interested in education.
-NCTM Home Page
The National Council of Teachers of Mathematics offers an array of services and a catalog of materials for teachers and specialists in mathematics education.
-Mathematical Sciences Education Board
This National Research Council site offers general information, publications, and news releases. Useful to teachers and math specialists.
-Institute for Mathematics and Science Education
Devoted to the Teaching Integrated Math and Science (TIMS) project for elementary school teaching and related programs. For mathematics teachers and specialists at all levels.
-A Math Website! For Middle School Students
Provides an annotated listing of fun and interesting web sites. A great place to search for resources for children.
-Kids Web--Mathematics
A meta-index of quality resources and links. This site provides a broad range of articles, games, puzzles, and demonstrations to explore math, from the origins of algebra and geometry to new
innovations in chaos theory and fractals.
-Mathematics to Prepare Our Children for the 21st Century
An electronic booklet that includes activities for parents to do with their children, ways of helping children with mathematics, and more.
-Treasure Trove of Mathematics
One person's virtual "encyclopedia" of mathematics information and tidbits arranged alphabetically. Fascinating to browse.
Updated to Here
-Appetizers and Lessons for Math and Reason
A massive collection of short items, sometimes funny and very clear. The site also offers reflections on teaching and using the site in lessons. Useful to high school math students and teachers.
-The Schoolhouse Classroom: Mathematics
The Activities Integrating Mathematics and Science Education Foundation publishes books and a magazine for educators. This site includes sample activities and lesson plans, puzzles for classroom use,
and links to other education sites.
-Busy Teacher's Website K-12: Mathematics
Fascinating collection of links to sites on fractals, games and toys, history, lesson plans, and classroom activities. Great resource for teachers.
-Secondary Mathematics Assessment and Resource Database
Offers a database of authentic assessment tools and activities submitted by teachers, searchable by keywords and menus. Also, tips for teachers, puzzles, software, and links to related Websites.
-Math Teacher Link
Web page for a professional development forum for mathematics teachers.
-Mathematics, Science, and Technology Education
Offers a directory of centers, graduate programs, and other internet resources. Of particular interest to teachers are the K-12 statistics page and the K-12 mathematics lessons.
-Science And Math Initiatives (SAMI)
Part of a project to improve math and science education and resource access in rural settings. This database serves as a clearinghouse of resources, funding, and curriculum for rural math and science
-Cornell Math and Science Gateway for Grades 9-12
Useful page for teachers with computers in the classroom. Most links are for teens, but some are on curriculum and teaching strategies. Useful to high school teachers, students, and parents.
-Midlands Improving Math and Science
Offers a searchable database of plans, puzzles, and other resources useful to teachers, K-12 administrators, and other specialists.
-The Mathematics On-Line Bookshelf
Offers a searchable database of math books.
-Mega Mathematics
Engages elementary school teachers and students with unusual and important ideas on the frontier of mathematics.
-Favorite Mathematical Constants
(later updated to)
-About Today's Date
Facts and events associated with the numbers of each date. Of interest to curious people of all ages.
-Fibonacci Numbers and Nature
http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibn at.html
References to technical papers, activities with vegetables and fruit, and links to other Websites.
-MathMol Home Page
Software, activities, hypermedia and more relating to the emerging field of molecular modeling. For teachers, developers of instructional materials, and others interested in molecular modeling.
-MacTutor History of Mathematics
Provides the biographies of over 550 mathematicians. Of particular interest to mathematicians, math educators, historians, and biographers.
-History of Mathematics
Over 1,000 important mathematicians, searchable by name, chronology, or geographic region. Hotlist of sites containing bibliographies and historical references. Useful to mathematicians, math
educators, historians, and enthusiasts.
-A Catalog of Mathematics Resources on WWW and the Internet
A massive list of links to mathematics and mathematics education sites worldwide. One section of the "catalog" is devoted to math teaching, math education, and math student servers.
-Yahoo! Mathematics Resources
Offers a searchable database of links of value to anyone searching for mathematics resources.
-Directory of Mathematics Resources
Part of the Galaxy guide to WWW information and services. Provides an extensive directory of web links to collections of materials, periodicals, organizations, and other directories.
-Blue Web'n Applications: Mathematics WebResource
Part of the WWW Virtual Library, this site offers an extensive collection of links to online materials, including high school servers, gopher servers, newsgroups, electronic journals, a software
index, bibliographies, and more.
-Mathematics Sources
Provides sources grouped into Preprint Archives, Database Gateways, Organizations, Mathematics Departments, and other categories. Useful for educators and specialists.
-Mathematics Information Servers
Perhaps the most comprehensive listing of electronic resources worldwide for mathematics. Particularly useful for mathematicians, educators, and specialists in mathematics education.
-Mathematics Education International Directory
http://acorn.educ.nottingham.ac.uk//SchEd/pages/gates/name s.html
Contact information for many mathematics educators worldwide, with e-mail addresses. Sponsored by the University of Nottingham, UK; searchable by country and organization.
David Haury is Director of the ERIC Clearinghouse for Science, Mathematics and Environmental Education, and Associate Professor of Mathematics, Science, and Technology Education at The Ohio State
University. Linda Milbourne is Associate Director of the ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
This digest was funded by the Office of Educational Research and Improvement, U.S. Department of Education under contract no. RR93002013. Opinions expressed in this digest do not necessarily reflect
the positions or policies of OERI or the Department of Education. | {"url":"http://www.stolaf.edu/people/wallace/Courses/MathEd/mathedresources.html","timestamp":"2014-04-17T07:53:28Z","content_type":null,"content_length":"16726","record_id":"<urn:uuid:4fe8dce5-7d06-4299-8f6e-b39c5e2040ba>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00474-ip-10-147-4-33.ec2.internal.warc.gz"} |
13 Mar 10:58 2013
Moving basic functions
David Luposchainsky <dluposchainsky <at> googlemail.com>
2013-03-13 09:58:41 GMT
Hello GHC HQ, hello mailing list,
there are a couple of basic functions that I think aren't where
they should be.
1. void is currently in Control.Monad. However, it is defined
only in terms of fmap (and therefore only has a Functor
constraint). Although this function is often used in a
monadic setting as ">> return ()", I really don't think
Control.Monad is the right place for it.
2. a) swap is the only function from Data.Tuple that is not
exported to Prelude. On #haskell, people are sometimes
even surprised there /is/ a Data.Tuple, and redefine
their own version of swap at need. I therefore suggest
including Data.Tuple.swap in the Prelude.
The obvious downside of this change would of course be
that it breaks code if there is a top-level user-defined
version of it. Fixing this is of course trivial, but
b) A related suggestion would be the addition of an
irrefutable swap, (swap'?), defined as
"swap ~(a,b) = (b,a)", and its addition to Prelude for
the same reasons.
3. $>, a flipped version of <$, currently resides in
Control.Comonad, but should be in Data.Functor. Applicative
has <* and *>, Monad has >>= and =<<, and I personally keep
(Continue reading) | {"url":"http://comments.gmane.org/gmane.comp.lang.haskell.libraries/18952","timestamp":"2014-04-18T06:07:13Z","content_type":null,"content_length":"49006","record_id":"<urn:uuid:9a3805a3-ca9b-4994-b93c-cba9339d9722>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00050-ip-10-147-4-33.ec2.internal.warc.gz"} |
How many grams are in one tablespoon?
In recipes, quantities of ingredients may be specified by mass (commonly called weight), by volume, or by count.
For most of history, most cookbooks did not specify quantities precisely, instead talking of "a nice leg of spring lamb", a "cupful" of lentils, a piece of butter "the size of a walnut", and
"sufficient" salt. Informal measurements such as a "pinch", a "drop", or a "hint" (soupçon) continue to be used from time to time. In the US, Fannie Farmer introduced the more exact specification
of quantities by volume in her 1896 Boston Cooking-School Cook Book.
In journalism, a human interest story is a feature story that discusses a person or people in an emotional way. It presents people and their problems, concerns, or achievements in a way that
brings about interest, sympathy or motivation in the reader or viewer.
Human interest stories may be "the story behind the story" about an event, organization, or otherwise faceless historical happening, such as about the life of an individual soldier during
wartime, an interview with a survivor of a natural disaster, a random act of kindness or profile of someone known for a career achievement.
United States customary units are a system of measurements commonly used in the United States. The U.S. customary system developed from English units which were in use in the British Empire before
American independence. Consequently most U.S. units are virtually identical to the British imperial units. However, the British system was overhauled in 1824, changing the definitions of some units
used there, so several differences exist between the two systems.
The majority of U.S. customary units were redefined in terms of the meter and the kilogram with the Mendenhall Order of 1893, and in practice, for many years before. These definitions were refined by
the international yard and pound agreement of 1959. The U.S. primarily uses customary units in its commercial activities, while science, medicine, government, and many sectors of industry use metric
units. The SI metric system, or International System of Units is preferred for many uses by NIST
The system of imperial units or the imperial system (also known as British Imperial) is the system of units first defined in the British Weights and Measures Act of 1824, which was later refined and
reduced. The system came into official use across the British Empire. By the late 20th century, most nations of the former empire had officially adopted the metric system as their main system of
measurement, but some Imperial units are still used in the United Kingdom and Canada.
Related Websites: | {"url":"http://answerparty.com/question/answer/how-many-grams-are-in-one-tablespoon","timestamp":"2014-04-19T14:35:05Z","content_type":null,"content_length":"25918","record_id":"<urn:uuid:a44254c2-6c3c-4244-9d68-3f6402eeb235>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00012-ip-10-147-4-33.ec2.internal.warc.gz"} |
December 7th 2011, 05:07 PM
Why is it obvious that attaching an (m+n)-cell to SmVSn is SmxSn?
Also, if you have a space X that is the real line, with two sphere's (S2) wedged at each integer, to determine its homology group, I figured I should use the Mayer–Vietoris sequence, so I split
it such that U = X - {odd integers}, V = X - {even integers}. Do you think that's what I should do?
December 8th 2011, 12:17 PM
Re: Topology
For your first question, note that $S^n \cong R^n \cup \{\infty\}$, that is, $S^n$ is the one point compactification of $R^n$.
Now $S^m \times S^n \cong (R^m \cup \{\infty\}) \times (R^n \cup \{\infty\})$
$= R^{m+n} \cup (R^m \times \{\infty\}) \cup (R^n \times \{\infty\}) \cup (\{\infty\} \times \{\infty\})$
$= R^{m+n} \sqcup R^m \sqcup R^n \sqcup \{\infty\}$(disjoint union)
First glue $R^m \sqcup R^n \sqcup \{\infty\}$ we get $S^m \vee S^n$, then glue $R^{m+n}$ by identifying it with $D^{m+n}$ the unit disk, then glue it to the $S^m \vee S^n$ frame. It's not easy to
express but you can always take m=n=1 as an example to study the procedure to glue (1 square)+(2 segments)+(1 point) together to get a torus. | {"url":"http://mathhelpforum.com/differential-geometry/193720-topology-print.html","timestamp":"2014-04-16T05:52:09Z","content_type":null,"content_length":"6400","record_id":"<urn:uuid:48863924-30b6-4777-bfe0-9ff74bdcc9ff>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00327-ip-10-147-4-33.ec2.internal.warc.gz"} |
Harmonic Function
Thanks lanedance, I didn't know about differentiating under the integral sign, good stuff!
I think I have the answer, would appreciate feedback as this is not 100% comfortable stuff for me.
So far i get:
1) Use the definition of [tex]\overline{f}(r)\equiv\frac{1}{2\pi}\int^{2\pi}_{0}f(r,\varphi)d\varphi[/tex]
to find [tex](r\frac{\partial \overline{f}}{\partial r}) = \frac{1}{2\pi}\int^{2\pi}_{0} r\frac{\partial f(r,\varphi)}{\partial r} d\varphi[/tex] in terms of [tex]f[/tex]
2) Note that
[tex]\left[r\frac{\partial\overline{f}}{\partial r}\right]}^{r=R}_{r=0}=\int^{R}_{0} \frac{\partial}{\partial r} (r\frac{\partial \overline{f}}{\partial r})dr = \frac{1}{2\pi}\int^{R}_{0}\int^{2\pi}_
{0} \frac{\partial}{\partial r} (r\frac{\partial f(r,\varphi)}{\partial r}) d\varphi dr[/tex]
3) Sub in the Laplacian expression [tex]\int^{R}_{0}\int^{2\pi}_{0} \left[\frac{\partial}{\partial r}(r\frac{\partial f}{\partial r})\right] d\varphi dr=0[/tex]
To find that
[tex]\left[r\frac{\partial\overline{f}}{\partial r}\right]}^{r=R}_{r=0}=0[/tex] | {"url":"http://www.physicsforums.com/showthread.php?t=421883","timestamp":"2014-04-20T00:50:01Z","content_type":null,"content_length":"33096","record_id":"<urn:uuid:5cb9bf66-48ee-4f2f-96fb-98803b0809e3>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00196-ip-10-147-4-33.ec2.internal.warc.gz"} |
Marica D. Presi\'c:
Representation of metamorphosis grammar in logic grammar: proof trees and their lengths
Marica D. Presi\'c and Slavisa B. Presi\'c:
A constructive proof of equivalence of formalism od DCG's with the formalism of type 0 phrase-structure grammars
A. Dobrynin and I. Gutman:
On a graph invariant related to the sum of all distances in a graph
Olga Bodroza-Panti\'c and Ratko Tos\'c:
On the number of 2-factors in rectangular lattice graphs
Miroslav Plo\v s\v cica:
Graphical compositions and weak congruences
Farid Bensherif et Guy Robin:
Sur l'itere de $\sin x$
Du\v sanka Peri\v si\'c:
Functional equations and tempered ultradistributions
Ivan Jovanovi\'c and Vladimir Rakocevi\'c:
Multipliers of mixed-norm sequence spaces and measures of noncompactness
Du\v san R\. Georgijevi\'c:
A note on approximation by Blascke--Potapov products
Stojan Radenovi\'c i Slavko Simi\'c:
Some classes of locally convex Riesz spaces
Pankaj Jain:
On imbeddings of weighted Sobolev spaces on an unbounded domain
Dragan Djordjevi\'c:
Regular and $T$-Fredholm elements in Banach algebras
Archana Roy and S.K. Singh:
Almost para-contact Finsler connections on vector bundle
Hiroshi Endo:
On the AC-contact Bochner curvature tensor field on almost cosymplectic manifolds
Milan Merkle and Ljiljana Petrovi\'c:
On Schur-convexity of some distribution functions
Drazen Panti\'c:
Stochastic calculus on one-dimensional diffusions
Bosko Jovanovi\'c:
On the convergence of a multicomponent alternating direction difference scheme
Miodrag Mateljevi\'c:
The dual of the Bergman space defined on a hyperbolic plane domain
Publication date for this issue: 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/070/index.html","timestamp":"2014-04-19T20:02:34Z","content_type":null,"content_length":"4569","record_id":"<urn:uuid:d5719598-7111-4bf6-99e5-ac41856751d7>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00645-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Doctoral Dissertation Defense of Yin Chang
Principal Component Models Applied to Confirmatory Factor Analysis
The Doctoral Dissertation Defense of Yin Chang
Starting at 3:10pm, Thursday, Dec 6th, 2012
1-134 Wilson Hall
Testing structural equation models, in practice, may not always go smoothly, and the solution that is printed in the output may be an improper solution. The term "improper solution" refers to several
possible problems with model estimation. Perhaps the most common problem that researchers have when they begin testing models is an error message that says something like "sigma matrix is not
positive definite" or "warning: negative psi matrix." Another improper solution involves "out of bounds" estimates, sometimes referred to as "Heywood cases," which are negative measurement error
variances, or negative disturbances. Heywood cases can occasionally be found in the output even without an error message.
The dissertation achieves the following goals: to develop a stable algorithm to estimate parameters, which would be robust to data set variability and converge with high probability; to construct an
algorithm which would detect the reason for failure to converge; to develop statistical theory to compute confidence regions for functions of parameters, under non-normality of responses; and to
develop statistical theory to perform hypotheses tests, such as goodness of fit tests and model comparison tests.
Based on the large simulation results, it can be demonstrated that the inference procedures for the proposed model work well enough to be used in practice and that the proposed model has advantages
over the conventional model, in terms of proportion of proper solutions; average coverage rates of upper one-sided nominal 95% confidence intervals, lower one-sided nominal 95% confidence intervals,
and two-sided nominal 95% confidence intervals; and average mean ratios of the width of two-sided nominal 95% confidence intervals. | {"url":"http://calendar.msu.montana.edu/events/8282&origin=msutoday","timestamp":"2014-04-17T09:35:51Z","content_type":null,"content_length":"20949","record_id":"<urn:uuid:eba98fb8-1c0e-45ac-8263-6f3d5b8c2e96>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00524-ip-10-147-4-33.ec2.internal.warc.gz"} |
Faculty Research Interests
Agronsky, Steve
Ph.D. University of California, Santa Barbara
Analysis, dynamical systems
Bonini, Vincent
Ph.D. University of California, Santa Cruz
Differential Geometry, Geometric Analysis, Conformal Geometry and Mathematical Relativity
Borzellino, Joe
Ph.D. University of California, Los Angeles
Riemannian geometry, differential topology of orbifolds
Brussel, Eric
Ph.D. University of California, Los Angeles
Algebraic geometry, cohomology, and division algebras
Camp, Charles D.
Ph.D. CalTech
Geophysical fluid dynamics, atmospheric dynamics, climate change, mathematical modeling, data analysis techniques
Champney, Danielle
Ph.D. University of California, Berkeley
Undergraduate mathematics education, students' use of images in mathematical sense-making, ongoing teacher preparation and education
Choboter, Paul
Ph.D. University of Alberta
Geophysical fluid dynamics, coastal ocean modeling
Easton, Rob
Ph.D. Stanford University
Algebraic Geometry and Tropical Geometry
Greenwald, Harvey
Ph.D. Washington University
Harmonic analysis
Grundmeier, Todd
Ph.D. University of New Hampshire
Mathematical problem posing and problem solving, pre-service teacher education, in-service professional development
Gu, Caixing
Ph.D. Indiana University, Bloomington
Operator theory, matrix analysis, system and control theory
Hamilton, Emily
Ph.D. University of California, Los Angeles
low-dimensional topology, hyperbolic geometry, geometric group theory
Hartig, Donald
Ph.D. University of California, Santa Barbara
Topological measure theory, spectral theory and its applications, the use of technology to enhance learning of mathematics
Kato, Goro
Ph.D. University of Rochester
Algebraic geometry (p-adic cohomology theory), D-modules, homological algebra
Kaul, Anton
Ph.D. Oregon State University
Geometric group theory
Kirk, Colleen
Ph.D. Northwestern University
Integral equations and nonlinear partial differential equations, with applications to combustion and quenching problems
Liese, Jeffrey
Ph.D. University of California, San Diego
Enumerative and Algebraic Combinatorics
Lin, Joyce
Ph.D. University of North Carolina at Chapel Hill
Applied math, math modeling, math biology, geophysical fluid dynamics
Medina, Elsa
Ph.D. University of Northern Colorado
Mathematics education
Mendes, Anthony
Ph.D. University of California, San Diego
Algebraic and enumerative combinatorics
Mueller, Jim
Ph.D. California Institute of Technology
Applied mathematics, asymptotic analysis, singular perturbation theory
Paquin, Dana
Ph.D. Stanford University
Mathematical modeling, applied mathematics, medical imaging
Patton, Linda
Ph.D. University of California, San Diego
Operator theory, complex analysis (one and several variables), Nevanlinna-Pick interpolation
Pearse, Erin
Ph.D. University of California, Riverside
Curvature and measurability questions for self-similar fractal sets, especially volume formulas for tubular neighbourhoods. As well as large networks, including boundary representations for infinite
graphs and the use of graph-theoretic techniques for analysis of large data sets, with applications to missing data.
Rawlings, Don
Ph.D. University of California, San Diego
Enumerative and algebraic combinatorics, discrete probabilities
Retsek, Dylan
Ph.D. Washington University, St. Louis
Complex analysis, functional analysis and composition operators
Richert, Ben
Ph.D. University of Illinois, Urbana-Champaign
Commutative algebra: free resolutions, the extremal behavior of Hilbert functions and (graded) Betti numbers, generic behavior, Gorenstein rings
Riley, Kate
Ph.D. Montana State University, Bozeman
Subject matter and pedagogical knowledge necessary for prospective teachers to become master teachers; undergraduates' conceptual knowledge in mathematical proof; how technology enhances the learning
of problem-solving, mathematical reasoning, and proof
Robbins, Marian
Ph.D. University of Virginia
Operator theory, functional analysis and complex function theory
Schinck-Mikel, Amelie
Ph.D. University of North Carolina, Charlotte
Socio-cultural issues in mathematics education, teacher education, language and mathematics learning, problem solving
Shapiro, Jonathan
Ph.D. University of California, Berkeley
Operator theory, complex analysis, and functional analysis
Sherman, Morgan
Ph.D. Columbia University
Algebraic and complex geometry; especially Hilbert schemes, balanced metrics
Stankus, Mark
Ph.D. University of California, San Diego
Operator theory, noncommutative Groebner basis, system engineering, computer science
Sze, Lawrence
Ph.D. Pennsylvania State University
Combinatorics and number theory
Todorov, Todor
Ph.D. University of Sofia and Bulgarian Academy of Sciences
Non-linear theory of generalized functions (Colombeau algebras), non-standard analysis, asymptotic analysis, coompactifications of ordered topological spaces, linear partial differential equations
with variable coefficients, and teaching calculus
White, Matthew
Ph.D. University of California, Santa Barbara
Yoshinobu, Stan
Ph.D. University of California, Los Angeles
Undergraduate Mathematics Education, Inservice and Preservice Teacher Preparation, Design and Implementation of inquiry-Based methods | {"url":"http://www.calpoly.edu/~math/faculty_research_interests.html","timestamp":"2014-04-19T15:22:56Z","content_type":null,"content_length":"14775","record_id":"<urn:uuid:f183973d-4998-4d35-83e8-02adea6408d2>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00027-ip-10-147-4-33.ec2.internal.warc.gz"} |
The purpose of this activity is to practice analyzing experiment results by applying the t test. The data you will analyze will be simulated data, randomly generated by a web service.
For this activity, you will need to use:
• a spreadsheet that can accept tab-separated-value data (such as Excel, OpenOffice, or Google Spreadsheet);
The data generator requires two inputs: an experiment and an experimental design. For this activity, we will use the Point-and-click experiment, which simulates the time it takes for a user to point
and click a particular target on screen using up to three different pointing devices (a mouse, a trackpad, and a trackball). The other input, the experiment design, is a string specifying the
conditions and trials made by each user in the experiment.
The output of the data generator is tab-separated-value data, which you can copy and paste into a spreadsheet to rearrange and compute statistics, and then transfer to the t test calculator to
perform the statistical test.
Generate data from the Point-and-Click experiment using the experiment design MMMM,MMMM,MMMM,PPPP,PPPP,PPPP. (How many users and trials is this?)
Move the data to your spreadsheet. It's a good idea to split the time column into two side-by-side columns, one for the mouse condition (M) and one for the trackpad condition (P).
Use the spreadsheet to compute the mean (called AVERAGE in most spreadsheets), standard deviation (STDEV), and standard error (STDEV/sqrt(n)) of each condition. If you graphed the two means with
error bars, would the error bars overlap?
Run a t test on this data at the 5% significance level. Determine the p value, the value of the t statistic, and the degrees of freedom.
Generate data from the Point-and-Click experiment using the experiment design MMPP,MMPP,MMPP,MMPP. (What's the risk of this experiment design?)
Move the data to your spreadsheet. Line up each user's M and P trials side-by-side.
Run a paired t test on this data at the 5% significance level. Determine the p value, the value of the t statistic, and the degrees of freedom. | {"url":"http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-831-user-interface-design-and-implementation-spring-2011/in-class-activities/experiment-analysis/","timestamp":"2014-04-18T16:19:02Z","content_type":null,"content_length":"36279","record_id":"<urn:uuid:68a0d5b0-8c11-4f28-981b-352123af8ad9>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00325-ip-10-147-4-33.ec2.internal.warc.gz"} |
London Number Theory Seminar
│ London Number Theory Seminar │ │ │
The London Number Theory Seminar is held weekly, on Wednesdays, during term time. The location of the seminar cycles between KCL, Imperial College and UCL.
This term (Spring 2014), the seminar will be hosted by UCL, and will be on Wednesdays at 4pm in UCL maths department, in room 707 ("the usual room") apart from Sarnak's talk which will be in 500. The
seminar will be preceded by tea and biscuits at 3.30 in room 606.
Sarah Zerbes is the organizer of the seminar this term; abstracts for this term's talks are available from her web pages here. Here are the titles:
15/01/14 Alex Bartel (Warwick)
Title: The Cohen-Lenstra heuristic revisited
22/01/14 James Newton (Cambridge)
Title: Local-global compatibility for Galois representations associated to Hilbert modular forms of low weight
29/01/14 Chris Williams (Warwick)
Title: Overconvergent modular symbols over imaginary quadratic fields
05/02/14 Victor Rotger (Univ. Polytècnica de Catalunya)
Title: The Birch and Swinnerton-Dyer conjecture for non-abelian twists of elliptic curves.
12/02/14 At 2:30 in 707: Masato Kurihara (Keio University)
Title: Higher Fitting ideals of arithmetic objects
12/02/14 At 4:00 in 707: Thanasis Bouganis (Durham)
Title: p-adic measures for Hermitian modular forms and the Rankin-Selberg method
19/02/14 Matthew Morrow (Nottingham)
Title: Deformation of algebraic cycles in characteristic p
26/02/14 Guido Kings (Regensburg)
Title: TBA
05/03/14 Malte Witte (Heidelberg)
Title: TBA
12/03/14 Chris Lazda (Imperial)
Title: Rigid rational homotopy theory and mixedness
19/03/14 Peter Sarnak (Princeton) ( NB: talk will be in room 500)
Title: TBA
26/03/14 Jenny Cooley (Warwick)
Title: TBA
The seminar is preceded by the Study Groups, which this term will be on Bhargava's work (1-2.15pm) and Perfectoid spaces (2.15-3.30pm).
A list of previous seminar talks is here.
There are two mailing lists for number theory in London:
This page is maintained, poorly, by Kevin Buzzard. If anyone thinks they can do better they're welcome to offer. | {"url":"http://www2.imperial.ac.uk/~buzzard/LNTS/lnts.html","timestamp":"2014-04-18T08:30:56Z","content_type":null,"content_length":"5836","record_id":"<urn:uuid:a5d02877-e3a9-43bf-94d9-ec88b0de6abe>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00225-ip-10-147-4-33.ec2.internal.warc.gz"} |
Post a reply
Before going on with a new idea I suggest a look at "Matrix Moves" in this thread.
Supposing we have the stochastic matrix
The central problems of Linear Algebra are the solution of a simultaneous set of linear equations or Ax = b and determining the eigenvalues of a matrix.
The eigenvalues are usually computed using a computer and we will not break with tradition, they are
There is a little theorem that says if a square matrix has distinct eigenvalues then it is diagonalizable. So this one is diagonalizable.
To do it we need the Eigenvectors of A:
To check whether we have diagonalized it we plug in to
Okay, so what? The useful fact is that to get A^k we only now need the following matrix equation.
Now D^k is easy to get because to raise a matrix with just diagonal elements like D to the kth power you just take each element and raise it to the kth power.
So if we wanted A^10 we would compute
And we are done! | {"url":"http://www.mathisfunforum.com/post.php?tid=18885","timestamp":"2014-04-21T05:28:08Z","content_type":null,"content_length":"20931","record_id":"<urn:uuid:85f3a414-c262-465f-ad37-067bcc24a73d>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00515-ip-10-147-4-33.ec2.internal.warc.gz"} |
Flipping Pancakes
April 7, 2009
This problem is a favorite of mathematicians, who are still searching for the formula that calculates the minimum number of flips required to sort a stack of pancakes. Before he dropped out of school
and became rich, Bill Gates wrote the only published paper of his career showing one solution to the problem (which has since been bettered). For a stack of n pancakes, our algorithm makes 2n flips,
which is far from optimal.
We begin with a function flip that reverses the top n pancakes on the stack; this is equivalent to inserting a spatula under the nth pancake and flipping:
(define (flip n xs)
(let loop ((n n) (xs xs) (ys '()))
(if (or (null? xs) (zero? n)) (append ys xs)
(loop (- n 1) (cdr xs) (cons (car xs) ys)))))
Our algorithm will work by finding the largest pancake in the stack, flipping it to the top, then flipping the entire stack to put the largest pancake on the bottom; the process is then repeated on
that diminishing portion of the stack that is not yet sorted. The find-max function finds the largest of the first n pancakes on the stack:
(define (find-max n xs)
(let loop ((n n) (xs xs) (k 0) (mx 0) (mk 0))
(cond ((or (zero? n) (null? xs)) mk)
((< mx (car xs)) (loop (- n 1) (cdr xs) (+ k 1) (car xs) (+ k 1)))
(else (loop (- n 1) (cdr xs) (+ k 1) mx mk)))))
Now we simply loop through the stack of pancakes:
(define (pancake xs)
(let loop ((k (length xs)) (xs xs))
(let* ((j k) (i (find-max j xs)))
(if (= j 1) xs
(loop (- k 1) (flip j (flip i xs)))))))
Here’s how the pancake sort looks:
> (pancake '(7 2 9 4 6 1 3 8 5))
(1 2 3 4 5 6 7 8 9)
You can run this program at http://programmingpraxis.codepad.org/HQFNnbnb.
Pages: 1 2
12 Responses to “Flipping Pancakes”
1. April 7, 2009 at 11:37 AM
This Haskell solution does some unnecessary flips if the bottom of the current subset is already in the correct position, but then again if you want efficiency just do a merge- or quicksort :)
main = print $ pancakeSort [7,2,9,4,6,1,3,8,5]
pancakeSort :: (Ord a) => [a] -> [a]
pancakeSort xs = foldr sort' xs [1..length xs]
sort' :: (Ord a) => Int -> [a] -> [a]
sort' n xs = flip' n $ flip' (snd . maximum . flip zip [1..] $ take n xs) xs
flip' :: Int -> [a] -> [a]
flip' n = uncurry ((++) . reverse) . splitAt n
2. April 7, 2009 at 7:34 PM
Man that Haskell solution is compact. It’s gorgeous. Too bad my Haskell is so rusty that it doesn’t mean anything to me.
Haskell, the twitter of programming languages…
3. April 8, 2009 at 12:25 PM
; Solution using fold
#lang scheme
(require srfi/1)
(define (flip n xs)
(let-values ([(top bottom) (split-at xs n)])
(append (reverse top) bottom)))
(define (find-max n xs)
(let ([m (apply max (take xs n))])
(list-index (λ (x) (= x m)) xs)))
(define (pancake xs)
(fold (λ (n xs) (flip n (flip (+ (find-max n xs) 1) xs)))
(reverse (iota (length xs) 1))))
(pancake ‘(7 2 9 4 6 11 1 3 8 5 13))
4. April 8, 2009 at 12:27 PM
; Solution without fold.
#lang scheme
(require srfi/1)
(define (flip n xs)
(let-values ([(top bottom) (split-at xs n)])
(append (reverse top) bottom)))
(define (find-max n xs)
(let ([m (apply max (take xs n))])
(list-index (λ (x) (= x m)) xs)))
(define (nest f n base)
(if (= n 0)
(nest f (- n 1) (f n base))))
(define (pancake xs)
(nest (λ (n xs) (flip n (flip (+ (find-max n xs) 1) xs)))
(length xs)
(pancake ‘(7 2 9 4 6 11 1 3 8 5 10 ))
5. April 8, 2009 at 4:48 PM
It’s not too hard to define Scheme functions so they resemble the curry-and-compose style of Haskell. I just uploaded a small library of higher-order functions to the Standard Prelude, including
compose and define-curried; fold-right is already there. That’s most of what you need to replicate the Haskell solution.
6. April 8, 2009 at 7:33 PM
I notice that the prelude contains this definition:
(define (second x y) y)
There is a strong tradition however to let second be:
(define (second xs) (list-ref xs 1))
I am not sure what the traditional name for
the projection is, but perhaps project2 ?
7. April 8, 2009 at 8:00 PM
I think an even older and stronger tradition is to spell your second “cadr”. I just looked; cadr appears on page 13 of the Lisp 1.5 Programmer’s Manual, in the definition of apply.
8. April 8, 2009 at 11:39 PM
In modern times second is used instead of cadr to stress that one is working with lists. The operation cadr is used for trees built using conses.
Both in SRFI 1 and the Common Lisp HyperSpec second is used to access the second element of a list.
9. April 21, 2009 at 2:44 AM
here’s my python:
def flipsort(pile):
for i in xrange(len(pile)):
max_index = pile.index(max(pile[i:]))
if max_index != i:
pile[max_index:] = pile[max_index:][::-1]
pile[i:] = pile[i:][::-1]
return pile
10. April 21, 2009 at 3:19 AM
fixed to work with duplicates:
def flipsort(pile):
for i in xrange(len(pile)):
slice = pile[i:]
max_index = slice.index(max(slice)) + i
if max_index != i:
step += 2
pile[max_index:] = pile[max_index:][::-1]
pile[i:] = pile[i:][::-1]
return pile | {"url":"http://programmingpraxis.com/2009/04/07/flipping-pancakes/2/","timestamp":"2014-04-21T12:37:36Z","content_type":null,"content_length":"71942","record_id":"<urn:uuid:065daf4e-273d-41b7-99cd-feec588d3cbe>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00196-ip-10-147-4-33.ec2.internal.warc.gz"} |
Trouble with Integration
July 11th 2011, 06:27 AM #1
Trouble with Integration
f:R ---> R is a continuous function and f(x) = f(2x) is true for all real numbers
If f(1)=3 then find the value of
$\int_{-1}^{1}f[f(x)] dx$
Re: Trouble with Integration
Necessary condition: Taking into account that $g(x)=3$ is continuous in $\mathbb{R}$ and satisfies $g(x)=g(2x)\;\wedge \;g(1)=3$ for all $x\in \mathbb{R}$ then, if the solution does not depend on
$f$ we necessarily have $\int_{-1}^{1}f[f(x)] \;dx=\int_{-1}^{1}3 \;dx=6$ .
Re: Trouble with Integration
We can show that f is necessarily constant.
Since $f(x) = f(2x)$ for all x,
$f(x/2) = f(x)$ .....(just substitute x/2 for x)
$f(x/(2^2)) = f(x)$ ..... similarly
$f(x/(2^n)) = f(x)$
Now let $n \to \infty$ and apply the continuity of f.
Re: Trouble with Integration
Another way: the solutions of the functional equation $f(2x)=f(x)$ are $f(x)=\phi(\log x)$ where $\phi$ is any periodic function with period $\log 2$ . According to the hypothesis, necessarily $f
(x)=3$ .
July 11th 2011, 11:02 AM #2
July 11th 2011, 04:09 PM #3
July 11th 2011, 11:42 PM #4 | {"url":"http://mathhelpforum.com/calculus/184410-trouble-integration.html","timestamp":"2014-04-17T13:56:46Z","content_type":null,"content_length":"43560","record_id":"<urn:uuid:d7599ffd-2ca7-4321-b699-00251a0160bd>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00482-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
Use the formula for the area of a trapezoid , where A is area, b1 and b2 are the length of the bases, and h is the height, to answer the question. How many square feet of grass are there on a
trapezoidal field with a height of 75 ft and bases of 125 ft and 81 ft?
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Well, do you know the formula for the area of a trapezoid?
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sorry about that it didnt copy.
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A= h (b1+b2/2)
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\[A={1 \over 2} \times (b1+b2) \times h\]
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yes. Now plug in the values. b1=125, b2=81, h=75
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what do you get?
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idk); i need the steps and all that with the response. ive been doing that problem for 2 hours and still havent got it
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You have A=1/2×(b1+b2)×h So just sub in your values of b1 and b2 (the lengths of the bases) and h (the height of the trapezoid). What can't you do?
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wait let me tell u so you can tell me if i got it right
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I'll give you one example. I hope you understand so you can plug in the rest into the formula. \[A=75({b1+b2 \over 2})\] I plugged in h=75 into the equation that you gave. Now, plug in b1 and b2
to solve.
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Go ahead. :)
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Basically, their saying that the field is a trapezoid and they want to find the area of the field. What we're doing now is finding the area of the field.
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A= h (b1+b2/2) A=75x(125x1+ 81x2/2) =125+6,561/2 =75x3,343 A=250,725 ehh im wrong huh? lol
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b1 and b2 are just names. You don't have to multiply
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it's short for 'base 1' and 'base 2'
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This should be your equation. \[A=75({125+81 \over 2})\]
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Can you solve it now? :)
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the answer is 7,500? ;)
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I can see why you were so confused. Solve the equation i gave. You're answer of 7,500 is close, but it's not correct.
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A=75x(125+81/2) A=75x(206/2) A=75x103 A=7,725 I added right but isomehow made the 6 look like a 0. oopps
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yes. 7,725 is right! :)
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No worries. We all make mistakes now and then. ;)
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Yay!!! Thankyou so much!!!
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You're welcome!
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CORDIC is an iterative algorithm for calculating trig functions including sine, cosine, magnitude and phase. It is particularly suited to hardware implementations because it does not require any
1. Basics
1.1 What does "CORDIC" mean?
COordinate Rotation DIgital Computer. (Doesn't help much, does it?!)
1.2 What does it do?
It calculates the trigonometric functions of sine, cosine, magnitude and phase (arctangent) to any desired precision. It can also calculate hyperbolic functions, but we don't cover that here.
1.3 How does it work?
CORDIC revolves around the idea of "rotating" the phase of a complex number, by multiplying it by a succession of constant values. However, the multiplies can all be powers of 2, so in binary
arithmetic they can be done using just shifts and adds; no actual multiplier is needed.
1.4 How does it compare to other approaches?
Compared to other approaches, CORDIC is a clear winner when a hardware multiplier is unavailable, e.g. in a microcontroller, or when you want to save the gates required to implement one, e.g. in
an FPGA. On the other hand, when a hardware multiplier is available, e.g. in a DSP microprocessor, table-lookup methods and good old-fashioned power series are generally faster than CORDIC.
2.0 Examples
2.1 Do you have an example in the form of an Excel spreadsheet?
Funny you should ask: I just happen to: cordic-xls.zip. I highly recommend you open this up and look it over a little before you ask any more questions.
2.2 Do you have an example as C code?
Yup, it's your lucky day: cordic.zip.
3.0 Principles
3.1 What symbols will we be using?
Given a complex value: C = Ic + jQc
we will create a rotated value: C' = Ic' + jQc'
by multiplying by a rotation value: R = Ir + jQr
3.2 What are the basic principles?
1. Recall that when you multiply a pair of complex numbers, their phases (angles) add and their magnitudes multiply. Similarly, when you multiply one complex number by the conjugate of the other,
the phase of the conjugated one is subtracted (though the magnitudes still multiply).
To add R's │ C' = C·R │ Ic' = Ic·Ir - Qc·Qr │
phase to C: │ │ Qc' = Qc·Ir + Ic·Qr │
To subtract R's │ C' = C·R* │ Ic' = Ic·Ir + Qc·Qr │
phase from C: │ │ Qc' = Qc·Ir - Ic·Qr │
2. To rotate by +90 degrees, multiply by R = 0 + j1. Similarly, to rotate by -90 degrees, multiply by R = 0 - j1. If you go through the Algebra above, the net effect is:
To add 90 degrees, │ Ic' = -Qc │ (negate Q, then swap) │
multiply by R = 0 + j1: │ Qc' = Ic │ │
To subtract 90 degrees, │ Ic' = Qc │ (negate I, then swap) │
multiply by R = 0 - j1: │ Qc' = -Ic │ │
3. To rotate by phases of less than 90 degrees, we will be multiplying by numbers of the form "R = 1 +/- jK". K will be decreasing powers of two, starting with 2^0 = 1.0. Therefore, K = 1.0, 0.5,
0.25, etc. (We use they symbol "L" to designate the power of two itself: 0, -1, -2, etc.)
Since the phase of a complex number "I + jQ" is atan(Q/I), the phase of "1 + jK" is atan(K). Likewise, the phase of "1 - jK" = atan(-K) = -atan(K).
To add phases we use "R = 1 + jK"; to subtract phases we use "R = 1 - jK". Since the real part of this, Ir, is equal to 1, we can simplify our table of equations to add and subtract phases for
the special case of CORDIC multiplications to:
To add a phase, │ Ic' = Ic - K·Qc = Ic - (2^-L)·Qc = Ic - (Qc >> L) │
multiply by R = 1 + jK: │ Qc' = Qc + K·Ic = Qc + (2^-L)·Ic = Qc + (Ic >> L) │
To subtract a phase, │ Ic' = Ic + K·Qc = Ic + (2^-L)·Qc = Ic + (Qc >> L) │
multiply by R = 1 - jK: │ Qc' = Qc - K·Ic = Qc - (2^-L)·Ic = Qc - (Ic >> L) │
4. Let's look at the phases and magnitudes of each of these multiplier values to get more of a feel for it. The table below lists values of L, starting with 0, and shows the corresponding values of
K, phase, magnitude, and CORDIC Gain (described below):
│ L │ K = 2^-L │ R = 1 + jK │ Phase of R │ Magnitude of R │ CORDIC Gain │
│ │ │ │ in degrees │ │ │
│ │ │ │ = atan(K) │ │ │
│ 0 │ 1.0 │ 1 + j1.0 │ 45.00000 │ 1.41421356 │ 1.414213562 │
│ 1 │ 0.5 │ 1 + j0.5 │ 26.56505 │ 1.11803399 │ 1.581138830 │
│ 2 │ 0.25 │ 1 + j0.25 │ 14.03624 │ 1.03077641 │ 1.629800601 │
│ 3 │ 0.125 │ 1 + j0.125 │ 7.12502 │ 1.00778222 │ 1.642484066 │
│ 4 │ 0.0625 │ 1 + j0.0625 │ 3.57633 │ 1.00195122 │ 1.645688916 │
│ 5 │ 0.03125 │ 1 + j0.031250 │ 1.78991 │ 1.00048816 │ 1.646492279 │
│ 6 │ 0.015625 │ 1 + j0.015625 │ 0.89517 │ 1.00012206 │ 1.646693254 │
│ 7 │ 0.007813 │ 1 + j0.007813 │ 0.44761 │ 1.00003052 │ 1.646743507 │
│ ... │ ... │ ... │ ... │ ... │ ... │
A few observations:
□ Since we're using powers of two for the K values, we can just shift and add our binary numbers. That's why the CORDIC algorithm doesn't need any multiplies!
□ You can see that starting with a phase of 45 degrees, the phase of each successive R multiplier is a little over half of the phase of the previous R. That's the key to understanding CORDIC:
we will be doing a "binary search" on phase by adding or subtracting successively smaller phases to reach some target phase.
□ The sum of the phases in the table up to L = 3 exceeds 92 degrees, so we can rotate a complex number by +/- 90 degrees as long as we do four or more "R = 1 +/- jK" rotations. Put that
together with the ability to rotate +/-90 degrees using "R = 0 +/- j1", and you can rotate a full +/-180 degrees.
□ Each rotation has a magnitude greater than 1.0. That isn't desirable, but it's the price we pay for using rotations of the form 1 + jK. The "CORDIC Gain" column in the table is simply a
cumulative magnitude calculated by multiplying the current magnitude by the previous magnitude. Notice that it converges to about 1.647; however, the actual CORDIC Gain depends on how many
iterations we do. (It doesn't depend on whether we add or subtract phases, because the magnitudes multiply either way.)
4.0 Applications
4.1 How can I use CORDIC to calculate magnitude?
You can calculate the magnitude of a complex number C = Ic + jQc if you can rotate it to have a phase of zero; then its new Qc value would be zero, so the magnitude would be given entirely by the
new Ic value.
"So how do I rotate it to zero," you ask? Well, I thought you might ask:
1. You can determine whether or not the complex number "C" has a positive phase just by looking at the sign of the Qc value: positive Qc means positive phase. As the very first step, if the
phase is positive, rotate it by -90 degrees; if it's negative, rotate it by +90 degrees. To rotate by +90 degrees, just negate Qc, then swap Ic and Qc; to rotate by -90 degrees, just negate
Ic, then swap. The phase of C is now less than +/- 90 degrees, so the "1 +/- jK" rotations to follow can rotate it to zero.
2. Next, do a series of iterations with successively smaller values of K, starting with K=1 (45 degrees). For each iteration, simply look at the sign of Qc to decide whether to add or subtract
phase; if Qc is negative, add a phase (by multiplying by "1 + jK"); if Qc is positive, subtract a phase (by multiplying by "1 - jK"). The accuracy of the result converges with each iteration:
the more iterations you do, the more accurate it becomes.
[Editorial Aside: Since each phase is a little more than half the previous phase, this algorithm is slightly underdamped. It could be made slightly more accurate, on average, for a given
number of iterations, by using "ideal" K values which would add/subtract phases of 45.0, 22.5, 11.25 degrees, etc. However, then the K values wouldn't be of the form 2^-L, they'd be 1.0,
0.414, 0.199, etc., and you couldn't multiply using just shift/add's (which would eliminate the major benefit of the algorithm). In practice, the difference in accuracy between the ideal K's
and these binary K's is generally negligible; therefore, for a multiplier-less CORDIC, go ahead and use the binary Ks, and if you need more accuracy, just do more iterations.]
Now, having rotated our complex number to have a phase of zero, we end up with "C = Ic + j0". The magnitude of this complex value is just Ic, since Qc is zero. However, in the rotation process, C
has been multiplied by a CORDIC Gain (cumulative magnitude) of about 1.647. Therefore, to get the true value of magnitude we must multiply by the reciprocal of 1.647, which is 0.607. (Remember,
the exact CORDIC Gain is a function of the how many iterations you do.) Unfortunately, we can't do this gain-adjustment multiply using a simple shift/add; however, in many applications this
factor can be compensated for in some other part of the system. Or, when relative magnitude is all that counts (e.g. AM demodulation), it can simply be neglected.
4.2 How can I use CORDIC to calculate phase?
To calculate phase, just rotate the complex number to have zero phase, as you did to calculate magnitude. Just a couple of details are different.
1. For each phase-addition/subtraction step, accumulate the actual number of degrees (or radians) you have rotated. The actuals will come from a table of "atan(K)" values like the "Phase of R"
column in the table above. The phase of the complex input value is the negative of the accumulated rotation required to bring it to a phase of zero.
2. Of course, you can skip compensating for the CORDIC Gain if you are interested only in phase.
4.3 Since magnitude and phase are both calculated by rotating to a phase of zero, can I calculate both simultaneously?
Yes--you're very astute.
4.4 How can I use CORDIC to calculate sine and cosine?
You basically do the inverse of calculating magnitude/phase by adding/subtracting phases so as to "accumulate" a rotation equal to the given phase. Specifically:
1. Start with a unity-magnitude value of C = Ic + jQc. The exact value depends on the given phase. For angles greater than +90, start with C = 0 + j1 (that is, +90 degrees); for angles less than
-90, start with C = 0 - j1 (that is, -90 degrees); for other angles, start with C = 1 + j0 (that is, zero degrees). Initialize an "accumulated rotation" variable to +90, -90, or 0
accordingly. (Of course, you also could do all this in terms of radians.)
2. Do a series of iterations. If the desired phase minus the accumulated rotation is less than zero, add the next angle in the table; otherwise, subtract the next angle. Do this using each value
in the table.
3. The "cosine" output is in "Ic"; the "sine" output is in "Qc".
A couple of notes:
1. Again, the accuracy improves by about a factor of two with each iteration; use as many iterations as your application's accuracy requires.
2. This algorithm gives you both cosine (Ic) and sine (Qc). Since CORDIC uses complex values to do its magic, it's not possible to calculate sine and cosine separately.
5.0 References
5.1 Where can I learn more about CORDIC on-line?
5.2 What are some article references for CORDIC?
1. Jack E. Volder, The CORDIC Trigonometric Computing Technique, IRE Transactions on Electronic Computers, September 1959.
2. J. E. Meggitt, Pseudo Division and Pseudo Multiplication Processes, IBM Journal, April 1962.
5.3 What are some book references for CORDIC?
1. M. E. Frerking , Digital Signal Processing in Communication Systems [Fre94].
2. Henry Briggs, Arithmetica Logarithmica, 1624.
6.0 Acknowledgments
Thanks to Ray Andraka for helping me understand. Thanks to Rick Lyons for review. Thanks to Mark Brown and Bill Wiese for providing links. | {"url":"http://www.dspguru.com/dsp/faqs/cordic","timestamp":"2014-04-19T17:10:14Z","content_type":null,"content_length":"39981","record_id":"<urn:uuid:6b91fdd7-5b89-41b4-8082-25548e0b5ac8>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00490-ip-10-147-4-33.ec2.internal.warc.gz"} |
Expansion of the null zone
To see the null zone of a valve clearly, a simulation was run for a range of ±6% of spool shift. Performance parameters from previous simulation runs were not changed, so Figure 1 and several that
follow show a magnified look at the null zone.
Figure 1 gives a detailed look at pressure metering because it covers such a small amount of spool shift. Again, it confirms the rule of thumb that pressure metering is confined to the overlap (0.85%
in this case) plus about 5% of spool travel.
Figure 1 — When the null zone is magnified, pressure metering is more obvious along with the fact that it all takes place in the center 5% or so of spool travel.
The two port curves crossing exactly at zero spool shift show that this valve is perfectly nulled. The laps on all the lands being identical is indicated by the two port curves crossing at exactly
500 psi. Later, the consequences of asymmetrical and imperfect flow grinding will be simulated by setting the overlaps on the individual lands to unequal values. But for now, the null performance of
this valve is very good, albeit simulated.
Flow metering and land-to-land leakage in Figure 2 are shown in expanded form around valve null. Here, the peak leakage is about 1.85 in.3/sec, compared to the inputted target of 2.75 in.3/sec.
Perhaps more important, though, is that a slight amount of non-straightness occurs in the flow metering curve between about ±1% around zero. This is a consequence of the transition from the
hyperbolic function to the straight line function. In other words, if perfectly constant flow gain is required through the null zone, it is not likely with this model.
Figure 2 — Expansion of the null zone gives a clearer and more detailed picture of the land-to-land leakage and the flow metering aberrations that can occur.
However, the amount of non-linearity is so slight that it will have no effect on the valve’s performance in any application. This is true whether the application is simulated or from a real valve.
The slight waviness will be imperceptible in the application.
Comparing simulation to reality
Real valves are likely to have some “lumps and bumps” if they are examined with the intensity, magnification, and total lack of instrumentation error that is possible with this simulation. That is,
the irregularity in the simulation is probably less than the noise that would be expected in any electronic data acquisition system. The question remains, however, “Can any real flow grinding process
produce a null flow metering that is straighter than that of the simulation?” No answer exists at this time.
A better view of the valve coefficient can be seen when it, too, is subjected to expansion of the null zone as depicted in Figure 3. Here, the curvature and the gradualness are noticeable, however,
the mathematical coincidence of both the values and the slopes at the transition point makes for a smooth characteristic.
Figure 3 — The curvature of the valve’s flow coefficients can be seen clearly when the null zone is expanded around zero. Key parameter values are also shown.
This graph has three key parameter values included and identified with the vertical dashed lines. The values are xX, xOL and xI. Here is a bit of review: The first is the transition point, xX, which
is where the hyperbola and straight line come together. The second, xOL, lies almost on top of the first because the two are very close in value. The third, xI, is that point where the hyperbolic
function would go to infinity if it were allowed to. The computer program has been coded so that to the left of the transition point, the hyperbola is used, but at and to the right of the transition
point, the straight line is used. Thus, the program uses only the KV values that correspond to the composite curves shown here.
Effects of changing overlap
Instead of having just one overlap for all four valve lands, four different overlaps can be modeled. The four different overlaps are 3%, 1.1%, 0% and –1%. Of course, the –1% value is normally
referred to as an underlay condition. One way of looking at the results is that we are simulating the effects of progressive flow grinding steps, starting with a spool blank that has 3% overlap,
testing for flow, pressure, and leakage metering. We would then grind off a bit more material from the lands to get to 1.1% overlap, test again, and so on, through 0% and -1% overlap values.
The results will closely approximate the values that can be expected from a real valve, except that in the simulation, each land will be given exactly the same amount of lap as all other lands. This
is a practical impossibility in any flow grinding process on a real valve. However, it easily succumbs to the magic of modeling and simulation.
Flow metering results
Valve manufacturers are eager to show off the flow metering curves of their valves because they are thought to be the most important characteristic. Their point is arguable and will be accepted as
true in the interest of harmonious simulations.
Figure 4 shows results of the four different amounts of overlap applied to the same valve. It shows the very typical and expected results if we were to flow grind a servovalve in the manner
simulated. With 3% overlap, a flow gain reduction is seen at the zero-crossing point. This is expected. But more importantly, the flow gain does not go to zero, there is no flow shut off, and there
is no real dead zone.
Figure 4 — Simulated flow metering curves near the null zone show the effects of different amounts of overlap in a servovalve.
Flow metering takes place over the entire overlap range, albeit at reduced gain. This is how real valves behave, and it is the reason that ISO 5598 defines a servovalve as a valve with less than 3%
center overlap. It is also the reason why some valve manufacturers state that their “zero-lapped valve” will have a region of reduced flow gain that is not less than 25% of the flow gain just outside
the null zone. This is not, necessarily, a disqualifying factor because most applications will not be adversely affected by the slight overlap and its accompanying flow gain reduction.
An evaluation of the flow gain at the zero-crossing in Figure 4 reveals that there is very close to a 25% reduction in gain relative to that measured between 6% and 8% of spool travel. However,
because of difficulties in interpreting data, competent people will not all arrive at 25%. The working group responsible for ISO 10770-1 refused to take up the methods for assessing and reporting the
amount of overlap.
At the other overlap extreme, that is, at –1%, flow gain increases at the origin. Again, this is totally expected and closely mimics the actions of a real valve that is under lapped. It is well
known, and has been widely reported. In fact, the flow gain doubles at the origin when an under lap exists, and the spool travel range over which gain doubling occurs follows the amount of underlap.
The phenomenon is realizable in even the most rudimentary valve models. The price paid for under lapping a valve is increased land-to-land leakage. For the 0% overlap case, an increase in flow gain
occurs at the origin. Not surprisingly, though, it does not extend very far beyond the zero crossing.
At 1.1% overlap, the flow gain is slightly decreased at null, which tells us that critical gain is somewhere between 0% (slightly under lapped) and 1.1% (slightly overlapped). It was shown in the
previous simulation run that 0.85% overlap resulted in flow metering that was critically lapped. This is quite typical, with the rule of thumb being that a physical overlap of about 0.5% to 1% is
required to achieve flow metering with the least gain change at the zero-crossing.
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