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Matrices and linear transformations
Let $A$ be a $2 \times 3$ matrix, say \begin{align*} A = \left[ \begin{array}{rrr} 1 & 0 & -1\\ 3 & 1 & 2 \end{array} \right]. \end{align*} What do you get if you multiply $A$ by the vector $\vc{x}=
(x,y,z)$? Remembering matrix multiplication, we see that \begin{align*} A\vc{x} = \left[ \begin{array}{rrr} 1 & 0 & -1\\ 3 & 1 & 2 \end{array} \right] \left[ \begin{array}{c} x\\ y\\ z \end{array} \
right] = \left[ \begin{array}{c} x - z\\ 3x + y +2z \end{array} \right] =(x-z,3x+y+2z). \end{align*}
If we define a function $\vc{f}(\vc{x}) = A\vc{x}$, we have created a function of three variables $(x,y,z)$ whose output is a two-dimensional vector $(x-z,3x+y+2z)$. Using function notation, we can
write $\vc{f} : \R^3 \to \R^2$. We have created a vector-valued function of three variables. So, for example, $\vc{f}(1,2,3) = (1-3,3\cdot 1 + 2 + 2 \cdot 3) = (-2, 11)$.
Given any $m \times n$ matrix $B$, we can define a function $\vc{g}: \R^n \to \R^m$ (note the order of $m$ and $n$ switched) by $\vc{g}(\vc{x}) = B \vc{x}$, where $\vc{x}$ is an $n$-dimensional
vector. As another example, if \begin{align*} C = \left[ \begin{array}{rr} 5 & -3\\ 1 & 0\\ -7 & 4\\ 0 & -2 \end{array} \right], \end{align*} then the function $\vc{h}(\vc{y}) = C \vc{y}$, where $y=
(y_1, y_2)$, is $\vc{h}(\vc{y}) = (5y_1-3y_2,y_1,-7y_1+4y_2,-2y_2)$.
In this way, we can associate with every matrix a function. What about going the other way around? Given some function, say $\vc{g}: \R^n \to \R^m$, can we associate with $\vc{g}(\vc{x})$ some
matrix? We can only if $\vc{g}(\vc{x})$ is a special kind of function called a linear transformation. The function $\vc{g}(\vc{x})$ is a linear transformation if each term of each component of $\vc
{g}(\vc{x})$ is a number times one of the variables. So, for example, the functions $\vc{f}(x,y)=(2x+y,y/2)$ and $\vc{g}(x,y,z)=(z,0,1.2 x)$ are linear transformation, but none of the following
functions are: $\vc{f}(x,y)=(x^2,y,x)$, $\,\vc{g}(x,y,z)=(y,xyz)$, or $\vc{h}(x,y,z)=(x+1,y,z)$. Note that both functions we obtained from matrices above were linear transformations.
Let's take the function $\vc{f}(x,y)=(2x+y,y,x-3y)$, which is a linear transformation from $\R^2$ to $\R^3$. The matrix $A$ associated with $\vc{f}$ will be a $3 \times 2$ matrix, which we'll write
as \begin{align*} A = \left[ \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22}\\ a_{31} & a_{32} \end{array} \right]. \end{align*} We need $A$ to satisfy $\vc{f}(\vc{x})=A\vc{x}$, where $\vc{x}=
The easiest way to find $A$ is the following. If we let $\vc{x}=(1,0)$, then $f(\vc{x})= A\vc{x}$ is the first column of $A$. (Can you see that?) So we know the first column of $A$ is simply \begin
{align*} f(1,0)=(2,0,1) = \left[ \begin{array}{r} 2\\0\\1 \end{array} \right]. \end{align*}
Similarly, if $\vc{x}=(0,1)$, then $f(\vc{x})=A\vc{x}$ is the second column of $A$, which is \begin{align*} f(0,1) = (1,1,-3) = \left[ \begin{array}{r} 1\\1\\-3 \end{array} \right]. \end{align*}
Putting these together, we see that the linear transformation $\vc{f}(\vc{x})$ is associated with the matrix \begin{align*} A= \left[ \begin{array}{rr} 2 & 1\\ 0 & 1\\ 1 & -3 \end{array} \right]. \
The important conclusion is that every linear transformation is associated with a matrix and vice versa. | {"url":"http://mathinsight.org/matrices_linear_transformations","timestamp":"2014-04-16T18:56:31Z","content_type":null,"content_length":"19425","record_id":"<urn:uuid:7c564254-42ab-4e05-97c5-bee5241bf3b6>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00307-ip-10-147-4-33.ec2.internal.warc.gz"} |
Initial value problems
April 29th 2007, 11:56 AM #1
Sep 2006
Initial value problems
I have been working on this hw set all weekend and it just isn't clicking!! Any help would be greatly appreciated.
Solve the given initial value problems by Laplace transforms. Show the details of your work.
y" + 9y = {
6 sin t if 0 <= t <= pi
0 if t > pi, y(0) = 0, y'(0) = 0
y" - 2y' + 2y = 8e^(-t)cos t ; y(0) = 16; y'(0) = -16
Last edited by CaptainBlack; April 29th 2007 at 12:26 PM.
[quote=Hollysti;49174]I have been working on this hw set all weekend and it just isn't clicking!! Any help would be greatly appreciated.
Solve the given initial value problems by Laplace transforms. Show the details of your work.
y" + 9y = {
6 sin t if 0 <= t <= pi
0 if t > pi, y(0) = 0, y'(0) = 0
Take the LT of the DE:
(s^2 + 9)Y = 6(e^{-pi s}+1)/(s^2+1)
(note the implicit assumption that y(0)=0, and y'(0)=0 here since
Ly'' = s^2 Y - s y(0) - y'(0) )
Y = 6(e^{-pi s}+1)/[(s^2+1)(s^2+9)
.. = (6/8)(e^{-pi s}+1)/(s^2+1) - (6/8)(e^{-pi s}+1)/(s^2+9)
Both of the terms on the right are LT's of single sin arches, so:
y_1(t) = (6/8) f(t) - (6/8) g(t)
f(t) = sin(t) 0<=t<=pi
f(t) = 0 otherwise.
ERROR, this is not the inverse LT of (e^{-pi s}+1)/(s^2+9)
g(t) = (1/3) sin(3 t) 0<= t <= pi/3
g(t) = 0 otherwise
This y is a particular integral of the DE (which as I assumed that y(0)=0 and y'(0)=0
when I tool the LT is actualy the answere required the following is not strikly necessary)
the general solution is this plus a solution of the homogeneous equation:
y'' + 9y = 0
which is Asin(3t) + Bcos(3t).
Hence the general solution is:
y(t) = y_1(t) + Asin(3t) + Bcos(3t).
Now y(0)=0, forces B=0, and:
y'(0) means that:
cos(0) - (1/3) 3 cos(0) + A 3 cos(0) = 0
so A = 0.
(check this very carefully there is almost certainly an error in the algebra)
Last edited by CaptainBlack; April 29th 2007 at 11:27 PM.
I'm sorry, but would you please show me how you got that very first step?? I can follow what you did after that... Thank you.
That would be:
Take the LT of the DE:
(s^2 + 9)Y = 6(e^{-pi s}+1)/(s^2+1)
The left hand side is the LT of y''+9y (with the implicit assumption that
y(0)=0 and y'(0)=0, which is why this will be a particular integral of the
DE rather than a general one, the initial conditions could be left in here
but it will complicate the subsequent algebra, and its the initial conditions
that we need any way).
The right hand side is the LT of the right hand side of the DE, you can
find this by looking up the LT in a table or by doing the required integration
Note there is a mistake in my algebra later which I will correct by the time
you read this.
Looking at the solution again there is something about it I don't like, I
will be looking at it again later today
Last edited by CaptainBlack; April 29th 2007 at 08:38 PM.
I left this problem with the partial result that the Laplace Transform Y(s) of
the required solution y(t) satisfying the required initial conditions was:
Y(s) = 6(e^{-pi s}+1)/[(s^2+1)(s^2+9)]
Now if we put:
F(s) = 6(e^{-pi s}+1)/(s^2+1),
G(s) = 1/(s^2+9)
we gave that F(s) is the Laplace Transform of:
f(t) = 6.sin(t) 0<=t<=pi
f(t) = 0 otherwise.
and G(s) is the Laplace Transform of:
g(t) = (1/3) sin(3t).
Now we may invoke the convolution theorem to conclude that:
y(t) = integral_{tau=0 to t} f(tau) g(t - tau) dtau.
..... = integral_{tau=0 to min(t, pi)} 2 sin(tau)sin(3(t-tau)) dtau
y(t) = integral_{tau=0 to t} 2 sin(tau)sin(3(t-tau)) dtau for 0<t<pi
..... = integral_{tau=0 to pi} 2 sin(tau)sin(3(t-tau)) dtau for t>=pi
and I will leave these integrals as an exercise for the reader (at least untill I have time
to do them myself)
I left us with the solution in the form:
y(t) = integral_{tau=0 to t} 2 sin(tau)sin(3(t-tau)) dtau for 0<t<pi
..... = integral_{tau=0 to pi} 2 sin(tau)sin(3(t-tau)) dtau for t>=pi
Now using QuickMath to do these integrals (QM does not like multi-character
variable names so we have to use something like u in place of tau there) we get:
integral_{tau=0 to t} 2 sin(tau)sin(3(t-tau)) dtau = sin^3(t)
integral_{tau=0 to pi} 2 sin(tau)sin(3(t-tau)) dtau = 0
y(t) = sin^3(t) for 0<t<pi
..... = 0 for t>=pi,
Which is very satisfying as that agrees to a high level of precission with
the result of my numerical integration of the DE.
April 29th 2007, 01:09 PM #2
Grand Panjandrum
Nov 2005
April 29th 2007, 07:26 PM #3
Sep 2006
April 29th 2007, 07:46 PM #4
Grand Panjandrum
Nov 2005
April 30th 2007, 10:44 PM #5
Grand Panjandrum
Nov 2005
April 30th 2007, 10:58 PM #6
Grand Panjandrum
Nov 2005 | {"url":"http://mathhelpforum.com/calculus/14303-initial-value-problems.html","timestamp":"2014-04-17T21:26:16Z","content_type":null,"content_length":"49823","record_id":"<urn:uuid:5e7a0542-ddf5-42d0-9b68-32e34aed9100>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00054-ip-10-147-4-33.ec2.internal.warc.gz"} |
Voltage Dividers
≡ Pages
Ideal Voltage Divider
There are two important parts to the voltage divider: the circuit and the equation.
The Circuit
A voltage divider involves applying a voltage source across a series of two resistors. You may see it drawn a few different ways, but they should always essentially be the same circuit.
Examples of voltage divider schematics. Shorthand, longhand, resistors at same/different angles, etc.
We’ll call the resistor closest to the input voltage (V[in]) R[1], and the resistor closest to ground R[2]. The voltage drop across R[2] is called V[out], that’s the divided voltage our circuit
exists to make.
That’s all there is to the circuit! V[out] is our divided voltage. That’s what’ll end up being a fraction of the input voltage.
The Equation
The voltage divider equation assumes that you know three values of the above circuit: the input voltage (V[in]), and both resistor values (R[1] and R[2]). Given those values, we can use this equation
to find the output voltage (V[out]):
Memorize that equation!
This equation states that the output voltage is directly proportional to the input voltage and the ratio of R[1] and R[2]. If you’d like to find out where this comes from, check out this section
where the equaion is derived. But for now, just write it down and remember it!
Have some fun experimenting with inputs and outputs to the voltage divider equation! Below, you can plug in numbers for V[in] and both resistors and see what kind of output voltage they produce.
Or, if you adjust V[out], you’ll see what resistance value at R[2] is required (given a V[in] and R[1]).
There are a few generalizations that are good to keep in mind when using voltage dividers. These are simplifications that make evaluating a voltage dividing circuit just a little easier.
First, if R2 and R1 are equal then the output voltage is half that of the input. This is true regardless of the resistors' values.
If R[2] is much larger (at least an order of magnitude) than R[1], then the output voltage will be very close to the input. There will be very little voltage across R[1].
Conversely, if R[2] is much smaller than R[1], the output voltage will be tiny compared to the input. Most of the input voltage will be across R[1] | {"url":"https://learn.sparkfun.com/tutorials/voltage-dividers/ideal-voltage-divider","timestamp":"2014-04-18T20:43:32Z","content_type":null,"content_length":"20330","record_id":"<urn:uuid:f8b2d702-3262-47f0-b6f9-25e2cbe9f1ba>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00054-ip-10-147-4-33.ec2.internal.warc.gz"} |
Four Identical Particles Of Mass M Each Are Placed ... | Chegg.com
Four identical particles of mass m each are placed at the vertices of a square with side length a and held there by four massless rods, which form the sides of the square. What is the rotational
inertia of this rigid body about an axis that (a) passes through the midpoints of opposite sides and lies in the plane of the square, (b) passes through the midpoint of one of the sides and is
perpendicular to the plane of the square, and (c) lies in the plane of the square and passes through two diagonally opposite particles? State your answer in terms of the given variables.
PLEASE SHOW ALL WORK, highlight the answer for a) b) and c) | {"url":"http://www.chegg.com/homework-help/questions-and-answers/four-identical-particles-mass-m-placed-vertices-square-side-length-held-four-massless-rods-q2232863","timestamp":"2014-04-20T14:54:23Z","content_type":null,"content_length":"21693","record_id":"<urn:uuid:d07a9073-a3f6-4287-8867-e9401b25484f>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00347-ip-10-147-4-33.ec2.internal.warc.gz"} |
Quick tangent problem
February 12th 2011, 12:58 PM #1
Nov 2010
Quick tangent problem
The tangent to the function $y=x^3-6x^2+8x$ at point $(3,-3)$intersects the curve at another point. How would I find that point?
What I did was: $f'(3)=\frac{y+3}{x-3}$
Where am I going wrong?
You have put "solved" on this. What solution did you get? Yes, what you did, saying that the slope of the line from (3,-3) to (x,y) must be the same as the derivative at 3, is correct. Although
you shouldn't use "f" where only "y" was used before. Other than that, you are not going wrong at all.
You have put "solved" on this. What solution did you get? Yes, what you did, saying that the slope of the line from (3,-3) to (x,y) must be the same as the derivative at 3, is correct. Although
you shouldn't use "f" where only "y" was used before. Other than that, you are not going wrong at all.
Thanks for the pointers! And yes, I was able to figure it out. I set the equation of the tangent (using the formula above) with the equation of the function to find where the other intersection
point was.
February 13th 2011, 04:10 AM #2
MHF Contributor
Apr 2005
February 13th 2011, 06:29 AM #3
Nov 2010 | {"url":"http://mathhelpforum.com/calculus/171023-quick-tangent-problem.html","timestamp":"2014-04-18T08:51:42Z","content_type":null,"content_length":"35503","record_id":"<urn:uuid:a5d03027-d74b-4ff8-b5de-95dca2851666>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00529-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hypothesis Testing: Power Function
Billy Bob
Replace [tex]\overline{X}[/tex] with [tex]\mu_a+4Z/\sqrt{n}[/tex]
Um...why can we replace X bar by mu_a + 4Z/sqrt n ? What is Z here? I don't understand this step...can you please explain a little bit more?
X bar has the distribution N(μ, 16/n). Here, we know the distribution of X bar, so I believe that we can express P(X bar > 13 + [(1.645 x 4)/(sqrt n)] when μ = μ_a) as an integral of the normal
density and replace μ by μ_a, but how can I integrate it and get a
Thank you very much! | {"url":"http://www.physicsforums.com/showthread.php?t=307504","timestamp":"2014-04-18T00:36:41Z","content_type":null,"content_length":"52192","record_id":"<urn:uuid:6b28acdc-62ee-409c-8020-ececb090ce1b>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00595-ip-10-147-4-33.ec2.internal.warc.gz"} |
= Preview Document = Member Document = Pin to Pinterest
single digits, common denominators; 5 pages.
A red-highlighted set of 27 labeled flash cards, 4 per page, each with a pie graph that illustrates the value to the whole of each fraction. Common Core: Fractions: 3.NF.A.1
A set of 28 labeled flashcards, with a pie graph illustrating the value of each to the whole. Common Core: Fractions: 3.NF.A.1
Fraction action includes three pages of fractions that need to be reduced to the lowest terms. Common Core Math: Fractions and Operations
Practice lessons to help students learn how to create fractions by grouping rectangles. Includes answer key.
Collect pairs by matching cards with equivalent fractions in this fun game, played like 'Go Fish'. CC: Math: 3.NF.3a, 3.NF.3b, 4.NF.1 | {"url":"http://www.abcteach.com/directory/subjects-math-fractions-643-2-3","timestamp":"2014-04-17T07:11:26Z","content_type":null,"content_length":"58140","record_id":"<urn:uuid:511735a8-6303-4032-9a55-ba3e4cd724a8>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00178-ip-10-147-4-33.ec2.internal.warc.gz"} |
Critical power for asymptotic connectivity
Results 1 - 10 of 80
- IEEE Journal on Selected Areas in Communications , 2003
"... We consider dynamic routing and power allocation for a wireless network with time varying channels. The network consists of power constrained nodes which transmit over wireless links with
adaptive transmission rates. Packets randomly enter the system at each node and wait in output queues to be tran ..."
Cited by 211 (50 self)
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We consider dynamic routing and power allocation for a wireless network with time varying channels. The network consists of power constrained nodes which transmit over wireless links with adaptive
transmission rates. Packets randomly enter the system at each node and wait in output queues to be transmitted through the network to their destinations. We establish the capacity region of all rate
matrices (# ij ) that the system can stably support---where (# ij ) represents the rate of traffic originating at node i and destined for node j. A joint routing and power allocation policy is
developed which stabilizes the system and provides bounded average delay guarantees whenever the input rates are within this capacity region. Such performance holds for general arrival and channel
state processes, even if these processes are unknown to the network controller. We then apply this control algorithm to an ad-hoc wireless network where channel variations are due to user mobility,
and compare its performance with the Grossglauser-Tse relay model developed in [13].
, 2002
"... Sensor networks pose a number of challenging conceptual and optimization problems such as location, deployment, and tracking [1]. One of the fundamental problems in sensor networks is the
calculation of the coverage. In [1], it is assumed that the sensor has the uniform sensing ability. In this pape ..."
Cited by 113 (9 self)
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Sensor networks pose a number of challenging conceptual and optimization problems such as location, deployment, and tracking [1]. One of the fundamental problems in sensor networks is the calculation
of the coverage. In [1], it is assumed that the sensor has the uniform sensing ability. In this paper, we give efficient distributed algorithms to optimally solve the best-coverage problem raised in
[1]. Here, we consider the sensing model: the sensing ability diminishes as the distance increases. As energy conservation is a major concern in wireless (or sensor) networks, we also consider how to
find an optimum bestcoverage -path with the least energy consumption. We also consider how to find an optimum best-coverage-path that travels a small distance. In addition, we justify the correctness
of the method proposed in [1] that uses the Delaunay triangulation to solve the best coverage problem. Moreover, we show that the search space of the best coverage problem can be confined to the
relative neighborhood graph, which can be constructed locally.
, 2003
"... In this paper, we analyze the critical transmitting range for connectivity in wireless ad hoc networks. More specifically, we consider the following problem: assume n nodes, each capable of
communicating with nodes within a radius of r, are randomly and uniformly distributed in a d-dimensional re ..."
Cited by 100 (12 self)
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In this paper, we analyze the critical transmitting range for connectivity in wireless ad hoc networks. More specifically, we consider the following problem: assume n nodes, each capable of
communicating with nodes within a radius of r, are randomly and uniformly distributed in a d-dimensional region with a side of length l; how large must the transmitting range r be to ensure that the
resulting network is connected with high probability? First, we consider this problem for stationary networks, and we provide tight upper and lower bounds on the critical transmitting range for
one-dimensional networks, and non-tight bounds for two and three-dimensional networks. Due to the presence of the geometric parameter l in the model, our results can be applied to dense as well as
sparse ad hoc networks, contrary to existing theoretical results that apply only to dense networks. We also investigate several related questions through extensive simulations. First, we evaluate the
relationship between the critical transmitting range and the minimum transmitting range that ensures formation of a connected component containing a large fraction (e.g. 90%) of the nodes. Then, we
consider the mobile version of the
- in Wireless Mesh Networks,” in Proc. ACM MobiCom , 2006
"... Wireless mesh networks are expected to be widely used to provide Internet access in the near future. In order to fulfill the expectations, these networks should provide high throughput
simultaneously to many users. Recent research has indicated that, due to its conservative CSMA/CA channel access sc ..."
Cited by 90 (9 self)
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Wireless mesh networks are expected to be widely used to provide Internet access in the near future. In order to fulfill the expectations, these networks should provide high throughput simultaneously
to many users. Recent research has indicated that, due to its conservative CSMA/CA channel access scheme and RTS/CTS mechanism, 802.11 is not suitable to achieve this goal. In this paper, we
investigate throughput improvements achievable by replacing CSMA/CA with an STDMA scheme where transmissions are scheduled according to the physical interference model. To this end, we present a
computationally efficient heuristic for computing a feasible schedule under the physical interference model and we prove, under uniform random node distribution, an approximation factor for the
length of this schedule relative to the shortest schedule possible with physical interference. This represents the first known polynomial-time algorithm for this problem with a proven approximation
factor. We also evaluate the throughput and execution time of this algorithm on representative wireless mesh network scenarios through packet-level simulations. The results show that throughput with
STDMA and physical-interferencebased scheduling can be up to three times higher than 802.11 for the parameter values simulated. The results also show that our scheduling algorithm can schedule
networks with 2000 nodes in about 2.5 minutes.
, 2003
"... Topology control, wherein nodes adjust their transmitting ranges to conserve energy, is an important feature in wireless ad hoc networks. In this paper, we present a topology control protocol
that is fully distributed, asynchronous, and localized. This protocol, referred to as the k-NEIGH protocol, ..."
Cited by 65 (0 self)
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Topology control, wherein nodes adjust their transmitting ranges to conserve energy, is an important feature in wireless ad hoc networks. In this paper, we present a topology control protocol that is
fully distributed, asynchronous, and localized. This protocol, referred to as the k-NEIGH protocol, maintains the number of neighbors of every node equal to or slightly below a specific value k.
Furthermore, the protocol ensures that the resulting communication graph is symmetric, thereby easing the operation of higher protocol layers. To evaluate the performance of the protocol, the value
of k that ensures a connected communication graph with high probability is evaluated. It is also shown that, with n nodes in the network, the protocol terminates on every node after exactly 2n
messages total and within strictly bounded time. Finally, extensive simulations are carried out, which show that the k-NEIGH protocol is about 20% more energy-efficient than the most widely-studied
existing protocol.
- Control in Wireless Multihop Networks,” Proc. IEEE INFOCOM , 2004
"... Abstract—In this paper, we investigate the impact of variable-range transmission power control on the physical and network connectivity, on network capacity, and on power savings in wireless
multihop networks. First, using previous work by Steele [18], we show that, for a path attenuation factor 2, ..."
Cited by 47 (2 self)
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Abstract—In this paper, we investigate the impact of variable-range transmission power control on the physical and network connectivity, on network capacity, and on power savings in wireless multihop
networks. First, using previous work by Steele [18], we show that, for a path attenuation factor 2, the average range of links in a planar random network of Am2 having n nodes is c ffiffiffi p A n 1.
We show that this average range is approximately half the range obtained when common-range transmission control is used. Combining this result and previous work by Gupta and Kumar [8], we derive an
expression for the average traffic carrying capacity of variable-range-based multihop networks. For 2, we show that this capacity remains constant even when more nodes are added to the network.
Second, we derive a model that approximates the signaling overhead of a routing protocol as a function of the transmission range and node mobility for both route discovery and route maintenance. We
show that there is an optimum setting for the transmission range, not necessarily the minimum, which maximizes the capacity available to nodes in the presence of node mobility. The results presented
in this paper highlight the need to design future MAC and routing protocols for wireless ad hoc and sensor networks based, not on common-range which is prevalent today, but on variable-range power
control. Index Terms—Multihop networks, ad hoc networks, traffic capacity, network connectivity, power savings. Ç 1
, 2001
"... There are many contexts in distributed wireless networks where there is a critical threshold, corresponding to a minimum amount of the communication effort or power expenditure by individual
nodes, above which a desirable global property exists with high probability. When this individual node ef ..."
Cited by 41 (0 self)
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There are many contexts in distributed wireless networks where there is a critical threshold, corresponding to a minimum amount of the communication effort or power expenditure by individual nodes,
above which a desirable global property exists with high probability. When this individual node effort is below the threshold the desired global property exists with a low probability. This "phase
transition" is typically seen to become sharper as the number of nodes in the network increases. We discuss in this paper some examples of properties that exhibit such critical behavior: node
reachability with probabilistic flooding, ad-hoc network connectivity, and sensor network coordination. We discuss the connections between these phenomena and the phase transitions that have been
shown to arise in random graphs. We argue that a good understanding of these phase transition phenomena can provide useful design principles for engineering distributed wireless networks.
, 2004
"... This paper addresses the problem of localized scatternet formation for multihop Bluetooth-based personal area ad hoc networks. Nodes are assumed to know their positions and are able to establish
connections with any of their neighboring nodes, located within their transmission radius, in the neighbo ..."
Cited by 40 (8 self)
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This paper addresses the problem of localized scatternet formation for multihop Bluetooth-based personal area ad hoc networks. Nodes are assumed to know their positions and are able to establish
connections with any of their neighboring nodes, located within their transmission radius, in the neighbor discovery phase. The next phase of the proposed formation algorithm is optional and can be
applied to construct a sparse geometric structure in a localized manner. We propose here a new sparse planar structure, namely, partial Delaunay triangulation (PDT), which can be constructed locally
and is denser than other known localized planar structures. In the next mandatory phase, the degree of each node is limited to seven by applying the Yao structure, and the master-slave relations in
piconets are formed in created subgraphs. This phase consists of several iterations. In each iteration, undecided nodes with higher keys than any of their undecided neighbors apply the Yao structure
to bound the degrees, decide master-slave relations on the remaining edges, and inform all neighbors about either deleting edges or master-slave decisions. To the best of our knowledge, our schemes
are the first schemes that construct degree limited (a node has at most seven slaves) and connected piconets in multihop networks, without parking any node. The creation and maintenance require small
overhead in addition to maintaining accurate location information for one-hop neighbors. The experiments confirm good functionality of created Bluetooth networks in addition to their fast creation
and straightforward maintenance.
- IEEE Transactions on Mobile Computing , 2005
"... this paper, we study the critical transmitting range (CTR) for connectivity in mobile ad hoc networks. We prove that ln n rM c n for some constant c 1, where rM is the CTR in the presence of
M-like node mobility and n is the number of network nodes. Our result holds for an arbitrary mobility model ..."
Cited by 30 (1 self)
Add to MetaCart
this paper, we study the critical transmitting range (CTR) for connectivity in mobile ad hoc networks. We prove that ln n rM c n for some constant c 1, where rM is the CTR in the presence of M-like
node mobility and n is the number of network nodes. Our result holds for an arbitrary mobility model M such that: 1) M is obstacle free and 2) nodes are allowed to move only within a certain bounded
area. We also investigate in detail the case of random waypoint mobility, which is the most common mobility model used in the simulation of ad hoc networks. Denoting with rw p the CTR with random
waypoint mobility when the pause time is set to p and node velocity is set to v, we prove that rw qffiffiffiffiffi pþ0:521405 v ln n p p n if p>0 and that rw qffiffiffiffiffi ln n 0 n. The results of
our simulations also suggest that if n is large enough (n 50), rw r 0 is well approximated by 4 ln n, where r is the critical range in case of uniformly distributed nodes. The results presented in
this paper provide a better understanding of the behavior of a fundamental network parameter in the presence of mobility and can be used to improve the accuracy of mobile ad hoc network simulations.
Index Terms—Critical transmitting range, connectivity, random waypoint model, mobility modeling, ad hoc networks. æ 1
- in Proc. IEEE INFOCOM 2005 , 2005
"... In this paper, we present an analytical procedure for the computation of the node isolation probability in an ad hoc network in the presence of channel randomness, with applications to shadowing
and fading phenomena. Such a probability coincides with the complement of the coverage probability, given ..."
Cited by 29 (4 self)
Add to MetaCart
In this paper, we present an analytical procedure for the computation of the node isolation probability in an ad hoc network in the presence of channel randomness, with applications to shadowing and
fading phenomena. Such a probability coincides with the complement of the coverage probability, given that nodes are distributed according to a Poisson point process. These results are used to obtain
an estimate of the connectivity features for very dense networks. For the case of superimposed lognormal shadowing and Rayleigh fading, the connectivity improvements achievable by means of diversity
schemes are investigated. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=199864","timestamp":"2014-04-17T06:31:59Z","content_type":null,"content_length":"42466","record_id":"<urn:uuid:78f37479-4ebe-401d-accf-c3a15e2c8e89>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00004-ip-10-147-4-33.ec2.internal.warc.gz"} |
Srinivasa Ramanujan
Ramanujan’s life is full of strange contrasts. He had no formal training in mathematics but yet “he was a natural mathematical genius, in the class of Gauss and Euler.” Probably Ramanujan’s life has
no parallel in the history of human thought. Godfrey Harold Hardy, (1877-1947), who made it possible for Ramanujan to go to Cambridge and give formal shape to his works, said in one of his lectures
given at Harvard Universty (which later came out as a book entitled Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work): “I have to form myself, as I have never really formed
before, and try to help you to form, some of the reasoned estimate of the most romantic figure in the recent history of mathematics, a man whose career seems full of paradoxes and contradictions, who
defies all cannons by which we are accustomed to judge one another and about whom all of us will probably agree in one judgement only, that he was in some sense a very great mathematician.”
Srinivasa Ramanujan Iyengar (best known as Srinivasa Ramanujan) was born on December 22, 1887, in Erode about 400 km from Chennai, formerly known as Madras where his mother’s parents lived. After one
year he was brought to his father’s town, Kumbakonam. His parents were K. Srinivasa Iyengar and Komalatammal. He passed his primary examination in 1897, scoring first in the district and then he
joined the Town High School. In 1904 he entered Kumbakonam’s Government College as F.A. student. He was awarded a scholarship. However, after school, Ramanujan’s total concentration was focussed on
mathematics. The result was that his formal education did not continue for long. He first failed in Kumbakonam’s Government College. He tried once again in Madras from Pachaiyappa’s College but he
failed again.
While at school he came across a book entitled A Synopsis of Elementary Results in Pure and Applied Mathematics by George Shoobridge Carr. The title of the book does not reflect its contents. It was
a compilation of about 5000 equations in algebra, calculus, trigonometry and analytical geometry with abridged demonstrations of the propositions. Carr had compressed a huge mass of mathematics that
was known in the late nineteenth century within two volumes. Ramanujan had the first one. It was certainly not a classic. But it had its positive features. According to Kanigel, “one strength of
Carr’s book was a movement, a flow to the formulas seemingly laid down one after another in artless profusion that gave the book a sly seductive logic of its own.” Thisbook had a great influence on
Ramanujan’s career. However, the book itself was not very great. Thus Hardy wrote about the book: “He (Carr) is now completely forgotten, even in his college, except in so far as Ramanujan kept his
name alive”. He further continued, “The book is not in any sense a great one, but Ramanujan made it famous and there is no doubt it influenced him (Ramanujan) profoundly”. We do not know how exactly
Carr’s book influenced Ramanujan but it certainly gave him a direction. `It had ignited a burst of fiercely single-minded intellectual activity’. Carr did not provide elaborate demonstration or step
by step proofs. He simply gave some hints to proceed in the right way. Ramanujan took it upon himself to solve all the problems in Carr’s Synopsis. And as E. H. Neville, an English mathematician,
wrote : “In proving one formula, as he worked through Carr’s synopsis, he discovered many others, and he began the practice of compiling a notebook.” Between 1903 and 1914 he had three notebooks.
While Ramanujan made up his mind to pursue mathematics forgetting everything else but then he had to work under extreme hardship. He could not even buy enough paper to record the proofs of his
results. Once he said to one of his friends, “when food is problem, how can I find money for paper? I may require four reams of paper every month.” In fact Ramanujan was in a very precarious
situation. He had lost his scholarship. He had failed in examination. What is more, he failed to prove a good tutor in the subject which he loved most.
At this juncture, Ramanujan was helped by R. Ramachandra Rao, then Collector of Nellore. Ramchandra Rao was educated at Madras Presidency College and had joined the Provincial Civil Service in 1890.
He also served as Secretary of the Indian Mathematical Society and even contributed solution to problem posed in its Journal. The Indian Mathematical Society was founded by V. Ramaswami Iyer, a
middle-level Government servant, in 1906. Its Journal put Ramanujan on the world’s mathematical map. Ramaswami Iyer met Ramanujan sometime late in 1910. Ramaswami Iyer gave Ramanujan notes of
introduction to his mathematical friends in Chennai (then Madras). One of them was P.V. Seshu Iyer, who earlier taught Ramanujan at the Government College. For a short period (14 months) Ramanujan
worked as clerk in the Madras Port Trust which he joined on March 1, 1912. This job he got with the help of S. Narayana Iyer.
Ramanujan’s name will always be linked to Godfrey Harold Hardy, a British mathematician. It is not because Ramanujan worked with Hardy at Cambridge but it was Hardy who made it possible for Ramanujan
to go to Cambridge. Hardy, widely recognised as the leading mathematician of his time, championed pure mathematics and had no interest in applied aspects. He discovered one of the fundamental results
in population genetics which explains the properties of dominant, and recessive genes in large mixed population, but he regarded the work as unimportant.
Encouraged by his well-wishers, Ramanujan, then 25 years old and had no formal education, wrote a letter to Hardy on January 16, 1913. The letter ran into eleven pages and it was filled with theorems
in divergent series. Ramanujan did not send proofs for his theorems. He requested Hardy for his advice and to help getting his results published. Ramanujan wrote : “I beg to introduce myself to you
as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £ 20 per annum. I have had no university education but I have undergone the ordinary school course. After
leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I
am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling“… I would request
you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the
expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me “. The letter has become an important historical
document. In fact, ‘this letter is one of the most important and exciting mathematical letters ever written’. At the first glance Hardy was not impressed with the contents of the letter. So Hardy
left it aside and got himself engaged in his daily routine work. But then he could not forget about it. In the evening Hardy again started examining the theorems sent by Ramanujan. He also requested
his colleague and a distinguished mathematician, John Edensor Littlewood (1885-1977) to come and examine the theorems. After examining closely they realized the importance of Ramanujan’s work. As
C.P. Snow recounted, ‘before mid-night they knew and knew for certain’ that the writer of the manuscripts was a man of genius’. Everyone in Cambridge concerned with mathematics came to know about the
letter. Many of them thought `at least another Jacobi in making had been found out’. Bertrand Arthur William Russell (1872-1970) wrote to Lady Ottoline Morell. “I found Hardy and Littlewood in a
state of wild excitement because they believe, they have discovered a second Newton, a Hindu Clerk in Madras … He wrote to Hardy telling of some results he has got, which Hardy thinks quite
Fortunately for Ramanujan, Hardy realised that the letter was the work of a genius. In the next three months Ramanujan received another three letters from Hardy. However, in the beginning Hardy
responded cautiously. He wrote on 8 February 1913. To quote from the letter. “I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before
I can judge properly of the value of what you have done it is essential that I should see proofs of some of your assertions … I hope very much that you will send me as quickly as possible at any rate
a few of your proofs, and follow this more at your leisure by more detailed account of your work on primer and divergent series. It seems to me quite likely that you have done a good deal of work
worth publication; and if you can produce satisfactory demonstration I should be very glad to do what I can to secure it” .
In the meantime Hardy started taking steps for bringing Ramanujan to England. He contacted the Indian Office in London to this effect. Ramanujan was awarded the first research scholarship by the
Madras University. This was possible by the recommendation of Gilbert Walker, then Head of the Indian Meteorological Department in Simla. Gilbert was not a pure mathematician but he was a former
Fellow and mathematical lecturer at Trinity College, Cambridge. Walker, who was prevailed upon by Francis Spring to look through Ramanujan’s notebooks wrote to the Registrar of the Madras University
: “The character of the work that I saw impressed me as comparable in originality with that of a Mathematical Fellow in a Cambridge College; it appears to lack, however, as might be expected in the
circumstances, the completeness and precision necessary before the universal validity of the results could be accepted. I have not specialised in the branches of pure mathematics at which he worked,
and could not therefore form a reliable estimate of his abilities, which might be of an order to bring him a European reputation. But it was perfectly clear to me that the University would be
justified in enabling S. Ramanujan for a few years at least to spend the whole of his time on mathematics without any anxiety as to his livelihood.”
Ramanujan was not very eager to travel abroad. In fact he was quite apprehensive. However, many of his well-wishers prevailed upon him and finally Ramanujan left Madras by S.S. Navesa on March 17,
1914. Ramanujan reached Cambridge on April 18, 1914. When Ramanujan reached England he was fully abreast of the recent developments in his field. This was described by J. R. Newman in 1968:
“Ramanujan arrived in England abreast and often ahead of contemporary mathematical knowledge. Thus, in a lone mighty sweep, he had succeeded in recreating in his field, through his own unaided
powers, a rich half century of European mathematics. One may doubt whether so prodigious a feat had ever been accomplished in the history of thought.”
Today it is simply futile to speculate about what would have happened if Ramanujan had not come in contact with Hardy. It could happen either way. But then Hardy should be given due credit for
recognizing Ramanujan’s originality and helping him to carry out his work. Hardy himself was very clear about his role. “Ramanujan was”, Hardy wrote, “my discovery. I did not invent him — like other
great men, he invented himself — but I was the first really competent person who had the chance to see some of his work, and I can still remember with satisfaction that I could recognize at once what
I treasure I had found.”
It may be noted that before writing to Hardy, Ramanujan had written to two well-known Cambridge mathematicians viz., H.F. Baker and E.W. Hobson. But both of them had expressed their inability to help
Ramanujan was awarded the B.A. degree in March 1916 for his work on ‘Highly composite Numbers’ which was published as a paper in the Journal of the London Mathematical Society. He was the second
Indian to become a Fellow of the Royal Society in 1918 and he became one of the youngest Fellows in the entire history of the Royal Society. He was elected “for his investigation in Elliptic
Functions and the Theory of Numbers.” On 13 October 1918 he was the first Indian to be elected a Fellow of Trinity College, Cambridge.
Much of Ramanujan’s mathematics comes under the heading of number theory — a purest realm of mathematics. The number theory is the abstract study of the structure of number systems and properties of
positive integers. It includes various theorems about prime numbers (a prime number is an integer greater than one that has not integral factor). Number theory includes analytic number theory,
originated by Leonhard Euler (1707-89); geometric theory - which uses such geometrical methods of analysis as Cartesian co-ordinates, vectors and matrices; and probabilistic number theory based on
probability theory. What Ramanujan did will be fully understood by a very few. In this connection it is worthwhile to note what Hardy had to say of the work of pure mathematicians: “What we do may be
small, but it has certain character of permanence and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something
beyond the powers of the vast majority of men.” In spite of abstract nature of his work Ramanujan is widely known.
Ramanujan was a mathematical genius in his own right on the basis of his work alone. He worked hard like any other great mathematician. He had no special, unexplained power. As Hardy, wrote: “I have
often been asked whether Ramanujan had any special secret; whether his methods differed in kind from those of other mathematicians; whether there was anything really abnormal in his mode of thought.
I cannot answer these questions with any confidence or conviction; but I do not believe it. My belief that all mathematicians think, at bottom, in the same kind of way, and that Ramanujan was no
Of course, as Hardy observed Ramanujan “combined a power of generalization, a feeling for form and a capacity for rapid modification of his hypotheses, that were often really startling, and made him,
in his peculiar field, without a rival in his day.
Here we do not attempt to describe what Ramanujan achieved. But let us note what Hardy had to say about the importance of Ramanujan’s work. “Opinions may differ as to the importance of Ramanujan’s
work, the kind of standard by which it should be judged and the influence which it is likely to have on the mathematics of the future. It has not the simplicity and the inevitableness of the greatest
work; it would be greater if it were less strange. One gift it shows which no one will deny—profound and invincible originality.”
The Norwegian mathematician Atle Selberg, one of the great number theorists of this century wrote : “Ramanujan’s recognition of the multiplicative properties of the coefficients of modular forms that
we now refer to as cusp forms and his conjectures formulated in this connection and their later generalization, have come to play a more central role in the mathematics of today, serving as a kind of
focus for the attention of quite a large group of the best mathematicians of our time. Other discoveries like the mock-theta functions are only in the very early stages of being understood and no one
can yet assess their real importance. So the final verdict is certainly not in, and it may not be in for a long time, but the estimates of Ramanujan’s nature in mathematics certainly have been
growing over the years. There is doubt no about that.”
Often people tend to speculate what Ramanujan would have achieved if he had not died or if his exceptional qualities were recognised at the very beginning. There are many instances of such untimely
death of gifted persons, or rejection of gifted persons by the society or the rigid educational system. In mathematics | {"url":"http://www.vigyanprasar.gov.in/scientists/Ramanujan.HTM","timestamp":"2014-04-19T17:15:09Z","content_type":null,"content_length":"28728","record_id":"<urn:uuid:221deb2a-97a0-4e86-9735-c5b3a3d147e7>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00647-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-tickets] [NumPy] #924: problem with summing large array of float32
[Numpy-tickets] [NumPy] #924: problem with summing large array of float32
NumPy numpy-tickets@scipy....
Thu Oct 2 23:09:34 CDT 2008
#924: problem with summing large array of float32
Reporter: emil | Owner: somebody
Type: defect | Status: new
Priority: high | Milestone:
Component: numpy.core | Version: 1.0.1
Severity: critical | Keywords: sum, dot, float32
In medical imaging large arrays of single floats are often used. In
summing over such an
array I discovered the following problem:
sum(a.flatten()) yields 16777216 (4096^2) ???
However, sum(sum(sum(a,axis=0),axis=0),axis=0) yields 110880000 the
correct result.
The problem doesn't occur with float64
dot(a.flatten() , a.flatten() ) also gives 16777216
I don't think the problem lies only with numpy. I tried to circumvent the
problem by
writing my own summation in fortran (gfortran) and using f2py. The
following fortran snippet
gave me also incorrect results
Real*4 sum
sum = 0.
do i = 1,840
do j = 1,2200
do k = 1,60
sum = sum +1.0
end do
end do
end do
sum ends up being 16777216 again! the incorrect result!
on the other hand the following works:
Real*4 sum1, sum
sum = 0.
do i = 1,840
sum1 = 0.
do j = 1,2200
do k = 1,60
sum1 = sum1 +1.0
end do
end do
sum = sum+sum1
end do
The same problem seems to come up in matlab.
My computer has amd64's (opteron) and is running suse linux
Ticket URL: <http://scipy.org/scipy/numpy/ticket/924>
NumPy <http://projects.scipy.org/scipy/numpy>
The fundamental package needed for scientific computing with Python.
More information about the Numpy-tickets mailing list | {"url":"http://mail.scipy.org/pipermail/numpy-tickets/2008-October/002362.html","timestamp":"2014-04-16T19:12:49Z","content_type":null,"content_length":"4884","record_id":"<urn:uuid:95c88ee7-9048-49a4-b68d-8bce21c7faf7>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00437-ip-10-147-4-33.ec2.internal.warc.gz"} |
WyzAnt Resources
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How do I find the remainder theorem and factor theorem of : f(x)=4x^6-64x^4+x^3-19?? | {"url":"http://www.wyzant.com/resources/answers/questions?f=active&pagesize=20&pagenum=242","timestamp":"2014-04-19T16:01:30Z","content_type":null,"content_length":"58316","record_id":"<urn:uuid:ddeef647-b114-40fa-ba06-5ce332e3acbf>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00137-ip-10-147-4-33.ec2.internal.warc.gz"} |
Power, speed, and torque... AGH - Chief Delphi
Ah, there come a point in a student's search of knownledge about speed, torque, and power of motor when he finally ask the question, "what is this power thing?"... I know this because I asked the
same question last year.
As you said it, power is work over time. Its a unit of how much work can be done over certain amount of time. But Power, is also equal to force times velocity, and torque times angular velocity. Play
with the equations, and you will see how those two are the same.
So, power is a measurement of how force and speed you have. Its a combination of both things. It is a measurement of how much work you are doing at each condition of speed/torque. Just look at the
speed torque curve. At the end point of the curve, where the stall torque is or free speed is, you can calculate that both ends have power of zero, simply by seeing that Stall torque * zero speed and
free speed * zero torque = to zero power.
In the middle is where you find maximum amount of power you can get out of a motor working in its best condition.
So what's important about power you ask? Well, with the calculation you showed, you can clearly figure out how you want to gear the motor... (you should gear it to had a load of about 1/2 stall
torque for max power). But in general, when you are faced with a problem, you should figure out how much power you need to accomplish a task, and see if the motor have enough power to do it. After
all, if FIRST switch to different motors, how do you know which one to use?
In your example. You know the torque you need to do the task... But at the load the motor is facing, is it moving at a speed fast enough? You have to go through the calculation and speed torque curve
just to see how fast its spinning. If you do this with power, you first figure out how much torque you need to do this, and how fast you want the thing to move... Multiply those numbers, and you get
the amount of power you need. Then you look up which motor have enough power for that, and you can see which motors you can and cannot use.
The important about the power curve, is that you can see where you should gear your motor at. Near the peak of the power curve is what load you want your motor feeling. The further you are away from
the peak, the more power you are wasting. The wasted power usually goes into Heat and heat up your motor.
I have a lecture note from the motor workshop I give for WRRF teams in the white paper. Go through the calculation at the end, and you will see my process of figuring out the motor and gearing.
Its on the fifth page @ | {"url":"http://www.chiefdelphi.com/forums/showthread.php?t=15050","timestamp":"2014-04-18T10:38:56Z","content_type":null,"content_length":"85827","record_id":"<urn:uuid:6f8293bd-1cda-44f9-a252-a96c66d1404b>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00204-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Quasicrystals as sums of waves in the plane
On the suggestion of a friend, I rendered this animation:
This quasicrystal is full of emergent patterns, but it can be described in a simple way. Imagine that every point in the plane is shaded according to the cosine of its y coordinate. The result would
look like this:
Now we can rotate this image to get other waves, like these:
Each frame of the animation is a summation of such waves at evenly-spaced rotations. The animation occurs as each wave moves forward.
I recommend viewing it up close, and then from a few feet back. There are different patterns at each spatial scale.
The code
To render this animation I wrote a Haskell program, using the Repa array library. For my purposes, the advantages of Repa are:
• Immutable arrays, supporting clean, expressive code
• A fast implementation, including automatic parallelization
• Easy output to image files, via repa-devil
Here is a simplified (but complete!) program, which renders a single still image.
import Data.Array.Repa ( Array, DIM2, DIM3, Z(..), (:.)(..) )
import qualified Data.Array.Repa as R
import qualified Data.Array.Repa.IO.DevIL as D
import Data.Word ( Word8 )
import Data.Fixed ( divMod' )
For clarity, we define a few type synonyms:
type R = Float
type R2 = (R, R)
type Angle = R
We'll convert pixel indices to coordinates in the real plane, with origin at the image center. We have to decide how many pixels to draw, and how much of the plane to show.
pixels :: Int
pixels = 800
scale :: R
scale = 128
Repa's array indices are "snoc lists" of the form (Z :. x :. y). By contrast, our planar coordinates are conventional tuples.
point :: DIM2 -> R2
point = \(Z :. x :. y) -> (adj x, adj y) where
adj n = scale * ((2 * fromIntegral n / denom) - 1)
denom = fromIntegral pixels - 1
A single wave is a cosine depending on x and y coordinates in some proportion, determined by the wave's orientation angle.
wave :: Angle -> R2 -> R
wave th = f where
(cth, sth) = (cos th, sin th)
f (x,y) = (cos (cth*x + sth*y) + 1) / 2
To combine several functions, we sum their outputs, and wrap to produce a result between 0 and 1. As n increases, (wrap n) will rise to 1, fall back to 0, rise again, and so on. sequence converts a
list of functions to a function returning a list, using the monad instance for ((->) r).
combine :: [R2 -> R] -> (R2 -> R)
combine xs = wrap . sum . sequence xs where
wrap n = case divMod' n 1 of
(k, v) | odd k -> 1-v
| otherwise -> v
To draw the quasicrystal, we combine waves at 7 angles evenly spaced between 0 and π.
angles :: Int -> [Angle]
angles n = take n $ enumFromThen 0 (pi / fromIntegral n)
quasicrystal :: DIM2 -> R
quasicrystal = combine (map wave (angles 7)) . point
We convert an array of floating-point values to an image in two steps. First, we map floats in [0,1] to bytes in [0,255]. Then we copy this to every color channel. The result is a 3-dimensional
array, indexed by (row, column, channel). repa-devil takes such an array and outputs a PNG image file.
toImage :: Array DIM2 R -> Array DIM3 Word8
toImage arr = R.traverse arr8 (:. 4) chans where
arr8 = R.map (floor . (*255) . min 1 . max 0) arr
chans _ (Z :. _ :. _ :. 3) = 255 -- alpha channel
chans a (Z :. x :. y :. _) = a (Z :. x :. y)
main :: IO ()
main = do
let arr = R.fromFunction (Z :. pixels :. pixels) quasicrystal
D.runIL $ D.writeImage "out.png" (toImage arr)
Running it
Repa's array operations automatically run in parallel. We just need to enable GHC's threaded runtime.
$ ghc -O2 -rtsopts -threaded quasicrystal.lhs
$ ./quasicrystal +RTS -N
$ xview out.png
And it looks like this:
Note that repa-devil silently refuses to overwrite an existing file, so you may need to rm out.png first.
On my 6-core machine, this parallel code ran in 3.72 seconds of wall-clock time, at a CPU utilization of 474%. The same code compiled without -threaded took 14.20 seconds, so the net efficiency of
parallelization is 382%. This is a good result; what's better is how little work it required on my part. Cutting a mere 10 seconds from a single run is not a big deal. But it starts to matter when
rendering many frames of animation, and trying out variations on the algorithm.
As a side note, switching from Float to Double increased the run time by about 30%. I suspect this is due to increased demand for memory bandwidth and cache space.
You can grab the Literate Haskell source and try it out on your own machine. This is my first Repa program ever, so I'd much appreciate feedback on improving the code.
Be sure to check out Michael Rule's work on animating quasicrystals.
22 comments:
1. How can you be doing both
> import qualified Data.Array.Repa as R
> type R = Float
at the same time? My Haskell is very rusty, but I can't see how that would work...
2. Should be fine... "R" will refer to the type, since "R" by itself as a module qualifier makes no sense.
3. Ported to HTML5 canvas: http://www.jasondavies.com/animated-quasicrystals/
4. nice idea! reminds me of the old plasmas in the demoscene...
ported to webgl via shadertoy - realtime in your browser!
5. The resultant image isn't a true structure of a quasicrystal, is it? Doesn't the pattern repeat beyond the bounds of the image?
6. It has 7-fold symmetry about the origin, so it can't be periodic.
7. These patterns never truly repeat, try it for yourself. You can render as much of the crystal as you like, and drive yourself crazy looking for alignment, but all you'll ever find are local
regions that approximately align.
8. Also ported to WebGL via PhiloGL. Can toggle fullscreen :)
9. Does not work on nvidia because you define "float v" twice, just rename it to some other value and it works fine.
10. // paste this into http://www.iquilezles.org/apps/shadertoy/ for pictures!
// this is a gpu/webgl version of http://mainisusuallyafunction.blogspot.com/2011/10/quasicrystals-as-sums-of-waves-in-plane.html
// 'implementation' by @mmalex for http://news.ycombinator.com/item?id=3153835
// fixed up by colski
#ifdef GL_ES
precision highp float;
uniform float time;
uniform vec2 resolution;
void main(void)
vec2 p = gl_FragCoord.xy / resolution.xy;
float t=time*5.0; // change this for more speed
vec2 up=vec2(256.0,0.0); // change this for finer/coarser stripes
const float c = 0.90096886790241, s = 0.43388373911755; // sin and cos of pi / 7
const vec2 rx=vec2(c, s), ry=vec2(-s, c);
float sum = 0.0;
for (int i = 0; i < 7; ++i)
sum += cos(dot(p,up)+t);
up = vec2(dot(up,rx), dot(up,ry));
float a1=(cos(sum+2.0)+1.0) * 0.5; // the important wrap around term. play with this for different effects
gl_FragColor = vec4(a1,a1,a1,1.0);
11. Ported to Processing: http://openprocessing.org/visuals/?visualID=43954
12. What about an R version?
frame <- function(N,Scale,Sym,FS,time) {
Upx=FS; Upy=0;
for (i in 1:Sym) {
for (i in 0:49) {
13. I found some code on the HaskellWiki for writing PNGs. I used it here:
To use my code you create a BMP, write to it (it's just a mutable array under the hood), and then write that to PNG when you're done. No need for 3rd party libraries that are hard to install as
it's just pure Haskell.
The code is BSD3 so feel free to use it if you like it.
14. Nice !! Same logic can be applied for CG water reflection using multiple level of normal maps moving in different direction and velocities.
15. Interactive, color mapped version implemented in Processing.
1. wrong broken or removed?
16. Great work. Can's say that I care for the Haskell though. Here's some Matlab code to recreate the image at the end of your post (which, by the way, I think is rotated by 90 degrees):
N=800; scale=1/2.5; angles=7; t=0;
for i=2:angles
set(gcf,'Units','pixels','Position',[0 0 N N]);
set(gca,'Position',[0 0 1 1]);
The 800x800 image takes 0.2 sec. on my old 2.4 GHz Intel Core 2 Duo MacBook Pro. The R version from @fermin above was helpful. However, it used repeated multiplication of rotation matrices by
each other in order to get the sequence of rotated waves, which can lead to numeric issues. Here's a more general Matlab function that outputs an arbitrarily-sized movie (takes under 3 sec. for a
512x512 25-frame movie like the animated GIF above):
function quasicrystal(N,scale,angles,frames)
if ~isvector(N) || isempty(N) || ~isnumeric(N) || length(N) > 2
error('First input must be one or two element numeric vector.');
if N(1) < 1 || mod(N(1),1) || (length(N) > 2 && (N(2) < 1 || mod(N(2),1)))
error('First input must be vector >= 1.');
if ~isscalar(scale) || isempty(scale) || ~isnumeric(scale) || scale <0
error('Second input must be positive scalar numeric value.');
if ~isscalar(angles) || isempty(angles) || ~isnumeric(angles)
error('Third input must be scalar numeric value.');
if angles < 1 || mod(angles,1)
error('Third input must be integer >= 1.');
if nargin == 3
if ~isscalar(frames) || isempty(frames) || ~isnumeric(frames)
error('Fourth input must be scalar numeric value.');
if frames < 1 || mod(frames,1)
error('Fourth input must be integer >= 1.');
if length(N) == 2 && N(1) ~= N(2)
for t=2*pi/frames*(0:frames-1)
for i=2:angles
17. Here's an updated and improved version of my Matlab code that can output movies, animated GIFs, or PNGs in Cartesian or log-polar coordinates (inspired by Michael Rule) with the ability to adjust
contrast and apply a colormap:
There are also some example animated GIFs and AVIs. The code will be particularly useful to anyone working in R or numPy/sciPy.
Also @madan's comment above about water reflection made me think of when I tried outputting in just polar coordinates, as opposed to log-polar. When the scale is 1, the resulting animation is a
very pleasing circular ripple pattern. It's especially nice if you have a full color colormap.
Cheers, -- Andy
18. Joining in the choir of reimplementations, here it is in GLSL (WebGL): http://glsl.heroku.com/e#5384.1
It lets you muck about with the code interactively, so that's neat :)
19. Hi, you make mind blowing ideas and a spectacular article hereanimated video production
20. Is there a java equivalent for this code? :(
21. see also this Gerasimov fractal | {"url":"http://mainisusuallyafunction.blogspot.com/2011/10/quasicrystals-as-sums-of-waves-in-plane.html?showComment=1319823103328","timestamp":"2014-04-21T01:03:36Z","content_type":null,"content_length":"138800","record_id":"<urn:uuid:a54fd7cd-7a32-45d6-98ef-0417199a2381>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00140-ip-10-147-4-33.ec2.internal.warc.gz"} |
Two charged wires and a cylinder
Required math: calculus, algebra
Required physics: electrostatics
Reference: Griffiths, David J. (2007) Introduction to Electrodynamics, 3rd Edition; Prentice Hall – Problem 3.36.
Another example of the method of images. We are given two infinite charged wires, one with linear charge density ${+\lambda}$ and the other with ${-\lambda}$. These wires lie in the ${x-y}$ plane
along the lines ${y=\pm a}$. Between them is an infinite conducting cylinder of radius ${R<a}$, with its axis on the ${x}$ axis. The cylinder carries no net charge. We are to find the potential
everywhere outside the cylinder.
The solution to this problem relies on an earlier example in which we worked out the potential due to two charged wires on their own. We saw there that the equipotential surfaces for two charged
wires were circular cylinders, so we should be able to set up an image configuration in which the surface of the cylinder can be replaced by image wires inside the cylinder.
In the charged wires problem, we had two wires at ${y=\pm a}$ and found that the potential could be written as
$\displaystyle V=\frac{\lambda}{4\pi\epsilon_{0}}\ln\frac{(y+a)^{2}+z^{2}}{(y-a)^{2}+z^{2}}\ \ \ \ \ (1)$
We also found that for a constant potential ${K}$, if we define the value ${A\equiv e^{4\pi\epsilon_{0}K/\lambda}}$ we could get expressions for the radius ${R}$ and axis ${y_{0}}$ of the
equipotential cylinders:
$\displaystyle y_{0}$ $\displaystyle =$ $\displaystyle \frac{A+1}{A-1}a$
$\displaystyle R$ $\displaystyle =$ $\displaystyle \sqrt{\frac{4a^{2}A}{(A-1)^{2}}}$
A useful relation between these two is
$\displaystyle y_{0}^{2}-R^{2}=a^{2}\ \ \ \ \ (2)$
To apply the method of images and the above solution, we need to create one image for each wire. Consider first the wire on the left, at ${y=-a}$. We want to create an image of this wire at ${y=-b}$,
where this location is inside the cylinder. We can map this image problem onto the two-wire problem by noting that the midpoint between the wire and its image is at ${y=-b-(a-b)/2}$. If we define $
{y_{1}=y+b+(a-b)/2}$, then the potential due to this wire-image configuration is
$\displaystyle V_{1}$ $\displaystyle =$ $\displaystyle \frac{\lambda}{4\pi\epsilon_{0}}\ln\frac{(y_{1}+\frac{a-b}{2})^{2}+z^{2}}{(y_{1}-\frac{a-b}{2})^{2}+z^{2}}$
$\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\lambda}{4\pi\epsilon_{0}}\ln\frac{(y+a)^{2}+z^{2}}{(y+b)^{2}+z^{2}}$
On the other side, the wire at ${y=+a}$ has an image at ${y=+b}$. Defining ${y_{2}=y-b-(a-b)/2}$, we get for the potential of this pair:
$\displaystyle V_{2}$ $\displaystyle =$ $\displaystyle \frac{\lambda}{4\pi\epsilon_{0}}\ln\frac{(y_{2}+\frac{a-b}{2})^{2}+z^{2}}{(y_{2}-\frac{a-b}{2})^{2}+z^{2}}$
$\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\lambda}{4\pi\epsilon_{0}}\ln\frac{(y-b)^{2}+z^{2}}{(y-a)^{2}+z^{2}}$
The only remaining problem is to find ${b}$. We can use 2 for this. Consider the first wire-image pair. The distance ${y_{0}}$ is the distance from the midpoint between the wire and image to the axis
of the cylinder, which is ${y_{0}=\frac{a-b}{2}+b=\frac{a+b}{2}}$. The quantity ${a}$ in this relation is half the distance between the wire and its image, so in this case must be replaced by ${\frac
{a-b}{2}}$. We therefore have
$\displaystyle \left(\frac{a+b}{2}\right)^{2}-R^{2}$ $\displaystyle =$ $\displaystyle \left(\frac{a-b}{2}\right)^{2}$
$\displaystyle b$ $\displaystyle =$ $\displaystyle \frac{R^{2}}{a}$
Since potentials obey the superposition principle, we can just add the two potentials from the two image-wire pairs to get the total potential:
$\displaystyle V$ $\displaystyle =$ $\displaystyle V_{1}+V_{2}$
$\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\lambda}{4\pi\epsilon_{0}}\left[\ln\frac{(y+a)^{2}+z^{2}}{(y+b)^{2}+z^{2}}+\ln\frac{(y-b)^{2}+z^{2}}{(y-a)^{2}+z^{2}}\right]$
$\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\lambda}{4\pi\epsilon_{0}}\ln\left[\frac{(y+a)^{2}+z^{2}}{(y+b)^{2}+z^{2}}\cdot\frac{(y-b)^{2}+z^{2}}{(y-a)^{2}+z^{2}}\right]$
$\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\lambda}{4\pi\epsilon_{0}}\ln\left[\frac{(y+a)^{2}+z^{2}}{(y+R^{2}/a)^{2}+z^{2}}\cdot\frac{(y-R^{2}/a)^{2}+z^{2}}{(y-a)^{2}+z^{2}}\right]$
To convert this to cylindrical coordinates, we can use ${y^{2}+z^{2}=r^{2}}$ and ${y=r\cos\phi}$:
$\displaystyle V(r,\ $\displaystyle $\displaystyle \frac{\lambda}{4\pi\epsilon_{0}}\ln\left[\frac{r^{2}+a^{2}+2ar\cos\phi}{r^{2}+2R^{2}r\cos\phi/a+R^{4}/a^{2}}\cdot\frac{r^{2}-2R^{2}r\cos\phi/a+R^{4}
phi)$ =$ /a^{2}}{r^{2}+a^{2}-2ar\cos\phi}\right]$
$\displaystyle$ $\displaystyle $\displaystyle \frac{\lambda}{4\pi\epsilon_{0}}\ln\left[\frac{r^{2}+a^{2}+2ar\cos\phi}{a^{2}r^{2}/R^{2}+2ar\cos\phi+R^{2}}\cdot\frac{a^{2}r^{2}/R^{2}-2ar\cos\phi+R
=$ ^{2}}{r^{2}+a^{2}-2ar\cos\phi}\right]$
Note that if ${r=R}$, then the argument of the logarithm becomes 1, so ${V=0}$ on the surface of the cylinder. This happens even though the potential of each wire-image pair separately is not zero at
${r=R}$. In fact, for the first pair, we have
$\displaystyle V_{1}(R,\phi)=\frac{\lambda}{4\pi\epsilon_{0}}\ln\frac{a^{2}}{R^{2}}$
and for the second pair, we get
$\displaystyle V_{2}(R,\phi)=\frac{\lambda}{4\pi\epsilon_{0}}\ln\frac{R^{2}}{a^{2}}$
Also, if ${\phi=\pi/2}$ or ${3\pi/2}$, ${V=0}$. This corresponds to the ${xz}$ plane, which is the plane of symmetry in the problem.
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By growescience, on Monday, 5 March 2012 at 16:35, under Electrodynamics, Physics. Tags: method of images. No Comments
Post a comment or leave a trackback: Trackback URL. | {"url":"http://physicspages.com/2012/03/05/two-charged-wires-and-a-cylinder/","timestamp":"2014-04-18T13:13:22Z","content_type":null,"content_length":"96238","record_id":"<urn:uuid:b2568f9f-7c86-4c4e-a770-32185e0ade1d>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00258-ip-10-147-4-33.ec2.internal.warc.gz"} |
help velocity and mass given
so V2 = 0
and now I just have to find T
so according to your equation it would look like this:
v2 = v1 + a*t
and if I fill it in I get
0 = 5.00 + (-.3924) * t
-5 = -.3924 * t
(-5/-.3924) = t
t = 12.42 sec (is that correct?)
Probably a typo, t = 12.74 should be your value.
thanks a lot for helping, but there's a second part to the problem
How long will it take for the disk to stop, and how far does it travel? ( do not neglect friction )
how do I begin the second part now that I have the time? Is there some equation?
I'm rather surprised that you don't seem to have had any introduction to the basic kinematic formulas. Yes there is an equation (there's
an equation :smile ). In this case you want one that relates distance traveled to initial velocity, acceleration, and time. Any ideas as to which of the standard kinematic equations might fit? | {"url":"http://www.physicsforums.com/showthread.php?p=3675790","timestamp":"2014-04-17T21:40:38Z","content_type":null,"content_length":"65018","record_id":"<urn:uuid:43ee9fd8-3d2c-4762-80dc-b9a776c07d88>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00061-ip-10-147-4-33.ec2.internal.warc.gz"} |
AutoCAD Tutorial: Calculate scale and sheet size - AutoCAD Tips Blog
AutoCAD Tutorial: Calculate scale and sheet size
This tutorial is unusual, because you don’t need AutoCAD to do it.
You draw full size in AutoCAD. But before long, you may need to set the scale of text, dimensions, and other objects that need to be the right size after you plot on a sheet of paper. In a large
drawing, such as a drawing of a house, you obviously need to scale down to fit it on a sheet of paper. For small objects, you might scale up.
You can add text and dimensions in paper space and avoid scaling. You can also use annotative text and dimensions, but you still need to choose a scale.
Let’s say you have a drawing of a house. The drawing is 175 feet wide by 120 feet high. Some typical scales for an architectural drawing of a house in the United States are 1/4″=1′ and 1/8″=1′.
Follow these steps:
1. Calculate the plotted size of the width at 1/4″=1′. 175′ x 1/4 = 43-1/4″.
2. Calculate the plotted size of the height at the same scale. 120′ x 1/4 = 30″.
3. Test this size on a size D sheet (34″x22″). It’s too small. Test it on a size E sheet (44″x34″). It would just fit, but allowing for a title block and margin would make it too tight.
4. Recalculate the width at 1/8″=1′. 175′ x 1/8 = 21-7/8″.
5. Recalculate the height at 1/8″=1′. 120′ x 1/8 = 15″.
6. A size C sheet (22″x17″) is close but you might have problems with the margins. If you can move some objects in the drawing to make it slightly narrower, you’ll do fine. Otherwise, you’ll need a
size D sheet.
7. Sometimes, you need the scale factor, which is just how many times bigger the full size is compared to the plotted size. For a 1/8″=1′ scale. multiply both sides of the equation by 8 to get 1″=
8′. Then convert to inches. 8″ x 12 = 96.
What about DIMSCALE? I use this command to enlarge or reduce all of the properties of dimensions so that they are always relative to the printed size of the titleblock. In this way, there is no
calculation involved (other than the ratio of what the current dimscale is and what i want the new value to be)and no scale to worry about. It all comes down to presentation and legibility. I haven’t
searched your site completely so i don’t know if you have covered this topic but it is worth looking into.
please send me all the shortcuts in cad and also some tutorials please
Thank You for all.
please send me the details of making a scale
Salam Every on i hv need Auto Scale Ant Text Size . And Arrow Size. i m working i m meter so i need in meter if i got it in Excel Sheet so i will be very thank full to u
Can you give this example in metric units . | {"url":"http://www.ellenfinkelstein.com/acadblog/autocad-tutorial-calculate-scale-and-sheet-size/","timestamp":"2014-04-17T06:41:55Z","content_type":null,"content_length":"127548","record_id":"<urn:uuid:14e02029-e4bc-4a12-a9bd-694d84e2e865>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00415-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Determine whether the graph of y = 3x^2 + 2x − 10 opens up or down and whether it has a maximum or minimum point. Up; Minimum Up; Maximum Down; Minimum Down; Maximum
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someone just asked this .. here was my ans Easiest if you graph it. it opens up bc coefft of x^2 is positive And has a min. Differentiate, get slope, see where slope = 0.
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Thank You! Can you help me some more please :)
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Standard from: y = ax^2 + bx + c Now, the a will tell you everything. When the a is negative, it will go down. Therefore, it has a maximum. if a is positive, it will go up. Therefore it has a
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okay thanks. can you help me some more :)
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graph it on yr phone - youll see it opens up. and maybe also plot some points
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
You haven't written a testimonial for Owlfred. | {"url":"http://openstudy.com/updates/50253889e4b0d38989b7386e","timestamp":"2014-04-20T01:00:29Z","content_type":null,"content_length":"39980","record_id":"<urn:uuid:8b3f52be-5812-4096-a517-0953f66af2d3>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00270-ip-10-147-4-33.ec2.internal.warc.gz"} |
Inscribe a circle into a sector?
Sector is the area of a circle bounded by two radii and the corresponding arc. In case if semi-circle one can draw a circle with the radius of the circle perpendicular to the diameter as diameter. In
case of major sector draw a diameter in the sector and draw the circle as drawn earlier. In case of a minor sector draw in circle of the triangle formed by the two radii and the corresponding chord. | {"url":"http://mathhelpforum.com/geometry/213153-inscribe-circle-into-sector.html","timestamp":"2014-04-19T04:47:59Z","content_type":null,"content_length":"33874","record_id":"<urn:uuid:a5f636c8-284a-47b0-a434-762008a8c283>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00295-ip-10-147-4-33.ec2.internal.warc.gz"} |
differential equation??
09-01-2008 #1
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Join Date
Jul 2008
differential equation??
I started to learn differential equation after learning calculus,self study.After few days,I couldn't get beyond the concept.
I know differential equation have some part in algorithms to construct fluid simulation and other area of computer graphics but that I believe have something to do with simulation stuff ....
But in practice,do you encounter any instance where you need to understand differential equation for use in building games.
Right now,I can't think of anything hence my low motivation right now in learning differential equation.
Differential equations are exceptionally useful in real world simulations.
They're usually encountered at undergraduate level, and are used extensively afterwards, which should give you some indication of their complexity. You could fill (and indeed there are) many
tomes on differential equations. It is not surprising you are having difficulty if you are trying to learn them through self study, although don't let me put you off.
I can't say how much they're used in games design, although I did hear that the games design industry is short on maths graduates (good news for me, should I ever choose to pursue a career in
that area), perhaps this is one of the reasons. But I'm just guessing at that.
Whilst differential equations are indeed important in all sorts of different maths-related areas, for most purposes (particularly in games), you can usually just use an already existing formula
to derive the result where a derivation is needed.
Of course, games programming covers MANY different areas. You can't be expected to be an expert in ALL those areas.
Compilers can produce warnings - make the compiler programmers happy: Use them!
Please don't PM me for help - and no, I don't do help over instant messengers.
Thanks for your answers
I fail my maths in high school
Btw,This message board is awesome,the replies came not even 1 hour mark after I post my original question.Hell yeah,maybe cboard banner should have the same slogan as pizza deliveries XD
Last edited by kypronite; 09-01-2008 at 11:06 AM. Reason: fix my spelling
Well I don't want to put you off, but if you failed maths at high school then you're going to struggle.
There is some good news though. DE's are a massive area of maths, it's equivalent to learning "programming", in fact even more so (they've been around a lot longer than computers). You can break
it down into so many sub categories. My advice would be to forgo them for now, and if you find you do need them in the future, just learn what you need.
I'm not a games programmer, but I've heard that a good area of maths to study is linear algebra, especially for 3d programming.
In addition, there are also books you can buy which specifically teach you the maths needed for games design and 3d programming. I haven't read them myself, but they would be a good place to
Good luck
I must mention that you do not absolutely have to know everything about differential equations in order to use them and make games. The biggest usage of DEs in game programming would be physics
simulation, in which case you can use a physics library.
I'm not immature, I'm refined in the opposite direction.
Thanks for all your replies and encouragement.
I was scouring the bookstore today and after half an hour,I found differential equation for dummies,just one of a few books on differential equation.I was hesistant in buying because of it
titles,but the author of the dummies book is Steve Holzner.I remember enjoying his vtc screencasts.
Yeah,so now I'm learning...not giving up yet
Well, if you do this:
pos.x += vel.x * frameTime;
pos.y += vel.y * frameTime;
you are solving a differential equation numerically.
Last edited by Sang-drax : Tomorrow at 02:21 AM. Reason: Time travelling
I didn't get a great maths grade despite having been into coding since I was 8. But I have quite a 'mathematical' mind in the sense that I can solve a fairly simple mathematical problem just by
sitting down and reasoning, despite lacking the parlance and tools of traditional maths.
But now I'm finding myself more interested in the concepts I found boring at school so I'm always looking things up on the net and learning more.
MSVC++ 6.0
Game programming is like any other field that uses applied mathematics. In that, I am simply saying that you are applying rules, and formulas for the most part. You are not usually deriving them,
you are not usually reworking them, you are just taking something that is well studied and putting it to use. It is nice to be the guy in the corner of your office who can actually explain how
these equations are derived when someone invariably asks though.
I often find complex math formulas and things such as DE's to flesh themselves out much clearer in the realm of computers than on paper. Essentially they are simpler with a computer as long as
you understand the basic premise. Actually implementing them is quite simple as Sang-drax has illustrated.
And unless you are someone like Bob I highly doubt you will ever be deriving complex formulas for a game since most of the equations are computed by a third party library or have been derived and
figured out long before you ever see the code line. Funny thing is that the hardest part about a physics engine is not really the math for computing mass moment of inertia, impulses, etc., but
actually detecting the collision between the objects. So in the end collision detection, at least for me, becomes the most complex and difficult part of game programming.
These days entire books and libraries (that is, as in compiled code libraries, not the ones that contains the books) are devoted to teaching the end result of a lot of the mathematical theories
that are used in modern computer science. Some stuff seems trivial enough to figure out on your own, while other stuff may not be so easily derived without doing some reading.
Collision is one of the harder things to get exactly right. Like most game developers, you can probably even see flawed collision detection bugs in almost every game you play (btw, I notice that
writing your own code either adds a fun level of constructive critisism to playing games, or an evil level of picking them apart).
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P.Mean: Filler material for future newsletters (created 2010-02-04)
P.Mean >> The Monthly Mean >> Filler material for future newsletters (created 2010-02-04)
The following is filler material which I plan to use in future newsletters.
Feedback for November 2008 newsletter. I got some very positive feedback from three people last month. Two people cited the article on ANOVA versus regression as being the most helpful. The one
suggestion for improvement was to drop the section on Wikipedia. I agree and in the next newsletter I might replace it with a "current events" entry that highlights a recent newspaper article
touching on issues in Statistics. some suggested topics for future newsletters are principles for good graphical displays (such as the guidance from Edward Tufte) and important events in the
history of Statistics.
Feedback for December 2008 newsletter. I only received feedback from two people (that's okay, since feedback is totally optional). One person had an interesting take on my newsletter. I don't
think I read your newsletter to learn specific things as much as to be reminded of a way of thinking. I see so much use of statistics that I suspect is really lousy that it is useful to just
immerse myself in the words of someone who knows what he's talking about, loves the field, and is able to teach/communicate well. I think I read your newsletter for inspiration ... for the fact
that it returns me to a place of clarity and dare I say hope?? I'm not sure I always know what I'm talking about, but I do try to get people to think about how statistics are used in the real
world. You don't have to be a rocket scientist or a brain surgeon to be able to critically evaluate the use of statistics in the real world. If one person finds this inspiring, then I'm happy.
Another respondent liked my discussion on overfitting, but found my discussion on combining measures on different scales of measurement confusing. I'll see if I can clarify the latter point in a
future webpage or newsletter entry. Both respondents had interesting suggestions for future discussion. One suggestion was about the use of propensity score, and whether it should used as a
matching factor, a weighting factor, or a categorizing factor. This is something I am actively working on for a client, and I hope to put some of this up soon. Another suggestion was an
explanation of adjusted odds ratios in logistic regression. I have some material on this at my old website and I'll try to elaborate further on this. A third suggestion was for "basic statistics"
although there was some concern that this might be boring for other readers. I generally have found that people who are interested in help will invariably use the term "statistics for beginners."
Actually, they will use a more perjorative term like "idiots" or "complete dummies" but this is not true. The people who ask for help are almost always very well educated and well read. They just
lack experience in a particular area that I happen to know a few things about. In all the classes I've taught recently and all the webpages I've written, I've only received one request to make
the discussion more technical. Even in people who I recognize understand statistics well, their desire is still for more basic information. I'm all in favor of getting a more solid understanding
of the fundamentals. A final request came in more of the form of a complaint. In some studies, a single patient may have more than one procedure. This can lead to some confusing situations and
the researchers themselves often fail to distinguish between patients and procedures. I think it is critical to always know how much data you have and of what type. The classic example that I
encountered was a breastfeeding study with a data set of 84 infants born to 72 mothers because 12 of the mothers had given birth to twins. I'll try to write up something about this and about a
closely related topic, pseudo-replication, for a future newsletter.
Feedback for January 2009 newsletter. January's newsletter produced three comments on the web survey and two emails. Someone was nice enough to point out a broken link in the January newsletter.
When I link to my own website, iContact gets confused, so I need to manually insert the proper URLs for these links. I forgot one, which was a plug for my book. The correct link is http://
www.pmean.com/Evidence.html. The same person also pointed out that when adults fall, it's not only the greater mass, but also the greater velocity due to the fact that we are up higher than kids.
Greater mass and greater velocity is a double whammy. Cathy's pretty much recovered from the skating experience, by the way. I got a compliment for my section on crude versus adjusted
comparisons. I was worried that it was a bit too technical. That person also wanted to see a discussion of generalized estimating equations in a future newsletter. I'm not sure I understand GEE
models well enough to explain them clearly, but I will try to place them in the general context of repeated measures designs as a starting point. It may take a while for me to get something
coherent written about this. This person also thought that "Nick News" was cute. Thanks! Two people emphasized a desire for basic statistics, how-to tutorials, and general concepts. That's my
general goal. I'm not writing for people who already know Statistics inside and out. I also got a nice email from someone who liked my description of CART and the reference on overlapping
confidence intervals. This person suggested some material on Bayesian models (more specifically, the Bayes factor). I had just come back from a conference on Bayesian Biostatistics, so the
material is fresh in my mind. The hard part will be trying to make it accessible.
Feedback for Feburary 2009 newsletter. I got only one response to my feedback webpage. The respondent was relieved by my comments about small sample sizes, a reaction I had not expected. "The
small samples problem isn't a big as i thought it was -- I worry about paediatric trials with 20-30 patients, but shouldn't so much." The entry was "Is there a sample size too small to allow
efficient randomization?" and it is only one of several considerations. I elaborate on this in more detail in this month's newsletter. Still, if you're able to conduct a research study with 20-30
patients and get good precision on your confidence intervals, don't let anyone convince you that your sample size is too small. That person also asked for more material on generalized estimating
equations (GEE), and other closely related topics. I've added a definition of GEE in this newsletter and hope to show some examples of various models in this area in future newsletters. I did get
some nice compliments by email. Thanks! I'd brag about all the nice things you've said, but my email folders are in limbo thanks to a malfunctioning laptop.
Feedback for March/April 2009 newsletter. I got several nice comments about the March/April newsletter. There was appreciation for the explanation of GEE models, the rule of 15 for logistic
regression, and the loss of power caused by dichotomization. Someone pointed out that the Java application on dichotomization is also available at www.bolderstats.com/jmsl/doc/medianSplit.html. I
probably need to explain GEE models in more detail, though, and further clarify the Bayesian approach. There was a suggestion to explain what heterogeneity is. I presume this is heterogeneity in
Feedback for May/June 2009 newsletter. Only two people sent feedback (that's okay, I know that you all love my newsletter). I got compliments on my description of mosaic plots. I also got
compliments on the material on covariate adjustment from one person, but another person found that material confusing. Suggestions of future topics included meta-analysis statistics, Bayesian
statistics, various methods for smoothing and checking for interactions with applications to survival analysis, and the problem with ROC curves. That's quite an ambitious list.
Feedback for July/August 2009 newsletter. Two people provided feedback to the last newsletter. I got positive comments about my article on weighted means and the JAMA article about changes to the
primary outcome measure. Both were confused by the article on Bayesian data analysis. One had difficulty with interpreting conditional probability, especially with the order (A given B versus B
given A). In spite of this, I was encouraged to write more about Bayesian statistics.
Feedback for September/October 2009 newsletter. Three people provided feedback to the last newsletter. I got praise for the article on rounding and re-ordering and the article on early stopping
of a study and general praise for the newsletter. The only concern was about the basic formatting. I don't know what this means and I'd appreciate additional information from the person who left
this comment or anyone else. Topics suggested for future newsletters include longitudinal analysis and robust analysis. I have been intending to write up a series of webpages about longitudinal
analyses, but it's a complicated area and takes a lot of time to develop. I do hope to have something soon about this.
Feedback for November 2009 newsletter. Only two people provided feedback to the last newsletter. The Stephen Jay Gould article "The median is not the message" and the description of Kaplan-Meier
curves drew praise. One person liked the simple explanation about the case mix index but the other person thought it was confusing. I got the excellent suggestion to talk about problems
associated with missing data. It will take some effort, but I'll see what I can do.
Feedback for December 2009 newsletter. Two people provided feedback to the last newsletter. Both liked the description of the ROC curve. For one person, it was more a review and a clarification.
I'm glad to do this. Sometimes the first time you learn something, it doesn't sink in. It's when you hear a second explanation, from a slightly different perspective, that it takes for an
abstract idea to take hold. One person found the description about power and the minimum clinically relevant difference to be tough to follow. There were no suggestions this month for new topics.
I do want to consider earlier suggestions about mixed models, generalized estimating equations, and multlilevel models in future newsletters, but these topics take a lot of time to prepare well.
I'm also going to try to talk about missing data. It's a very important topic, but again one that takes some work to prepare well.
Feedback for January 2010 newsletter. Two people provided feedback to the last newsletter. One liked the description of the stem and leaf diagram, especially the explanation of how you might use
two digits in the stems. The other liked the recent entry in the seven deadly sins of researchers (wrath) and the comments about heterogeneity in clinical trials. There was nothing unclear to
either respondent. One suggestion for future topics was some of the issues associated with random effects meta-analysis, such as the use of confidence intervals versus prediction intervals. The
other suggestion was how to do power calculations when you have the wrong standard deviation.
Feedback for February/March 2010 newsletter. Three people provided feedback to the last newsletter. Two liked my description of sensitivity and specificity and the other liked my explanation
about weights. One person wanted more explanation of how to use control charts. I'm thinking of putting together a more elaborate set of webpages about quality improvement in general and that
would include several different variations of the control chart. Suggestions for future newsletters include fixed versus random effects models and survival analysis. For the latter topic, I would
note that my definition in the November 2009 newsletter was for the Kaplan-Meier plot. There is more to survival analysis than just Kaplan-Meier, of course, so I'll see what I can do. Let me
apologize in advance though for this and earlier suggestions that I have not found the time to write just yet. These webpages take quite a bit of effort, especially the ones that describe some of
the more advanced perspective.
Feedback for April 2010 newsletter. Only one person provided feedback to the last newsletter. That person liked the articles I had highlighted (Can we rely on the best trial? A comparison of
individual trials and systematic reviews, and Convincing the Public to Accept New Medical Guidelines). That person also suggested that I review some more basic review topics like Type I and Type
II errors. I do have definitions of these two terms at my old website but I still like the suggestions and will try to elaborate more on these definitions in a future newsletter.
Feedback for May/June 2010 newsletter. Three people provided feedback to the last newsletter. I got compliments on the article about why randomization doesn't always work, about Simpson's paradox
(I think this person was thinking about my interaction article), and the link to the journal article about overdiagnosis of cancer. Several people were confused, though, about the description of
interactions among two continuous variables in a linear regression model. I'll see if I can simplify it. There weren't a lot of suggestions about future topics and one person liked being
surprised. A comment about the advantages of meta-analysis over simply counting the number of positive/negative studies was offered, though.
Feedback for July 2010 newsletter. Three people provided feedback to the last newsletter. I got compliments on the article about why randomization doesn't always work, about Simpson's paradox (I
think this person was thinking about my interaction article), and the link to the journal article about overdiagnosis of cancer. Several people were confused, though, about the description of
interactions among two continuous variables in a linear regression model. I'll see if I can simplify it. There weren't a lot of suggestions about future topics and one person liked being
surprised. A comment about the advantages of meta-analysis over simply counting the number of positive/negative studies was offered, though.
Feedback for August 2010 newsletter. Three people provided feedback to the last newsletter. Two people liked the material about Structural Equations Modeling. One person challenged my approach to
applying data from a t-test because I ignored the problems that non-normality could cause. There's not a clear consensus on how to best approach non-normality, but my major beef was with a
different test, the test that checked the assumption of equality of variances. I'll try to explain in greater detail why I dislike this test in a later newsletter. One person suggested that I
explain the special issues associated with equivalence and non-inferiority trials. This is an excellent suggestion. Another asked for examples of R scripts to solve certain problems. That might
be a bit advanced for many of the readers of the newsletter, but I could always link to my website for people who would like to see this.
Feedback for the September/October 2010 newsletter. Two people provided feedback to the last newsletter. I got positive feedback on the link on poster preparation tips. There was some confusion
about the section on unequal sample sizes and on the log transformation. Suggestions about future topics included selective outcome reporting and multivariate outcomes in systematic overviews.
Feedback for the November 2010 newsletter. One person provided feedback to the last newsletter. That person liked the description of an internal pilot study and the link to the Emily Rosa study
(which was not on the newsletter, but could be found through my website). This person suggested an article about subgroup analysis, which I was able to accomodate in the current newsletter.
Feedback for December 2010 newsletter. One person provided feedback to the last newsletter. That person liked the description of an internal pilot study and the link to the Emily Rosa study. This
person suggested an article about subgroup analysis, which I was able to accomodate in the current newsletter.
Feedback for January/February 2011 newsletter. Two people provided feedback to the last newsletter. I got compliments on my discussion of the Poisson distribution and the video on the
relationship between wealth and health. I also got a compliment for stressing the need to worry about false positives when evaluating a p-value. The section on the Bonferroni adjustment and other
adjustments for multiple comparisons caused more than a bit of confusion. I also got some helpful comments on how to improve the article on subgroup findings. I was encouraged to write more about
subgroup analysis. I also got a request to write about "using stats for population management." I'm not quite sure what this last comment means.
Feedback for March/April 2011 newsletter.
Seven deadly sins summary:
1. Pride (done!) / Humility
2. Sloth (done!) / Diligence
3. Gluttony (done!) / Temperance
4. Lust (done!) / Chastity or Purity
5. Wrath (done!) / Patience
6. Envy (done!) / Kindness
7. Greed / Charity
Ideas for me to write about
Sample size for a propensity score adjusted model. Someone asked about how to calculate sample size in model where the outcome was adjusted using propensity scores. It's a tricky question, and
your answer depends on exactly how you would use the propensity score.
One way to use the propensity score is to match observations with similar propensity score values. This gives you a matched pairs t-test rather than an independent samples t-test. So you might
think, aha, I gain power because a matched pairs t-test is more powerful than an independent samples t-test. But not so fast. The process of matching will leave some of your observations
unmatched. So there's a loss in power due to the unmatched observations. In fact, matching on the propensity score probably only makes sense when you have a large number of control patients
compared to treatment patients, so it isn't hard to find a match for each treated patient. You don't lose so much in this setting because the unmatched control subjects were in the surplus
anyway. In fact, leaving a bunch of the treated observations unmatched has the potential to bias your results.
So your power calculation has to account for the expected degree of correlation within a matched pair and the expected decline in sample size because of unmatched observations.
Another way to use the propensity score is to create strata based on cutpoints of the propensity score. Values within each strata have similar covariates, so should be more homogenous as a
result. Again, you're thinking, aha, more power because of the stratification. But again it doesn't work that way. The samples sizes in the strata are likely to be unequal and seriously unequal
if there is serious covariate imbalance. You'll have far too many treatment patients for strata representing a propensity score on one end of the scale and far too many control patients for
strata representing a propensity score on the other end of the scale.
So in this setting your power calculation has to account for the expected amount of imbalance within each strata (which is related to the expected amount of covariate imbalance).
Finally, you can use the propensity score as a covariate in your model. One more time, you're thinking, aha, the covariate, if it is correlated with the outcome, will reduce variation and improve
power. Yes, but there is multi-collinearity that you need to factor in. The propensity score is going to be correlated with your binary indicator of treatment versus control and it is going to be
more strongly correlated if there is serious covariate imbalance.
So in this setting, you power calcuation has to account for the reduction in variation when the propensity score is added to the model, and it has to account for the collinearity induced by the
propensity score.
The calculations described here are tedious, but not difficult. The hard part is figuring out reasonable estimates for loss of observation, imbalance within the strata, and collinearity. If you
find this daunting, then just calculate power without accounting for the propensity score and then add a fudge factor that depends on how badly you expect the covariates to be imbalanced.
Propensity scores produce an unbiased comparison, but in most cases they do so at a cost in terms of lost power.
What is a descriptive study? (Distinguish between descriptive = no hypothesis versus descriptive = no experimental intervention0
What is the Hawthorne effect?
What is Simpson's paradox?
What is analysis of means (ANOM)?
Analytic versus enumerative studies.
Conflict of interest
I plan to be involved in a project where 100 cases are coded in terms of clincial and offence features. To enhance the design, I wanted to have a sample of cases rated by more than one coder. I
was hoping for advice regarding either established guidlines (e.g 10-20% of cases) or literature on this subject. Most papers I have seen, where a sample is reviewed, the figures seem rather
To answer this question properly, you should be more specific about why you are collecting the data and what you plan to do with it. "Enhancing the design" is a bit vague.
In my experience, why most people include a sub-sample of observations with a second rater is to establish that the correlation between the two raters is sufficiently high to overcome any
objections about the subjectivity of the coding. So select a sample size so that the 95% confidence interval is reasonably narrow. A correlation with a confidence interval from 0.1 to 0.8 does
little to establish that your reliability is high. Try to get a sample size so that the lower limit of your confidence interval is above 0.3, 0.5, or 0.7, perhaps, assuming that the true
correlation among raters is fairly strong (0.8, say).
A philosophical question about the extrapolation of a research finding
Someone posed a philosophical question on EDSTAT-L. Let's assume, he said, for the sake of argument that you've run a perfect randomized experiement at a single location. Let's assume, for sake
of argument, that you have carefully defined the population of interest and have all of the "noise" factors. The experiment was done as well as could be done. If you find a statistically
significant difference, to whom can you say your results apply to? There were several good responses. Here's what I wrote.
In theory no one, because the population of interest was sampled in the past, and even at the same site the present set of students is different than the previous set of students. Even if they
are the exact same students, they are a tiny bit older and thus are different than they were when the experiment was done.
In practice, though, we extrapolate findings all the time. Our ability to extrapolate depends largely on a set of unprovable assumptions. It's impossible to get completely away from those
assumptions, though repeated experimentation gives us some level of confidence about whether those assumptions are reasonable or unreasonable.
Clearly similarity of the two populations would be of help. They will never match perfectly, but a similar demographic mix, especially on those factors strongly associated with the outcome, is
helpful. There are no hard and fast rules about this, of course. It just takes careful judgment.
If you want to see how well an experiment extrapolates to a different population, run that same experiment on the new population. That's been done many times and you do get a rough feel for when
it is safe or unsafe to extrapolate. But you'll never know this with absolute certainty.
If you never tried to extrapolate without first having absolute certainty, then you would continually plunge your hand into a fire, because getting burned 100 times does not prove that you will
get burned the 101st time. Maybe the first 100 times were just bad luck.
German tank problem: http://en.wikipedia.org/wiki/German_tank_problem#Exposition
Power for comparison of two Poisson means
What's a good value for Cronbach's alpha? http://www.childrensmercy.org/stats/weblog2004/CronbachAlpha.asp
Nick news
I remain utterly convinced that posting images from publicly available documents, questioning their integrity when there is sufficient evidence to suggest a problem, is in no way grounds for a
libel or defamation suit. In short—don't shoot the messenger. If you didn't want your scientific data to be questioned, you shouldn't have published it! —Paul Brookes, an associate professor at
the University of Rochester Medical Center and a self-identified owner of the now-defunct Science Fraud blog, in a post explaining why libel threats had led him to shut down the site (as reported
on Retraction Watch, Jan. 3, 2013) as quoted at http://www.the-scientist.com//?articles.view/articleNo/34187/title/Speaking-of-Science/
There's a cute quote by Stephen Senn in Statistical Issues in Drug Development (1997): Medical statistician: one who will not accept that Columbus discovered America ... because he said he was
looking for India in the trial plan.
"It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong" Richard Feynman, as quoted at http://thinkexist.com/
"Statistics are no substitute for judgment." Clay, Henry Read more: http://quotationsbook.com/quote/37336/#ixzz1PsnVuHcj on Quotations Book
Counting is the religion of this generation it is its hope and its salvation. - Gertrude Stein, as quoted in The Emperor of All Maladies: A Biography of Cancer, Siddhartha Mukherjee.
Fiction is about the suspension of disbelief; science is about the suspension of belief. - James Porter
"In God we trust. All others must bring data."
David Cox: "There are no routine statistical questions, only questionable statistical routines"
"... no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the
light of his evidence and his ideas." - Sir Ronald A. Fisher (1956)
"All scientific work is incomplete - whether it be observational or experimental. All scientific work is liable to be upset or modified by advancing knowledge. That does not confer upon us a
freedom to ignore the knowledge we already have or postpone the action that it appears to demand at a given time." Sir Austin Bradford Hill, as quoted at Toxipedia http://toxipedia.org/display/
"On two occasions I have been asked [by members of Parliament!], 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able rightly to
apprehend the kind of confusion of ideas that could provoke such a question." Charles Babbage, as quoted at http://stackoverflow.com/questions/17512/computer-language-puns-and-jokes
"If you're a politician, admitting you're wrong is a weakness, but if you're an engineer, you essentially want to be wrong half the time. If you do experiments and you're always right, then you
aren't getting enough information out of those experiments. You want your experiment to be like the flip of a coin: You have no idea if it is going to come up heads or tails. You want to not know
what the results are going to be." Peter Norvig, as quoted at http://www.slate.com/blogs/blogs/thewrongstuff/archive/2010/08/03/
To call in the statistician after the experiment is done may be no more than asking him to perform a post-mortem examination: he may be able to say what the experiment died of. ~ Sir Ronald
Aylmer Fisher
The plural of anecdote is not data. ~ Roger Brinner
The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data. ~ John Tukey
In addition to what John (and others) have said, don't forget the good point raised by Bland & Altman (BMJ 2009;338:a3166). --- start excerpt --- The aversion to parametric methods for small
samples may arise from the inability to assess the distribution shape when there are so few observations. How can we tell whether data follow a normal distribution if we have only a few
observations? The answer is that we have not only the data to be analysed, but usually also experience of other sets of measurements of the same thing. In addition, general experience tells us
that body size measurements are usually approximately normal, as are the logarithms of many blood concentrations and the square roots of counts. --- end excerpt ---
Cancer causes cell phones. http://xkcd.com/925/ or http://imgs.xkcd.com/comics/cell_phones.png
A statistical lament (after Joni Mitchell), Robert Dawson. http://www.lablit.com/article/646.
A frightening new study proves that new studies are frightening. Charles Fielding, as quoted on his Facebook page.
My bad joke about actuaries is that they're the ones who decided Statistics wasn't boring enough by itself, so they decided to mix it with insurance.
There's a story about two doctors who are floating above the countryside in a hot air balloon. They are drifting with the wind and enjoying the scenery, but after a couple of hours, they realize
that they are totally lost. They see someone down on the ground, and shout down "Hello! Can you tell us where we are?" The person on the ground replies, "you're fifty feet up in the air, in a hot
air balloon." One doctor turns to the other and says, "That person on the ground must be a statistician." "How did you know?" came astonished reply from the ground. "Only a statistician would
provide an answer that was totally accurate and totally useless at the same time." In my stories, of course, the statistician always has the last word. "Very good. But I can also tell that you
two are doctors." It was the doctors' turn to be astonished. The statistician explained. "Only a doctor would have such a good view of the area and still not have any idea they were."
Doctors love to use big words. These are the folks who take a simple ear ache and call it "otitis media." To them, a runny nose is "rhinorhea" and a tummy ache is "gastrointestinal distress."
It's enough to make me produce lacrimal secretions.
R Through Excel. A Spreadsheet Interface for Statistics, Data Analysis, and Graphics. Heiberger, Richard M., Neuwirth, Erich.
Modeling Survival Data: Extending the Cox Model. Terry M. Therneau, Patricia M. Grambsch
The Truth Wears Off http://www.newyorker.com/reporting/2010/12/13/101213fa_fact_lehrer
Results Unproven, Robotic Surgery Wins Converts. http://www.nytimes.com/2010/02/14/health/14robot.html
In the City’s Vital Statistics, So Many Ways to Die by Accident http://www.nytimes.com/2010/02/16/nyregion/16accidents.html
Khamis, H. and Kepler, M.: Sample size in multiple regressions: 20 + 5k, Journal of Applied Statistical Science, to appear in Vol. 17. Also see http://ablejec.nib.si/as2007/
appliedstatistics2007booklet.pdf (page 64).
It's not too recent, but an outstanding overview is: Biases in the interpretation and use of research results. RJ MacCoun. Annu Rev Psychol
1998: 49; 259-87.
* http://socrates.berkeley.edu/~maccoun/ar_bias.html
* http://ist-socrates.berkeley.edu/~maccoun/MacCoun_AnnualReview98.pdf
Among the fascinating findings summarized in this paper:
"partisans on both sides of a dispute tend to see the exact same
media coverage as favoring their opponents� position"
"24 students favoring capital punishment and 24 opposing it were recruited; each group believed the existing evidence favored their views. They were then given descriptions of two fictitious
studies, one supporting the deterrence hypothesis, the other failing to support it. For half the respondents, the prodeterrence paper used a cross-sectional methodology (cross-state homicide
rates) and the antideterrent paper used a longitudinal methodology (within-state rates before and after capital punishment was adopted); for the remaining respondents, the methodologies were
reversed. Each description contained a defense of the particular methodology and a critique of the opposing approach. Students received and provided initial reactions to each study's results
before being given methodological details to evaluate. Analyses of student ratings of the quality and persuasiveness of these studies revealed a biased assimilation effect: students more
favorably evaluated whichever study supported their initial views on the deterrent effect, irrespective of research methodology. Students' open-ended comments reveal how either methodology,
cross-sectional or longitudinal, could be seen as superior or inferior, depending on how well its results accorded with one's initial views."
"people will feel less safe after a noncatastrophic technological breakdown if they already oppose the particular technology, but will feel more safe after such a breakdown if they support the
I think that MacCoun also mentions somewhere about a study that showed people two contradictory articles on the same political topic. After reading both articles people became STRONGER in their
beliefs. After all, they found a study that supported what they knew all along, and that other study must have been flawed somehow.
This paper should be required reading for anyone in Evidence-Based Medicine.
John P. A. Ioannidis. Why Most Published Research Findings Are False. PLoS Med. 2005;2(8):e124. Abstract: "There is increasing concern that most current published research findings are false. The
probability that a research claim is true may depend on study power and bias, the number of other studies on the same question, and, importantly, the ratio of true to no relationships among the
relationships probed in each scientific field. In this framework, a research finding is less likely to be true when the studies conducted in a field are smaller; when effect sizes are smaller;
when there is a greater number and lesser preselection of tested relationships; where there is greater flexibility in designs, definitions, outcomes, and analytical modes; when there is greater
financial and other interest and prejudice; and when more teams are involved in a scientific field in chase of statistical significance. Simulations show that for most study designs and settings,
it is more likely for a research claim to be false than true. Moreover, for many current scientific fields, claimed research findings may often be simply accurate measures of the prevailing bias.
In this essay, I discuss the implications of these problems for the conduct and interpretation of research." [Accessed January 6, 2009]. Available at: http://www.pubmedcentral.nih.gov/
Susan A. Peters. Engaging with the Art and Science of Statistics. Mathematics Teacher. 2010;103(7):496. Abstract: "Statistics uses scientific tools but also requires the art of flexible and
creative reasoning." [Accessed September 24, 2010]. Available at: http://www.nctm.org/eresources/view_media.asp?article_id=9145.
The Christmas carol, The Twelve Days of Christmas, is the musical equivalent of the 99 bottles of beer song. It involves a true love who is extraordinarily generous and gives one present on the
first day of Chirstimas (a partridge in a pear tree), three presents on the second day of Christmas (two turtle doves and a partridge in a pear tree), six presents on the third day of Christmas
(three French hens, two turtle doves, and a partridge in a pear tree). So after only three days, the true love has given 1+3+6 = 10 gifts. Two part question: 1. What is the total number of gifts
given on the first through twelfth days of Christmas. 2. Give a formula for the number of gifts given by the nth day of Christmas.
Unsung heroes
John F. Hall -- http://surveyresearch.weebly.com
Jerry Dallal -- http://www.jerrydallal.com/
Frank Harrell
Edward Tufte, http://www.edwardtufte.com/tufte
Trend and variation. Animated short on statistics from Norwegian infotainment program Siffer. Produced by TeddyTV for NRK. Animation by Ole Christoffer Haga. http://www.youtube.com/watch?v=
Statistics - Dream Job of the next decade. From a Keynote Presentation by Hal Varian - Chief Economist, Google, to the 2008 Almaden Institute - "Innovating with Information". The full
presentation (as well as all the other presentations at the very interesting meeting) is available at http://www.almaden.ibm.com/institute/agenda.shtml Hal Varian makes the argument that with
data in huge supply and statisticians in short supply, being a statistician has to be the 'really sexy job for the 2010s'. http://www.youtube.com/watch?v=D4FQsYTbLoI
Also see http://www.youtube.com/watch?v=tm3lZJdEvCc
Phil Murfet, Statistician: I Make Medicines. Phil Murfet explains how Pfizer is using the science of statistics to make better medicines for patients. http://www.youtube.com/watch?v=N89473ZLgOM
Important Massage from the Chief Statistician. Dr. Nussbaum has run the numbers, assessed the percentages, and evaluated the norms. He offers, with a small amount of confidence, 'whats in and
whats out' for 2009. http://www.youtube.com/watch?v=cizJgk5SpmY
Statistician for The Unofficial Guide to Walt Disney World. Fred Hazelton, Statistician for the Unofficial Guide to Walt Disney World discusses his work with the guide and with touringplans.com
on an Ottawa TV Morning Show. http://www.youtube.com/watch?v=8VWh7V-rwKM
Myth surrounding statisticians. Interview with Pravin Shekar, Chief-BizDev and New Initiatives, Dexterity, Chennai (www.dexterity.in), March 27, 2009, 1 pm. http://www.youtube.com/watch?v=
Inspirational statistician. High school girl creates her own stat keeping software. http://www.youtube.com/watch?v=v9DpxkhphWM
World Statistics Day: Statistics All Around Us. US Census Bureau. Help the US Census Bureau celebrate World Statistics Day with the United Nations and statistical agencies across the globe. This
video highlights some of the many benefits we receive from the statistical information provided by the U.S. federal statistical community. The key question: "What would our country -- our world
-- be like, without statistics?" http://www.youtube.com/watch?v=piSCkkSvoMo
Why Statistics is Important: Saves lives, money and time. Doug Edwards. This story illustrates how important Statistics is to human life. It is presented in a true story format. I got the idea
from reading the Book "Super Freakonomics" by S. Levitt and S. Dubner. The video illustrates the events that took place to solve the over 2000 early deaths of new mothers at the Vienna General
Hospital between 1841 and 1846. http://www.youtube.com/watch?v=SBYBcENWZc4
Lies, Damned Lies, and Statistics (About TED Talks) "Excerpt: In a brilliantly tongue-in-cheek analysis, Sebastian Wernicke turns the tools of statistical analysis on TEDTalks, to come up with a
metric for creating "the optimum TEDTalk" based on user ratings. How do you rate it?" Available at http://www.youtube.com/watch?v=1Totz8aa2Gg
Open-mindedness.; 2009. A look at some of the flawed thinking that prompts people who believe in certain non-scientific concepts to advise others who don't to be more open-minded. music ©
QualiaSoup . A note on spelling: A number of people have commented that 'close-minded' should be spelt 'closeD-minded'. This is incorrect. It's one of those cases in English where what you might
suspect to be the case, isn't. The term is 'close-minded', in the same way that when one is reticent, one is said to be 'close-mouthed', not 'closeD-mouthed'. [Accessed August 24, 2010].
Available at: http://www.youtube.com/watch?v=T69TOuqaqXI&feature=youtube_gdata_player.
Dr. Andrew Newberg - Why God Doesn't Use Biostatistics.; 2008. An excerpt from Dr. Andrew Newberg's keynote lecture "Why God Doesn't Use Biostatistics: Science and the Study of the Mind, the
Body, and Spirituality" presented on September 11, 2008 at the United Nations & Nour Foundation symposium, "Beyond the Mind-Body Problem: New Paradigms in the Science of Consciousness," inspired
by the philosophy of Ostad Elahi. http://www.mindbodysymposium.com [Accessed August 25, 2010]. Available at: http://www.youtube.com/watch?v=V6iWazXDTps&feature=youtube_gdata_player.
http://www.youtube.com/watch?v=_sHa82UPPes&feature=email Rod Jackson is an epidemiologist in New Zealand and this video is a response from some of the medical students to his comments on risks of
heart disease.
Gordon Guyatt. McMaster University > Evidence-Based Clinical Practice Workshop. Description: This website provides details on how to teach evidence-based clinical practice, through workshops
offered at McMaster University, as well as through videos, such as "Making Sense of Likelihood Ratios" and "Understanding Odds Ratios." The site also offers some of the teaching handouts as PDF
files. [Accessed January 24, 2010]. Available at: http://ebm.mcmaster.ca/.
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Ahh please help!
April 8th 2008, 02:40 PM #1
Junior Member
Apr 2008
Ahh please help!
Car A and Car B leave the origin at noon. Car A travels north at 66 mph and car B travels east at 112 mph.
(a) How far apart are they after 1 hour?
(b) How far apart are they after
(c) When are they 1300 miles apart?
I know (a) is 130 miles. But I can't seem to get b, and therefore i can't do c. If anyone can help I'd greatly appreciate it =)
What you are looking for is distance (D) as a function of time (t).
Well, we know the equations that give us the distance of Car A and Car B from the origin:
$<br /> D_A(t)=66t$
$<br /> D_B(t)=112t$
And we know that to get the distance between the two we use Pythagoras Theorem.
So, we have that:
$D_{AB}(t)=\sqrt {(D_A)^2+(D_B)^2}$
We then add in our knowledge of $D_A$ and $D_B$ to get:
$D_{AB}(t)=\sqrt {(66t)^2+(112t)^2}$
Thank you so much!
Hello, lilikoipssn!
Car A and Car B leave the origin at noon.
Car A travels north at 66 mph and car B travels east at 112 mph.
(a) How far apart are they after 1 hour?
(b) How far apart are they after $t$ hours?
(c) When are they 1300 miles apart?
We can solve all the problems at once . . .
P *
| *
| * d
66t | *
| *
| *
* - - - - - - - - *
O 112t
Car $A$ drives north at 66 mph.
. . In $t$ hours, it has gone $66t$ miles to point $P$
Car $B$ drives east at 112 mph.
. . In $t$ hours, it has gone $112t$ miles to point $Q.$
Their distance $d$ is the hypotenuse of right triangle $POQ.$
. . $d \;=\;\sqrt{(66t)^2 + (112t)^2} \;=\;\sqrt{16900t^2} \;=\;130t$ miles.
(a) When $t = 1\!:\;\;d \:=\:130(1) \:=\:130$ miles.
(b) At time $t$, their distance is: . $d \:=\:130t$ miles.
(c) If $d = 1300$, we have: . $130t \:=\:1300 \quad\Rightarrow\quad t \:=\: 10$ hours.
April 8th 2008, 02:59 PM #2
April 8th 2008, 03:25 PM #3
Junior Member
Apr 2008
April 8th 2008, 03:26 PM #4
Super Member
May 2006
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Mr. Long's Math Blog
January 3rd, 2011
Credit Recovery: If you failed a class the previous term, you still have a chance to earn credit using the Credit Recovery Program.
You must apply within 2 weeks of the end of the failed term to participate. The usual time requirement to earn the credit is 2-3 weeks after you start on the program. While each teacher’s
requirement to recover credit may be slightly different, mine is fairly straight forward.
I allow you to turn in any late work in (more than the 3 I allow during the term) and to retake (or take for the first time if you haven’t done so already) any tests. The goal is to get your grade to
a 60%. You do whatever work that will raise your grade to that point then you’re done. If turning in homework alone does not raise your grade to 60%, you will have to retake a test. If you’re
missing a test, you will have to take it because there’s really no way to not take a test and accumulate the required 60% for passing.
Once you have earned the credit will have an “F” on your transcript (for calculating your overall GPA), but you will be given credit for successfully passing the class (for calculating the number of
math credits for graduation). If you have any questions about this, please talk to me.
To take advantage of this:
(1) Go to the counselors’ office and get a form
(2) Take the form to the Book Keeper and pay $30
(3) Bring the form to me. I will tell you what you will have to do to earn the credit for the failed quarter.
Math Tutoring
August 10th, 2010
If you have a grade below a C, you will be required to attend your class period’s “Royal Review and Reward ( R & R)” period which follows your regularly schedule class period once per week (each of
the 8 periods has one R & R period once per week).
You may choose to stay to the R & R period even if you have a grade of C or better, just to use it for a study hall.
If the R & R period (once per week for your math class) is not enough to raise your grade, I expect you to attend after-school tutoring. Math tutoring is available every school day after school from
3 PM (not 2:30) to 4 PM.
Some students stay after school the same day they have math and complete their homework during tutoring time. The material is fresher in their minds and, if they need help, a tutor is available
right then.
You can also stop in to my classroom right after you arrive at school in the morning if you have a quick question or two. I’ll try to accommodate you if I have time.
MATH TUTORING
Everyday After School
3:00 to 4:00
Day Instructor Room
Mon Mr. Dick 221
Tue Mr. Long 100
Wed Mr. Long 100
Thur Mr. Long 100
Fri Mr. Dick 221
August 10th, 2010
I would like to welcome all of my Secondary Math 2 (Math-2), Secondary Math 3 (Math-3) and Pre-Calculus students to my class. I’m excited to help you learn what I think is a really cool subject. In
a world that is increasingly subjective, it’s comforting to know that with math you can get the straight unvarnished truth.
Day #1: I will hand out your books (to Pre-Calculus students only). I will then discuss the disclosure statement (class rules and policies) which is available in the Rules and Policies section of
this blog. I will expect each of you to read it since you will be required to conform to my classroom rules. Your parents/guardians can just skim through it quickly. I just want them to know that if
they have any questions about my policies, they can easily find the disclosure statement in my blog. I would also like you to print out the acknowledgement form and have you and your parents/
guardians sign it. Please return the form for extra extra credit.
If your or your parents/guardians have any questions, I will promptly respond to email (jelong@wsd.net) within a day or two at the most.
Day #2: Begin learning mathematics.
Math-2 HW #2 Core Test Prep
April 17th, 2014
lesson date: Thurs 17 April
HW is due: Tues 22 April
Math-2 #2 Core Test Prep Notes
April 17th, 2014
lesson date: Thurs 17 April
HW is due: Tues 22 April
Pre-Cal Lesson 6-4 Notes (Polar Coordinates)
April 16th, 2014
Lesson date: Wed 16 April
HW is due: Mon 21 April
Pre-Cal HW #6-4
April 16th, 2014
Lesson date: Wed 16 April
HW is due: Mon 21 April
Section 6-4 page 539 even problems: 2, 4, 8, 14-22, 28a, 30a, 32, 34
Section P-2 page 72 even problems: 42-48 (circles)
Math-3 HW #3 Core test Prep
April 16th, 2014
Lesson date:
1st period: Wed 16 April
2nd period: Mon 21 April
8th period: Tues 22 April
HW due:
1st period: Mon 21 April
2nd period: Wed 23 April
8th period: Thurs 24 April
Math-3 Core Test Prep #3 Notes
April 16th, 2014
Lesson date:
1st period: Wed 16 April
2nd period: Mon 21 April
8th period: Tues 22 April
HW due:
1st period: Mon 21 April
2nd period: Wed 23 April
8th period: Thurs 24 April
Math-3 (honors) HW #1 Core Test Prep
April 15th, 2014
Lesson date:
2nd period: Mon 14 April
8th period: Tues 15 April
HW is due:
2nd period: Wed 16 April
8th period: Thurs 17 April
Math-2 HW #10-3
April 15th, 2014
Lesson date:
Fri 11 April
HW is due:
Tues 15 April
Math-2 Lesson 10-3 Notes
April 15th, 2014
Lesson date:
Fri 11 April
HW is due:
Tues 15 April
Math-2 Core Test Prep HW #1
April 15th, 2014
Lesson date: Tues 15 April
HW is due: Thurs 17 April | {"url":"http://blog.wsd.net/jelong/","timestamp":"2014-04-19T10:13:09Z","content_type":null,"content_length":"36336","record_id":"<urn:uuid:359a08db-ea87-4fa0-9007-7d39951e92f9>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00101-ip-10-147-4-33.ec2.internal.warc.gz"} |
search results
Expand all Collapse all Results 76 - 100 of 147
76. CJM 2007 (vol 59 pp. 1098)
Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions
In this paper we study ruled surfaces which appear as an exceptional surface in a succession of blowing-ups. In particular we prove that the $e$-invariant of such a ruled exceptional surface $E$ is
strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of $E$). This fact immediately enables us to resolve an open problem concerning an
intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of $e$ to the study of the
poles of the well-known topological, Hodge and motivic zeta functions.
Categories:14E15, 14J26, 14B05, 14J17, 32S45
77. CJM 2007 (vol 59 pp. 1069)
Quotients jacobiens : une approche algébrique
Le diagramme d'Eisenbud et Neumann d'un germe est un arbre qui repr\'esente ce germe et permet d'en calculer les invariants. On donne une d\'emonstration alg\'ebrique d'un r\'esultat caract\
'erisant l'ensemble des quotients jacobiens d'un germe d'application $(f,g)$ \`a partir du diagramme d'Eisenbud et Neumann de $fg$.
Keywords:Singularité, jacobien, quotient jacobien, polygone de Newton
Categories:14B05, 32S05, 32S50
78. CJM 2007 (vol 59 pp. 742)
Geometry and Spectra of Closed Extensions of Elliptic Cone Operators
We study the geometry of the set of closed extensions of index $0$ of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a
necessary and sufficient condition for the existence of rays of minimal growth for such operators.
Keywords:resolvents, manifolds with conical singularities, spectral theor, boundary value problems, Grassmannians
Categories:58J50, 35J70, 14M15
79. CJM 2007 (vol 59 pp. 488)
Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties
We consider the $k$-osculating varieties $O_{k,n.d}$ to the (Veronese) $d$-uple embeddings of $\PP^n$. We study the dimension of their higher secant varieties via inverse systems (apolarity). By
associating certain 0-dimensional schemes $Y\subset \PP^n$ to $O^s_{k,n,d}$ and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are
defective or not.
Categories:14N15, 15A69
80. CJM 2007 (vol 59 pp. 372)
Zeta Functions of Supersingular Curves of Genus 2
We determine which isogeny classes of supersingular abelian surfaces over a finite field $k$ of characteristic $2$ contain jacobians. We deal with this problem in a direct way by computing
explicitly the zeta function of all supersingular curves of genus $2$. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the
number of curves, up to $k$-isomorphism, leading to the same zeta function.
Categories:11G20, 14G15, 11G10
81. CJM 2007 (vol 59 pp. 36)
Classification of Ding's Schubert Varieties: Finer Rook Equivalence
K.~Ding studied a class of Schubert varieties $X_\lambda$ in type A partial flag manifolds, indexed by integer partitions $\lambda$ and in bijection with dominant permutations. He observed that the
Schubert cell structure of $X_\lambda$ is indexed by maximal rook placements on the Ferrers board $B_\lambda$, and that the integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are
additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$ satisfy the combinatorial condition of \emph{rook-equivalence}. We classify the varieties $X_\lambda$ up to isomorphism,
distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.
Keywords:Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotence
Categories:14M15, 05E05
82. CJM 2006 (vol 58 pp. 1000)
On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of $\operatorname{SL}_n$
We compute some Hodge and Betti numbers of the moduli space of stable rank $r$, degree $d$ vector bundles on a smooth projective curve. We do not assume $r$ and $d$ are coprime. In the process we
equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank $r$, degree $d$ vector bundles
on a curve. Our methods also yield a formula for the Poincar\'e polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof
that the Tamagawa number of $\sln$ is one.
Categories:14H, 14L
83. CJM 2006 (vol 58 pp. 476)
Apolar Schemes of Algebraic Forms
This is a note on the classical Waring's problem for algebraic forms. Fix integers $(n,d,r,s)$, and let $\Lambda$ be a general $r$-dimensional subspace of degree $d$ homogeneous polynomials in
$n+1$ variables. Let $\mathcal{A}$ denote the variety of $s$-sided polar polyhedra of $\Lambda$. We carry out a case-by-case study of the structure of $\mathcal{A}$ for several specific values of $
(n,d,r,s)$. In the first batch of examples, $\mathcal{A}$ is shown to be a rational variety. In the second batch, $\mathcal{A}$ is a finite set of which we calculate the cardinality.}
Keywords:Waring's problem, apolarity, polar polyhedron
Categories:14N05, 14N15
84. CJM 2006 (vol 58 pp. 262)
Connections on a Parabolic Principal Bundle Over a Curve
The aim here is to define connections on a parabolic principal bundle. Some applications are given.
Keywords:parabolic bundle, holomorphic connection, unitary connection
Categories:53C07, 32L05, 14F05
85. CJM 2006 (vol 58 pp. 93)
Motivic Haar Measure on Reductive Groups
We define a motivic analogue of the Haar measure for groups of the form $G(k\llp t\rrp)$, where~$k$ is an algebraically closed field of characteristic zero, and $G$ is a reductive algebraic group
defined over $k$. A classical Haar measure on such groups does not exist since they are not locally compact. We use the theory of motivic integration introduced by M.~Kontsevich to define an
additive function on a certain natural Boolean algebra of subsets of $G(k\llp t\rrp)$. This function takes values in the so-called dimensional completion of the Grothendieck ring of the category of
varieties over the base field. It is invariant under translations by all elements of $G(k\llp t\rrp)$, and therefore we call it a motivic analogue of Haar measure. We give an explicit construction
of the motivic Haar measure, and then prove that the result is independent of all the choices that are made in the process.
Keywords:motivic integration, reductive group
Categories:14A15, 14L15
86. CJM 2005 (vol 57 pp. 1178)
Asymptotic Behavior of the Length of Local Cohomology
Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring, and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in $R$. Let $\lambda(M)$
denote the length of an $R$-module $M$. In this paper, we show that $$ \lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d} =\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^
n,R(-d)\bigr)\bigr)}{n^d} $$ always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$-primary ideals $I$ in a local Cohen--Macaulay ring, where $e(I)$ denotes the multiplicity of $I$.
But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have
polynomial growth.
Keywords:powers of ideals, local cohomology, Hilbert function, linear growth
Categories:13D40, 14B15, 13D45
87. CJM 2005 (vol 57 pp. 1314)
Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra
In 1999 V. Arnol'd introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure
(invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference
between the geometric and (defined in this paper) algebraic restriction of a $1$-form to a singular submanifold. We prove that a germ of any subset $N$ of a contact manifold is well defined, up to
contactomorphisms, by the algebraic restriction to $N$ of the contact structure. This is a generalization of the Darboux-Givental' theorem for smooth submanifolds of a contact manifold. Studying
the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete
solution of three classification problems, including a simple explanation of V.~Arnold's results and further classification results for singular curves in a contact space. We also prove several
results on the external geometry of a singular submanifold $N$ in terms of the algebraic restriction of the contact structure to $N$. In particular, the algebraic restriction is zero if and only if
$N$ is contained in a smooth Legendrian submanifold of $M$.
Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves
Categories:53D10, 14B05, 58K50
88. CJM 2005 (vol 57 pp. 724)
Some Results on Surfaces of General Type
In this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series.
Some of the corollaries of more general results include: results on property $N_p$ associated to $K_S \otimes B^{\otimes n}$ where $B$ is base-point free and ample divisor with $B\otimes K^*$ {\it
nef}, results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are
Categories:13D02, 14C20, 14J29
89. CJM 2005 (vol 57 pp. 338)
Certain Exponential Sums and Random Walks on Elliptic Curves
For a given elliptic curve $\E$, we obtain an upper bound on the discrepancy of sets of multiples $z_sG$ where $z_s$ runs through a sequence $\cZ=\(z_1, \dots, z_T\)$ such that $k z_1,\dots, kz_T $
is a permutation of $z_1, \dots, z_T$, both sequences taken modulo $t$, for sufficiently many distinct values of $k$ modulo $t$. We apply this result to studying an analogue of the power generator
over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings.
Categories:11L07, 11T23, 11T71, 14H52, 94A60
90. CJM 2005 (vol 57 pp. 400)
Generalized $k$-Configurations
In this paper, we find configurations of points in $n$-dimensional projective space ($\proj ^n$) which simultaneously generalize both $k$-configurations and reduced 0-dimensional complete
intersections. Recall that $k$-configurations in $\proj ^2$ are disjoint unions of distinct points on lines and in $\proj ^n$ are inductively disjoint unions of $k$-configurations on hyperplanes,
subject to certain conditions. Furthermore, the Hilbert function of a $k$-configuration is determined from those of the smaller $k$-configurations. We call our generalized constructions
$k_D$-configurations, where $D=\{ d_1, \ldots ,d_r\}$ (a set of $r$ positive integers with repetition allowed) is the type of a given complete intersection in $\proj ^n$. We show that the Hilbert
function of any $k_D$-configuration can be obtained from those of smaller $k_D$-configurations. We then provide applications of this result in two different directions, both of which are motivated
by corresponding results about $k$-configurations.
Categories:13D40, 14M10
91. CJM 2005 (vol 57 pp. 3)
Enriques Diagrams and Adjacency of Planar Curve Singularities
We study adjacency of equisingularity types of planar complex curve singularities in terms of their Enriques diagrams. The goal is, given two equisingularity types, to determine whether one of them
is adjacent to the other. For linear adjacency a complete answer is obtained, whereas for arbitrary (analytic) adjacency a necessary condition and a sufficient condition are proved. We also obtain
new examples of exceptional deformations, {\em i.e.,} singular curves of type $\mathcal{D}'$ that can be deformed to a curve of type $\mathcal{D}$ without $\mathcal{D}'$ being adjacent to $\mathcal
92. CJM 2004 (vol 56 pp. 1145)
On Log $\mathbb Q$-Homology Planes and Weighted Projective Planes
We classify normal affine surfaces with trivial Makar-Limanov invariant and finite Picard group of the smooth locus, realizing them as open subsets of weighted projective planes. We also show that
such a surface admits, up to conjugacy, one or two $G_a$-actions.
Categories:14R05, 14J26, 14R20
93. CJM 2004 (vol 56 pp. 1308)
Variations of Mixed Hodge Structures of Multiple Polylogarithms
It is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall {\em explicitly} determine these structures related to
multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications: First we study the limit of mixed
Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple {\em poly}\/logarithms. Then following Deligne
and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical
polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight $k$ when $k=2$ and 3. At the end of this paper, motivated by
Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.
Categories:14D07, 14D05, 33B30
94. CJM 2004 (vol 56 pp. 1094)
Cycle-Level Intersection Theory for Toric Varieties
This paper addresses the problem of constructing a cycle-level intersection theory for toric varieties. We show that by making one global choice, we can determine a cycle representative for the
intersection of an equivariant Cartier divisor with an invariant cycle on a toric variety. For a toric variety defined by a fan in $N$, the choice consists of giving an inner product or a complete
flag for $M_\Q= \Qt \Hom(N,\mathbb{Z})$, or more generally giving for each cone $\s$ in the fan a linear subspace of $M_\Q$ complementary to $\s^\perp$, satisfying certain compatibility conditions.
We show that these intersection cycles have properties analogous to the usual intersections modulo rational equivalence. If $X$ is simplicial (for instance, if $X$ is non-singular), we obtain a
commutative ring structure to the invariant cycles of $X$ with rational coefficients. This ring structure determines cycles representing certain characteristic classes of the toric variety. We also
discuss how to define intersection cycles that require no choices, at the expense of increasing the size of the coefficient field.
Keywords:toric varieties, intersection theory
Categories:14M25, 14C17
95. CJM 2004 (vol 56 pp. 716)
Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions
We study the Hilbert functions of fat points in $\popo$. If $Z \subseteq \popo$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function
$_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for $l \gg 0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of
the points in $\popo$. This enables us to compute all but a finite number values of $H_{Z}$ without using the coordinates of points. We also characterize the ACM fat point schemes sing our
description of the eventual behaviour. In fact, n the case that $Z \subseteq \popo$ is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual
values of the Hilbert function.
Keywords:Hilbert function, points, fat points, Cohen-Macaulay, multi-projective space
Categories:13D40, 13D02, 13H10, 14A15
96. CJM 2004 (vol 56 pp. 495)
Coinvariant Algebras of Finite Subgroups of $\SL(3,C)$
For most of the finite subgroups of $\SL(3,\mathbf{C})$, we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae \cite{M99} for subgroups of $\SU
(2)$. We also study the $G$-orbit Hilbert scheme $\Hilb^G(\mathbf{C}^3)$ for any finite subgroup $G$ of $\SO(3)$, which is known to be a minimal (crepant) resolution of the orbit space $\mathbf{C}^
3/G$. In this case the fiber over the origin of the Hilbert-Chow morphism from $\Hilb^G(\mathbf{C}^3)$ to $\mathbf{C}^3/G$ consists of finitely many smooth rational curves, whose planar dual graph
is identified with a certain subgraph of the representation graph of $G$. This is an $\SO(3)$ version of the McKay correspondence in the $\SU(2)$ case.
Keywords:Hilbert scheme, Invariant theory, Coinvariant algebra,, McKay quiver, McKay correspondence
Categories:14J30, 14J17
97. CJM 2004 (vol 56 pp. 612)
Solvable Points on Projective Algebraic Curves
We examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective
curves without solvable points of genus $g$, when $g$ is at least $40$, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus $0$,
$2$, $3$ or $4$ defined over any field has a solvable point. Finally we prove that every genus $1$ curve defined over a local field of characteristic zero with residue field of characteristic $p$
has a divisor of degree prime to $6p$ defined over a solvable extension.
Categories:14H25, 11D88
98. CJM 2004 (vol 56 pp. 310)
The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order
In this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th
problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of
divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued
affine invariants. According to the possible values taken by them in this family we obtain a total of $18$ topologically distinct phase portraits. We show that inside the class of all quadratic
systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the
sense of \cite{DRR} and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.
Categories:34C40, 51F14, 14D05, 14D25
99. CJM 2003 (vol 55 pp. 1100)
Polar Homology
For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar $k$-chains are subvarieties of complex dimension
$k$ with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincar\'e residue on it. One can also define the corresponding analogues for the
intersection and linking numbers of complex submanifolds, which have the properties similar to those of the corresponding topological notions.
Keywords:Poincar\' e residue, holomorphic linking
Categories:14C10, 14F10, 58A14
100. CJM 2003 (vol 55 pp. 897)
Hypergeometric Abelian Varieties
In this paper, we construct abelian varieties associated to Gauss' and Appell--Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct them
were considered by several authors in settings ranging from monodromy groups (Deligne, Mostow), exceptional sets (Cohen, Wolfart, W\"ustholz), modular embeddings (Cohen, Wolfart) to CM-type (Cohen,
Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction.
Categories:11, 14
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Posts about Promoting OR on O.R. by the Beach
Due to a number of things that have been keeping me busy, including a 20-day fight against a kidney stone that is now finally over, I haven’t had much time to post. I have two ideas that I plan to
turn into posts soon, but in the meantime I’d like to suggest that everyone write a post about “santa claus” and “reindeer” (and properly tag it with those words). As the picture below indicates,
I’ve been getting a lot of hits lately (way above average) and, digging deeper, I see that my post entitled “How Should Santa Pair Up His Reindeer?” has had 2,209 views this past week. Yay for
As a matter of fact, I suggest that everyone write a post about each special date of the year (Easter, Fourth of July, Valentine’s Day, Summer Camp, etc.). Those will keep bringing you recurring
visits, without any extra effort, year after year. Of course not all (in fact, most) of those visits will be here for O.R. But that’s the point! If we want to spread the word about O.R. we’ve got to
take people by surprise. “I was just looking for a cute reindeer picture and this guy blew my mind by showing me that math and reindeer have something to do with each other! How cool is that!”
BTW, I just realized I do not have an Easter post. I need to take care of this…
Did You See Any OR During Apple’s iPhone 5 Announcement? I did!
On September 12, Apple finally announced its much-awaited iPhone 5. I didn’t have time to watch the keynote speech, but I watched the shorter 7-minute video that’s posted on Apple’s web site
featuring Jony Ive, Apple’s Senior Vice President, Design. In that video, at around the 5-minute, 26-second mark, something they said caught my attention: the way they put parts together during the
assembly process. I encourage you to watch that part of the video before reading on.
Jony Ive says:
Never before, have we built a product with this extraordinary level of fit and finish. We’ve developed manufacturing processes that are our most complex and ambitious.
And on Apple’s web site, they say this:
During manufacturing, each iPhone 5 aluminum housing is photographed by two high-powered 29MP cameras. A machine then examines the images and compares them against 725 unique inlays to find the
most precise match for every single iPhone.
So let’s see if I understood this correctly. In a typical manufacturing operation, the multiple parts that get put together to create a product are put together without much fuss. A machine makes
part A, another machine makes part B, and perhaps a robotic arm or a third machine takes any one of the many part A’s that are coming down a conveyor belt and attaches it to any one of the many part
B’s that are coming down another conveyor belt. What Apple did was to improve on the “any one” choice. I don’t know if Apple pioneered this idea, I’d say probably not, but this is the first time I
hear about something like this. If you’ve seen this before, let me know in the comments.
Before OR comes into play, Computer Science does its job in the form of computer vision / image processing algorithms. The photographs of the parts are analyzed and (I’m guessing) a fitness score is
calculated for every possible matching pair of parts A (the housing) and B (the inlay). What happens next? How do they pick the winning match? Here are some possibilities:
1. Each part A is matched with the part B, among the 725 candidates, that produces the best matching score.
2. A 725 by 725 matrix of fitness scores is created between 725 parts of type A and 725 parts of type B, and the best 725 matches are chosen so as to maximize the overall fitness score (i.e. the sum
of the fitness scores of all the chosen matches).
3. Proceed as in the previous case, but pick the 725 matches that maximize the minimum fitness score. That is, we worry about the worst case and don’t let the worst match be too bad when compared to
the best match.
After these 725 pairs are put together, new sets of parts A and B come down the conveyor belt and the matching process is repeated. Possibility number 1 is the fastest (e.g. do a binary search, or
build a priority queue), but not necessarily the best because every now and then a bad match will have to be made. Possibilities 2 (an assignment problem) and 3 (assignment problem with a max-min
objective) are better, in my opinion, with the third one being my favorite. They are, however, more time consuming than possibility 1. Jony Ive says the choice is made “instantaneously”, which
doesn’t preclude something fancier than possibility 1 from being used given the assignment problems are pretty small.
The result? In the words of Jony Ive:
The variances from product to product, we now measure in microns.
It is well-known that OR plays a very important role in manufacturing (facility layout, machine/job scheduling, etc.) but it’s not every day that people stop to think about what happens in a
manufacturing plant. This highly-popular announcement being watched by so many people around the world painted a very clear picture of the kinds of problems high-tech manufacturing facilities face. I
think it’s a great example of what OR can do, and how relevant it is to our companies and our lives.
Filed under Applications, iPhone, Motivation, Promoting OR
Fourth of July Logistics in Coral Gables: No OR, No Glory
After a six-year hiatus, the city of Coral Gables and the Biltmore Hotel decided to host the Fourth of July celebrations once again including, of course, a very nice fireworks display on the
Biltmore 18-hole golf course. My wife and I had watched the Independence Day fireworks at Biscayne bay and on the beach the past two years, so we thought this would be a nice change.
At the outset, the event seemed to be very well organized with buses and trolleys departing from four different places in the city to take people to the hotel, as shown in the map below.
So we parked our car at the Andalusia garage (Garage 4 on the map) and took the 6pm trolley. There was going to be a concert starting at 7pm, while the fireworks would go off at 9pm. We found a nice
spot to place our chairs and my wife’s camera tripod, so we sat down and relaxed. Numerous food trucks offered plenty of tasty choices, the concert was entertaining and, most importantly, we loved
the fireworks. All in all, we were very pleased with the whole thing. The problems started once the fireworks ended. Take a look at this map.
The red arrows indicate the flow of people trying to exit the golf course through a single narrow path (people coming from all directions were converging to that point). The yellow arrows start at
the trolley/bus stop (a single stop) and show the path the trolleys/buses would take to go back to the garages in the previous map.
By now you’ve already guessed what happened, but I’ll list some of the main problems: (1) large congestion to exit the golf course (bottleneck); (2) no organized lines were formed by the police;
people simply aggregated as a large mass at the bus stop (forget about FIFO); (3) tons of people actually drove their cars and parked not only in the parking lot depicted above, but also all around
the neighborhood surrounding the hotel. Therefore, the yellow bus path was full of pedestrians walking to their cars (or walking home) and the police did not allow trolleys/buses to come in or out
while there were pedestrians on the road (that is, forever); (4) we were given no indication as to which would be the destination of the incoming trolley/bus until they were parked at the stop (crowd
left in the dark = annoyed crowd).
After standing there for a while, my wife and I decided that it would be much faster and less stressful if we simply walked back to Garage 4 (a 1.3-mile, 25-minute walk). Yes, it was very hot that
day, and we had to carry some heavy chairs and equipment, but it was better than suffering through the chaos.
As an Operations Research person, I couldn’t stop thinking of all the bad decisions that were made by the organization of this event. I know they meant well, but everyone’s experience would have been
much more enjoyable if they did a few things differently. Some of my suggestions below require conveying information to the attendees ahead of time, but this could have been accomplished by handing
out flyers to people as they arrived. (Arrivals were not a problem because they were spread out over 3.5 hours, between 5 and 8:30pm.)
• Divide the crowd by telling people to exit the golf course through different paths depending on where they’re headed: those walking home exit through gate A, those walking to the Biltmore parking
lot exit through gate B, those wishing to catch a trolley/bus, exit through gate C, etc.
• Have multiple bus stops, reasonably away from each other.
• Have barricades set up so that: (1) lines are properly formed at the bus stops, (2) pedestrians do not walk on the road and impede the flow of trolleys/buses.
• Schedule the return trips of trolleys/buses in advance and tell people to come to the bus stop at their assigned time based on desired destination (à la Disney fast pass).
These are just some ideas that came to mind right away, but I bet more improvements are possible (what would you have done, dear reader?). Judging by how many of my friends who did not attend the
event already knew it had had a chaotic ending even before I told them, I’m sure the city received plenty of feedback. I expect next year’s event to run much more smoothly. However, just in case they
need a little extra help, I’d like to write a quick letter to the City of Coral Gables:
Dear City of Coral Gables:
I’m a professor at the University of Miami who specializes in using advanced analytical methods to help with decision making. If you need help with the logistics of your Fourth of July Fireworks
or any other city-sponsored activity, I’m available. Here’s my contact information.
Tallys Yunes.
To end this post on a happy note, here are some beautiful photos of the fireworks taken by my favorite photographer. Enjoy!
Filed under Applications, Holidays, Promoting OR
The “Real” Reason Bill Cook Created the TSP App
By now, most people are aware of the latest Internet meme Texts from Hillary which is, by the way, hilarious. You’re also probably aware that Bill Cook created an iPhone App that allows one to solve
traveling salesman problems (TSP) on a mobile phone! If you like optimization, you have to give this App a try; and make sure to check out the Traveling Salesman book too!
Inspired by Texts from Hillary I finally figured out the “real” reason why Bill Cook created the App. Here it is:
Filed under Applications, Books, iPhone, Meme, People, Promoting OR, Traveling Salesman Problem
Operations Research Memes
In the spirit of bringing awareness about O.R. to the masses, I created the memes below. Perhaps they’ll gain some traction or at least get a few people to wonder about what O.R. is. Who knows, they
may even motivate someone to Google the term! If you end up making your own O.R.-inspired meme, please send me a link to it via the comments section. To create mine, I used the quickmeme.com web
UPDATE: A few other OR bloggers and tweeps joined the meme crusade! Here are their creations (in chronological order of my becoming aware of them):
Laura McLay created the memes below:
Michael Trick created these:
Paul Rubin suggested the creation of this one:
Guido Diepen created this one:
Bill Cook made this cool TSP meme:
Paul Rubin made this one, western style:
My MBA student William Bucciero got inspired by these O.R. memes and made some of his own. He was kind enough to share them with me. I think he did a great job! Here they are:
Another one of my MBA students, Jason Siem, also joined the O.R. meme bandwagon. Here’s one of his (pretty funny and true):
Filed under Meme, Promoting OR
Choosing Summer Camps for Your Kids
Today I’m going to write about a decision that’s made by many American families each year: how to pick summer camps for our kids. There are several issues to take into account, such as cost,
benefit, hours, and kids’ preferences. I’ll introduce an optimization model for summer camp selection through a numerical example. The example portrays a large family, but the same ideas apply if a
few smaller families want to get together and solve this problem. This way they can take advantage of the discounts and take turns driving the kids around.
The Joneses have six kids: Amy, Beth, Cathy, David, Earl and Fred (yes, their first names are alphabetically sorted, matching their increasing order of age; Mr. and Mrs. Jones always knew they’d have
six kids and hence named their firstborn with an ‘F’ name). This year, they’ve narrowed down their list of potential summer camps to the following ten: Math, Chess, Nature, Crafts, Cooking,
Gymnastics, Soccer, Tennis, Diving, and Fishing. The Nature camp takes kids on a hike through the woods with the guidance of a biologist; they make frequent stops upon encountering specific plants
and animals, during which a mini science lecture is delivered (pretty cool!). The Cooking camp involves cooking chemistry instruction, à la Alton Brown (also pretty cool).
The following table contains some data related to each camp:
The Cost column indicates the cost per child. The Discount column indicates the percentage discount that each child enrolled after the first would receive on the cost of each camp. For instance, if
three children are enrolled in Math camp, the first would cost $1100, and the second and third would cost $770 each (30% less). The Hours column is self-explanatory and the last two columns indicate
whether or not that particular camp develops mental and physical abilities, respectively (a value of one = yes, zero=no).
The next table shows some of the child-specific requirements:
For example, the Joneses want Fred to attend at least 3 camps that develop mental abilities, and at least 1 camp that develops physical abilities. The last two columns in the above table indicate the
minimum and maximum number of camp hours for each child over the 9-week summer break.
The next thing parents need to take into account are their children’s preferences. So here they are:
The smaller the number in the above table, the more desirable a particular camp is. For example, Amy is a bit of a math nerd, and if we were to flip David’s preference scores for Math and Tennis, he
could be classified as a bit of a jock. Some conflicts exist, in the sense that not all camps are compatible with each other in terms of time schedules. In this particular case, let’s assume that no
child can attend both the Soccer and Tennis camps, or both the Nature and Soccer camps. Here’s how we are going to use this preference table to create a sense of fairness among the children: whenever
a child that prefers camp X to camp Y goes to camp Y and doesn’t go to camp X, nobody else gets to go to camp X either. For example, if Amy goes to Nature camp and isn’t sent to either Math or Chess
camp, none of her siblings are allowed to go to Math or Chess either. Conversely, if the Joneses decide to send Earl to Chess camp and Fred to Tennis camp, they must also send Earl to Tennis camp
(because Earl prefers Tennis to Chess, and “Fred is going! Why can’t I go too!”). Clearly, there are other ways to use/interpret this table, such as trying to send everyone to at least one of their
top N choices, but we won’t consider those alternatives here.
After taking all of the above issues and conditions into account, here’s a solution that satisfies all the requirements while resulting in the minimum cost of $22,180.00 (You guessed it…the Joneses
are probably *not* among the 99%):
Amy goes to Math, Crafts, Cooking, and Tennis; Beth goes to Math, Cooking, Tennis, and Fishing; Cathy goes to Math, Crafts, Tennis, and Fishing; David goes to Math, Cooking, and Fishing; Earl goes to
Math, Tennis, and Fishing; and Fred goes to Math, Crafts, Cooking, and Tennis. Mmm…interestingly, everyone goes to Math camp. I think the Joneses are on to something…
Depending on your own requirements, preferences, and costs your solution may differ, of course. But this should give you an idea of how this simple problem can easily become very complicated to
solve. No need to fear, though! Operations Research is here!
Food for Thought: Here’s an interesting question that helps illustrate how high-quality solutions can be counterintuitive: by looking at the preferences table, we see that everyone prefers Soccer to
Tennis. In addition, Soccer camp is less expensive than Tennis camp. So how come we send almost everyone to Tennis camp? Isn’t that strange? Let me know what you think in the comments below! That’s
one of the advantages of using an analytical approach to decision making: it helps us find solutions we wouldn’t even consider otherwise because they don’t seem to make sense (at least not at first).
If you’re curious about how I managed to find the optimal solution, read on!
Details of the Analysis:
To find the minimum-cost solution, we can create a mathematical representation of the problem, a.k.a. a model, and then solve this model with the help of a computer. Let’s see how.
The first obvious decision to make is who goes where. So let the binary variable $x_{ij}$ equal $1$ when child $i$ goes to camp $j$, and equal to $0$ otherwise. We’ll also need another binary
variable $y_j$ that is equal to $1$ when at least one child goes to camp $j$ and equal to $0$ when none of the children go to camp $j$. We are now ready to write our objective function and
constraints. I’ll refer to the problem data using the column headings of the tables above. The subscript $i$ will always refer to a child, and the subscript $j$ will always refer to a camp.
To minimize the total cost, we write the following objective function:
$\displaystyle \min \sum_i \sum_j (1-\mathrm{Discount}_j)\mathrm{Cost}_j x_{ij} + \sum_j \mathrm{Discount}_j \mathrm{Cost}_j y_j$
Note how we are using the $y_j$ variable to handle the discount for sending more than one child to camp $j$: we charge every child the discounted price in the double summation and add the discount
back in only once if $y_j=1$.
Now we have to deal with the four requirements: minimum and maximum hours, mental activity, and physical activity. For every child $i$, we have to write the following four constraints:
$\displaystyle \sum_j \mathrm{Hours}_j x_{ij} \geq \mathrm{MinTimeReq}_i$
$\displaystyle \sum_j \mathrm{Hours}_j x_{ij} \leq \mathrm{MaxTimeReq}_i$
$\displaystyle \sum_j \mathrm{IsMental}_j x_{ij} \geq \mathrm{MentalReq}_i$
$\displaystyle \sum_j \mathrm{IsPhysical}_j x_{ij} \geq \mathrm{PhysicalReq}_i$
Next, we enforce the preference rules. Let’s recall the example involving Earl and Fred: if Earl goes to Chess camp and someone else (it doesn’t matter who) goes to Tennis camp, then Earl has to go
to Tennis camp as well. Here’s what this constraint would look like:
$x_{\mathrm{Earl},\mathrm{Chess}} + y_{\mathrm{Tennis}} - x_{\mathrm{Earl},\mathrm{Tennis}} \leq 1$
Of course, we have to repeat this constraint for every child $i$ and every pair of camps $j_1$ and $j_2$ such that child $i$ prefers $j_1$ to $j_2$ in the following way:
$x_{ij_2} + y_{j_1} - x_{ij_1} \leq 1$
The camp compatibility constraints say that no child $i$ can attend both Soccer and Tennis, or both Nature and Soccer, therefore:
$x_{i,\mathrm{Soccer}} + x_{i,\mathrm{Tennis}} \leq 1$
$x_{i,\mathrm{Nature}} + x_{i,\mathrm{Soccer}} \leq 1$
Finally, we need to relate the $x_{ij}$ and $y_j$ variables by stating that unless $y_j=1$, no $x_{ij}$ can be equal to $1$. So we write the following constraint for all values of $i$ and $j$:
$x_{ij} \leq y_j$
And that’s the end of our model. Here’s a representation of this mathematical model in AMPL in case you want to play with it yourself. This is the model I used to obtain the numerical results
reported above. Enjoy!
Filed under Applications, INFORMS Monthly Blog Challenge, Integer Programming, Mathematical Programming, Modeling, Motivation, Promoting OR, Summer camp
A Conversation with Mr. X
This year I’ll be blogging during the INFORMS conference in Charlotte. My first post is already up, and it’s entitled A Conversation with Mr. X. Make sure to check it out!
I’m also looking forward to eating some delicious southern food (I hope they’ll have enough vegetarian options like they did in Austin). I’ll make sure to add barbecue sauce to my potato salad again;
an act that drives my wife completely nuts :-)
See you in Charlotte!
An O.R. Vocabulary Test for Non-Experts
I was talking to my wife the other day recalling how much fun she has while overhearing words from some of my research-related phone calls. We started to think about what comes to people’s minds when
they hear an OR-related term whose definition is not obvious to them. I’m not talking about obscure and technical mathematical terms such as a “contrapolymatroid“, but terms at which a non-expert
would actually be able to take an educated guess, such as “large-neighborhood search“. So I made a list of ten such terms and asked three friends (named A, B, and C) to define them to the best of
their ability. The only rule was that they had to do it on the spot, off the top of their heads; no Googling allowed. Because none of them have training in OR, some of the answers turned out to be
pretty interesting.
1. A global constraint.
A) All the stuff in the world that’s holding us back.
B) All the factors that prevent the open market from being truly open: laws, politics, foreign/domestic policy, national borders, etc.
C) Gravity.
2. Complementary slackness.
A) A dude who hangs out in a bar with no job, but complements the decor and vibe perfectly.
B) An equal and opposite reaction to whatever sectors are experiencing growth in the marketplace.
C) Time off from performing a task or responsibility granted by a superior or by oneself.
3. An odd cycle.
A) When you get your period unexpectedly; or that cycle on the washing machine that no one ever uses.
B) An economic cycle (quarter, fiscal year, etc.) which displays characteristics unlike the ones that preceded or succeeded it. In other words, in a sustained period of economic growth, it’s the
one segment that shows recession.
C) A phenomenon with awkward tendencies and characteristics that is repeated every so often.
4. A spanning tree.
A) A tree that creeps from your neighbor’s yard to yours. Usually makes a huge mess in yours.
B) Has something to do with Ethernet networks.
C) A rather large plant with either a long branch span or time span on planet Earth.
5. A cutting plane.
A) A wood working tool that both cuts and planes.
B) No freaking clue.
C) A slice that intersects a 3D object in order to provide another viewpoint.
6. A shadow price.
A) The hidden cost of owning things. Like the extra cost of owning and maintaining a house or a luxury car.
B) The true representative value of goods and services, compared to the value dictated by the supply/demand of the marketplace.
C) A value for an item or service which can be obtained but that requires the buyer to perform an extensive search.
7. A comb inequality.
A) When you have a better comb than I do.
B) huh?
C) Inadequacies that persist despite efforts to eliminate them.
8. Duality gap.
A) The gap between personalities in someone with multiple personality disorder.
B) Again no idea.
C) A two-faced abyss. In other words, an alternative that may seem unfortunate but that possesses some advantages.
9. A feasible region.
A) The region where it is possible for you to live given your income, wants, and available houses.
B) Sounds like agriculture. Sorry, I got nothing.
C) An area or scope which could be a viable alternative for several purposes.
10. The first-fail principle.
A) When you get to repeat a class the first time you fail it, if approved by your high-school principal.
B) The idea that early adopters in a new sector of the market who fail will provide secondary adopters guidance through their failure. Not literal guidance, of course, but the secondary adopters
will come into the marketplace and make decisions based upon others’ prior failures.
C) If you fail miserably the first time, don’t try again.
The first lesson I learned from this very non-scientific experiment is: if you’re at a party and somebody asks you what you do, you’re probably better off using an example. For instance: “Do you ever
wonder how hurricane paths are estimated? That’s what I do.” You’d of course replace “hurricane paths” with your favorite problem. If the example comes before “scary” words, I believe the end result
will be much better. If things go well, the ideal reaction by other person will be: “That’s so cool! What kind of training do you need to do that?” From that point on, you proceed to convince them
that math is cool.
Secondly, the amusing nature of the answers above notwithstanding, this experiment got me thinking about how to make OR more visible and accessible to the general audience. That’s one of the goals of
the INFORMS Public Information Committee (PIC), of which I’ve recently become a member. We already have some ideas and initiatives lined up, but I’m open to your comments and suggestions. Feel free
to send me your thoughts by e-mail or via the comments section below. By the way, if you feel like doing this experiment with your own friends, feel free to send me their answers and I’ll add them to
the bunch.
Filed under INFORMS Public Information Committee, People, Promoting OR
MLB Umpire Scheduling
There are two purposes to this post. First, I’d like to follow-up on Michael Trick’s post on the importance of teaching and its relationship with research. One of the points Mike makes is that your
next exciting research or consulting project may come from current or former students. Those of us who teach undergraduate and MBA classes have the opportunity to network with (future) managers and
practitioners who will eventually put their training to the test, producing actual answers to real-life problems. And if one of those problems requires more OR knowledge than what they had the
opportunity to learn in school, they might remember their friendly neighborhood OR professor. Another way research can come out of teaching is during hands-on projects. Back in 2006, Mike was in
charge of an elective OR-project class that allows MBA students to try their hands on a real-life problem; in that case umpire scheduling. To my delight, Mike invited me to be the TA for that course
and I gladly accepted. The rest is history.
The second purpose of this post is to help myself keep track of the recent news stories about our umpire scheduling paper. Thanks to an excellent job by the PR departments at the University of Miami
School of Business (thanks, Catharine!) and Michigan State University, the story has appeared in numerous outlets. As a matter of fact, I’m very excited to report that Scientific American had a
60-second science podcast about our work:
August 18, Scientific American: Researchers Tell Umpires Where to Go (PDF version)
Here are a few other news outlets that covered the story (I’m trying to keep this list up-to-date for my own sake). I’m also providing a link to a PDF version of each story in case the web pages are
taken offline:
April 2012, The Spring issue of Business Miami Magazine has an article about our work entitled Road Trip (PDF version).
October 19, WAMC Northeast Public Radio Academic Minute. I recorded a 1:45-minute explanation of the problem, approach, and results which aired as one of WAMC’s Academic Minutes on the same day of
the first game of the World Series. That was a lot of fun! Click on the link to listen. If the link doesn’t work, here’s the MP3 file.
September 6, Miami New Times: Tallys Yunes, UM Professor, Solves MLB’s Umpire Scheduling Dilemma (PDF version). This article also appeared in print, in the September 8-14 issue of Miami New Times.
Here’s a PDF scan of that.
August 3, PhysOrg: University of Miami Business Professor Helps Create a Successful Scheduling Method for Umpires in Major League Baseball (PDF version)
August 3, HPCwire: Business Prof Solves Traveling Umpire Problem for Major League Baseball (PDF version)
July 31, University of Miami School of Business: School’s Management Science Research Resolves Major League Baseball’s Umpire Scheduling Challenges (PDF version)
July 21, ScienceDaily: Scholar Helps Make Major League Baseball Umpire Schedule a Hit (PDF version)
July 21, ThePostGame: MLB Umpires Have a Turkish Secret Weapon (PDF version)
July 20, PhysOrg: Michigan State Scholar Helps Make MLB Umpire Schedule a Hit (PDF version)
July 20, Michigan State University News: Michigan State Scholar Helps Make MLB Umpire Schedule a Hit (PDF version)
I greatly enjoy the teaching side of my job because I believe it complements the research side quite well. I’m looking forward to bringing articles like the ones above to my classes in the Spring and
I’m sure they’ll be well received.
Further acknowledgments: thanks to those who also helped spread the word about the umpire scheduling problem on Twitter, especially Paul Rubin (@parubin), Aurélie Thiele (@aureliethiele), and
@INFORMS (is that you, Mary Leszczynski? :-).
Filed under Applications, Heuristics, Promoting OR, Research, Sports, Teaching, Traveling Umpire Problem
Rescue Mission, Part 3 of 3
Make sure to read part 1 and part 2 first!
The Edelman Award presentations are available here.
The INFORMS Public Information Committee (PIC) web page is here.
A long list of real-life applications of Analytics and O.R. is available here.
The topographic maps above are actual maps of Mars.
Filed under Applications, Heuristics, INFORMS Monthly Blog Challenge, Promoting OR | {"url":"http://orbythebeach.wordpress.com/category/promoting-or/","timestamp":"2014-04-18T08:01:53Z","content_type":null,"content_length":"102502","record_id":"<urn:uuid:b23f8e03-0eaf-4fec-9339-03fe854f1e9f>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00495-ip-10-147-4-33.ec2.internal.warc.gz"} |
Dividing Two Numbers with and without Units
Date: 09/15/2004 at 14:12:07
From: Kenneth
Subject: Division With and Without Units
It makes sense to divide a number or an amount, for example $500.00,
by another number having no units (dollars in this example), such as
10, as in $500.00/10 = $50.00, and it also makes sense to divide
$500.00 by $10.00, $500.00/$10.00 = 50, both amounts having units.
However, what does it indicate when the divisor has the units
(dollars) but not the dividend, as in 500 is divided by $10.00? Does
500/$10.00 make sense?
500/$10 as a ratio makes sense. This ratio could represent the number
of 500 items for the cost of $10.00. 500 divided by $10.00 equals
what? 50 or $50.00 will not provide a correct quotient. 500/$10.00
cannot equal either $50.00 or 50.
I believe that $500.00/$10.00 represent measurement division and that
$500.00/10 represent partition division. 500/$10.00 does not match
either one of these definitions for division.
Date: 09/15/2004 at 15:36:25
From: Doctor Rick
Subject: Re: Division With and Without Units
Hi, Kenneth.
Your observations are correct. It does sometimes make sense to divide
a dimensionless quantity by a quantity that has units. What are the
units of the quotient in this case? We see it sometimes in stores
(especially dollar stores): "4 for $1". We can also say "4 per
dollar". In science, such units are often written as "dollar^-1"
(that's dollar with an exponent of -1) because 1/x = x^(-1). I can't
say I've seen dollar^(-1) in use anywhere, but I'd understand it.
What we have here is essentially a rate--which can perhaps be called a
third kind of division according to your thinking. For instance, if a
car goes 100 miles in 2 hours, its average speed is 50 miles per hour,
or miles/hour. The result of dividing miles by hours is the new unit
miles/hour. Similarly, if a factory produces 4,000 widgets in an
8-hour shift, its average rate of production is 500 widgets/hour: when
we divide a number of widgets by a number of hours, we get
widgets/hour. We don't usually think of "widgets" as a unit, so we
can say instead that the production rate of widgets is 500 units/hour.
Here, "units" stands for a dimensionless quantity--a number of
widgets, or of anything else.
One example of a real-life unit of the form we're discussing is the
Hertz. This unit of frequency represents the rate at which something
happens, such as one period of an electromagnetic wave; thus the old
name was "cycles per second", but you would often see the units as
"sec^(-1)" (1/second, or "units per second") because cycles are not
really a unit.
- Doctor Rick, The Math Forum
Date: 09/15/2004 at 21:01:26
From: Kenneth
Subject: Thank you (Division With and Without Units)
Hello Doctor Rick:
I want to thank you for the reply and information. The Math Forum
provides a great service for those seeking help and assistance! | {"url":"http://mathforum.org/library/drmath/view/65582.html","timestamp":"2014-04-16T17:02:24Z","content_type":null,"content_length":"8153","record_id":"<urn:uuid:acf44d5c-ef44-4e78-99bf-8d83a1cc71a9>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00551-ip-10-147-4-33.ec2.internal.warc.gz"} |
How close is a 2D Gaussian to an Airy disk?
0) If by "how large" you mean the angular diameter of the Airy disk, looking at the
wikipedia article
, the intensity of the Airy disk [itex]I(\theta)[/itex] is related to the radius of the aperture [itex]a[/itex] ( in this case the radius of the satellite dish? ), using the same notation as the
wikipedia article. ( I'm using the term Airy "disk" and not "function" here, because it seems different from an
Airy function
1) If detector (LBNF) has full sensitivity to an angle larger than the angle of the dish edge, I think we can take ignore the angular sensitivity distribution of the detector, because then the
aperture is going to be the whole dish ( if the background behind the dish emits little radio signals of the relevent frequency) . If the sensitivity of the detector falls off with increasing angle
from the central axis with such a rapid rate that it drops off before it reaches the dish edge, then I guess we have to take that in to account.
2) This is certainly an issue.
3) We were detecting a geocentric satellite. So it did not move in relation to our position.
4) So I guess considering 1) and 2), it is possible that our plot can NOT be the cross section of an Airy disk.
When I told the professor in charge of the experiment that I fitted a Gaussian, he said "Why did you fit a Gaussian? It's supposed to be an Airy disk." The reason why did not fit one yet is because
the fitting software I am using right now does not have Bessel functions ( the wikipedia Airy disk equation is given in terms of Bessel functions ). I'm going to find another fitting software which
has Bessel functions and fit it. So this observation might be premature, but I can't see any secondary peaks.
The other two members in our group have the same problem. How to find a fitting software that can fit an Airy disk. But we think we found one (
), and we are going to try it out.
This link
lists the functions in the NASA IDL Astronomy User's Library for IDL (which is very widely used in the Astronomy community) , and even they seem to be fitting
Point Spread Functions
with Gaussians and not Airy disks ( functions FIND and GETPSF ). | {"url":"http://www.physicsforums.com/showthread.php?p=3806961","timestamp":"2014-04-16T10:23:26Z","content_type":null,"content_length":"37242","record_id":"<urn:uuid:8c607a11-c7ec-4115-9c80-0b08a3046994>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00121-ip-10-147-4-33.ec2.internal.warc.gz"} |
the first resource for mathematics
Optimization of transit route network, vehicle headways and timetables for large-scale transit networks.
(English) Zbl 1138.90350
Summary: This paper presents a metaheuristic method for optimizing transit networks, including route network design, vehicle headway, and timetable assignment. Given information on transit demand,
the street network of the transit service area, and total fleet size, the goal is to identify a transit network that minimizes a passenger cost function. Transit network optimization is a complex
combinatorial problem due to huge search spaces of route network, vehicle headways, and timetables. The methodology described in this paper includes a representation of transit network variable
search spaces (route network, headway, and timetable); a user cost function based on passenger random arrival times, route network, vehicle headways, and timetables; and a metaheuristic search scheme
that combines simulated annealing, tabu, and greedy search methods. This methodology has been tested with problems reported in the existing literature, and applied to a large-scale realistic network
optimization problem. The results show that the methodology is capable of producing improved solutions to large-scale transit network design problems in reasonable amounts of time and computing
90B10 Network models, deterministic (optimization)
90B06 Transportation, logistics
90C27 Combinatorial optimization | {"url":"http://zbmath.org/?q=an:1138.90350","timestamp":"2014-04-16T16:38:03Z","content_type":null,"content_length":"21495","record_id":"<urn:uuid:a9ac3259-3468-4186-8198-c15fb7c41792>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00560-ip-10-147-4-33.ec2.internal.warc.gz"} |
Comments on A Neighborhood of Infinity: The Essence of Quantum ComputingReally neat, thank you!
To those insisting on &qu...Really neat, thank you!
To those insisting on &qu...Neat post, thanksAnatoly,Suppose nature is keeps track of all of th...Hi Anatoly --- I'm not sure you can gain anything ...Can you explain what you mean further?Yes. Althoug...I agree that the existing quantum algorithms emplo...Could your entire post be summarized by the phrase...Anatoly,Some of what you're saying is metaphysical...The link to the "old post" just goes to your blog....I don't see the contradiction between what you sai...You might be interested in the work of Scott Aaron...There is no real contradiction. Bennett wrote:"......
tag:blogger.com,1999:blog-11295132.post1319866402436769779..comments2014-04-16T10:57:46.206-07:00Dan Piponihttps://plus.google.com/
107913314994758123748noreply@blogger.comBlogger13125tag:blogger.com,1999:blog-11295132.post-84228458139850568362012-11-18T10:33:32.707-08:002012-11-18T10:33:32.707-08:00Really neat, thank you!<br />
<br />To those insisting on "coherent superposition" as an accurate description of quantum computing, I must ask: Which supersedes, the noun or the adjective?<br /><br />Non-convexity, FTW!
Kim-Ee Yeohnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-12118572717104116652012-11-18T10:33:09.271-08:002012-11-18T10:33:09.271-08:00Really neat, thank you!<br /><br />To those
insisting on "coherent superposition" as an accurate description of quantum computing, I must ask: Which supersedes, the noun or the adjective?<br /><br />Non-convexity, FTW!Kim-Ee
Yeohnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-56503663040502349002008-01-08T00:31:00.000-08:002008-01-08T00:31:00.000-08:00Neat post,
thanksbshanksnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-48712775517598653892007-02-21T07:11:00.000-08:002007-02-21T07:11:00.000-08:00Anatoly,<BR/><BR/>Suppose nature is keeps track of
all of the superposed states simultaneously. Now suppose we have N+1 entangled qubits in a row. You claim that nature is tracking 2^(N+1) states. If this were the case then we could get nature to
store 2^N bits in N+1 qubits. Here's the scheme I propose: list your 2^N bits in a row. Label the first one 0, the next one 1 and so on. So we have 2^N pairs (i,f(i)) for i = 1,...,2^N-1. We can
encode the 'i' part as N qubits in the obvious way and we can store f(i) as the (N+1)-th qubit. Write this as |i,f(i)>. Now form the state psi that is sum_i |i;f(i)>. This encodes our entire set of 2
^N qubits in one state. For example, suppose we wish to store the 4 bits of data (0,1,1,1) then we could encode these as the state<BR/>|0,0;0>+|0,1;1>+|1,0;1>+|1,1;1><BR/>For example, the term |
0,1,1> tells us that the 01th (in binary) element in the sequence is 1. So if you claim that nature tracks N+1 superposed states then you must also agree that nature must be using at least 2^N bits
to store this state. So we have a pretty cool database here that allows 2^N bits to be stored in N+1 qubits.<BR/><BR/>But now consider how to use this database. For any i in the range 0,...,2^N-1 we
expect to be able to extract f(i). Let's reduce it to the simple case: extracting f(0) from psi. We'd like f(0) to be an observable, so we need a Hermitian operator, H, for which f(0) is the
eigenvalue. In other words, we want H such that H psi = f(0) psi.<BR/><BR/>Let's take the simplest case N=1. Then our state is |0;f(0)>+|1;f(1)>. We want<BR/>H(|0;0>+|1;0>) = 0<BR/>H(|0;1>+|1;0>) = |
0;1>+|1;0><BR/>H(|0;0>+|1;1>) = 0<BR/>H(|0;1>+|1;1>) = |0;1>+|1;1><BR/>From the first and third line we get<BR/>H(|1;0>-|1;1>) = 0<BR/>From the second and fourth lines we get<BR/>H(|1;0>-|1;1>) = |1;
0>-|1;1><BR/>So |1;0>=|1;1> which is clearly nonsense.<BR/>So there is no such Hermitian operator, f(0) isn't an observable and the database idea fails.<BR/><BR/>So either (1) nature isn't really
keeping track of our 2^N superposed states or (2) nature keeps track of our 2^N states but when we make an observation it throws that information away. I don't know about you, but entities that
magically disappear when you observe them give me a bad feeling. At the very least, I'd refrain from calling them 'real'.<BR/><BR/>This isn't really my original point. But it does show that a quantum
computer in a superposition of N states falls a long way short of being anything like N computers running in parallel.sigfpehttp://www.blogger.com/profile/
08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-15950857422376091712007-02-20T18:49:00.000-08:002007-02-20T18:49:00.000-08:00Hi Anatoly --- I'm not sure you can gain
anything though thinking of the "evolution" of a pure state through time.<BR/><BR/>Whether you take your measurement after one millimeter or one meter, the interference pattern will be precisely
analogous to the one as described by sigfpe's linear superposition. No extra "processing" is going on<BR/><BR/>Indeed it could well be wrong to assume that there is a background metric of time
ticking away alongside pure quantum states. This is probably an example of what Jaynes would call "the Mind Projection Fallicy"<BR/><BR/>Here's a nice bombastic Jaynes quote, just to get everyone
riled up :)<BR/><BR/>"A standard of logic that would be considered a psychiatric disorder in other fields, is the accepted norm in quantum theory"Alhttp://www.blogger.com/profile/
06113096184512859955noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-19288082941904051652007-02-20T17:15:00.000-08:002007-02-20T17:15:00.000-08:00<I>Can you explain what you mean further?</
I><BR/><BR/>Yes. Althought I spoke of states being "real" in a quantum superposition, there was less metaphysics in that then the words perhaps implied. The claim has physical rather than
metaphysical content, and it is the situation of a quantum interference that shows this, especially vividly in the case of a single particle.<BR/><BR/><A HREF="http://en.wikipedia.org/wiki/
Double-slit_experiment" REL="nofollow">In the double-slit experiment</A>, a narrow beam of light falls on a black surface with two very narrow vertical slits in it close to each other. After passing
through the slits the light falls onto a screen parallel to the surface. On the screen there is an interference pictureL periodic bands of darker and lighter regions, rather than two separate
identical lit regions<BR/>with darkness everywhere else, which is what you would expect if light behaved like a lot of very small particles (like bullets). In the classical model, the interference is
explained by looking at the light as a wave that is split into two waves by the slits and those interfere with each other.<BR/><BR/>When, however, the beam of light is more and more diluted until it
consists of a single photon, something happens that cannot be explained in the classical picture: if we repeatedly send in single photons and record the places they illuminate the screen, over time
exactly the same interference picture emerges as in the case of a beam of light. In the classical model, a photon, being a particle, must pass through one of the slits<BR/>if it passes at all; we can
set up the light emissions so that the probability of it passing through each of the slits is equal. A single particle cannot 'interfere' with itself in the classical model; if it passes through one
slit, it'll strike the screen somewhere in one band; if the other - in another; if the two bands overlap, it can even contribute to the them both at the same time (but this isn't interference as
understood in the classical physics model, where it is studied in terms of waves). <BR/><BR/>In reality, however, what we see is that a single photon appears to "know" about two slits simultaneously,
even though, being a particle, it cannot pass through both at the same time. If we place a detector on one of the slits to tell us when photons pass by, the interference picture on the screen
immediately changes to a bullet-like picture of two bands - that is, even photons that pass through the <I>other</I> slit, without the detector in it, are somehow affected by the fact that we are
observing a different slit. There is simply no way to explain coherently in the classical model how a single particle, passing through one slit at a time, can build up an interference picture. The
unescapable conclusion is that the theory must allow for some kind of interference of the particle with itself. That is the physical content, and the formulas give the exact numbers. The metaphysical
explanations (the so-called interpretations of quantum mechanics) differ: some say there is no photon as an entity, there is only the wavefunction that evolves, interferes and then collapses when we
make an observation; some say there are parallel universes in one of which the photon passed through one slit, in another through the other, and the only way those universes can interact is through
interference of those "copies" of the same photon with each other. And so on. But the essential difference with the classical probabilistic model of a single particle moving around is physical, not
metaphysical. <BR/><BR/>When you say that in the quantum case the formalism is the same as i n the classical probabilistic case, merely the "weightings" are complex and not just real positive, that
misses the picture of how probabilities are in fact used in both models. In the classical probabilistic case, the probabilities of differnet outcomes are "distributed" at the moment of probabilistic
choice, even if we don't know those distributions yet. The photon passes through one slit with probability 50% and the other with probability 50%, and at that moment the probabilities are final.
Sure, we can construct our final result sets in an overlapping manner so that sometimes either possibility contributes to a particular result, and its probability is thereby increased. But that's
just an example of additivity of probabilities. In the quantum case, the probabilities only 'get assigned' at the moment an observation happens - the photon strikes the screen. You are correct in
saying that a complicated picture of interference is possible at that time because the amplitudes are complex and not just real positive; but not less important is the fact that the probabilities are
calculated from those complex amplitudes only at the moment of observation. If the amplitudes were complex, but the probabilities were calculated at the slits, interference wouldn't happen. In fact,
this is precisely what happens when we put a detector on one of the slits (that detects a photon but still lets it pass through). In the classical probabilistic case, placing a detector that tells us
which slit the photon went through but otherwise leaves the photon be cannot change the picture on the screen, quite obviously. <BR/><BR/>In the double slit experiment, in the fraction of time the
photon flies from the surface with the slits to the screen is time in which the photon exists in a superposition of two states: one of having passed through the first slit, the other - through the
second. Because we know that the final calculation of probabilities, a.k.a. the collapse of the wavefunction, happens only at the moment of observation on the screen, we are forced to conclude that
each of those states "evolves" independently and simultaneously with the other. In the two states the photon has different position and vector of movement, and each state changes according to the
physical laws (here a simple movement in a straight line with the speed of light towards the screen) simultaneously with the other, at the same time also interfering with the other according to
particular formulas. Another way to phrase it is to say that the universe is forced to keep track of each state's information (position and vector of movement) and process that information in
parallel. In a system with N photons arranged carefully into a suitable superposition, the universe will be forced to keep track of information about an exponential number of states, simultaneously,
until at some point we observe and the wavefunction collapses. By carefully guiding the states through some physical changes (that apply to them simultaneously in parallel) that affect interfernce
between them in a direction suitable to us, we increase the chances of an interesting, to us, outcome when we collapse the wavefunction later by observing. Thus Shor's algorithm etc.Anatolyhttp://
www.lovestwell.orgnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-62710930326927704902007-02-20T07:30:00.000-08:002007-02-20T07:30:00.000-08:00I agree that the existing quantum algorithms
employ superposition in a clever way to enhance/decrease the amplitudes for the wanted/unwanted states respectively. For that we need probability amplitudes. However, the point is that to the extent
that quantum computers will eventually be built, they will rely on these superpositions actually existing and evolving during execution. As far as I understand classical computation, such
superpositions do not exist classically (with neither real nor complex number coefficients). I mean they do not exist as real physical states even though we could perhaps formally write down linear
superposition of states. I guess you see this as soon as you try to run (simulate really) a probabilistic or non-determinsitic computer (or program) - in the general case you get exponential slowdown
just as you get when you simulate a quantum computer classically. You simply have to go down all the paths in the computational tree (in a breadth-first search and/or using many parallell processes).
Hope I have not missunderstood your post!Anders Bengtssonhttp://www.blogger.com/profile/
14139508499015816145noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-30960304253286251982007-02-19T20:45:00.000-08:002007-02-19T20:45:00.000-08:00Could your entire post be summarized by the
phrase, "Bells's Theorem" ? <BR/><BR/>In other words, doesn't Bell's Theorem provide the underpinning of the difference between probabilistic and quantum computing, by showing that quantum mechanics
cannot be simulated by any <BR/>"hidden variable" theory ?Sureshhttp://www.blogger.com/profile/
15898357513326041822noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-29254895809465617962007-02-19T18:40:00.000-08:002007-02-19T18:40:00.000-08:00Anatoly,<BR/><BR/>Some of what you're
saying is metaphysical speculation, especially your claim that "only one of these states is actually real". I tend to ignore metaphysical speculation and concentrate on what we can actually measure
in experiments.<BR/><BR/>> You can call that state of affairs a superposition of states if you wish<BR/><BR/>I do. It seems like the same sort of thing to me, and the mathematics is very similar.<BR
/><BR/>> but the fact is, only one of these states is actually real, and the non-real ones have no way of influencing the real one. No interference can happen.<BR/><BR/>Interference happens all the
time in classical probabilistic systems - interference is nothing but the linear combination of vectors. I don't see why you think there's some magic whereby classical systems have one "real" state
and quantum systems somehow have all states real. What's different about quantum mechanics is that the weightings in the linear combinations aren't all directed along the positive real axis. I don't
see how this makes one state "real". My branching diagram was a perfectly good example of interference in a classical system. The only difference from a quantum system is that in a classical system
the interference is always 'constructive' because of the positive weights.<BR/><BR/>Can you explain what you mean further?sigfpehttp://www.blogger.com/profile/
08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-2982728865431125182007-02-19T18:11:00.000-08:002007-02-19T18:11:00.000-08:00The link to the "old post" just goes to your
blog.Dylannoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-91055503306801156112007-02-19T18:05:00.000-08:002007-02-19T18:05:00.000-08:00I don't see the contradiction between what you said
and the "usual" claims about why quantum computing's speedup works.<BR/><BR/>You are correct to put the emphasis on "we can paradoxically make something less likely by adding a new way to achieve
it". But the <I>reason</I> we are able to do it, in terms of quantum mechanics, is the fact that the system exists in a superposition of states which are interfering with each other. Quantum
interference is the reason that we can paradoxically etc.<BR/><BR/>The crucial difference between a probabilistic classical computer and a quantum computer is not merely the fact that in the latter
the amplitudes are complex numbers. In the classical case, you know that the computer exists in some particular state, you just don't know which one it is. You can call that state of affairs a
superposition of states if you wish, but the fact is, only one of these states is actually real, and the non-real ones have no way of influencing the real one. No interference can happen. In the
quantum case, saying that the computer exists in a superposition of states is not merely saying that it'll be discovered to be in one of them with such and such probabilities when we look under the
hood later. It means more than that: it means that all of those states are simultaneously "real"; until the wave collapses they are all "there" co-existing simultaneously in the superposition and <I>
that</I> is why they can influence each other and create quantum interference.<BR/><BR/>In the classical case, "nature" needs to keep track of one state only, it's just that we don't know yet which
one it is. In the quantum state, "nature" is forced to keep track of exponentially many different states at once - otherwise interference wouldn't happen. As we subject the system to various
operators, all the states undergo the operations in parallel; and while we can't reach that huge freebie exponential store of information directly, we can manipulate interference into revealing some
of it to us, hence Shor's algorithm etc.<BR/><BR/>So when Bennett et al. say that the power of quantum computers lies in following a superposition through a computation, they refer to a "real"
superposition rather than a "formal" one in the classical probabilistic case, and the difference between them is manifested precisely in the "real" superposition's ability to create interference
between its states - the "paradoxical" property that you stress. So you're saying more or less the same thing, in different words, it seems to me.Anatolyhttp://
www.lovestwell.orgnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-10346684872970467242007-02-19T15:09:00.000-08:002007-02-19T15:09:00.000-08:00You might be interested in the work of <A
HREF="http://scottaaronson.com/blog" REL="nofollow">Scott Aaronsen</A>. He discusses the limitations und nature of Quantum Computing very well. Don't miss his his Quantum Computing <A HREF="http://
www.scottaaronson.com/democritus/" REL="nofollow">lectures</A> <BR/>I especially like his connection of the intractibility of NP-complete problems as conjectured <A HREF="http://scottaaronson.com/
talks/colloq.ppt" REL="nofollow">physical principle</A>!Harald
Holtmannnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-470241716340709282007-02-19T15:04:00.000-08:002007-02-19T15:04:00.000-08:00There is no real contradiction. Bennett wrote:<BR/><BR/>
"... follow a <I>coherent</I> superposition of computation paths."<BR/><BR/>Coherence means phase correct superposition, or: |a + b|^2.<BR/><BR/>Probability superposition is superposition of the
absolute square of the wave functions: |a|^2 + |b|^2. <BR/><BR/>Quantum computing of course only works with the first kind of superposition. This is well known and of course also known by Bennet et.
al.Karsten Wagnerhttp://www.blogger.com/profile/09652404623625038743noreply@blogger.com | {"url":"http://blog.sigfpe.com/feeds/1319866402436769779/comments/default","timestamp":"2014-04-17T04:49:38Z","content_type":null,"content_length":"39041","record_id":"<urn:uuid:eb48f561-3695-4d4c-a147-e90f01b85b42>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00264-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Differential geometry
1. The problem statement, all variables and given/known data
On R^3 with the usual coordinates (x,y,z), consider the pairs of vector fields X,Y given below. For each pair, determine if there is a function f:R^3-->R with non-vanishing derivative df satisfying
Xf=Yf=0, and either find such a function or prove that there is none.
(a) X=(e^x)d/dx - ((e^x)z + 2y)d/dz, Y=(e^x)d/dy - (2y)d/dz
(b) X=(e^x)d/dx - ((e^x)z + 2x)d/dz, Y=(e^x)d/dy - (2y)d/dz
2. Relevant equations
Could you help me start this problem.
3. The attempt at a solution
Sorry I don't know how to start this problem. | {"url":"http://www.physicsforums.com/showpost.php?p=1381856&postcount=1","timestamp":"2014-04-17T18:37:55Z","content_type":null,"content_length":"8968","record_id":"<urn:uuid:95894f87-eb9f-47b1-b9ca-118f5acf816b>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00465-ip-10-147-4-33.ec2.internal.warc.gz"} |
math module for Decimals
Gabriel Genellina gagsl-py2 at yahoo.com.ar
Sun Dec 28 03:33:36 CET 2008
En Sat, 27 Dec 2008 22:02:51 -0200, <jerry.carl.mi at gmail.com> escribió:
> I have been looking for a Python module with math functions that would
> both eat and spit Decimals. The standard math module eats Decimals
> allright but spits floats... herefore exp(sin(Decimal())) produces exp
> () of a float :-(
Which math functions? ln, log10, exp, sqrt already exist as methods of
Decimal instances. At the end of the Decimal docs there are a few
examples, including computing sin and cos (but apparently they naïvely use
a McLaurin series like you noticed in other module).
There is a way to convert a float into a Decimal object with no loss in
precision, see the FAQ (using float.as_integer_ratio(), I think such
function was implemented for exactly this use case)
So you might convert to float, operate, and go back -- if the precision of
"double" operands is enough for you.
Gabriel Genellina
More information about the Python-list mailing list | {"url":"https://mail.python.org/pipermail/python-list/2008-December/482029.html","timestamp":"2014-04-21T00:05:03Z","content_type":null,"content_length":"3441","record_id":"<urn:uuid:d4386a4f-ab23-42af-92be-91b1ab724e76>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00435-ip-10-147-4-33.ec2.internal.warc.gz"} |
Tuesday, July 11
The George Pólya Prize:
Polynomials in Discrete Mathematics
3:00 PM-3:30 PM
Room: Rio Mar 5
Chair: Jeffry N. Kahn, Rutgers University, USA
The speaker will describe a general algebraic technique that can be called "Combinatorial Nullstellensatz" and discuss some of its applications in Combinatorial Number Theory, in Graph Theory, and in
Combinatorics. These applications include results in additive number theory and in the study of graph coloring problems. The method is based on some elementary properties of polynomials. A closely
related technique is useful in settling a problem of Shannon in Information Theory.
Noga Alon
Department of Mathematics
Tel-Aviv University, Israel | {"url":"http://www.siam.org/meetings/an00/polya.htm","timestamp":"2014-04-20T19:57:46Z","content_type":null,"content_length":"2214","record_id":"<urn:uuid:0a97dd7e-170f-43ad-abbe-785e39dcc5d0>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00438-ip-10-147-4-33.ec2.internal.warc.gz"} |
In mathematics, a logarithm of a given number to a given base is the power to which you need to raise the base in order to get the number. For example, the logarithm of 1000 to the common base 10 is
3, because 10 raised to a power of 3 is 1000. More generally, for a number x, a base b and a power n,
$\mbox{if}~~ b^n = x, ~~\mbox{then}~~ \log_b (x) = n. \,$
This result is extended to powers of b that are not integers by requiring the logarithm with base b of a number x is the exponent y that satisfies x = b^y. It is written log[b](x) or, if the base is
implicit, as log(x).
In other words, for a number x, a base b and an exponent y,
$\mbox{if}~~ b^y = x, ~~\mbox{then}~~ \log_b (x) = y. \,$
A major utility of logarithms is that they reduce multiplication to addition, by the formula:
$\log (x \times y) = \log x + \log y \,$
That is, the log of a product of two numbers is the sum of the logarithms of those numbers. Consequently, to multiply two numbers, find their logarithms, with the help of tables of logarithms, add
them, and then find the inverse logarithm of the sum, using the same tables.
Properties of the logarithm
When x and b are restricted to positive real numbers, log[b](x) is a unique real number. The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are
defined for real numbers and for complex numbers.
The major property of logarithms is that they map multiplication to addition. This ability stems from the following identity:
$b^x \times b^y = b^{x+y} \ ,$
which by taking logarithms becomes
$\log_b \left(b^x \times b^y \right) = \log_b \left( b^{x+y} \right)$$\ = x + y = \log_b \left(b^x \right) + \log_b \left(b^y \right). \$
A related property is reduction of exponentiation to multiplication. Using the identity:
$c = b^{\log_b (c )} \ ,$
it follows that c to the power p (exponentiation) is:
$c^p = \left(b^{\log_b (c )}\right)^p = b^{p \log_b (c )} \ ,$
or, taking logarithms:
$\log_b \left(c^p \right) = p \log_b (c ) \ .$
In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = b^
Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. Logarithms make lengthy numerical
operations easier to perform. The whole process is made easy by using tables of logarithms, or a slide rule, antiquated now that calculators are available. Although the above practical advantages are
not important for numerical work today, they are used in graphical analysis (see Bode plot).
The logarithm as a function
Though logarithms have been traditionally thought of as arithmetic sequences of numbers corresponding to geometric sequences of other (positive real) numbers, as in the 1797 Britannica definition,
they are also the result of applying an analytic function. The function can therefore be meaningfully extended to complex numbers.
The function log[b](x) depends on both b and x, but the term logarithm function (or logarithmic function) in standard usage refers to a function of the form log[b](x) in which the base b is fixed and
so the only argument is x. Thus there is one logarithm function for each value of the base b (which must be positive and must differ from 1). Viewed in this way, the base-b logarithm function is the
inverse function of the exponential function b^x. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.
The base can also be a complex number; the evaluation of the log is just slightly more complicated in this case. See imaginary base.
Logarithm of a complex number
When the base b is real and z is a complex number, say z = x + i y, the logarithm of z is found easily by putting z in polar form that is, z = (x^2 + y^2)^1/2 exp (i tan^−1 (y / x) ). If the base of
the logarithm is chosen as e , that is, using log[e] (denoted by ln and called the natural logarithm), the logarithm becomes:
$\ln(z) = \ln \left[ \left( x^2 + y^2 \right) ^{1/2} e^{i \tan^{-1}y/x}\right]$
$= \ln \left[ \left( x^2 + y^2 \right) ^{1/2}\right] + \ln \left[ e^{i \tan^{-1}y/x}\right]$
$=\frac {1}{2} \ln \left( x^2 + y^2 \right) + i \tan^{-1}\left( \frac {y} {x} \right) \ .$
This evaluation uses the properties of all logarithms (see above), regardless of choice of base: log[b] (c d ) = log[b] (c ) + log[b] (d ) and its generalization to arbitrary products log[b] b^z = z.
Because the inverse tangent is a multiple valued function of its argument, the logarithm of a complex number is not unique either. See article on complex logarithm.
Group theory
From the pure mathematical perspective, the identity
$\log(cd) = \log(c) + \log(d) \,$
is fundamental in two senses. First, the remaining three arithmetic properties can be derived from it. Furthermore, it expresses an isomorphism between the multiplicative group of the positive real
numbers and the additive group of all the reals.
Logarithmic functions are the only continuous isomorphisms from the multiplicative group of positive real numbers to the additive group of real numbers.
The most widely used bases for logarithms are 10, the mathematical constant e ≈ 2.71828... and 2. When "log" is written without a base (b missing from log[b]), the intent can usually be determined
from context:
• natural logarithm (log[ e], ln, log, or Ln) in mathematical analysis, statistics, economics and some engineering fields. The reasons to consider e the natural base for logarithms, though perhaps
not obvious, are numerous and compelling.
• common logarithm (log[10] or simply log; sometimes lg) in various engineering fields, especially for power levels and power ratios, such as acoustical sound pressure, and in logarithm tables to
be used to simplify hand calculations
• binary logarithm (log[2]; sometimes lg, lb, or ld) in information theory and computer-related fields
• indefinite logarithm when the base is irrelevant, e.g. in complexity theory when describing the asymptotic behaviour of algorithms in big O notation.
To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.
Other notations
The notation "ln(x)" invariably means log[e](x), i.e., the natural logarithm of x, but the implied base for "log(x)" varies by discipline:
• Mathematicians generally understand both "ln(x)" and "log(x)" to mean log[e](x) and write "log[10](x)" when the base-10 logarithm of x is intended. Calculus textbooks will occasionally write "lg(
x)" to represent "log[10](x)".
• Many engineers, biologists, astronomers, and some others write only "ln(x)" or "log[e](x)" when they mean the natural logarithm of x, and take "log(x)" to mean log[10](x) or, sometimes in the
context of computing, log[2](x).
• On most calculators, the LOG button is log[10](x) and LN is log[e](x).
• In most commonly used computer programming languages, including C, C++, Java, Fortran, Ruby, and BASIC, the "log" function returns the natural logarithm. The base-10 function, if it is available,
is generally "log10."
• Some people use Log(x) (capital L) to mean log[10](x), and use log(x) with a lowercase l to mean log[e](x).
• The notation Log(x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.
• A notation frequently used in some European countries is the notation ^blog(x) instead of log[b](x).
This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/x, and having a simple definition), while the base 10
logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus natural logarithms were only extensively used in fields like
calculus while decimal logarithms were widely used elsewhere.
As recently as 1984, Paul Halmos in his "automathography" I Want to Be a Mathematician heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used.
The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley. As of 2005, many mathematicians have adopted the "ln" notation, but most use "log".
In computer science, the base 2 logarithm is sometimes written as lg(x), as suggested by Edward Reingold and popularized by Donald Knuth. However, lg(x) is also sometimes used for the common log, and
lb(x) for the binary log. In Russian literature, the notation lg(x) is also generally used for the base 10 logarithm. In German, lg(x) also denotes the base 10 logarithm, while sometimes ld(x) or lb(
x) is used for the base 2 logarithm.
The clear advice of the United States Department of Commerce National Institute of Standards and Technology is to follow the ISO standard Mathematical signs and symbols for use in physical sciences
and technology, ISO 31-11:1992, which suggests these notations:
• The notation "ln(x)" means log[e](x);
• The notation "lg(x)" means log[10](x);
• The notation "lb(x)" means log[2](x).
As the difference between logarithms to different bases is one of scale, it is possible to consider all logarithm functions to be the same, merely giving the answer in different units, such as dB,
neper, bits or bytes; see the section Science and engineering below.
Change of base
While there are several useful identities, the most important for calculator use lets one find logarithms with bases other than those built into the calculator (usually log[e] and log[10]). To find a
logarithm with base b, using any other base k:
$\log_b(x) = \frac{\log_k(x)}{\log_k(b)}.$
Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other. So to calculate the log with base 2 of the number 16 with a calculator:
$\log_2(16) = \frac{\log(16)}{\log(2)}.$
Uses of logarithms
Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used in the solution of integrals. The logarithm is one of three closely
related functions. In the equation b^n = x, b can be determined with radicals, n with logarithms, and x with exponentials. See logarithmic identities for several rules governing the logarithm
Various quantities in science are expressed as logarithms of other quantities; see logarithmic scale for an explanation and a more complete list.
• In chemistry, the negative of the base-10 logarithm of the concentration of hydronium ions (H[3]O^+, the form H^+ takes in water) is the measure known as pH. The concentration of hydronium ions
in neutral water is 10^−7 mol/L at 25 °C, hence a pH of 7.
• The bel (symbol B) is a unit of measure which is the base-10 logarithm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics, and acoustics. The
Bel is named after telecommunications pioneer Alexander Graham Bell. The decibel (dB), equal to 0.1 bel, is more commonly used. The neper is a similar unit which uses the natural logarithm of a
• The Richter scale measures earthquake intensity on a base-10 logarithmic scale.
• In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −1 B.
• In astronomy, the apparent magnitude measures the brightness of stars logarithmically, since the eye also responds logarithmically to brightness.
• In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation.
• In computer science, logarithms often appear in bounds for computational complexity. For example, to sort N items using comparison can require time proportional to the product N × log N.
Similarly, base-2 logarithms are used to express the amount of storage space or memory required for a binary representation of a number—with k bits (each a 0 or a 1) one can represent 2^k
distinct values, so any natural number N can be represented in no more than (log[2] N) + 1 bits.
• Similarly, in information theory logarithms are used as a measure of quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the
amount of information conveyed by any one such message is quantified as log[2] N bits.
• In geometry the logarithm is used to form the metric for the half-plane model of hyperbolic geometry.
• Many types of engineering and scientific data are typically graphed on log-log or semilog axes, in order to most clearly show the form of the data.
• In inferential statistics, the logarithm of the data in a dataset can be used for parametric statistical testing if the original data does not meet the assumption of normality.
• Musical intervals are measured logarithmically as semitones. The interval between two notes in semitones is the base-2^1/12 logarithm of the frequency ratio (or equivalently, 12 times the base-2
logarithm). Fractional semitones are used for non-equal temperaments. Especially to measure deviations from the equal tempered scale, intervals are also expressed in cents (hundredths of an
equally-tempered semitone). The interval between two notes in cents is the base-2^1/1200 logarithm of the frequency ratio (or 1200 times the base-2 logarithm). In MIDI, notes are numbered on the
semitone scale (logarithmic absolute nominal pitch with middle C at 60). For microtuning to other tuning systems, a logarithmic scale is defined filling in the ranges between the semitones of the
equal tempered scale in a compatible way. This scale corresponds to the note numbers for whole semitones. (see microtuning in MIDI).
Exponential functions
One way of defining the exponential function e^x, also written as exp(x), is as the inverse of the natural logarithm. It is positive for every real argument x.
The operation of "raising b to a power p" for positive arguments b and all real exponents p is defined by
$b^p = \left( e^{\ln b} \right) ^p = e^{p \ln b }.\,$
The antilogarithm function is another name for the inverse of the logarithmic function. It is written antilog[b](n) and means the same as b^n.
Easier computations
Logarithms can be used to replace difficult operations on numbers by easier operations on their logs (in any base), as the following table summarizes. In the table, upper-case variables represent
logs of corresponding lower-case variables:
Operation with numbers Operation with exponents Logarithmic identity
$\!\, c d$ $\!\, C + D$ $\!\, \log(c d) = \log(c) + \log(d)$
$\!\, c / d$ $\!\, C - D$ $\!\, \log(c / d) = \log(c) - \log(d)$
$\!\, c ^ d$ $\!\, Cd$ $\!\, \log(c ^ d) = d \log(c)$
$\!\, \sqrt[d]{c}$ $\!\, C / d$ $\!\, \log(\sqrt[d]{c}) = \frac{\log(c)}{d}$
These arithmetic properties of logarithms make such calculations much faster. The use of logarithms was an essential skill until electronic computers and calculators became available. Indeed the
discovery of logarithms, just before Newton's era, had an impact in the scientific world that can be compared with that of the advent of computers in the 20th century because it made feasible many
calculations that had previously been too laborious.
As an example, to approximate the product of two numbers one can look up their logarithms in a table, add them, and, using the table again, proceed from that sum to its antilogarithm, which is the
desired product. The precision of the approximation can be increased by interpolating between table entries. For manual calculations that demand any appreciable precision, this process, requiring
three lookups and a sum, is much faster than performing the multiplication. To achieve seven decimal places of accuracy requires a table that fills a single large volume; a table for nine-decimal
accuracy occupies a few shelves. Similarly, to approximate a power c^d one can look up log c in the table, look up the log of that, and add to it the log of d; roots can be approximated in much the
same way.
One key application of these techniques was celestial navigation. Once the invention of the chronometer made possible the accurate measurement of longitude at sea, mariners had everything necessary
to reduce their navigational computations to mere additions. A five-digit table of logarithms and a table of the logarithms of trigonometric functions sufficed for most purposes, and those tables
could fit in a small book. Another critical application with even broader impact was the slide rule, an essential calculating tool for engineers. Many of the powerful capabilities of the slide rule
derive from a clever but simple design that relies on the arithmetic properties of logarithms. The slide rule allows computation much faster still than the techniques based on tables, but provides
much less precision.
The natural logarithm of a positive number x can be defined as
$\ln (x) \equiv \int_{1}^{x} \frac{dt}{t}.$
The derivative of the natural logarithm function is
$\frac{d}{dx} \ln(x) = \frac{1}{x}.$
By applying the change-of-base rule, the derivative for other bases is
$\frac{d}{dx} \log_b(x) = \frac{d}{dx} \frac {\ln(x)}{\ln(b)} = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}.$
The antiderivative of the natural logarithm ln(x) is
$\int \ln(x) \,dx = x \ln(x) - x + C,$
and so the antiderivative of the logarithm for other bases is
$\int \log_b(x) \,dx = x \log_b(x) - \frac{x}{\ln(b)} + C = x \log_b \left(\frac{x}{e}\right) + C.$
See also: Table of limits, list of integrals of logarithmic functions.
Series for calculating the natural logarithm
There are several series for calculating natural logarithms. The simplest, though inefficient, is:
$\ln (z) = \sum_{n=1}^\infty \frac{-{(-1)}^n}{n} (z-1)^n$ when $|z-1|<1 \!.$
To derive this series, start with ($|x|<1 \!.$)
$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots.$
Integrate both sides to obtain
$-\ln(1-x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots$
$\ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \cdots.$
Letting $z = 1-x \!$ and thus $x = -(z-1) \!$, we get
$\ln z = (z-1) - \frac{(z-1)^2}{2} + \frac{(z-1)^3}{3} - \frac{(z-1)^4}{4} + \cdots$
A more efficient series is
$\ln (z) = 2 \sum_{n=0}^\infty \frac{1}{2n+1} {\left ( \frac{z-1}{z+1} \right ) }^{2n+1}$
for z with positive real part.
To derive this series, we begin by substituting −x for x and get
$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots.$
Subtracting, we get
$\ln \frac{1+x}{1-x} = \ln(1+x) - \ln(1-x) = 2x + 2\frac{x^3}{3} + 2\frac{x^5}{5} + \cdots.$
Letting $z = \frac{1+x}{1-x} \!$ and thus $x = \frac{z-1}{z+1} \!$, we get
$\ln z = 2 \left ( \frac{z-1}{z+1} + \frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3 + \frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5 + \cdots \right ).$
For example, applying this series to
$z = \frac{11}{9},$
we get
$\frac{z-1}{z+1} = \frac{\frac{11}{9} - 1}{\frac{11}{9} + 1} = \frac{1}{10},$
and thus
$\ln (1.2222222\dots) = \frac{2}{10} \left (1 + \frac{1}{3\cdot 100} + \frac{1}{5 \cdot 10000} + \frac{1}{7 \cdot 1000000} + \cdots \right )$
$= 0.2 \cdot (1.0000000\dots + 0.0033333\dots + 0.0000200\dots + 0.0000001\dots + \cdots)$
$= 0.2 \cdot 1.0033535\dots = 0.2006707\dots$
where we factored 1/10 out of the sum in the first line.
For any other base b, we use
$\log_b (x) = \frac{\ln (x)}{\ln (b)}.$
Most computer languages use log(x) for the natural logarithm, while the common log is typically denoted log10(x). The argument and return values are typically a floating point (or double precision)
data type.
As the argument is floating point, it can be useful to consider the following:
A floating point value x is represented by a mantissa m and exponent n to form
$x = m2^n.\,$
$\ln(x) = \ln(m) + n\ln(2).\,$
Thus, instead of computing ln(x) we compute ln(m) for some m such that 1 ≤ m < 2. Having m in this range means that the value $u = \frac{m - 1}{m+1}$ is always in the range $0 \le u < \frac13$. Some
machines use the mantissa in the range $0.5 \le m < 1$ and in that case the value for u will be in the range $-\frac13 < u \le 0$ In either case, the series is even easier to compute.
To compute a base 2 logarithm on a number between 1 and 2 in an alternate way, square it repeatedly. Every time it goes over 2, divide it by 2 and write a "1" bit, else just write a "0" bit. This is
because squaring doubles the logarithm of a number.
The integer part of the logarithm to base 2 of an unsigned integer is given by the position of the left-most bit, and can be computed in O(n) steps using the following algorithm:
int log2(int x){
int r = 0;
while( (x >> r) != 0){
return r-1; // returns -1 for x==0, floor(log2(x)) otherwise
However, it can also be computed in O(log n) steps by trying to shift by powers of 2 and checking that the result stays nonzero: for example, first >>16, then >>8, ... (Each step reveals one bit of
the result)
The ordinary logarithm of positive reals generalizes to negative and complex arguments, though it is a multivalued function that needs a branch cut terminating at the branch point at 0 to make an
ordinary function or principal branch. The logarithm (to base e) of a complex number z is the complex number ln(|z|) + i arg(z), where |z| is the modulus of z, arg(z) is the argument, and i is the
imaginary unit; see complex logarithm for details.
The discrete logarithm is a related notion in the theory of finite groups. It involves solving the equation b^n = x, where b and x are elements of the group, and n is an integer specifying a power in
the group operation. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in
public key cryptography.
The logarithm of a matrix is the inverse of the matrix exponential.
It is possible to take the logarithm of a quaternions and octonions.
A double logarithm, ln(ln(x)), is the inverse function of the double exponential function. A super-logarithm or hyper-4-logarithm is the inverse function of tetration. The super-logarithm of x grows
even more slowly than the double logarithm for large x.
For each positive b not equal to 1, the function log[b ] (x) is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. They are
the only such isomorphisms that are continuous. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.
The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston in Scotland ( Joost Bürgi
independently discovered logarithms; however, he did not publish his discovery until four years after Napier). This method contributed to the advance of science, and especially of astronomy, by
making some difficult calculations possible. Prior to the advent of calculators and computers, it was used constantly in surveying, navigation, and other branches of practical mathematics. It
supplanted the more involved method of prosthaphaeresis, which relied on trigonometric identities as a quick method of computing products. Besides the utility of the logarithm concept in computation,
the natural logarithm presented a solution to the problem of quadrature of a hyperbolic sector at the hand of Gregoire de Saint-Vincent in 1647.
At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm to mean a number that indicates a ratio: λόγος ( logos) meaning
proportion, and ἀριθμός (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers they represent, so that an arithmetic series of
logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of
tables until they fell into disuse.
Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it is useful to make
the ratio r in the geometric series close to 1. Napier chose r = 1 - 10^−7 = 0.999999 (Bürgi chose r = 1 + 10^−4 = 1.0001). Napier's original logarithms did not have log 1 = 0 but rather log 10^7 =
0. Thus if N is a number and L is its logarithm as calculated by Napier, N = 10^7(1 − 10^−7)^L. Since (1 − 10^−7)^10^7 is approximately 1/e, this makes L/10^7 approximately equal to log[1/e] N/10^7.
Tables of logarithms
Prior to the advent of computers and calculators, using logarithms meant using tables of logarithms, which had to be created manually. Base-10 logarithms are useful in computations when electronic
means are not available. See common logarithm for details, including the use of characteristics and mantissas of common (i.e., base-10) logarithms.
In 1617, Henry Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. This he followed, in 1624,
by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which
the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 was filled up by Adriaan Vlacq, a Dutch mathematician; but in his table, which appeared in 1628, the
logarithms were given to only ten places of decimals.
Vlacq's table was later found to contain 603 errors, but "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than
2,100,000 printed figures are liable to error." An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega.
François Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of
interpolation, which were greatest in the early part of the table; and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by
Mr. Sang in 1871, whose table contained the seven-place logarithms of all numbers below 200,000.
Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions.
Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony, by an original computation, under the auspices of the
French republican government of the 1700s. This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twenty-four
places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris. It was begun in 1792; and "the whole of the calculations, which to secure greater accuracy were
performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years." Cubic interpolation could be used to find the logarithm of any number
to a similar accuracy. | {"url":"http://www.pustakalaya.org/wiki/wp/l/Logarithm.htm","timestamp":"2014-04-24T22:47:32Z","content_type":null,"content_length":"56211","record_id":"<urn:uuid:14fc9cb6-c526-4586-8c3a-a548d2678f18>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00195-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: trigo
Replies: 1 Last Post: Nov 26, 2004 12:36 PM
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vinc trigo
Posted: Nov 26, 2004 1:09 AM
Posts: 1
Registered: 1/25/05 How do you do, i turn some circle as of hours.
Of how to write [sin](x+pi/3)+[sin](x-5pi/6) under the form
Thank you of the sub for all forms from help.
submissions: post to k12.ed.math or e-mail to k12math@k12groups.org
private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org
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Fonts for Mozilla's MathML engine
Quick Instructions
Try this test page to quickly determine whether you need to install additional fonts and to follow the instructions below. You may also want to check out the Mozilla MathML torture test to test your
installation against concrete examples.
Mathematical formulas make use of various symbols represented by specific Unicode characters. Mozilla can display any of these symbols provided suitable Unicode fonts are installed. Furthermore, in
accordance with the W3C CSS2 recommendation on fonts, authors can specify an ordered list of particular fonts which they prefer (using the font-family property of CSS), with the assurance that
Mozilla's font engine will hunt for alternate fonts whenever their specified fonts are not found on a particular user's system. If no appropriate font is ultimately found for a given character,
Mozilla will instead display a box containing the hexadecimal representation of the Unicode code point for the character.
Some mathematical operators are also drawn by using glyphs for larger variants of the character (e.g. summation symbol) or by assembling glyphs for a few pieces of the character (e.g braces). These
variant or part glyphs do not in general have Unicode assignments, and so font-specific support is required for some characters. Mozilla has general support for the common delimiters and arrows that
have the necessary Unicode assignments. For other characters, Mozilla has font-specific support. In general, if the necessary glyphs are not available on the system then Mozilla tries to stretch
symbols using scale transforms, but that may result in a less pleasant rendering.
To see MathML as intended, you need sufficient font support, which may mean installing some fonts. Note that some Web sites may provide the required mathematical fonts as Web fonts. In that case, you
will see the correct rendering on these pages if you use Gecko 15.0 (Firefox 15.0 / Thunderbird 15.0 / SeaMonkey 2.12) or greater.
Recommended Fonts
The STIX Fonts Project has been developing a comprehensive set of fonts that cover all the symbols in MathML and also contains glyphs to stretch mathematical operators. They are available under the
SIL Open Font License and should be installed for best MathML rendering. Note that currently, only the fonts in /Fonts/STIX-General/ are used for stretching mathematical operators with STIX fonts.
Asana Math is also freely available under the SIL Open Font License and has good support for mathematical characters. Starting in Gecko 7.0 (Firefox 7.0 / Thunderbird 7.0 / SeaMonkey 2.4), Mozilla
can also use the glyphs contained in that font for stretchy operators. Starting in Gecko 13.0 (Firefox 13.0 / Thunderbird 13.0 / SeaMonkey 2.10), MathJax fonts are used as the default fonts for
mathematical text and symbol stretching. These fonts are freely available under the Apache License. They are close to LaTeX rendering and should look more familiar to scientists. MathJax fonts also
improves support for mathvariant attribute (e.g. to write double-struck, script or fraktur text).
All the fonts mentioned above may already be installed by default on your system or can easily be installed via your software manager. For convenience we provide a zip archive of MathML fonts. The
fonts may be installed by extracting the font files (which have the .otf extension) from the zip archive, then following instructions for Microsoft Windows, or copying the files to a ~/Library/Fonts
folder on Mac OS X, or to a ~/.fonts/ directory (which may need to be created) on Unix/Linux systems. Alternatively, follow dafont's "How to install a font". These fonts are OpenType PostScript
fonts, which are supported by most modern operating systems. However some operating systems may not support these fonts. On such systems, it will be necessary to convert the fonts to a supported
format. For example Asana Math is available in TrueType format and there is an unofficial TrueType conversion for STIX fonts.
The following fonts are not necessary when the above fonts are installed, but are worth noting because of their Unicode support for many mathematical characters.
Experimental Fonts with a MATH table
We are currently adding support for fonts with a MATH table. MathJax and STIX-General fonts are expected to be deprecated in future versions. If you have the right builds and wish to test the new
fonts then here is a list of such fonts. Note that this list is not exhaustive, any font with a MATH table will be supported and you can even create your own fonts.
Future work (please help!)
Reset old preferences
If users have previously changed the "font.mathfont-family" preference for a previous version of Mozilla, then it is best to reset this to the default value. To do this, enter the URL "about:config",
"Filter" for "mathfont", and "Reset" to the default value through the context menu on the preference.
Document Tags and Contributors | {"url":"https://developer.mozilla.org/en-US/docs/Mozilla/MathML_Project/fonts","timestamp":"2014-04-18T22:58:13Z","content_type":null,"content_length":"38022","record_id":"<urn:uuid:45dfab12-bdce-441a-b012-9d429b384e8f>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00423-ip-10-147-4-33.ec2.internal.warc.gz"} |
Olney, MD SAT Math Tutor
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The Hodge decomposition theorem for general three-dimensional vector fields, without cuts
Ghiloni, Riccardo (2010) The Hodge decomposition theorem for general three-dimensional vector fields, without cuts. UNSPECIFIED. (Unpublished)
Download (400Kb) | Preview
Preprint. The Hodge decomposition of the space $L^2(\Omega)^3$ of square sommable vector fields defined on a domain $\Omega$ of the ordinary three–dimensional space is a tool used in a wide theoretic
and applied literature concerning Electromagnetism and Fluid dynamics. In order to obtain an effective Hodge decomposition, suitable also for numerical computations, the classical strategy, which
goes back to Helmholtz and which is used nowadays as well, is to consider only bounded domains with sufficiently regular boundary, which contain connected and pairwise disjoint surfaces, called cut
surfaces of $\Omega$, whose complement $C$ in $\Omega$ is simply connected. The advantage of this situation is that the union of such cut surfaces does not have singularities, every curl–free vector
field in $\Omega$ admits potential in $C$ and hence it is possible to apply standard variational methods to obtain the Hodge decomposition via scalar potential formulations. Recently, R. Benedetti,
R. Frigerio and the author gave an exhaustive topological description of bounded domains, called Helmholtz domains, which admit such cut surfaces. We proved that the Helmholtz domains are truly
special three–dimensional domains. In fact, their topology is elementary. An evidence of this claim is given, for example, by the fact that the complement of any non-trivial thickened knot in a box
domain is not a Helmholtz domain. In this way, we realize that the range of application of the classical strategy is quite limited. In this paper, we announce several results concerning an effective
Hodge decomposition of $L^2(\Omega)^3$ valid for every bounded domain $\Omega$ with Lipschitz–regular boundary, where $\Omega$ describes a spatial medium formed by a possible inhomogeneous and/or
anisotropic material.
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Sample Lesson Plan
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Complementary Angle Theorem
August 6th 2006, 09:29 AM #1
Junior Member
Aug 2006
Complementary Angle Theorem
There is an effect of the complementary angle theorom that I don't understand. For example:
sin squared(40) + sin squared(50) = sin squared(40) + cos squared(40) = 1
This previous statement's purpose was to show how sin(50) = cos(40), but what got me was the answer being equal to 1. Doing some experimenting, to see if this is a rule:
sin squared(10) + sin squared(80) = sin squared(10) + cos squared(10) = 1
and so on:
cos squared(10) + cos squared(80) = 1
sin squared(30) + sin squared(60) = 1
Why is it, when these functions are squared, that the answer = 1?
There is an effect of the complementary angle theorom that I don't understand. For example:
sin squared(40) + sin squared(50) = sin squared(40) + cos squared(40) = 1
This previous statement's purpose was to show how sin(50) = cos(40), but what got me was the answer being equal to 1. Doing some experimenting, to see if this is a rule:
sin squared(10) + sin squared(80) = sin squared(10) + cos squared(10) = 1
and so on:
cos squared(10) + cos squared(80) = 1
sin squared(30) + sin squared(60) = 1
Why is it, when these functions are squared, that the answer = 1?
Pythagoras' theorem in terms of trig functions is:
$<br /> \sin^2(x)+\cos^2(x)=1<br />$
Okay, I see it now. Thanks.
There is an effect of the complementary angle theorom that I don't understand. For example:
sin squared(40) + sin squared(50) = sin squared(40) + cos squared(40) = 1
I've attached a diagram to show you what I've calculated.
If the sum of the 2 angles is 90° you are dealing with a right triangle.
$\alpha+\beta=90^\circ \Longrightarrow \beta=90^\circ-\alpha$
The red line corresponds to $\sin(\alpha)$ and the blue line corrsponds to $\sin(\beta)=sin(90^\circ-\alpha)$
According to the diagram is: $(\mbox{red})^2+(\mbox{blue})^2=1$
Plug in the sine values and you'll get the property you have detected.
The blue line corresponds to $\cos(\alpha)$. So now you can easily complete the Pythagoran rule of the Sine and Cosine function.
earboth, thanks. What program did you use to draw that?
earboth, thanks. What program did you use to draw that?
have a look here: http://www.dynageo.de
August 6th 2006, 10:03 AM #2
Grand Panjandrum
Nov 2005
August 6th 2006, 10:11 AM #3
Junior Member
Aug 2006
August 6th 2006, 10:23 AM #4
August 6th 2006, 10:58 AM #5
Junior Member
Aug 2006
August 7th 2006, 01:38 AM #6 | {"url":"http://mathhelpforum.com/trigonometry/4687-complementary-angle-theorem.html","timestamp":"2014-04-18T15:02:59Z","content_type":null,"content_length":"48774","record_id":"<urn:uuid:898d2686-90f0-4289-ae9d-270e5dfb0f4f>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00347-ip-10-147-4-33.ec2.internal.warc.gz"} |
Spring Quarter I will teach Math 143 (Calculus 3) and Math 341 (Number Theory). Math 143 is the third quarter of our Calculus sequence for Engineering and Science Majors. We will study infinite
series, parametric equations, polar coordinates, and vectors and vector-valued functions. Math 341 is an introduction to Elementary Number Theory. We will discuss properties of numbers, Euclid’s
Algorithm, greatest common divisors, diophantine equations, prime numbers, congruences, number theoretic functions, the quadratic reciprocity laws, and primitive roots and indices.
Go here for homework assignments in Math 143.
Go here for homework assignments in Math 341. | {"url":"http://www.calpoly.edu/~dhartig/Classes.html","timestamp":"2014-04-19T09:25:08Z","content_type":null,"content_length":"4368","record_id":"<urn:uuid:bdc7ad16-e681-4e0e-b16f-a6fbdb5b4209>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00166-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Modeling high-z output in verilog-A
1. 18th March 2006, 07:37 #1
Junior Member level 2
Join Date
Nov 2005
3 / 3
high-z output
How do i model high-z output in verilog-A ?
Currently I'm modeling it to output = 0, but it is causing conflict when i cascode them at top level (error = output connected to another output).
Currently the method i'm using is
vout = 0;
V(OUT) <+ vout ;
With what value should i replace the '0' with in order to get a high-z output?
Thanks a million for ur help !!!!!
2. 18th March 2006, 13:26 #2
Advanced Member level 3
Join Date
Jun 2003
106 / 106
verilog high z output
Zout = 1e+15;
V(OUT) <+ I(OUT) * Zout;
[size=2]Best Regards,[/size]
3. 18th March 2006, 13:49 #3
Junior Member level 2
Join Date
Nov 2005
3 / 3
switch modeling verilog-a
Thanks for ur advice. But I'm still having the same problem. It seems that the formula u gave still yeilds a voltage value. How do i model it so that the circuit sees a disconnected output
(i.e. high-z state) ?
4. 19th March 2006, 02:31 #4
Advanced Member level 3
Join Date
Jun 2003
106 / 106
switch modeling veriloga
A disconnected output also has a voltage value. To see the high-z effect, you should connect another low-z driving source to this node.
[size=2]Best Regards,[/size]
5. 19th March 2006, 09:19 #5
Advanced Member level 4
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211 / 211
verilog high z
as far as i understand u need to model open circuit , so i think u should make input current equal zero and no conditions on voltage "u confused me a little bit, as what i understand is that
V(out)=0 is short cicuit not open circuit ,so plz post the solution that u will find".
btw i think if u just make the output node as a voltage node and not electrical it wont draw any current "not sure".
6. 19th March 2006, 15:42 #6
Junior Member level 2
Join Date
Nov 2005
3 / 3
Re: Modeling high-z output in verilog-A
A disconnected output also has a voltage value. To see the high-z effect, you should connect another low-z driving source to this node.
Can you advise me how do i code this in VerilogA? Thanks :)
Added after 2 hours 34 minutes:
then again, is there any method to disable output in verilogA?
7. 20th March 2006, 05:21 #7
Advanced Member level 3
Join Date
Jun 2003
106 / 106
Re: Modeling high-z output in verilog-A
A disconnected output also has a voltage value. To see the high-z effect, you should connect another low-z driving source to this node.
Can you advise me how do i code this in VerilogA? Thanks :)
Add another module with low Zout. Short the ouput of these two module. If one (and only one in this case) output is in High-Z mode, the voltage at the common output node will be set by the
another module. If both output are in low-z mode, the voltage at the common output node will be different with any output voltage when their are no short-connected.
then again, is there any method to disable output in verilogA?
What do you expect by disabling an output terminal. If you put a mechanical switch before the output terminal, when you open the switch, what VOLTAGE VALUE do you expect at the output? It can
be any value! For an ideal switch (the off state resistance is infinite) and a non-ideal output terminal (it has parasitic capacitance), the output voltage when the output terminal is
disabled (by open the switch) will permantly remain to the value before it is disabled.
[size=2]Best Regards,[/size] | {"url":"http://www.edaboard.com/thread59504.html","timestamp":"2014-04-21T10:22:12Z","content_type":null,"content_length":"78931","record_id":"<urn:uuid:d485cf0b-4e1a-4911-90ba-a67cc0eaac86>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00334-ip-10-147-4-33.ec2.internal.warc.gz"} |
Missing answers - sanity check needed
August 10th 2008, 07:51 AM #1
Jul 2008
Missing answers - sanity check needed
I have a worksheet with answers missing. The questions are simple uniform motion in a circle ones but I would like to run through two of them here and just have you more experienced folks confirm
my answers, if you wouldn't mind.
Question 1
Find the speed of a particle of dust, 5cm from the center of a disc which is revolving at 45rpm. Also find the acceleration of the particle. Answers to 2 decimal places.
$<br /> \omega = 45\text{ rpm}<br /> = \frac{45(360)}{60} \text{ deg s}^{-1}<br /> = \frac{45(360)(\pi)}{60(180)} \text{ rad s}^{-1}<br /> = \frac{3}{2}\pi \text{ rad s}^{-1}<br />$
For the speed of the particle
$<br /> v = r\omega<br /> = 0.05\bigg(\frac{3}{2}\pi\bigg)<br /> = { \color{blue}0.34 \text{ ms}^{-1} } \text{ (2 d.p.)}<br />$
For the acceleration of the particle
$<br /> a = r\omega^2<br /> = 0.05\bigg(\frac{3}{2}\pi\bigg)^2<br /> = {\color{blue}1.11 \text{ ms}^{-2} } \text{ (2 d.p.)}<br />$
Question 2
A particle P of mass 20 grams is connected by a taut inextensible string of length 2m to a fixed point O of a smooth horizontal table on which the particle rests. The string will break if the
tension exceeds 7.84N. Calculate the greatest possible speed at which P can describe a circle of 2m about O.
$\text{mass, } m = 0.02 \text{ kg}$
For a maximum string tension of 7.84N
$F = ma$
$7.84 = 0.02a$
$a = 392 \text{ ms}^{-2}$
For the speed
$v^2 = ra^2$
$v = \sqrt{ 2(392)^2 }$
$v = {\color{blue}554.37 \text{ ms}^{-1} } \text{ (2 d.p.)}$
I have a worksheet with answers missing. The questions are simple uniform motion in a circle ones but I would like to run through two of them here and just have you more experienced folks confirm
my answers, if you wouldn't mind.
Question 1
Find the speed of a particle of dust, 5cm from the center of a disc which is revolving at 45rpm. Also find the acceleration of the particle. Answers to 2 decimal places.
$<br /> \omega = 45\text{ rpm}<br /> = \frac{45(360)}{60} \text{ deg s}^{-1}<br /> = \frac{45(360)(\pi)}{60(180)} \text{ rad s}^{-1}<br /> = \frac{3}{2}\pi \text{ rad s}^{-1}<br />$
For the speed of the particle
$<br /> v = r\omega<br /> = 0.05\bigg(\frac{3}{2}\pi\bigg)<br /> = { \color{blue}0.34 \text{ ms}^{-1} } \text{ (2 d.p.)}<br />$
For the acceleration of the particle
$<br /> a = r\omega^2<br /> = 0.05\bigg(\frac{3}{2}\pi\bigg)^2<br /> = {\color{blue}1.11 \text{ ms}^{-2} } \text{ (2 d.p.)}<br />$
Question 2
A particle P of mass 20 grams is connected by a taut inextensible string of length 2m to a fixed point O of a smooth horizontal table on which the particle rests. The string will break if the
tension exceeds 7.84N. Calculate the greatest possible speed at which P can describe a circle of 2m about O.
$\text{mass, } m = 0.02 \text{ kg}$
For a maximum string tension of 7.84N
$F = ma$
$7.84 = 0.02a$
$a = 392 \text{ ms}^{-2}$
For the speed
$v^2 = ra^2$
$v = \sqrt{ 2(392)^2 }$
$v = {\color{blue}554.37 \text{ ms}^{-1} } \text{ (2 d.p.)}$
I didn't check the numbers, but the Physics is correct.
Just as a note for problem 2 I would specify that both the force and acceleration are "centripetal" for the sake of clarity.
Acting tension question
Thanks for taking a look Dan.
I have a diagram in my working which I didn't include in this thread which shows the tension acting along the string from particle P towards O. My question is, given Newton's Third Law (equal and
opposite forces), should tension act in equally along both $\overrightarrow{PO}$and $\overrightarrow{OP}$?
This should probably be obvious but it is something I've never been absolutely clear on and never thought to ask...
Last edited by Wonkihead; August 10th 2008 at 06:12 PM.
August 10th 2008, 03:31 PM #2
August 10th 2008, 05:49 PM #3
Jul 2008 | {"url":"http://mathhelpforum.com/advanced-applied-math/45669-missing-answers-sanity-check-needed.html","timestamp":"2014-04-20T07:00:24Z","content_type":null,"content_length":"45595","record_id":"<urn:uuid:5e40401a-4ce8-4250-b979-f45961e1f118>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00173-ip-10-147-4-33.ec2.internal.warc.gz"} |
User Justin Moore
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I'm a set theorist working at Cornell. Most of my work has been with forcing axioms and the combinatorics and Ramsey theory of the first uncountable cardinal. I am also interested in ways in which
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SparkNotes: SAT Physics: The Kinetic Theory of Gases & the Ideal Gas Law
12.1 Heat and Temperature 12.5 Key Formulas
12.2 The Kinetic Theory of Gases & the Ideal Gas Law 12.6 Practice Questions
12.3 The Laws of Thermodynamics 12.7 Explanations
12.4 Heat Engines
The Kinetic Theory of Gases & the Ideal Gas Law
We said earlier that temperature is a measure of the kinetic energy of the molecules in a material, but we didn’t elaborate on that remark. Because individual molecules are so small, and because
there are so many molecules in most substances, it would be impossible to study their behavior individually. However, if we know the basic rules that govern the behavior of individual molecules, we
can make statistical calculations that tell us roughly how a collection of millions of molecules would behave. This, essentially, is what thermal physics is: the study of the macroscopic effects of
the microscopic molecules that make up the world of everyday things.
The kinetic theory of gases makes the transition between the microscopic world of molecules and the macroscopic world of quantities like temperature and pressure. It starts out with a few basic
postulates regarding molecular behavior, and infers how this behavior manifests itself on a macroscopic level. One of the most important results of the kinetic theory is the derivation of the ideal
gas law, which not only is very useful and important, it’s also almost certain to be tested on SAT II Physics.
The Kinetic Theory of Gases
We can summarize the kinetic theory of gases with four basic postulates:
1. Gases are made up of molecules: We can treat molecules as point masses that are perfect spheres. Molecules in a gas are very far apart, so that the space between each individual molecule is many
orders of magnitude greater than the diameter of the molecule.
2. Molecules are in constant random motion: There is no general pattern governing either the magnitude or direction of the velocity of the molecules in a gas. At any given time, molecules are moving
in many different directions at many different speeds.
3. The movement of molecules is governed by Newton’s Laws: In accordance with Newton’s First Law, each molecule moves in a straight line at a steady velocity, not interacting with any of the other
molecules except in a collision. In a collision, molecules exert equal and opposite forces on one another.
4. Molecular collisions are perfectly elastic: Molecules do not lose any kinetic energy when they collide with one another.
The kinetic theory projects a picture of gases as tiny balls that bounce off one another whenever they come into contact. This is, of course, only an approximation, but it turns out to be a
remarkably accurate approximation for how gases behave in the real world.
These assumptions allow us to build definitions of temperature and pressure that are based on the mass movement of molecules.
The kinetic theory explains why temperature should be a measure of the average kinetic energy of molecules. According to the kinetic theory, any given molecule has a certain mass, m; a certain
velocity, v; and a kinetic energy of ^1/ [2 ]mv^2. As we said, molecules in any system move at a wide variety of different velocities, but the average of these velocities reflects the total amount of
energy in that system.
We know from experience that substances are solids at lower temperatures and liquids and gases at higher temperatures. This accords with our definition of temperature as average kinetic energy: since
the molecules in gases and liquids have more freedom of movement, they have a higher average velocity.
In physics, pressure, P, is the measure of the force exerted over a certain area. We generally say something exerts a lot of pressure on an object if it exerts a great amount of force on that object,
and if that force is exerted over a small area. Mathematically:
Pressure is measured in units of pascals (Pa), where 1 Pa = 1 N/m^2.
Pressure comes into play whenever force is exerted on a certain area, but it plays a particularly important role with regard to gases. The kinetic theory tells us that gas molecules obey Newton’s
Laws: they travel with a constant velocity until they collide, exerting a force on the object with which they collide. If we imagine gas molecules in a closed container, the molecules will collide
with the walls of the container with some frequency, each time exerting a small force on the walls of the container. The more frequently these molecules collide with the walls of the container, the
greater the net force and hence the greater the pressure they exert on the walls of the container.
Balloons provide an example of how pressure works. By forcing more and more air into an enclosed space, a great deal of pressure builds up inside the balloon. In the meantime, the rubber walls of the
balloon stretch out more and more, becoming increasingly weak. The balloon will pop when the force of pressure exerted on the rubber walls is greater than the walls can withstand.
The Ideal Gas Law
The ideal gas law relates temperature, volume, and pressure, so that we can calculate any one of these quantities in terms of the others. This law stands in relation to gases in the same way that
Newton’s Second Law stands in relation to dynamics: if you master this, you’ve mastered all the math you’re going to need to know. Ready for it? Here it is:
Effectively, this equation tells us that temperature, T, is directly proportional to volume, V, and pressure, P. In metric units, volume is measured in m^3, where 1m^3 = 10^6cm^2.
The n stands for the number of moles of gas molecules. One mole (mol) is just a big number—
The R in the law is a constant of proportionality called the universal gas constant, set at 8.31 J/mol · K. This constant effectively relates temperature to kinetic energy. If we think of RT as the
kinetic energy of an average molecule, then nRT is the total kinetic energy of all the gas molecules put together.
Deriving the Ideal Gas Law
Imagine a gas in a cylinder of base A, with one moving wall. The pressure of the gas exerts a force of F = PA on the moving wall of the cylinder. This force is sufficient to move the cylinder’s wall
back a distance L, meaning that the volume of the cylinder increases by AL. In terms of A, this equation reads A = L. If we now substitute in L for A in the equation F = PA, we get F = PL, or
If you recall in the chapter on work, energy, and power, we defined work as force multiplied by displacement. By pushing the movable wall of the container a distance L by exerting a force F, the gas
molecules have done an amount of work equal to FL, which in turn is equal to P
The work done by a gas signifies a change in energy: as the gas increases in energy, it does a certain amount of work on the cylinder. If a change in the value of PV signifies a change in energy,
then PV itself should signify the total energy of the gas. In other words, both PV and nRT are expressions for the total kinetic energy of the molecules of a gas.
Boyle’s Law and Charles’s Law
SAT II Physics will not expect you to plug a series of numbers into the ideal gas law equation. The value of n is usually constant, and the value of R is always constant. In most problems, either T,
P, or V will also be held constant, so that you will only need to consider how changes in one of those values affects another of those values. There are a couple of simplifications of the ideal gas
law that deal with just these situations.
Boyle’s Law
Boyle’s Law deals with gases at a constant temperature. It tells us that an increase in pressure is accompanied by a decrease in volume, and vice versa:
Charles’s Law
Charles’s Law deals with gases at a constant pressure. In such cases, volume and temperature are directly proportional:
Gases in a Closed Container
You may also encounter problems that deal with “gases in a closed container,” which is another way of saying that the volume remains constant. For such problems, pressure and temperature are directly
Example 1
A gas in a cylinder is kept at a constant temperature while a piston compresses it to half its original volume. What is the effect of this compression on the pressure the gas exerts on the walls of
the cylinder?
Questions like this come up all the time on SAT II Physics. Answering it is a simple matter of applying Boyle’s Law, or remembering that pressure and volume are inversely proportional in the ideal
gas law. If volume is halved, pressure is doubled.
Example 2
A gas in a closed container is heated from 0ºC to 273ºC. How does this affect the pressure of the gas on the walls of the container?
First, we have to remember that in the ideal gas law, temperature is measured in Kelvins. In those terms, the temperature goes from 273 K to 546 K; in other words, the temperature doubles. Because we
are dealing with a closed container, we know the volume remains constant. Because pressure and temperature are directly proportional, we know that if the temperature is doubled, then the pressure is
doubled as well. This is why it’s a really bad idea to heat an aerosol canister. | {"url":"http://www.sparknotes.com/testprep/books/sat2/physics/chapter12section2.rhtml","timestamp":"2014-04-18T00:35:39Z","content_type":null,"content_length":"60052","record_id":"<urn:uuid:4a05a3f4-55ca-49d6-983a-852bd9b21162>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00436-ip-10-147-4-33.ec2.internal.warc.gz"} |
Cobordism and Topological Field Theories Weeks 6 & 7
Posted by Alexander Hoffnung
Welcome back to the cobordism and TFT seminar at UCR! We are posting weeks 6 and 7 of the seminar simultaneously because it flows nicely and the new quarter will start before we know it. So we want
to be all caught up by the first week of the second quarter.
Week 6 starts back with the regular lecture series by Julie Bergner. These lectures are especially nice as they begin to introduce $n$-dimensional topological field theories in Lecture 6 and the
higher category theory needed to formulate these concepts in Lecture 7.
You can click here for the previous week’s lecture and keep reading below the fold for a link to the notes for Weeks 6 and 7…
In Lecture 6, Julie reminds us of some of facts about $2$-dimensional topological field theories stated in Chris Carlson’s lecture in Week 4. She comes to a sketch of a $2$-extended TFT by chopping
up manifolds into pieces and considering the problems encountered when one considers TFT’s of dimension higher than 2.
Following the realization that we no longer have a symmetric monoidal category of cobordisms, in Lecture 7, Julie then introduces $2$-categories of cobordisms. She then hints at the coming need for $
(\infty,2)$-categories and gives a definition for extended $\mathcal{C}$-valued TFT’s.
Here are the lecture notes for Weeks 6 and 7.
Next week, John Baez will fill in some details regarding symmetric monoidal weak $2$-categories…
Posted at December 21, 2009 4:15 PM UTC
Re: Cobordism and Topological Field Theories Weeks 6 & 7
It is a very minor point, but I suggest that the arrows labeled “left” be put into mboxes.
Posted by: Scott Carter on December 22, 2009 9:22 PM | Permalink | Reply to this
Re: Cobordism and Topological Field Theories Weeks 6 & 7
Hi Scott,
There is certainly no point too minor for me, except that I do not see the arrows you are referring to. I am pretty sure I will regret this comment later when I am being less dense, but can you point
the way for me?
Posted by: Alex Hoffnung on December 22, 2009 10:14 PM | Permalink | Reply to this
Re: Cobordism and Topological Field Theories Weeks 6 & 7
Alex, Yes I see why. I was speaking about lecture 4 about a 3rd of the way down. Beta has a subscript that is not boxed. The subscript is “left.” In definition 4, it looks to me as if “Map” is dollar
M a p dollar, rather than “dollar left bracket backslash mbox left bracket backslash it Map right bracket right bracket […] dollar.” At the bottom of that page, there is a subscript on hom that looks
kind of weird.
Right before lecture 7, morphisms has an “n” at the end.
This is not to say I am reading carefully — quite the contrary. Just these things stick out to me.
Posted by: Scott Carter on December 23, 2009 12:17 AM | Permalink | Reply to this
Re: Cobordism and Topological Field Theories Weeks 6 & 7
Thanks Scott.
I will go in and clean up shortly. I also noticed that extra `n’ while looking for the arrows.
Posted by: Alex Hoffnung on December 23, 2009 12:27 AM | Permalink | Reply to this
Re: Cobordism and Topological Field Theories Weeks 6 & 7
Actually, I have never used \mbox. I gave it a quick try and it seemed to make matters worse. I will experiment with it a bit more later.
Posted by: Alex Hoffnung on December 23, 2009 2:38 AM | Permalink | Reply to this
Re: Cobordism and Topological Field Theories Weeks 6 & 7
Posted by: Toby Bartels on December 23, 2009 9:11 PM | Permalink | Reply to this
Re: Cobordism and Topological Field Theories Weeks 6 & 7
Ah, thanks! That seems to work well.
I also cleaned up Lecture 7 as Scott suggested. Instances of “Map” and “Cob” should look much better now. Now that I have seen the nicer version the old one sticks out like a sore thumb.
Posted by: Alex Hoffnung on December 23, 2009 9:48 PM | Permalink | Reply to this
Re: Cobordism and Topological Field Theories Weeks 6 & 7
I’m not sure whether this is still active but were there ever any lecture notes for the seminars after this one? I’m trying to rap my head around the cobordism hypothesis and so they would be very
useful to me.
Posted by: Tom on February 23, 2012 6:02 PM | Permalink | Reply to this | {"url":"http://golem.ph.utexas.edu/category/2009/12/cobordism_and_topological_fiel_4.html","timestamp":"2014-04-19T22:14:26Z","content_type":null,"content_length":"22335","record_id":"<urn:uuid:6a495033-c89d-45e8-8060-2530038bc969>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00444-ip-10-147-4-33.ec2.internal.warc.gz"} |
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[R] Creating and editing networks
Uwe Ligges ligges at statistik.uni-dortmund.de
Sun May 9 14:35:02 CEST 2004
Gabor Grothendieck wrote:
> I want to draw and edit networks. Currently I will do something like this --
> my actual networks are larger and more complex:
> 1. Plot network
> # use polygon with nodes as example network
> N <- 20
> t. <- 2*pi*seq(N+1)/N
> plot(cos(t.), sin(t.), type="b", pch=19, cex=5, col="blue", axes=F, ann=F)
> 2. copy and paste the graphic as a metafile into Word.
> 3. edit the network
> there. The editing typically just involves moving nodes and the attached
> edges. Of course, Word does not know the edges are attached to the nodes so I
> have to move the nodes and then move the attached edges, as well.
> The networks are too complex to create the diagrams by hand which is why
> I want to create them in R and then edit them by hand.
> The above strategy is marginally acceptable but
> I was wondering if anyone had a better strategy which retains the
> simplicity of the present approach but gives me the capability to
> have the edges move along with attached nodes when editing them.
> (Note that the strategy must consist of generating the network under program
> control, preferably in R, and then editing it afterwards. I am not looking
> to create the entire network by hand since the networks are too complex for
> that. I had tried specifying the networks in dot from Bell Labs but 1. its
> still more work than generating the network in R 2. the automatic
> layout it chooses is so far from what I need that its a lot of work to
> fix it up by hand in dotty and 3. I could never get them to look just how
> I like. I have also tried Excel and Powerpoint, both
> of which support connectors, which are edges that move along with attached
> nodes, but to use them I have to create the entire network by hand and
> the networks have grown too complex to do that.)
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
Have you tried the CRAN package "dynamicGraph" by Jens Henrik Badsberg?
Uwe Ligges
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The Number of Chains of Subgroups in the Lattice of Subgroups of the Dicyclic Group
Discrete Dynamics in Nature and Society
VolumeΒ 2012Β (2012), Article IDΒ 760246, 17 pages
Research Article
The Number of Chains of Subgroups in the Lattice of Subgroups of the Dicyclic Group
^1Department of Mathematics, Kangnung-Wonju National University, Kangnung 210-702, Republic of Korea
^2Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea
Received 9 May 2012; Accepted 25 July 2012
Academic Editor: Prasanta K.Β Panigrahi
Copyright Β© 2012 Ju-Mok Oh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
We give an explicit formula for the number of chains of subgroups in the lattice of subgroups of the dicyclic group of order by finding its generating function of multivariables.
1. Introduction
Throughout this paper, all groups are assumed to be finite. The lattice of subgroups of a given group is the lattice where is the set of all subgroups of and the partial order is the set inclusion.
In this lattice , a chain of subgroups of is a subset of linearly ordered by set inclusion. A chain of subgroups of is called -rooted (or rooted) if it contains . Otherwise, it is called unrooted.
The problem of counting chains of subgroups of a given group has received attention by researchers with related to classifying fuzzy subgroups of under a certain type of equivalence relation. Some
works have been done on the particular families of finite abelian groups (e.g., see [1β 4]). As a step of this problem toward non-abelian groups, the first author [5] has found an explicit formula
for the number of chains of subgroups in the lattice of subgroups of the dihedral group of order where is an arbitrary positive integer. As a continuation of this work, we give an explicit formula
for the number of chains of subgroups in the lattice of subgroups of the dicyclic group of order by finding its generating function of multivariables where is an arbitrary integer.
2. Preliminaries
Given a group , let , , and be the collection of chains of subgroups of , of unrooted chains of subgroups of , and of -rooted chains of subgroups of , respectively. Let , , and .
The following simple observation is useful for enumerating chains of subgroups of a given group.
Proposition 2.1. Let be a finite group. Then and .
For a fixed positive integer , we define a function as follows: for any .
Proposition 2.2 (see [5]). Let be the cyclic group of order where are distinct prime numbers and are positive integers. Then the number of rooted chains of subgroups in the lattice of subgroups of is
the coefficient of of
Let be the set of all integer numbers. Given distinct positive integers , we define a function where
Most of our notations are standard and for undefined group theoretical terminologies we refer the reader to [6, 7]. For a general theory of solving a recurrence relation using a generating function,
we refer the reader to [8, 9].
3. The Number of Chains of Subgroups of the Dicyclic Group
Throughout the section, we assume that is a positive integer, where are distinct prime numbers and are nonnegative integers and the dicyclic group of order is defined by the following presentation:
where is the identity element.
By the elementary group theory, the following is wellknown.
Lemma 3.1. The dicyclic group has an index subgroup , which is isomorphic to , and has index subgroups which are isomorphic to the dicyclic group of order where .
Lemma 3.2. (1) For any , where .
(2) For any distinct prime factors of , where are nonnegative integers.
Proof. (1) To the contrary suppose that Then for some integers and . This implies . Since , we have , a contradiction.
(2) We only give its proof when . The general case can be proved by the inductive process. Let Clearly, . Since , there exist integers and such that . Note that . On the other hand, Considering the
order of , one can see that . Since we have .
By Lemma 3.1, we have Using the inclusion-exclusion principle and Lemma 3.2, one can see that the number has the following form: for suitable integers and . In the following, we determine the numbers
and explicitly.
Lemma 3.3. (1) .
(2) .
Proof. (1) Clearly for any . For any integer , one can see by Lemma 3.2 that among intersections of the subgroups of the right-hand side of (3.10), the group isomorphic to only appears in
-intersection of the subgroups where and . Since there are such choices, we have .
(2) By Lemma 3.2, one can see that among intersections of the subgroups of the right-hand side of (3.10), the group isomorphic to only appears one of the following two forms: where and , and each
subgroup type in the first form must appear at least once, and it can appear more than once, while each subgroup type in the second form must appear at least once, and one of the subgroup types must
appear more than once. Let be the number of the groups isomorphic to obtained from the first form, and let be the number of the groups isomorphic to obtained from the second form. Then clearly . Note
that On the other hand, Therefore, we have .
By Proposition 2.1 and Lemma 3.3, (3.11) becomes Let and let . Then (3.16) becomes
Throughout the remaining part of the section, we solve the recurrence relation of (3.17) by using generating function technique. From now on, we allow each to be zero for computational convenience.
Let where .
For a fixed integer such that are distinct prime numbers and are non-negative integers, we define a function as follows. for any .
Lemma 3.4. Let be a positive integer. If , then If , then for any .
Proof. Assume first that . Then (3.17) with gives us that Taking to both sides of (3.22), we have because and by a direct computation.
From now on, we assume that . We prove (3.21) by double induction on and . Equation (3.17) with gives us that Taking of both sides of (3.24), we have because and by the definition, and by (3.17) with
. That is, Thus (3.21) holds for .
Assume now that (3.21) holds from to and consider the case for . Note that the last two terms of the right-hand side of (3.17) can be divided into three terms, respectively, as follows: Taking of
both sides of (3.17) and using (3.28), one can see that Further since by (3.17), we have Thus (3.21) holds for . Assume that (3.21) holds from to and consider the case for . Note that the last two
terms of the right-hand side of (3.21) can be divided into three terms, respectively, as follows: Taking of both sides of (3.21), we have Note that by induction hypothesis. Thus Therefore, (3.21)
holds for .
Equation (3.21) with gives us that Taking of both sides of (3.37), we get that
Lemma 3.5. If , then
Proof. If , then since and , the equation holds by (3.20). Assume now that (3.39) holds for . Then by (3.38) we get that which implies that Thus (3.39) holds for .
By Lemmas 3.4 and 3.5 and (3.38), we have We now need to find the function explicitly.
Lemma 3.6. If , then If for , then
Proof. We first assume that . Then by Proposition 2.2, is the coefficient of of which implies that is the coefficient of of and hence by the definition of we get that
Assume now that for . Since , by Proposition 2.2 is the coefficient of of Since by the definition, is the coefficient of of By changing the variables by , respectively, we get that is the coefficient
of of By the definition of , we have
By Proposition 2.1, (3.43), and Lemma 3.6, we have the following theorem.
Theorem 3.7. Let be a positive integer such that are distinct prime numbers and are positive integers. Let be the dicyclic group of order . Let be the number of rooted chains of subgroups in the
lattice of subgroups of . (1)If , then is the coefficient of of (2)If for , then is the coefficient of of
Furthermore, the number of chains of subgroups in the lattice of subgroups of is the coefficient of of
We now want to find the coefficient of of explicitly. Since by the definition, the coefficient of of is Thus the coefficient of of is Continuing this process, one can see that the coefficient of of
is Similarly one can see that the coefficient of of is the coefficient of of is and the coefficient of of is
Therefore, one can have the following.
Corollary 3.8. Let and be the positive integer and the dicyclic group, respectively, defined in Theorem 3.7. Let be the number of rooted chains of subgroups in the lattice of subgroups of . (1)If ,
then where if , then and if , then (2)If for , then where if , then
The first author was funded by the Korean Government (KRF-2009-353-C00040). In the case of the third author, this research was supported by Basic Science Research Program Through the National
Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0025252).
1. V. Murali and B. B. Makamba, β On an equivalence of fuzzy subgroups. II,β Fuzzy Sets and Systems, vol. 136, no. 1, pp. 93β 104, 2003. View at Publisher Β· View at Google Scholar Β· View at
Zentralblatt MATH
2. V. Murali and B. B. Makamba, β Counting the number of fuzzy subgroups of an abelian group of order ${p}^{n}{q}^{m}$,β Fuzzy Sets and Systems, vol. 144, no. 3, pp. 459β 470, 2004. View at
Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
3. J.-M. Oh, β The number of chains of subgroups of a finite cycle group,β European Journal of Combinatorics, vol. 33, no. 2, pp. 259β 266, 2012. View at Publisher Β· View at Google Scholar
4. M. Tărnăuceanu and L. Bentea, β On the number of fuzzy subgroups of finite abelian groups,β Fuzzy Sets and Systems, vol. 159, no. 9, pp. 1084β 1096, 2008. View at Publisher Β· View at Google
Scholar Β· View at Zentralblatt MATH
5. J. M. Oh, β The number of chains of subgroups of the dihedral group,β Submitted.
6. J. S. Rose, A Course on Group Theory, Dover Publications, New York, NY, USA, 1994.
7. W. R. Scott, Group Theory, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964.
8. M. Aigner, Combinatorial Theory, Springer, New York, NY, USA, 1979. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
9. A. Tucker, Applied Combinatorics, John Wiley & Sons, New York, NY, USA, 1995. | {"url":"http://www.hindawi.com/journals/ddns/2012/760246/","timestamp":"2014-04-16T22:59:55Z","content_type":null,"content_length":"1024058","record_id":"<urn:uuid:18147ea6-dd0d-49c3-b1d7-c6ed4d473905>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00635-ip-10-147-4-33.ec2.internal.warc.gz"} |
DOCUMENTA MATHEMATICA, Extra Vol. ICM II (1998), 423-432
DOCUMENTA MATHEMATICA
, Extra Volume ICM II (1998), 423-432
A. N. Dranishnikov
Title: Dimension Theory and Large Riemannian Manifolds
In this paper we discuss some recent applications of dimension theory to the Novikov and similar conjectures. We consider only geometrically finite groups i.e. groups $\Gamma$ that have a compact
classifying space $B\Gamma$. It is still unknown whether all such groups admit a sphere at infinity \cite{B}. In late 80s old Alexandroff's problem on the coincidence between covering and
cohomological dimensions was solved negatively \cite{Dr}. This brought to existence a locally nice homology sphere which is infinite dimensional. In the beginning of 90s S. Ferry conjectured that if
such homology sphere can be presented as a sphere at infinity of some group $\Gamma$, then the Novikov conjecture is false for $\Gamma$. Here we discuss the development of this idea. We outline a
reduction of the Novikov conjecture to dimension theoretic problems. The pionering work in this direction was done by G. Yu [Yu]. He found a reduction of the Novikov Conjecture to the problem of
finite asymptotic dimensionality of the fundamental group $\Gamma$. Our approach is based on the hypothetical equivalence between asymptotical dimension of a group and the covering dimension of its
Higson corona. The slogan here is that most of the asymptotic properties of $\Gamma$ can be expressed in terms of topological properties of the Higson corona $\nu\Gamma$. At the end of the paper we
compare existing reductions of the Novikov conjecture in terms of the Higson corona.
1991 Mathematics Subject Classification:
Keywords and Phrases:
Full text: dvi.gz 22 k, dvi 48 k, ps.gz 79 k.
Home Page of DOCUMENTA MATHEMATICA | {"url":"http://www.emis.de/journals/DMJDMV/xvol-icm/06/Dranishnikov.MAN.html","timestamp":"2014-04-16T04:11:42Z","content_type":null,"content_length":"2521","record_id":"<urn:uuid:98a8c40f-29a8-40dd-bf1e-08c44ec38aaa>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00508-ip-10-147-4-33.ec2.internal.warc.gz"} |
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reactOnNPC v.2.0.2 by hakore
here i've attached the log + translation and the reactOnNPC plugins in case the problem isn't in the code but in the plugins.
oo btw "You will need to input the accents!" you mean something like this ?
^3FBA11t dêlâpån y^029105 pülûh s^5E9D86f lïma g^000000.
^3FBA11t Eight y^029105 y s^5E9D86f lïma g^000000.
Old Openkore Manual | Macro Plugin Manual | Old Forum | My Perl Bible | Guide to figuring out server information
reactOnNPC talk num @eval(my $numerotxt = #2~1;my $answer;switch ($numerotxt) { case /seratus/ {$answer=1} case /duaratus/ {$answer=2} case /tigaratus/ {$answer=3} case /empatratus/ {$answer=4} case
/limaratus/ {$answer=5} case /enamratus/ {$answer=6} case /tujuhratus/ {$answer=7} case /delapanratus/ {$answer=8} case /sembilanratus/ {$answer=9}}switch ($numerotxt) { case /sepuluh|belas/
{$answer.=1} case /duapuluh/ {$answer.=2} case /tigapuluh/ {$answer.=3} case /empatpuluh/ {$answer.=4} case /limapuluh/ {$answer.=5} case /enampuluh/ {$answer.=6} case /tujuhpuluh/ {$answer.=7} case
/delapanpuluh/ {$answer.=8} case /sembilanpuluh/ {$answer.=9} else {$answer.=0}}switch ($numerotxt) { case / .?satu.? |sebelas/ {$answer.=1} case / .?dua.? |duabelas/ {$answer.=2} case / .?tiga.? |
tigabelas/ {$answer.=3} case / .?empat.? |empatbelas/ {$answer.=4} case / .?lima.? |limabelas/ {$answer.=5} case / .?enam.? |enambelas/ {$answer.=6} case / .?tujuh.? |tujuhbelas/ {$answer.=7} case /
.?delapan.? |delapanbelas/ {$answer.=8} case / .?sembilan.? |sembilanbelas/ {$answer.=9} else {$answer.=0}}return $answer;)
type number
msg_0 /\[Police\]/
msg_1 /Input This Number:/
msg_2 /(\w+)/
I've already translated it but not all of it
Old Openkore Manual | Macro Plugin Manual | Old Forum | My Perl Bible | Guide to figuring out server information
No problem...
About the test, I'm not playing or bot anymore... don't have much time. If someone is willing to test it, please do it...
As soon as it is OK, I'll post it!
Your english is fine... but you forgot to translate the file, I don't speak malasian (at least I think it is).
help ! i need a hand... and yess i'm a noob.i can't pass this anti bot since this is my first attempt!
I know nothing about PERL so what I can do is copas (Copy Paste) from this forum... every possible code and plugins in this forum to pass this anti bot system(at least that is what i can think of
You are using the plugin I changed to use just for nesting commands, to your case you need the original plugin, download it from the first post.
Do the follow to see the messages from the plugin without using debug:
1) Open your reactOnNPC.pl file
2) Find these lines: (they are not together)
debug "[reactOnNPC] NPC message saved ($reactOnNPC{index}): \"$msg\".\n", "reactOnNPC";
debug "[reactOnNPC] Conditions for reactOnNPC_$i not met.\n", "reactOnNPC";
debug "[reactOnNPC] One or more lines doesn't match for \"reactOnNPC_$i\" ($j).\n", "reactOnNPC";
debug "[reactOnNPC] Matching \"$subject\" to \"$pattern\" ($line)... ", "reactOnNPC";
debug "doesn't match.\n", "reactOnNPC";
3) Where is written
change it to
at all these lines.
Ex: message "doesn't match.\n", "reactOnNPC";
About the accents, I mean just this:
delapan -> dêlâpån
Just add the symbols above the vowels!
You don't need the color codes, like this:
Everything started with ^ and followed by 6 hexadecimal digits is a color code.
Right now the plugin isn't work because of the letters between the words.
Try this code:
reactOnNPC talk num @eval(my $numerotxt = #2~1;my $answer;switch ($numerotxt) { case /se.?.?ratus/ {$answer=1} case /dua.?.?ratus/ {$answer=2} case /tiga.?.?ratus/ {$answer=3} case /empat.?.?ratus/
{$answer=4} case /lima.?.?ratus/ {$answer=5} case /enam.?.?ratus/ {$answer=6} case /tujuh.?.?ratus/ {$answer=7} case /delapan.?.?ratus/ {$answer=8} case /sembilan.?.?ratus/ {$answer=9}}switch
($numerotxt) { case /se.?.?puluh|belas/ {$answer.=1} case /dua.?.?puluh/ {$answer.=2} case /tiga.?.?puluh/ {$answer.=3} case /empat.?.?puluh/ {$answer.=4} case /lima.?.?puluh/ {$answer.=5} case /
enam.?.?puluh/ {$answer.=6} case /tujuh.?.?puluh/ {$answer.=7} case /delapan.?.?puluh/ {$answer.=8} case /sembilan.?.?puluh/ {$answer.=9} else {$answer.=0}}switch ($numerotxt) { case /satu.{0,3}$|
se.?.?belas/ {$answer.=1} case /dua.{0,3}$|dua.?.?belas/ {$answer.=2} case /tiga.{0,3}$|tiga.?.?belas/ {$answer.=3} case /empat.{0,3}$|empat.?.?belas/ {$answer.=4} case /lima.{0,3}$|lima.?.?belas/
{$answer.=5} case /enam.{0,3}$|enam.?.?belas/ {$answer.=6} case /tujuh.{0,3}$|tujuh.?.?belas/ {$answer.=7} case /delapan.{0,3}$|delapan.?.?belas/ {$answer.=8} case /sembilan.{0,3}$|sembilan.?.?belas/
{$answer.=9} else {$answer.=0}}return $answer;)
type number
msg_0 /\[Police\]/
msg_1 /Input This Number:/
msg_2 /(\w+)/
But don't forget the accents!
Old Openkore Manual | Macro Plugin Manual | Old Forum | My Perl Bible | Guide to figuring out server information
I already added v.2.0.0 in the first page.
ok i've made some additions to the reactOnNPC plugin so it will catch guild messages and succeded, but i still can't get my bot to respond, can you point me in the right direction, here is my log:
Log with Spanish words
You are in Frozen state
NPC image: mets_alpha.bmp (0)
Unknown #110019081: [PolicÃa]
[reactOnNPC] NPC message saved (1): "[^0000FFPolicÃa^000000]".
[Guild] Red Plant: wagabo fapeyip mivuv
[reactOnNPC] NPC message saved (2): "Plant: wagabo fapeyip mivuv".
[Guild] Black Mushroom: hap 947 tajax
[reactOnNPC] NPC message saved (3): "k Mushroom: hap 947 tajax".
[Guild] Snake: yep otarot eh exovC
[reactOnNPC] NPC message saved (4): "e: yep otarot eh exovC".
[Guild] Blue Plant: gopiconoki idudox fehE
[reactOnNPC] NPC message saved (5): " Plant: gopiconoki idudox fehE".
Unknown #110019081:
[reactOnNPC] NPC message saved (6): " ".
Unknown #110019081: Un monstruo acaba de decir un número (ver chat log)
[reactOnNPC] NPC message saved (7): "^F51200Un monstruo acaba de decir un
número (ver chat log)^000000".
Unknown #110019081:
[reactOnNPC] NPC message saved (8): " ".
Unknown #110019081: Restan sesenta y cinco segundos
[reactOnNPC] NPC message saved (9): "^0C0542 Restan
^5D1E76^35B90Bsesenta^9522A3 y ^49BC8Bcinco^000000^924C80 segundos".
Unknown #110019081: Auto-continuing talking
[reactOnNPC] onNPCAction type is: continue.
Statistics posting failed: HTTP server returned error status.
NPC Exists: BotKillerFnc1 (93, 62) (ID 110019081) - (0)
BotKillerFnc1: Type 'talk num <number #>' to input a number.
[reactOnNPC] onNPCAction type is: number.
Calculating random route to: Inside Pyramid(moc_pryd01): 96, 152
On route to: Inside Pyramid(moc_pryd01): 96, 152
Route Solution Ready!
Translated log
You are in Frozen state
NPC image: mets_alpha.bmp (0)
Unknown #110019081: [Police]
[reactOnNPC] NPC message saved (1): "[^0000FFPolice^000000]".
[Guild] Red Plant: wagabo fapeyip mivuv
[reactOnNPC] NPC message saved (2): "Plant: wagabo fapeyip mivuv".
[Guild] Black Mushroom: hap 947 tajax
[reactOnNPC] NPC message saved (3): "k Mushroom: hap 947 tajax".
[Guild] Snake: yep otarot eh exovC
[reactOnNPC] NPC message saved (4): "e: yep otarot eh exovC".
[Guild] Blue Plant: gopiconoki idudox fehE
[reactOnNPC] NPC message saved (5): " Plant: gopiconoki idudox fehE".
Unknown #110019081:
[reactOnNPC] NPC message saved (6): " ".
Unknown #110019081: A monster said a number (see chat log)
[reactOnNPC] NPC message saved (7): "^F51200A monster said a number (see chat log)^000000".
Unknown #110019081:
[reactOnNPC] NPC message saved (8): " ".
Unknown #110019081: Remains sixty five seconds
[reactOnNPC] NPC message saved (9): "^0C0542 Remains
^5D1E76^35B90Bsixty^9522A3 y ^49BC8Bfive^000000^924C80 seconds".
Unknown #110019081: Auto-continuing talking
[reactOnNPC] onNPCAction type is: continue.
Statistics posting failed: HTTP server returned error status.
NPC Exists: BotKillerFnc1 (93, 62) (ID 110019081) - (0)
BotKillerFnc1: Type 'talk num <number #>' to input a number.
[reactOnNPC] onNPCAction type is: number.
Calculating random route to: Inside Pyramid(moc_pryd01): 96, 152
On route to: Inside Pyramid(moc_pryd01): 96, 152
Route Solution Ready!
At my config.txt i have this:
reactOnNPC talk num @resp(#1~2#1~3#1~4#1~5
type number
msg_0 [PolicÃa]
msg_1 /\d+/
msg_2 /\d+/
msg_3 /\d+/
msg_4 /\d+/
msg_5 /\d+/
I'am posting my modified reactOnNPC.pl so you can see waht i did.
a million thanks in advanced!!
Try this, and use the attached modified plugin. I need to fix some codes.
It's not tested though.
reactOnNPC talk num @eval("#g0~1" || "#g1~1" || "#g2~1" || "#g3~1") {
msg_0 /\[PolicÃa\]/
msg_1 /./
msg_2 /Un monstruo acaba de decir un número \(ver chat log\)/
msg_3 /./
msg_4 /Restan .* segundos/
guildMsg_0 /^\D*(\d+)?\D*$/
guildMsg_1 /^\D*(\d+)?\D*$/
guildMsg_2 /^\D*(\d+)?\D*$/
guildMsg_3 /^\D*(\d+)?\D*$/
type number
Million thanks Hakore. It did worked, the only funny thing is that somehow
i had to change the spanish words at the config.txt for the english version.
Here it's my log
You have look: GM Perfect Hide
You have look: Ruwach
You are in Frozen state
NPC image: mets_alpha.bmp (0)
[GM] Police: I really sorry but i need to know if you are Playing or Boting.
Unknown #110019081: [Police]
[reactOnNPC] NPC message saved (0): "[^0000FFPolice^000000]".
[Guild] Snake: axicoc 791 orovu
[reactOnNPC] Guild message saved (0): "Snake: axicoc 791 orovu".
[Guild] White Plant: ecilufi hurepek lehibu
[reactOnNPC] Guild message saved (1): "White Plant: ecilufi hurepek lehibu".
[Guild] Petite: iralul irenu bit
[reactOnNPC] Guild message saved (2): "Petite: iralul irenu bit".
[Guild] Eggyra: ugipixuf iyuc firepop
[reactOnNPC] Guild message saved (3): "Eggyra: ugipixuf iyuc firepop".
Unknown #110019081:
[reactOnNPC] NPC message saved (1): " ".
Unknown #110019081: Enter the number who said one of the monsters:
[reactOnNPC] NPC message saved (2): "^F51200Enter the number who said one of
the monsters:^000000".
Unknown #110019081:
[reactOnNPC] NPC message saved (3): " ".
Unknown #110019081: ninêtÿ seconds to respond
[reactOnNPC] NPC message saved (4): "^988737 ^8887BC^81582Bninêtÿ
^B019C0^000000^97A828 seconds to respond".
Unknown #110019081: Auto-continuing talking
[reactOnNPC] onNPCAction type is: continue.
[reactOnNPC] Conditions for reactOnNPC_0 not met.
Monster Died: Familiar (0)
Item Appeared: Red Herb (0) x 1 (59, 104)
Target died
NPC Exists: Unknown #110019081 (58, 103) (ID 110019081) - (0)
Unknown #110019081: Type 'talk num <number #>' to input a number.
[reactOnNPC] onNPCAction type is: number.
[reactOnNPC] Matching "[Police]" to "/\[Police\]/" (0)... regexp ok.
[reactOnNPC] Matching " " to "/./" (1)... regexp ok.
[reactOnNPC] Matching "Enter the number who said one of the monsters:" to
"/Enter the number who said one of the monsters:/" (2)... regexp ok.
[reactOnNPC] Matching " " to "/./" (3)... regexp ok.
[reactOnNPC] Matching " ninêtÿ seconds to respond" to "/ ninêtÿ seconds to
respond/" (4)... regexp ok.
[reactOnNPC] Matching "Snake: axicoc 791 orovu" to "/^\D*(\d+)?\D*$/" (0)... reg
exp ok.
[reactOnNPC] Matching "White Plant: ecilufi hurepek lehibu" to
"/^\D*(\d+)?\D*$/" (1)... regexp ok.
[reactOnNPC] Matching "Petite: iralul irenu bit" to "/^\D*(\d+)?\D*$/" (2)... re
gexp ok.
[reactOnNPC] Matching "Eggyra: ugipixuf iyuc firepop" to "/^\D*(\d+)?\D*$/"
(3)... regexp ok.
[reactOnNPC] Reacting to NPC. Executing command "talk num 791".
NPC image: (0)
You are out of look: Ruwach
You are out of Frozen state
You are out of look: GM Perfect Hide
Map Change: moc_pryd01.gat (58, 103)
Sending Map Loaded
This is what i have at the config.txt, had to change it a little bit:
reactOnNPC talk num @eval("#g0~1" || "#g1~1" || "#g2~1" || "#g3~1") {
msg_0 /\[Police\]/
msg_1 /./
msg_2 /Enter the number who said one of the monsters:/
msg_3 /./
msg_4 / seconds to respond/ #better leave it whitout specifiyng how many seconds cause that changes a lot
guildMsg_0 /^\D*(\d+)?\D*$/
guildMsg_1 /^\D*(\d+)?\D*$/
guildMsg_2 /^\D*(\d+)?\D*$/
guildMsg_3 /^\D*(\d+)?\D*$/
type number
So i guess if i'am going to use it on a mexican private server i need to change
the reactOnNPC accordingly to the spanish version.
Thanks again Hakore, i'am going to make sure i learn Pearl from now on. | {"url":"http://forums.openkore.com/viewtopic.php?f=34&t=198&start=110","timestamp":"2014-04-17T07:02:08Z","content_type":null,"content_length":"64453","record_id":"<urn:uuid:b7826162-26a7-4866-b3f7-3ef144cd13b8>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00579-ip-10-147-4-33.ec2.internal.warc.gz"} |
Logicism, Intuitionism, and Formalism: What Has Become of Them?
This book grew out of two conferences held in August 2004 at Uppsala University: “Logicism, Intuitionism, and Formalism” and “A Symposium on Constructive Mathematics”. Twenty-four mathematicians made
contributions to the book in three broad sections, namely: Logicism and Neo-Logicism; Intuitionalism and Constructive Mathematics; and Formalism. Generally, the book is about what has happened as a
result of the various attempts, since the publications of Gödel disturbed the thinking of mathematicians about basic mathematical concepts, to put mathematics on a more solid foundation.
I was pleased that almost all the authors gave interesting historical insights into the foundational approach of each school of discussion and then followed up with modern results and research
programs currently underway in those areas. The impression from the book is that various schools of philosophy of mathematics are alive and well, perhaps fragmenting into various sub-branches in a
kind of a revival of foundational mathematics. Intuitionism and Constructive mathematics is proving extremely useful in theoretical computer science problems and other domains of applications for the
other schools seems to be emerging.
At the core, the different mathematical schools differ mostly by what they consider constitutes a proof and what would best be taken as a foundation of mathematics. The schools also differ in the
fundamental properties that statements can have. Statement “properties” discussed in the book included not only truth values, but also provability, decidability, computability, constructibility and
allowability of statements. Thus, it is meaningful to talk about statements that are “true” but not “provable”, or statements that are “provable” but not “computable”, and so on.
Some topics covered in the book include abstraction principles, choice principles, FAN and BAR theorems, axiomatic set theory, reverse mathematics and category theory. Other topics of interest to
foundational mathematics include meta-mathematical statements; Gödel’s theorems, the Liar’s paradox and other statements that reference fundamental statement properties, alternatives to axiomatic set
theory for foundations; role of recursive definitions; role of recursive proofs; and the role of mathematical induction. One interesting section dealt with principle of totality in context of using
the operation of pairing as a foundation for mathematics something like set theory is often used.
The book comes nicely typeset, with a good solid binding, and a solid index. While the sections are somewhat independent the book flows nicely. Overall, I found the book highly readable and
enjoyable. My only caveat is the steep price; hopefully the publisher will create an e-book version to make it more affordable to a wider audience.
Collin Carbno is a specialist in process improvement and methodology. He holds a Master’s of Science Degree in theoretical physics and completed course work for Ph.D. in theoretical physics
(relativistic rotating stars) in 1979 at the University of Regina. He has been employed for nearly 30 years in various IT and process work at Saskatchewan Telecommunications and currently holds a
Professional Physics Designation from the Canadian Association of Physicists, and the Information System Professional designation from the Canadian Information Process Society. | {"url":"http://www.maa.org/publications/maa-reviews/logicism-intuitionism-and-formalism-what-has-become-of-them","timestamp":"2014-04-20T12:55:36Z","content_type":null,"content_length":"99813","record_id":"<urn:uuid:b1de87ae-b117-4d11-ab5e-f79a9f0d2dd5>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00440-ip-10-147-4-33.ec2.internal.warc.gz"} |
Decompose any linear transformation as a product of INJECTIVE and SURJECTIVE ones
January 4th 2011, 11:53 AM #1
Dec 2010
Decompose any linear transformation as a product of INJECTIVE and SURJECTIVE ones
I am solving problems from my linear algebra book and one of them asks me to prove that any linear transformation A: E ->F (between vector spaces E and F of any dimension, be it finite or not)
can be written as the composition (product) of a surjective linear transformation S and an injective linear transformation T, that is: A = TS. ( A(x) =T ( S(x) ) )
I have done this, this is in fact just the same proof for the general fact that ANY FUNCTION can be expressed as the composition of a surjective and injective function in the same order as above
HOWEVER, I was also asked to prove or disprove that any linear transformation can be also decomposed as the composition of an injective linear transformation T and a surjective linear
transformation S, that is, A= ST . The reverse order.
Now, I will sketch a proof for the validity of this assertion, however, I think it relies on finite-dimensional reasoning and that's why I'm not quite sure if there's some subtle assumption I am
overlooking there. I will in fact, assume that the vector spaces are finite-dimensional, but even with this assumption, I fail to see if there's any limitation to do the same for
infinite-dimensional spaces (interpreting that dim(E)<dim(F) means that there is an injective linear transformation from E to F).
So, Let's take the arbitrary linear function A: E->F and assume that dim(E)<dim(F). Use R^n with n= dim(E) + dim(Ker(A)) + (dim(F) - dim(E) ) = dim(Ker(A)) + dim(F).
Now, consider a basis B of E. A must be defined just by considering its values on the elements of this basis. Consider the set A(B) = {Av, v in B}. Take a maximal linearly independent subset of A
(B). (For finite-dimensional spaces this is just an inductive valid step, but I think I can do this in general by assuming the axiom of choice, in the form of Zorn's lemma, just like in the proof
that every vector space has a basis).
so, I can modify the original basis in case by swapping the elements that are not in this maximal linearly independent subset by a suitable linear combination of them such that their images all
go to 0 in F. So I can just assume that B is formed by elements whose images are L.I. and the others go to zero (they will generate the kernel).
Now, I define the a linear function T:E -> R^n by setting these elements to the first elements of the basis of R^n, which come a in number of dim(Ker(A)) (assuming finite dimensions) (because
they are LI, I can do this without any contradictions).
the remaining elements of the basis of E are mapped to the next elements of the basis of R^n (they are a total of dim(E) - dim(Ker(A))).
In this way i define an injective (not necessarily surjective, because there are dim(F) - dim(E) elements of the basis of R^n that were not reached) linear transformation T.
Now, define the linear transformation S:R^n -> F by setting the first elements of R^n to zero so that A(x) = TS(x) for x being the elements of thebasis of E that originally
went to zero.
for the rest of elements of the basis of E that correspond to the next elements of the basis of R^n we map them to what they were originally. In this way A(x) = TS(x) for all elements of the
basis of E, that is, A = TS.
But I still have room to cover all of F. I define the next dim(F)-dim(E) elements of the basis of R^n to be sent to the last corresponding elements of the basis of F. now we only need to cover
the dim(Ker(A)) elements of F that were not reached because their corresponding ones were sent to zero. this can be covered by the last elements of the basis of R^n we have.
I know this is very long to follow but it's quite easy to see in a picture. I am attaching one.
Now, I know that this cumbersome proof should be ok for finite dimensional spaces but I see too many assumptions and uglyness for the general case for infinite dimensional spaces(with R^n
replaced by a sufficiently big vector space). Does anyone know of a neat solution? Moreover, I did not address the cases where dim(E)>dim(F) (in the finite and infinite dimensional sense, even
when one is finite and the other is not).
Is this false in general? (although i think the proof I showed above works for finite dimensional spaces). I was checking the wikipedia on surjection and injection and they mention the validity
only for the decomposition A= TS, not the other way around (they say nothing, but if there were a symmetrical statement, they would mention it , right?)
any help regarding this would be very much appreciated.
Oh wait, I see what you do in the proof now. You lift it first into a larger space for injection.
How about this then:
$A: E\to F$
Consider the product space $E \times F$ (no need to worry about finite dimensions).
$B: E\to E \times F$
where $B(e) = (e, A(e))$ injective.
$C: E \times F \to F$
where $C(e, f) = f$ surjective.
$A = CB$
Now this seems to be general enough to work for all functions and not just linear ones. Did I make a mistake?
Last edited by snowtea; January 4th 2011 at 12:40 PM.
Oh wait, I see what you do in the proof now. You lift it first into a larger space for injection.
How about this then:
$A: E\to F$
Consider the product space $E \times F$ (no need to worry about finite dimensions).
$B: E\to E \times F$
where $B(e) = (e, A(e))$ injective.
$C: E \times F \to F$
where $C(e, f) = f$ surjective.
$A = CB$
Now this seems to be general enough to work for all functions and not just linear ones. Did I make a mistake?
Thank you! This is completely right as far as I can see. But I suspect that it needs the Axiom of choice at some point. I'm not comfortable invoking it because I know it can be dispensed with
sometimes when there is some rule to work with the infinities (not sure in which cases). Do you know how to make sure about its usage? In any case, this solves the problem. Thanks!
This argument is constructive - the injection and surjection are given explicitly. It looks like AC is not being used.
January 4th 2011, 12:23 PM #2
Senior Member
Dec 2010
January 11th 2011, 03:24 AM #3
Dec 2010
January 11th 2011, 06:46 AM #4
Senior Member
Nov 2010
Staten Island, NY | {"url":"http://mathhelpforum.com/advanced-algebra/167447-decompose-any-linear-transformation-product-injective-surjective-ones.html","timestamp":"2014-04-17T16:21:16Z","content_type":null,"content_length":"46946","record_id":"<urn:uuid:b6eae28f-2404-41ec-ae11-53b2e4271392>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00339-ip-10-147-4-33.ec2.internal.warc.gz"} |
Protein sequence design on the basis of topology optimization techniques -Using continuous modeling of discrete amino acid types
Ananthasuresh, GK (2006) Protein sequence design on the basis of topology optimization techniques -Using continuous modeling of discrete amino acid types. In: IUTAM Symposium on Topological Design
Optimization of Structures, Machines and Materials,, Oct 2005, Copenhagen, Denmark, pp. 455-466.
fulltext.pdf - Published Version
Restricted to Registered users only
Download (287Kb) | Request a copy
The notion of optimization is inherent in protein design. A long linear chain of twenty types of amino acid residues are known to fold to a 3-D conformation that minimizes the combined inter-residue
energy interactions. There are two distinct protein design problems, viz. predicting the folded structure from a given sequence of amino acid monomers (folding problem) and determining a sequence for
a given folded structure (inverse folding problem). These two problems have much similarity to engineering structural analysis and structural optimization problems respectively. In the folding
problem, a protein chain with a given sequence folds to a conformation, called a native state, which has a unique global minimum energy value when compared to all other unfolded conformations. This
involves a search in the conformation space. This is somewhat akin to the principle of minimum potential energy that determines the deformed static equilibrium configuration of an elastic structure
of given topology, shape, and size that is subjected to certain boundary conditions. In the inverse-folding problem, one has to design a sequence with some objectives (having a specific feature of
the folded structure, docking with another protein, etc.) and constraints (sequence being fixed in some portion, a particular composition of amino acid types, etc.) while obtaining a sequence that
would fold to the desired conformation satisfying the criteria of folding. This requires a search in the sequence space. This is similar to structural optimization in the design-variable space
wherein a certain feature of structural response is optimized subject to some constraints while satisfying the governing static or dynamic equilibrium equations. Based on this similarity, in this
work we apply the topology optimization methods to protein design, discuss modeling issues and present some initial results.
Item Type: Conference Paper
Additional Information: Copyright of this article belongs to Springer.
Keywords: Protein sequence design;continuous modeling of amino acid types;topology optimization;inverse folding.
Department/Centre: Division of Mechanical Sciences > Mechanical Engineering
Date Deposited: 26 Aug 2010 04:33
Last Modified: 19 Sep 2010 06:12
URI: http://eprints.iisc.ernet.in/id/eprint/30679
Actions (login required) | {"url":"http://eprints.iisc.ernet.in/30679/","timestamp":"2014-04-21T15:40:26Z","content_type":null,"content_length":"26112","record_id":"<urn:uuid:aa0ec85a-9ec6-4d42-bf38-1784f0ce097d>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00510-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sample Lesson Plan
Students must have the ability to add and subtract real numbers, and they ... Microsoft Word - Sample_Math_Project Lesson_Plan.doc Author: jjc Created Date
Mathematics Lesson Plan for 2 nd Grade
Mathematics Lesson Plan for 2nd Grade For the lesson on April 29, 2010 At DePaul ... IL.6.C.2c - Select appropriate methods and tools for computing with whole numbers
SAMPLE IPA LESSON PLAN Standard #4 (Real World Connections) Grade One Lesson: #2 of 4 ... and that the numbers on the thermometer are not just numbers, but ...
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Mathematics Lesson Plan for 2nd Grade For the lesson on April 29, 2010 At DePaul ... IL.6.C.2c - Select appropriate methods and tools for computing with whole numbers
Course Syllabus
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... and relevance of the lesson to real life. All ... In this section, we are going to see a sample lesson plan ... with Numbers Triangular Numbers 1. Lesson Plan
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Printed on 6/16/09US01_Square_Root_lesson_plan_w_reflection.doc 1 Mathematics Lesson ... We want students to understand that there exist certain numbers that are tangible ...
2002 Basic Education Curriculum Secondary Level
Lastly, sample lesson plans are provided to illustrate the mode of integration, ... integers, rational numbers, irrational numbers, real numbers 1.1 review operations ... | {"url":"http://www.cawnet.org/docid/sample+lesson+plan+msep+6/","timestamp":"2014-04-21T14:45:22Z","content_type":null,"content_length":"47404","record_id":"<urn:uuid:2468ecfc-5214-4d7f-966f-4526fc351cb3>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00124-ip-10-147-4-33.ec2.internal.warc.gz"} |
Variable resistance
1. The problem statement, all variables and given/known data
"A circuit has a variable resistance (total 100 ohms) across which is a voltage drop of 12 V supplied by a battery. What must be its resistance if the voltage is to be A) 12V and B) 6.2V?"
2. Relevant equations
3. The attempt at a solution
We haven't learned anything besides Ohm's Law for these equations, so I tried plugging in different values to try and dial in the answers, but it didn't work. Is there something I'm missing or am I
being asked to solve for two unknowns? Hints appreciated. | {"url":"http://www.physicsforums.com/showthread.php?p=1237702","timestamp":"2014-04-20T01:04:40Z","content_type":null,"content_length":"27671","record_id":"<urn:uuid:12b0deb9-2a37-4f5f-b42e-5e08896c6b9c>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00348-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Linear Algebra, Linear Transformations
Sci Advisor
PF Gold
P: 38,882
Saying there is a unique linear tranformation satisfying certain conditions, means there is one and only one such transformation.
For example, if we are only told that the linear transformation L maps (1, 0), in R^2 to (1, 1), then we know that L(x,y)= L(x(1, 0)+ y(0, 1))= xL(1,0)+ yL(0,1)= x(1,1)+ yL(0,1)= (x,x)+ yL(0,1). But
I don't know what L(0,1) is. Since (0,1) and (1, 0) are independent, I cannot get L(0,1) from "L(1, 0)= (1,1)". There are many such (actually an infinite number) of Linear transformation satisfying
that condition.
In general, a linear transformation on an n-dimensional vector space is completely determined by its action on n independent vectors (which then must be a basis for the space).
But if I am told that L maps (1, 0) to (1, 1) and maps (0, 1) to (2, 3), then I can say that L(x,y)= L(x(1,0)+ y(0,1))= xL(1,0)+ yL(0,1)=x(1,1)+ y(2,3)= (x+2y, x+ 3y). That is the only limear
transformation satisfying those conditions- it is unique.
In the example, T(9x+5) = (.1,.2) and T(7x+4) = (.3,.8), "9x+ 5" and "7x+ 4" are independent and, since the set of linear polynomials has dimension 2, form a basis for that space. Given any such
polynomial, ax+ b, it is easy to show that ax+ b= (4a- 7b)(9x+ 5)+ (9b-5a)(7x+ 4) so that T(ax+ b)= (4a- 7b)T(9x+5)+ (9b-5a)T(7x+4)= (4a+ 7b)(.1,.2)+ (9b-5a)(.3, .8)= (-1.1a+ 3.4b, -3.2a+ 8.6), a
unique linear transformation. | {"url":"http://www.physicsforums.com/showpost.php?p=3259983&postcount=3","timestamp":"2014-04-20T08:38:09Z","content_type":null,"content_length":"9005","record_id":"<urn:uuid:2bd19ab9-7dc7-4f66-81ff-e240d6521680>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00606-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sample Lesson Plan
Students must have the ability to add and subtract real numbers, and they ... Microsoft Word - Sample_Math_Project Lesson_Plan.doc Author: jjc Created Date
Mathematics Lesson Plan for 2 nd Grade
Mathematics Lesson Plan for 2nd Grade For the lesson on April 29, 2010 At DePaul ... IL.6.C.2c - Select appropriate methods and tools for computing with whole numbers
SAMPLE IPA LESSON PLAN Standard #4 (Real World Connections) Grade One Lesson: #2 of 4 ... and that the numbers on the thermometer are not just numbers, but ...
sample lesson plans overview
Sample Lesson Plans Overview This section contains a ... Lesson Plan Sequence Each lesson plan follows a common ... students rewrite the poem using real words ...
Mathematics lessons for Grade 2
1 Adding whole numbers 43 2 Subtracting whole ... Using these lesson plans These sample lessons for Grade 2 are ... Each lesson plan has sufficient material to support ...
Real World Lesson Plans
Rationale: This lesson plan is designed to get students considering a real-world ... Resources used for these lesson plans: www.google.com key words: sample ...
Math Lesson Plan 8th Grade Curriculum Total Activities: 345
Math Lesson Plan 8th Grade Curriculum Total Activities: 345 ... situations using real numbers. M8029 M8030 5 Properties-Apply ... Sample Bias-Identify whether a sample is biased.
Mathematics Lesson Plan for 2 nd Grade
Mathematics Lesson Plan for 2nd Grade For the lesson on April 29, 2010 At DePaul ... IL.6.C.2c - Select appropriate methods and tools for computing with whole numbers
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adding whole numbers? Present students with real-world problems that involve the ... Sample Lesson Plan Grade 1, Number Author: rebecca.pound Created Date
24 Lesson Plan: Other Trigonometric Functions Name
24 Lesson Plan: Other Trigonometric Functions Name_____ Due: Tuesday ... What other trigonometric function(s) have all real numbers as their range?
Math Lesson Plan 8th Grade Curriculum Total Activities: 345
Math Lesson Plan 8th Grade Curriculum Total Activities: 345 ... situations using real numbers. M8029 M8030 5 Properties-Apply ... Sample Bias-Identify whether a sample is biased.
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... and relevance of the lesson to real life. All ... In this section, we are going to see a sample lesson plan ... with Numbers Triangular Numbers 1. Lesson Plan
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Lastly, sample lesson plans are provided to illustrate the mode of integration, ... integers, rational numbers, irrational numbers, real numbers 1.1 review operations ... | {"url":"http://www.cawnet.org/docid/sample+lesson+plan+msep+6/","timestamp":"2014-04-21T14:45:22Z","content_type":null,"content_length":"47404","record_id":"<urn:uuid:2468ecfc-5214-4d7f-966f-4526fc351cb3>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00124-ip-10-147-4-33.ec2.internal.warc.gz"} |
Next: Arguments Up: Standard Nonsymmetric Eigenvalue Problems Previous: LA_GEESX   Contents   Index
LA_GEESX computes for a real/complex square matrix
Optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left, computes a reciprocal condition number for the average of the selected
eigenvalues, and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues. The leading columns of
A real matrix is in real-Schur form if it is block upper triangular with
A complex matrix is in complex-Schur form if it is upper triangular.
Next: Arguments Up: Standard Nonsymmetric Eigenvalue Problems Previous: LA_GEESX   Contents   Index Susan Blackford 2001-08-19 | {"url":"http://www.netlib.org/lapack95/lug95/node255.html","timestamp":"2014-04-20T23:36:05Z","content_type":null,"content_length":"6846","record_id":"<urn:uuid:aad8140e-a9f4-4490-8092-021962fcfc6f>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00322-ip-10-147-4-33.ec2.internal.warc.gz"} |
Question regarding 50 50 chance betting?
February 28th 2011, 11:36 AM #1
Feb 2011
Question regarding 50 50 chance betting?
Hey guys! So I really need some help on a theoretical problem that has been on my mind for a long time
Ok. So you have $200 and a 50 - 50 chance of winning a game (exactly 50 50, it is not 51 49 or anything - exactly equal chances like flipping a coin).
What is the best method of betting in order to profit in the end? Is there some successful model?
For example, would it be best to bet $50, and if you win bet another $50 but if you loose bet only $25? Or bet $50, and if you win only bet $25 so that you definitely end up on top?
Theoretically this would continue indefinitely (as in you wouldn't want to "walk away" at any time, just continue making thousands or whatever).
So for those who don't like to read:
How do you turn $200 into a lot of money when there is a 50-50 chance of winning? What is the best model for doing so?
You will need to know the odds of the events you're betting on before staking anything.
Once you have them, fractional Kelly staking is a good start.
Hey guys! So I really need some help on a theoretical problem that has been on my mind for a long time
Ok. So you have $200 and a 50 - 50 chance of winning a game (exactly 50 50, it is not 51 49 or anything - exactly equal chances like flipping a coin).
What is the best method of betting in order to profit in the end? Is there some successful model?
For example, would it be best to bet $50, and if you win bet another $50 but if you loose bet only $25? Or bet $50, and if you win only bet $25 so that you definitely end up on top?
Theoretically this would continue indefinitely (as in you wouldn't want to "walk away" at any time, just continue making thousands or whatever).
So for those who don't like to read:
How do you turn $200 into a lot of money when there is a 50-50 chance of winning? What is the best model for doing so?
Assuming fair bets, with probability 1 you will eventually go bust.
March 2nd 2011, 04:10 AM #2
Mar 2011
March 2nd 2011, 04:19 AM #3
Grand Panjandrum
Nov 2005 | {"url":"http://mathhelpforum.com/statistics/172943-question-regarding-50-50-chance-betting.html","timestamp":"2014-04-19T22:52:04Z","content_type":null,"content_length":"36846","record_id":"<urn:uuid:e0aaaa70-a131-4469-baeb-3ad7f4116f47>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00108-ip-10-147-4-33.ec2.internal.warc.gz"} |
M.S. in Mathematics: Applied Statistics
Admissions │ Courses │ Student Resources
This program is designed to increase the number of professionals with the broad training in statistical methodology that is suitable for applications in industry, medicine, business and government. A
typical student in this program will be exposed to a broad base of practical techniques and applications from a wide variety of fields. At the same time the student's background in theory is
developed so that the learning process will continue in any professional environment pursued after graduation.
The primary goal of the master’s degree program is to provide a basis for the skilled and competent application of modern statistical methods. In addition to the basic theoretical foundations, areas
of methodology include regression analysis, design of experiments, multivariate analysis, quality control, survival analysis, time series, sample surveys, categorical and nonparametric methodology.
All applied courses use and emphasize the importance of modern statistical computing software. Most courses are offered in the late afternoon or evening to accommodate students who are engaged in
professional development while maintaining full-time careers.
Applied Statistics Course Requirements
The plan of study must include the following five core courses: Applied Regression Analysis, Design of Experiments, Probability Theory, Applied Multivariate Analysis, and Mathematical Statistics
The remaining five courses are electives and should be chosen with the approval of the student's advisor, subject to one of the two options below:
• Option I- No Thesis: At least 9 credit hours must be taken in statistics beyond the core curriculum. The remaining courses may be taken in mathematics or in areas related to statistics, subject
to the approval of the academic advisor.
• Option II- Thesis:A 6-credit written thesis must be submitted for an oral defense. The thesis topic must be approved by the student's advisor. Also at least 6 credit hours must be taken in
statistics beyond the core curriculum. The remaining courses may be taken in mathematics or in areas related to statistics, subject to the approval of the academic advisor.
Comprehensive Examination
In addition to the above coursework, each student must pass a written and oral comprehensive examination over the material in the core courses. All students are strongly encouraged to complete the
comprehensive exam as soon as possible after completing the core courses and not at the very end of the program.
General Requirements and Policies
The Graduate Student Handbook contains the details of the program requirements, policies, and procedures.
Additional Information
For further information, contact Prof. Hanxiang Peng at 317-274-8070 or hpeng@math.iupui.edu. | {"url":"http://math.iupui.edu/graduate/degrees/ms/stats","timestamp":"2014-04-20T19:16:03Z","content_type":null,"content_length":"48810","record_id":"<urn:uuid:9f5a79ce-6838-4f45-b1aa-fd5d94f377cc>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00494-ip-10-147-4-33.ec2.internal.warc.gz"} |
Which year was Rivers Cuomo born in?
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Which year was Rivers Cuomo born in?
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Archives of the Caml mailing list > Message from oleg@p...
Re: (continuation monad) type problem...
Date: -- (:)
From: oleg@p...
Subject: Re: (continuation monad) type problem...
Pietro Abate wrote:
> but ContT is not a monad as the type inferred for bind is not
> polymorphic : bind : 'a k -> ('a -> 'a k) -> 'a k
> And I think this is true in general as soon as I define a monad type
> recursively.
That polymorphism problem is inherent; to avoid it we need to add
extra parameters to our continuation (as Jacques Carette pointed
out), or do something else. Namely, turn to delimited continuations
with multiple prompts. Truly undelimited continuations are hardly ever
called for. Multi-prompt delimited continuations are strictly more
expressive; they also have clear operational and denotational
semantics. The latter is the consequence of CPS transforms. Also,
multi-prompt delimited continuations have no typing problems and they
are available in the CC monad, which is truly a monad with no
> The point of the ContListM is to capture a list of computations ('a m),
> where each element (res * 'a k M.m) has the result of the computation up
> to a certain point (res in the example) and the list of all the
> continuations on that branch ('a k M.m). Each element of the list ('a m)
> corresponds to a computation branch. The idea is to create a fsa that
> stops at each state, asks for new input and resumes the computation
> following an external procedure. Alternation in the fsa gives the need
> to have an outer monad list to account different possible
> branches/computations that the fsa can choose.
I'm afraid I could not understand the problem. Why the conventional
FSA implementations won't work? I should point out that the
phrase ``result of the computation up to a certain point'' is
evocative of delimited continuations. Incidentally, delimited
continuations can be used to realize the LogicT monad (transformer,
actually) -- which is MonadPlus with additional operations supporting
a `look-ahead' in a non-deterministic computation. The latter is
needed to implement committed choice and "don't care"
nondeterminism. The implementation indeed maintains the tree of
continuations (each `mplus' builds a fork). The current
implementation is in Haskell
but it can be quite easily ported to OCaml, now that CC monad is
available here too. | {"url":"http://caml.inria.fr/pub/ml-archives/caml-list/2006/07/4b3eaebc6ef643648824bb78776c4a07.en.html","timestamp":"2014-04-17T09:51:24Z","content_type":null,"content_length":"7292","record_id":"<urn:uuid:f18fbd87-2c0e-453f-afc6-ba731a62d5fa>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00057-ip-10-147-4-33.ec2.internal.warc.gz"} |
RE : [ALGTOP-L] the (co)homology of \Sigma_n and O(n-1)
Joyal, André joyal.andre at uqam.ca
Thu Oct 16 00:06:14 EDT 2008
The theory of Q-rings can be used to answer your question.
See my paper with Terry Bisson
"Q-Rings and the Homology of the Symmetric Group"
(A Q-ring R is defined to be a commutative algebra equipped with a ring
homomorphism Q_t:R-->R[[t]] satisfing two simple conditions)
Your map
\phi_n: H_* B\Sigma_n \to H_*BO(n-1)
is closely related to the map
\psi_n: H_* B\Sigma_n \to H_*BO(n)
induced by the natural representation \Sigma_n \to BO(n)
I will describe the latter first.
It is easier to describe the maps phi_n globally by taking a direct sum:
\psi_*:H_* B\Sigma_* \to H_*BO(*)
where B\Sigma_* is the disjoint union of the classifying spaces B\Sigma_n
and BO(*) is the disjoint union of the classifying spaces BO(n).
The map psi_* is a map of Q-rings because it is induced by a map of E_\infty-spaces.
The Q-ring H_* B\Sigma_* is in fact the free Q-ring on one generator
Q<x> where x is the non-zero class in H_0(\Sigma_1).
Thus \psi_* is completely determined by \psi_*(x)=b_0 the non-zero class of H_0BO(1).
The Q-ring structure of H_*BO(*) can be described explicitly (by using a result of Priddy).
As an algebra, H_*BO(*) is a symmetric algebra on the homology of BO_1=RP^\infty.
It is thus a polynomial ring in an infinite sequences of variables b_0,b_1,b_3, ...
where b_n\in H_nRP^\infty is the non-zero class.
The Q-ring structure of H_*BO(*) is completely determined by the formula
where b(x) is the power series b_0+b_1x+b_2x^2+---
and where Q_t(b)(x)=Q_t(b_0)+Q_t(b_1)x+Q_t(b_2)x^2+---
Everything can be computed explicitly from these formulas.
For example, we have Q_t(b_0)=b_0b(t).
Finally, we have the relation
for every n>0, since the representation \Sigma_n-->B(n-1)
is obtained by cancelling the trivial representation from the
natural representation \Sigma_n-->B(n).
-------- Message d'origine--------
De: algtop-l-bounces+joyal.andre=uqam.ca at lists.lehigh.edu de la part de Dev Sinha
Date: mer. 15/10/2008 15:30
À: algtop-l at lists.lehigh.edu
Objet : [ALGTOP-L] the (co)homology of \Sigma_n and O(n-1)
I have a group cohomology question for the list. Let R: \Sigma_n \to
O(n-1) denote the homomorphism given by the reduced regular
representation. What is known about the effect of BR : B\Sigma_n \to
BO(n-1) on cohomology or homology?
I'm more interested in explicit information, in particular how
Stiefel-Whitney classes evaluate on the Dyer-Lashof-Kudo-Araki algebra,
but structural theorems would also be of interest.
ALGTOP-L mailing list
ALGTOP-L at lists.lehigh.edu
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How Does the Volume Change?
Date: 03/04/99 at 10:05:59
From: Caleb
Subject: How does the volume change when you change one side?
Recently in math class, we have been doing a project where we change
one side of an object and see how it affects the volume. If you are
doing this to a triangular prism and your formula is 1/2 the base
times length times height, could you cut the 1/2 in half instead of
the base, length, or height and get the same results?
Date: 03/04/99 at 13:06:10
From: Doctor Rick
Subject: Re: How does the volume change when you change one side?
It sounds as if you have noticed that whatever numbers you put in the
formula, when you make the base, length or height half as big, the
answer is the same as
1/4 * base * length * height
and you want to know if this will always work.
If so, the answer is yes. What you are asking about is algebra. If you
have a triangular prism and you reduce the length (for example) by a
factor of 2, the volume of the new prism is
Vnew = 1/2 base * (length/2) * height
You can rearrange the operations using the rules of algebra (the
commutative and associative principles, in particular). It does not
matter what numbers base, length, and height stand for; these will
always be equal:
1/2 * base * (length/2) * height
1/2 * base * (1/2 * length) * height (commutative principle)
1/2 * (base * 1/2) * length * height (associative principle)
1/2 * (1/2 * base) * length * height (commutative principle again)
(1/2 * 1/2) * base * length * height (associative principle again)
1/4 * base * length * height
This formula will work for any base, length, and height, because the
commutative and associative principles are true for any numbers.
If this is new to you, then you are discovering for yourself how
algebra works, and that is great. If you have seen it before, I hope
this helps you see how these principles can be put to work for you.
- Doctor Rick, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/57469.html","timestamp":"2014-04-19T03:08:58Z","content_type":null,"content_length":"6856","record_id":"<urn:uuid:aac7a9f5-acb4-4c5b-8582-342f89c4a126>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00290-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is the diffraction limit of a telescope?
The minimum angular separation of two sources that can be distinguished by a telescope depends on the wavelength of the light being observed and the diameter of the telescope. This angle is called
the DIFFRACTION LIMIT.
In this equation, the computed angle is in radians. Remember that there are 206,265 arcseconds in one radian.
Here are some examples of calculations. | {"url":"http://astro.cornell.edu/academics/courses/astro201/diff_limit.htm","timestamp":"2014-04-20T23:26:55Z","content_type":null,"content_length":"1223","record_id":"<urn:uuid:93add685-1ae2-40a8-9fed-6aa7dafa034d>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00283-ip-10-147-4-33.ec2.internal.warc.gz"} |
This collection of activities from Carleton College faculty is a subset of the larger set of examples available through the National Numeracy Network.
Current Search Limits
showing only
1 match
A Quantitative Analysis of Pausanias' Testimony on Athletic Statues at Olympia part of Examples
Students read an ancient account of the statues of athletes at Olympia and extract from it a set of data presented in a spreadsheet. After analyzing the data using tables, graphs, and maps, the
students use quantitative reasoning combined with qualitative arguments to write a paper about how Pausanias' account of the statues reflects both his own biases and the cultural norms of ancient
Greek athletics. | {"url":"http://serc.carleton.edu/quirk/quantitative_writing/examples.html?q1=sercvocabs__43%3A620","timestamp":"2014-04-16T22:06:29Z","content_type":null,"content_length":"20613","record_id":"<urn:uuid:29d95ab0-35f9-49ce-9c64-f1408f36b51c>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00454-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rethinking How We Use Calculators
May 16, 2013 | José Vilson
If ever you come across a set of math teachers, whether at a common planning meeting or a bar during happy hour, bring up the conversation of calculators and watch the sparks fly. The arguments for
and against calculators have the spirited vigor of a Red Sox vs. Yankees game without the animus. One side argues for the use of efficient and available technology in the classroom, while the other
argues for numeracy and fluency to the highest order.
In other words, are you old school or new school?
Two Schools of Thought
The old school suggests that, in order to develop a rich sense of numbers and fluency, we shouldn't allow students to use calculators. In a world over-equipped with technology tools, students must be
able to do operations without the calculator there. In this school of thought, calculators strip students of curiosity about how numbers work because they can arrive at the answer just by pressing a
few buttons, not by going through the long-established procedure of finding the answer. Calculators already spit out everything from long multiplication and division to graphs and solutions to
simultaneous equations. The old school crowd isn't completely anti-technology; many of them stand by
The Geometer's Sketchpad
. They just wouldn't want all the mysteries and intricacies of math unlocked so quickly.
The new school suggests that we take a different outlook on the calculator issue. If we can so readily solve problems with a calculator, then why not give one to our students? The old way of writing
out the multiplication table in both a table and list form is antiquated and tiresome. The calculator is also much more efficient, reducing the amount of time one spends on a problem. For instance,
imagine trying to divide a seven digit number by a two digit number higher than 12? Such a task seems tedious when a calculator can do it in a fraction of the time. Even operations with fractions can
be simplified with calculators, so finding things like the least common denominator or remainders feels pointless if the screen just told the student the answer.
My response to this one lies somewhere in the middle. I'm not totally against calculators. I use them frequently enough when creating answer keys for my exams -- after I take time to do the problems
myself. I use them when working on my taxes, and I've used one when trying to get a new couch for my mom to fit in her apartment. (Thank you, Pythagorean Theorem!)
The Right Tool Used the Right Way
Where I start siding with the old school mathematicians is in this: how do we
that the calculator is telling us the truth? Numbers don't lie, but humans make meaning of these numbers and hope to ascertain how they apply in the context given. If we rely solely on calculators
without giving much thought to the number we've put down, or simply assuming the calculator is always right, then we end up with everything from wrong answers to the financial collapse of 2008.
Calculators are tools to help solve problems, not the solver of the problem itself. Our students need to develop a sense of numeracy that allows them to estimate the distance between two items, for
example. We can't underestimate the usefulness of looking at numbers and making quick calculations by comparing those numbers without having to pull out a smartphone.
The calculations we make on the fly matter more than the ones we make in math or science class, yet these classes are where students get the most explicit reasons for using them. In larger math
problems, I can see the usefulness of a calculator. For instance, when finding the differences between planets in scientific notation, we shouldn't have to plug away at doing operations to the first
factors. However, if we insert the notation wrong in our calculator, we could end up with a number bigger than we bargained for.
We need to think about the way we use calculators, or any piece of technology. Assuming that the tool is doing exactly what we're asking it to do -- or actually has the answer to the question we've
asked -- is a dangerous proposition for the non-thinker. We need a healthy balance of working within the number system and doing more complex problems. We need to let calculators serve their purpose
in moderation.
What do you think?
see more see less
Comments (14)
Comment RSSSign in or register to post comments
I believe that students should not be able to use calculators in the beginning stages of building their foundation skills in mathematics.
Students need to build the confidence in doing simple arithmetic and calculations so when they achieve higher levels of mathematics they can then focus their learning on more complex concepts.
Students can then check their work via the calculator.
Again: I don't understand why this serious debate about calculators still rages on: This problem has already been solved by the QAMA calculator- retaining the indispensable benefits of calculators
but without the damage they do: This calculator shows the result only after the user also inputted a suitably reasonable mental estimate. See
I began teaching 6th-8th grade 4 years ago. I had a number of students in 6th grade very weak in math facts and they
transferred that weakness to their overall math skills. After a few months I insisted they use the calculator and would penalize if they did not. Over the course of the next three years their math
skills grew, their confidence grew and their reliance on the calculator decreased. If I noticed a student using the calculator in place of their brain as this point I would mimic the old cowboy quick
drawing his gun in a shoot-out. They would laugh and use their brain. I have noticed some students have excellent recall of math facts and that helps with computation but it does not also transfer to
high order thinking skills and vice versa. So basically I say get to know your students strengths and weaknesses and give the tools that will allow them to grow in both areas.
I have this exact debate with myself almost every day when I'm teaching. I'd say I'm a little bit of both old school and new school when dealing with the calculators. I allow my students to have the
calculators on the desk, however, there are many times where I will make the students put the calculator cover on and use scratch work to solve the problem. For example, I refused in the beginning to
allow students to use calculators for order of operations problems because they were putting something incorrectly into the calculator (whether it be forgetting a parenthesis or using 2 as a factor
instead of an exponent), and I was trying to stress the correct order in which to solve the problems. Even still now that we're at the end of the year, when a student tries to quickly solve an order
of operations problem on the calculator, and they don't see the answer there in their multiple choice, they know to try to solve it on paper with only their brains, and they then are able to get the
correct answer. Now on the flip side, calculators are good at catching small errors in the work, and I think as long as my kids write down their work and the process they are taking to solve the
problem (and WHY they're doing that such process) then I'm very okay with them using calculators. Ah! I'm just so back-and-forth with this every day!
One last thing: I used to play a game with my students back when I taught 5th grade to help them with times tables and division, but also to show them the power of their own brains. It was called
"The Brain vs The Calculator" (fancy title ha), and one student had a calculator, and the other didn't. I would ask a multiplication fact, and The Brain was able to yell it out whenever they had the
answer. The Calculator wasn't allowed to yell it out until they had the answer on their calculator screen. It most cases, The Brains were answering much quicker, while The Calculators were still
trying to type it in. I was just trying to show my students that yes, you can rely on your own brain, and it can sometimes be quicker for the smaller math problems!
I have yet to see secondary math curriculum that addresses decimal place accuracy and errors for numerical approximations made on the calculator. Round off, truncation, and discretization error
should be addressed when we use the calculator to calculate irrationals, non-terminating non-repeating decimals, root approximations, or any of the numerical differential or integration methods.
Additionally, I'm surprised that the debate about calculators hasn't been replaced by one about computer use in mathematics classrooms. Sadly and ironically, computers in math classrooms are rarely
used to compute. Instead, they are largely used for direct delivery of content, assessment, and for word processing (Powerpoints for presentations). I look forward to conversations about programing
and computational software use in our math classrooms.
I see that it may have come across that way, but I'm proud to be old school, too. Thanks for bringing up this discussion. I can only hope lots of primary school educators think more deeply about
As an element of "old school" classification, I take issue with your supposed reasons for being "old school." I'm not so worried about the richness, mystery, etc., I personally think the use of
calculators in early math classes diminishes students' ability to do math at higher levels. Students prefer to use calculators because it's easier. Well, maybe so in grade school, but in high school
students without number skills simply don't do the math--even though they have and are allowed to use calculators. Why? Because if you can't complete a trinomial square because you don't know the
numbers, you're not going to be motivated to take out your calculator at every step. How many problems would you last doing algebra this way, decimal approximations and all? It's simply too
cumbersome to be effective for algebra/calculus. (Of course, not if one is talking about graphing/solver calculators--but my thoughts are geared more toward elementary/middle school.)
Also, I'm amazed by this notion we've taught our kids that arriving at some "easy to understand" number as the answer is the most important thing. Why should a student desire to see an approximation
of 1.41 instead of the exactness of the square root of 2. (And, why 2 decimal places? Maybe we should just call it 1. Then 1^2=2. Ha.) But seriously, in mathematics we have the rare opportunity to be
exact. Where else in a person's life does this exist? In my opinion, mathematics hits the real world in measurement, science, statistics, economics, etc. Students can get plenty of exposure to this
in other classes. When does 1.41 serve better than the square root of 2? Yup, science lab. I get that. Let the science teachers worry about calculators and approximations. Let the mathematics
teachers teach mathematics. Then it's not a calculators vs. non-calculators debate--the student with the abilities to do it both ways can choose based on the situation.
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Posts by
Posts by jane
Total # Posts: 1,318
(6q^6)^-4 When I worked this I got 1/1296q^24. One of the choices given was 1/1296q^-24. The other choices were: 6q^1296, 1296q^2, 6q^-24. Is the answer any of these choices?
Thank you so much for taking the time to explain it.
Simplify: (-h^4)^5 Would it be -h^20 or h^20
Find the slope of the given line, if it is defined y = (x + 4)/7 I can't find the answer. I don't understand what went wrong with my calculation. Thank you.
I got it, its undefined
Find the slope of the given line. 2x + 3 = 0 Do I use the slope formula? I don't know how to do it. Thanks
When Romeo and Juliet meet they speak just fourteen lines before their first kiss. These fourteen lines make up a shared sonnet, A sonnet is a perfect, way of forming a poem writing about love. In
the fifth scene the lovers share a sonnet which use images of saints and pilgrim...
i wasn't going to right this down i was just going to wright ideas down on my page an use them
could you please check this!
Is this what i could right for it all is their anything else i could add sorry for all these questions thanks
A sonnet is a perfect, idealized poetic form often used to write about love.Romeo uses religious imagery to convey his feelings. The use of words such as "holy shrine, faith, pilgrims, sin, devotion"
give a feeling of deep sincerity to his first talk with Juliet. Lik...
no not really
Thank very much that's a great help,would this be the reason why their is sonnetWhen Romeo and Juliet meet they speak just fourteen lines before their first kiss. These fourteen lines make up a
shared sonnet, with a rhyme scheme. A sonnet is a perfect, idealized poetic for...
Love at first sight: The ball Romeo sees her first. Their first words are a shared sonnet: analyse the reason for this. Analyse the language and the imagery. What do we learn about her feeling's?
At the great hall of the Capulets Romeo sees Juliet from across the room, and asks a servingman who she is. The servingman does not know. Romeo is transfixed; Rosaline vanishes from his mind and he
declares that he has never been in love until this moment. Moving through the c...
i know i didn't want the answers i just wanted help Thanks
love at first sight the ball romeo sees her first. their first words are a shared sonnet analyse the reason for this. analyse the language and the imagery what do we learn about her feeling? ROMEO
A block (m = 38.0kg) sits on an inclined plane of 38 degrees. If there is no friction,k what magnitude force is needed to keep the block from sliding? If there is friction, with coefficients 0.580
and 0.630, what is the maximum magnitude force that can be applied before the bl...
there seems to be something wrong with the names a different one keeps coming up each time
wondering would you know this..what do we learn about juliet from this?
We first meet juliet (act 1 scene 3) she has a conversation with her mother about marriage. discuss this. what do we learn about juliet from this? ROMEO AND JULIET
Solve by factoring: 12 w(squared) = 28w +5
Reiny, thank you so much. You broke it down so it was easy to understand.
How do I set this up and how do I solve: The sum of two numbers is 20 The difference between 3 times the larger number and twice the smaller number is 40. What is the larger number?
Please show me how to set this up and how to solve: A ribbon with straight edges has an area of 24 inches squared. Its width is x and its length is 2x + 13. What is the width of the ribbon in inches?
A school is fencing in a rectangular area for a playground. It plans to enclose the playground using fencing on three sides (One length is a wall) The school has budgeted enough money for 75 ft of
fencing material and would like to make a playground with an area of 600 ft squa...
A car, mass - 1950 kg, accelerates at +2.33 m/s (squared). Find the magnitude of the normal force acting on the car. If later, when the car is cruising at 26.5 m/s, the driver applies a force of
magnitude 10,000 N to stop the car. What is the distance it takes for the car to c...
A 70kg person dives horizontally from a 200kg boat with a speed of 2 m/s. What is the recoil speed of the boat?
Please show me how to set it up and solve: You are making a rectangular table. The area of the table should be 10ft squared. You want the length of the table to be 1ft shorter than twice its width.
What should the dimensions of the able be?
Please show me how to set it up and solve: You are making a rectangular table. The area of the table should be 10ft squared. You want the length of the table to be 1ft shorter than twice its width.
What should the dimensions of the able be?
Please show me how to set it up and to solve: A cell phone company sells about 500 phones each week when it charges $75 per phone. It sells about 20 more phones per week for each $1 decrease in
price. The company's revenue is the product of the number of phones sold and th...
The area of a rectangle is 6n(to the power of 2) + n - 2. What is the expression that represents the perimeter of the rectangle?
I'm sorry, its r(x) = (4x^2 + 1)/(4x^2 − 1); x = −1, 0, 1, ..., 9
First, give the technology formula for the given function and then use technology to evaluate the function for the given values of x. (Round your answers to four decimal places.) r(x) = 4x2 + 1/4x2 −
1; x = −1, 0, 1, ..., 9 a)(4x^2 − 1)/(4x^2 + 1) b)(4x + 1)^...
In MLA (version 7), how do cite a textbook in a works cited page? Would it be the same as if you were citing an encyclopedia article or does it have a different citation?
Sometimes I get confused with the inequalities. You have $47 to spend on music and movie downloads. Each album down load cost $7 and each movie download costs $8. Write and graph a linear inequality
that represents this situation. Let x represent the number of albums and y the...
Reid and Maria both play soccer. This season, Reid scored 4 less than twice the number of goals that Maria scored. The difference in the number of goals they scored was 6. How many goals did each of
them score?
Solid mensuration
Find an equation of the circle tangent to x + y = 3 at (2,1) and with center on 3x - 2y - 6 = 0
How do I set this up and solve? You are planing a rectangular dining pavilion. Its length is 3 times its width x. You want a stone walkway that is 3 ft wide around the pavilion. You have enough
stones to cover 396 ft squared and want to use them all in the walkway. What should...
How does most state constitutions differ from the US Constitution?
The Bill of rights was ratifid by what fraction of all state legislatures?
What is the function of the Articles of the Constitution?
Please explain to me how to solve: A circular mirror is surrounded by a square metal frame. The radius of the mirror is 5x. The side length of the metal frame is 15x. What is the area of the metal
frame? Write your answer in factored form.
A circular table is painted yellow with a red square in the middle. The radius of the tabletop is 6x. The side length of the red square is 3x. What is the area of the yellow part of the tabletop?
Write your answer in factored form.
How do I set this up and how to solve it? Russ bought 3 medium and 2 large submarine sandwiches for $29.95. Stacy bought 4 medium and 1 large submarine sandwiches for $28.45. What is the price for
each medium and each large submarine sandwich?
Please explain how to solve: A circular table is painted yellow with a red square in the middle. The radius of the tabletop is 6x. The side length of the red square is 3x. What is the area of the
yellow part of the tabletop? Write your answer in factored form.
Please explain to me how to solve: A circular mirror is surrounded by a square metal frame. The radius of the mirror is 5x. The side length of the metal frame is 15x. What is the area of the metal
frame? Write your answer in factored form.
basic math
I need to convert 40mg/mL to 5mg/mL with distilled water. How much distilled water would I use to do this?
Please show me how to solve: Doctors can use radioactive iodine to treat some forms of cancer. The half-life of iodine - 131 is 8 days. A patient receives a treatment of 12 millicuries of iodine -
131. (A mullicurie is a unit of radioactivity.) How much iodine - 131 remains in...
Hydra are small freshwater animals. They can double in number every two days in a laboratory tank. Suppose one tank has an initial population of 60 hydra. When will there be more than 5000 hydra? How
can a table help you identify a pattern? What function models the situation ...
Hydra are small freshwater animals. They can double in number every two days in a laboratory tank. Suppose one tank has an initial population of 60 hydra. When will there be more than 5000 hydra? How
can a table help you identify a pattern? What function models the situation (...
social studies
who was the 8th US president
what is the square of 13
I've just begun to study Exponential functions: When you make a table starting with: -2 -4^x What are my coordinates? -1 0 1 2
Alg. I
I've just begun to study Exponential functions: When you make a table starting with: -2 -4^x What are my coordinates? -1 0 1 2
Finding a break-even point: How do I set up the equation and solve? Producing a musical costs $88,000 plus $59,000 per performance. One sold-out performance earns $7500 in revenue. If every
performance sells out, how many performances are needed to break even?
Sorry, Reiny, I copied the 2nd choice incorrectly. I had posted this problem with you before and the answer given that was not equalalent to A=1/2/bh was 2b =A/h. The problem is: A = 1/2bh, which is
not equivalent? Choices: 2A =bh, h=2A/b, 2b=A/h, b=2A/h When I got my test bac...
A recipe for trail mix calls for 3 cups of raisins and 2 cups of peanuts. If you plan on using 15 cups of raisins, how many cups of peanuts do you need?
I had posted this problem with you before and the answer given was 2b =A/h. The problem is: A = 1/2bh, which is not correct? Choices: 2A =bh, b=2A/b, 2b=A/h, b=2A/h When I got my test back 2b=A/h was
correct. Did I make a mistake or was their answer incorrect?
Algebra I
Exponential Function. Please show me how to solve: g(t) = -0.5 x 4w (that is 4 to the power of w) w=18
Which equation is not equivalent to A = 1/2bh 2A=bh h=2A/b 2b=A/h b=2A/h I think it is 2b=A/h. Is this correct?
The Venn diagram below shows the dinner orders for a local restaurant. This was submitted before, but no one answered. Thanks! How many people did NOT order steak for dinner? Steak, chicken & fish
were order. In the Venn circles, the total that did not order steak was 56. Ther...
The Venn diagram below shows the dinner orders for a local restaurant. How many people did NOT order steak for dinner? Steak, chicken & fish were order. In the Venn circles, the total that did not
order steak was 56. There is 43 in the right lower corner of the rectangle that ...
Please explain how to solve: You have 11 cups of flour. It takes 1 cup of flour to make 24 cookies. The function c(f) = 24f represents the number of cookies, c, that can be made with f cups of flour.
What domain and range are reasonable for the function? My choices are: Domain...
Given A = {2,4,6,8,10} B= {1,2,3,4,5,6,7,8,9,10} What is (A ¿B) ? Is it {2,4,6,8,10}?
Please explain how to solve: 1/3 = p-2/p+8
What equation do you get when you solve for x? -g = sx I got -g/s = x. Is this correct?
You have 11 cups of flour. It takes 1 cup of flour to make 24 cookies. The function c(f) = 24f represents the number of cookies, c, that can be made with f cups of flour. What domain and range are
reasonable for the function?
Sorry about the symbols before. Admission to the fair costs $7.75. Each ride costs you $0.50. You have $15 to spend at the fair including admission. Which inequality represents the number of rides
you can ride? r ¡Ý 15 (r is more than or equal to 15) r ¡&Uum...
Admission to the fair costs $7.75. Each ride costs you $0.50. You have $15 to spend at the fair including admission. Which inequality represents the number of rides you can ride? r ¡Ý 15 r ¡Ü 14 r <
14 r > 14
The table shows the cost of a ski rental package for a given number of people. Using the rate of change, what is the cost for 13 people? People Cost ($) 2 70 3 105 4 140 5 175 There are 4 answer
choices - 560, 440, 520, 480. I got 455. Am I wrong?
Given A = 2,4,6,8,10 B= 1,2,3,4,5,6,7,8,9,10 What is (A ¿B) ?
Explain how to solve? Which equation below shows the equation y = 1/4x + 4 written in standard form using integers? A) 2x + 3y = 21 B) 3x 2y = 21 C) 2x + 3y = 21 D) 2x 3y = 21
What is the value of p in the proportion below? 1/3 = P-2/p+8
Language - PYTHON Design a program that calculates the total amount of a meal purchased at a restaurant. The program should ask the user to enter the charge for the food, and then calculate the
amount of a 15% tip and 7% sales tax. Display each of these amounts and the total. ...
A copy center offers its customers two different pricing plans for black and white photocopies of 8.5 in. by 11 in. pages. Customers can either pay $0.08 per page or pay $7.50 for a discount card
that lowers the cost to $0.05 per page. Which equation can be used to solve for ...
The table shows the height of an elevator above ground level after a certain amount of time. Let y stand for the height of the elevator in feet and let x stand for the time in seconds. Which equation
models the data below? Time (s) Height (ft) 10 235 20 220 40 190 60 160
Please explain in detail on how to solve: In 2000, the U.S. government owed about $4.63 trillion to its creditors. The population of the US was282.4 million people. How much did the government owe
per person in 2000? Round to the nearest dollar. In 2005, the debt had grown to ...
Please explain how to solve: The wavelength of a radio wave is defined as speed divided by frequency. An FM radio station has a frequency of 9 x 10(power of 7) waves per second. The speed of the
waves is about 3 x 10 (power of 8) meters per second. What is the wavelength of th...
Please explain in detail how to solve this: During one year, about 163 million adults over 18 years old in the U. S. spent a total of about 93 billion hours online at home. On average, how many hours
per day did each adult spend online at home? How do you write each number in ...
Please explain in detail how to solve each question: At a certain high school, 350 students are taking an algebra course. The ratio of boys to girls taking algebra is 33:37. How many more girls are
taking algebra than boys? How can you write a system of equations to model the ...
An artist is going to sell two sizes of prints at an art fair. The artist will charge $20 for the small print and $45 for a large print. The artist would like to sell twice as many small prints as
large prints. The booth the artist is renting for the day cost $510. How many of...
During one year, about 13 million adults over 18 years old in the U.S. spent a total of about 93 billion hours online at home. On average how many hours per day did each adult spend on line at home?
How do you write each number in scientific notation? How do you convert the un...
During one year, people in the U.S. older than 18 years old watched a total of 342 billion hours of television. The population of the U.S. older than 18 years old was about 209 million people. On
average, how many hours of television did each person older than 18 years old wat...
During one year, about 13 million adults over 18 years old in the U.S. spent a total of about 93 billion hours online at home. On average how many hours per day did each adult spend on line at home?
How do you write each number in scientific notation? How do you convert the un...
You have $47 to spend on music and movie downloads. Each album download costs $7 and each movie download costs $8. Write and graph a linear inequality that represents this situation. Let x represent
the number of albums and y the number of movies.
Please explain how to solve: Reid and Maria both play soccer. This season, Reid scored 4 less than twice the number of goals that Maria scored. The difference in the number of goals they scored was
6. How many goals did each of them score?
I believe the answer is 5 months; however, I did it the long way. Explain how to find the answer. Tom has a collection of 30 CDs and Nita has a collection of 15 CDs. Tom is adding 1 CD a month to his
collection while Nita is adding 4 CDs a month to her collection. Find the num...
The vertices of quadrilateral ABCD are (A(1,1), B(1,5), C(5,5) and D(7,1). You want to transform ABCD into a parallelogram by only moving point B. A parallelogram is a four-sided figure with both
pairs of opposite sides parallel. What should be the new x-coordinate of point B?
Explain how to solve: Sam is ordering pizza. Tony's Pizza charges $7 for a large cheese pizza plus $.75 for each additional topping. Maria's Pizza charges $8 for a large cheese pizza plus $.50 for
each additional topping. For what number of toppings will the cost of a ...
Explain how to solve: A chemist has one solution containing 30% insecticide and another solution containing 50% insecticide. How much of each solution should the chemist mix to get 200 L of a 42%
A farmer grows only pumpkins and corn on her 420-acre farm. This year she wants to plant 250 more acres of corn than pumpkins. How many acres of each crop should the farmer plant?
Please show me how to solve: The length of a rectangle is 3 times the width. The perimeter is 44 cm. What are the dimensions of the rectangle?
A corner store sells two kinds of baked goods: cakes and pies. A cake costs $14 and a pie costs $8. In one day, the store sold 12 baked goods for a total of $144. How many cakes did they sell?
You have $47 to spend on music and movie downloads. Each album download costs $7 and each movie download costs $8. Write and graph a linear inequality that represents this situation. Let x represent
the number of albums and y the number of movies.
How many solutions does the system have? y - 5x = -6 3y - 15x = -12 A) infinitely many solutions B) no solution C) two solutions D) one solution
Please show me how to solve: Sharon has some one-dollar bills and some five-dollar bills. She has 14 bills. The value of the bills is $30. Solve a system of equations using elimination to find how
many of each kind of bill she has.
Please show me how to solve: At the local ballpark, the team charges $5 for each ticket and expects to make $1,300 in concessions. The team must pay its players $1,800 and pay all other workers
$1,500. Each fan gets a free bat that costs the team $3 per bat. How many tickets m...
A local citizen wants to fence a rectangular community garden. The length of the garden should be at least 110 ft, and the distance around should be no more than 380 ft. Write a system of
inequalities that models the possible dimensions of the garden. Graph the system to show ...
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Simon Kirwan Donaldson
Born: 20 August 1957 in Cambridge, England
Click the picture above
to see three larger pictures
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Simon Donaldson's secondary school education was at Sevenoaks School in Kent which he attended from 1970 to 1975. He then entered Pembroke College, Cambridge where he studied until 1980, receiving
his B.A. in 1979. One of his tutors at Cambridge described him as a very good student but certainly not the top student in his year. Apparently he would always come to his tutorials carrying a violin
In 1980 Donaldson began postgraduate work at Worcester College, Oxford, first under Nigel Hitchen's supervision and later under Atiyah's supervision. Atiyah writes in [2]:-
In 1982, when he was a second-year graduate student, Simon Donaldson proved a result that stunned the mathematical world.
This result was published by Donaldson in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in the Bulletin of the American Mathematical Society in 1983. Atiyah
continues his description of Donaldson's work [2]:-
Together with the important work of Michael Freedman ..., Donaldson's result implied that there are "exotic" 4-spaces, i.e. 4-dimensional differentiable manifolds which are topologically but not
differentiably equivalent to the standard Euclidean 4-space R^4. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces
have the remarkable property that (unlike R^4) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere !
After being awarded his doctorate from Oxford in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983-84 at the Institute for Advanced
Study at Princeton. After returning to Oxford he was appointed Wallis Professor of Mathematics in 1985, before moving to Imperial College, London in 1999.
Donaldson has received many honours for his work. He received the Junior Whitehead Prize from the London Mathematical Society in 1985. In the following year he was elected a Fellow of the Royal
Society and, also in 1986, he received a Fields Medal at the International Congress at Berkeley. In 1991 Donaldson received the William Hopkins Prize from the Cambridge Philosophical Society and, the
following year, the Royal Medal from the Royal Society. He was awarded the London Mathematical Society's Polya prize in 1999 and more recently, the 2006 King Faisal Prize. He also received the
Crafoord Prize from the Royal Swedish Academy of Sciences in 1994:-
... for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ...
Atiyah describes the contribution which led to Donaldson's award of a Fields Medal in [2]. He sums up Donaldson's contribution:-
When Donaldson produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration. Slowly the message
has gotten across and now Donaldson's ideas are beginning to be used by others in a variety of ways. ... Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about
the geometry of 4-dimensions have been discovered. Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is
firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry.
The article [3] is very interesting and provides both a collection of reminiscences by Donaldson on how he came to make his major discoveries while a graduate student at Oxford and also a survey of
areas which he has worked on in recent years. Donaldson writes in [3] that nearly all his work has all come under the headings:-
(1) Differential geometry of holomorphic vector bundles.
(2) Applications of gauge theory to 4-manifold topology.
and he relates his contribution to that of many others in the field.
Donaldson's work in summed up by R Stern in [6]:-
In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which
much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds.
Donaldson was elected to the National Academy of Sciences in 2000. In February 2006 he was awarded the King Faisal International Prize for science for:-
... seminal contributions to theories which have strengthened the links between mathematics and physics, and helped provide a rigorous foundation for physical theories giving a very good
description of the laws of matter at the sub-nuclear level.
In April 2008, he was awarded the Frederic Esser Nemmers Prize in Mathematics from Northwestern University. The Prize was given for his:-
... groundbreaking work in four-dimensional topology, symplectic geometry and gauge theory, and for his remarkable use of ideas from physics to advance pure mathematics.
John Franks, the Chair of Mathematics at Northwestern University, gave more details of Donaldson's contributions which led to the award:-
Donaldson's breakthrough work developed new techniques in the geometry of four-manifolds and the study of their smooth structures. His methods have been described as extremely subtle, using
difficult nonlinear partial differential equations. Using instantons, solutions to the equations of Yang-Mills gauge theory, he gained important insight into the structure of closed
four-manifolds. Gauge theory techniques also enabled him to show the existence of four-manifolds with no smooth structure and others with infinitely many. His work has provided the seminal steps
for the work of others in study of four-manifolds.
In 2009 Donaldson, together with Clifford H Taubes, was awarded the Shaw Prize in Mathematical Sciences. The Committee who made the award wrote that Donaldson and Taubes:-
... are the two geometers who have transformed the whole subject by pioneering techniques and ideas originating in theoretical physics, including quantum theory [and] have totally changed our
geometrical understanding of space and time.
Donaldson was knighted in 2012.
Article by: J J O'Connor and E F Robertson
Click on this link to see a list of the Glossary entries for this page
List of References (6 books/articles)
Mathematicians born in the same country
Honours awarded to Simon Donaldson
(Click below for those honoured in this way)
LMS Whitehead Prize 1984
BMC morning speaker 1986
Speaker at International Congress 1986
Fields Medal 1986
Fellow of the Royal Society 1986
Royal Society Royal Medal winner 1992
LMS Pólya Prize 1999
BMC plenary speaker 2000
Shaw Prize 2009
Cross-references in MacTutor
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JOC/EFR © September 2009 School of Mathematics and Statistics
Copyright information University of St Andrews, Scotland
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Randy Glasbergen
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How Many Lemons Does it Take?\
Utilization Strategies
Lesson Plans
Meet the Teachers
HOW MANY LEMONS DOES IT TAKE?
by April Olson
University of Idaho
ITV SERIES The Challenge of the Unknown
GRADES 5 - 7
Students will discuss and learn about different methods that can be used to estimate from the video. Students will then practice estimation skills by recreating a sample problem from the video. Web
sites containing other estimation activities are included.
• Students will be able to identify estimation problems in every day life.
• Students will practice estimation skills by recreating a Transect Sampling.
• Students will be introduced to characteristics of Lemon Sharks.
Students need specific responsibilities while viewing the video. Hand out the "Am I Close?" focus-video guide and read over with the students. Tell them these are the things they want to watch
closely for in the video.
However, they do not need to write the answers down as they hear them. The video will be paused several times, during which students will discuss answers and have time to write.
Also, tell students not to worry about the questions on the back of the handout; they will be dealt with later.
• Challenge of the Unknown teaching guide and video series
• Copies of the "Am I Close?" focus-video guide for the class
• "A Fact Sheet on Lemon Sharks"-included in teaching guide
• "Comparative Size" chart-included in teaching guide
• Overhead of Question 4 from the "Getting Started: Thinking Estimation" worksheet-information available in teaching guide(need to make into overhead)
• Overhead of the Bimini Islands drawing, taken off of "Transect Sampling: A Birds-Eye View"-information available in teaching guide.
• Masking tape
• Goldfish Crackers
• Lesson will take approximately 60-70 minutes-depending upon how much time is spent on discussion questions. May split into two sessions if needed
The teacher will introduce the lesson by posing an estimation problem.
Put up overhead of question #4 (triangle problem) and write the students estimations on the chalkboard.
Then ask them to count the actual number of triangles and decide if they made good estimates. *Answer: 80 triangles
Ask students if they think it is always easy to estimate. Can they think of some things that might be hard to estimate? Do they think it is possible to have to research a problem before estimating?
Ask if anyone has heard of Lemon Sharks. If so, what do they know about them? Then, (as a class) read over "A Fact Sheet on Lemon Sharks" and then show the "Comparative Sizes" chart.
Ask students what they can tell you about Lemon Sharks when comparing them to others.
The tape needs to be set on the first segment, "One If by Air. . .Two If by Sea," of the video section, "Estimation: Am I Close?"
PLAY the video. PAUSE right after the narrator says, "Lemon Sharks" (Visual: Samuel Gruber is hooking his ultralight to the boat). Ask students what Samuel Gruber's profession is. Discuss what a
marine biologist is and what they do. *Ask students if anyone has heard of the name of the lagoon-Bimini Lagoon-write the name on the board. Ask if they remember where Bimini Lagoon is located. What
state is it by? Then show students Bimini Lagoon (near Florida) on a map.
RESUME the video. PAUSE right after narrator says, "He still needed an accurate survey of the whole lagoon." Ask students if they can explain the tag-and-recapture method. Then ask why this method
wasn't accurate for Gruber's estimation.
RESUME the video. PAUSE right after narrator says, "In his ultralight Gruber can easily cover the whole lagoon." Ask the class what is an ultralight and what does it do? Do they think Gruber will
have better luck using the ultralight to make an estimation than he did with tag-and recapture?
RESUME the video. PAUSE right after the narrator says, "strip by strip" (Visual: aerial picture of lagoon being shown on screen). Ask students what system Gruber uses to survey the entire lagoon with
the ultralight.
RESUME the video. STOP the video after narrator says, "Two educated guesses are better than one." Ask students if Gruber's survey of the Lemon Sharks by air was problem free. Why or why not? Then ask
the class why they think Gruber decided to use two methods instead of one. Do they think this helped him make his estimation more accurate? Why?
Post Viewing Activities
· (5-10 minutes) Classroom Discussion: As a class, discuss the questions on the back of the "Am I Close?" handout. (20-25 minutes) Students are going to try out the transect sampling method by
recreating Gruber's situation from the video. Show the overhead of the Bimini Islands drawing from the "Transect Sampling: A Bird's-Eye View" activity sheet.
Point out that the transect strips are shaped like rectangles. Tell students that they are going to pretend the classroom is the Bimini Islands and lagoon. We have to estimate how many lemon sharks
are in the lagoon. We will be surveying the area by air using the transect method. In order to carry out the survey the water must be divided into transects. Tell the students they can work together
and do this by using tape.
*Note: If the floor is already divided by wooden beams or linoleum squares they can be used instead of tape. Before students start working, the area that represents the lagoon must be defined. Also,
for this project, students will need to create 20 transects. After the students have finished sectioning off the "water" disperse the goldfish crackers onto the floor (not evenly). Next, students
will take 5 of 20 transects and count the goldfish in each transect. Any goldfish that are on the lines between transects should be counted as part of the transect they are "swimming" into.
From this information, have the students make individual estimations (record them) of the total number of goldfish in the "lagoon." Then actually count the goldfish. How close were their estimations?
Repeat the experiment, this time count the fish in 10 transects. Were the estimations closer than before?
Afterwards pull the students together for a discussion. What did they learn from this? Was it better to count more transects? Did it give them a more accurate estimation? What does this tell them?
Students can start an Estimation Journal. The first page should consist of a list of all the possible things a person could use estimation skills for. Students should write in their journal (every
day) about all of the estimation they participated in that day.
The journal will help students see how often they actually use estimation and how important it is to practice their skills.
Students should also be encouraged to bring in things the whole class can estimate. Example: Jars filled with something unique.
If you are interested in finding more estimation activities you might want to check out these web sites: (for a variety of grades)
There is another great activity listed in The Challenge of the Unknown teaching guidebook. The activity is located on page 212 and is entitled, "Tag and Recapture in the Classroom." This is a fun
activity that allows students to practice using estimations. It is similar to the activity above but it recreates a tag-and-recapture scenario.
Give students the opportunity to practice estimation in all of their subject areas. For example, have students estimate how long they think it will take them to read a specific story or how many
words are on a page in a textbook. Students can also engage in estimation using "everyday" aspects of life. They can estimate how long it takes them to brush their teeth or how many fries are on
their plate. Hold a brainstorming session and let students come up with things they can estimate.
For additional lesson plans and ideas relating to this topic and many others try TeacherSource at PBS Online! You will find activities, lesson plans, teacher guides and links to other great
educational web sites! Search the database by keyword, grade level or subject area! Mathline and Scienceline are also great resources for teachers seeking teaching tips, lesson plans, assessment
methods, professional development, and much more!
The Idaho 2000 National Teacher Training Institute is made possible through the efforts of
Idaho Public Television | {"url":"http://www.idahoptv.org/ntti/nttilessons/lessons2000/olson.html","timestamp":"2014-04-18T21:09:56Z","content_type":null,"content_length":"19912","record_id":"<urn:uuid:adf21f2a-289a-4c0e-99db-31134e51238b>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00273-ip-10-147-4-33.ec2.internal.warc.gz"} |
December 11th 2008, 07:46 PM
Pre Lives
Inverse and 2 Word Problems (Algebra 2)
I'm having a hard time with this inverse problem and I'm not sure whether I have completely found the inverse or if I did something wrong.
Find inverse: y= 4x^2 - 10x
This is what I have done so far step by step
x= 4y^2 - 10y
0= 4y^2 - 10y - x
Then set up quadratic formula (used power of 1/2 because i don't know how to put sq root on here)
10 +(-10^2 - 4(4)(-x))^(1/2)/8
10 +(100 + 16x)^(1/2)/8 This is where I got stuck
Ran into more trouble with 2 word problems:
1st problem
Write an equation that can be used to solve the problem. The answer the question asked. A group of college students are volunteering for habitat for humanity during their spring break. They are
putting the finishing touches on a house they built. Working alone, Kathy can paint a certain room in 4 hours. Rhonda can paint the same room in 6 hours. How long will it take them working
together to paint the room?
2nd problem
The sum of the reciprocal of a number and the reciprocal of 6 less than the number is 8 times the reciprocal of the original number. Find the original number.
December 11th 2008, 11:00 PM
I'm having a hard time with this inverse problem and I'm not sure whether I have completely found the inverse or if I did something wrong.
Find inverse: y= 4x^2 - 10x
This is what I have done so far step by step
x= 4y^2 - 10y
0= 4y^2 - 10y - x
Then set up quadratic formula (used power of 1/2 because i don't know how to put sq root on here)
10 +(-10^2 - 4(4)(-x))^(1/2)/8
10 +(100 + 16x)^(1/2)/8 This is where I got stuck
There only exist an inverse function if the original function doesn't change its monotony. Since your function first is monotonically decreasing and after the vertex it is monotonically
increasing you have to split the domain into parts such that the function is monotonic.
1. Calculate the coordinates of the vertex by completing the square:
$y=4x^2-10x +\dfrac{25}4 - \dfrac{25}4 = 4 \left(x-\dfrac54 \right)^2-\dfrac{25}4$
Thus the vertex is at $V \left(\frac54\ ,\ -\frac{25}4 \right)$
2. Now you can split your function into two monotonic parts:
$f(x)=\left\{\begin{array}{l} 4 \left(x-\dfrac54 \right)^2-\dfrac{25}4\ ,\ x < \dfrac54\ ,\ f(x) \geq -\dfrac{25}4 \\ 4 \left(x-\dfrac54 \right)^2-\dfrac{25}4\ ,\ x \geq \dfrac54\ ,\ f(x) \geq -\
dfrac{25}4 \end{array}\right.$
For each of these parts exist an inverse function.
3. Swap the variables and solve the equation for y:
$x=4y^2-10y~\implies~y = \dfrac54\pm\ \dfrac14 \sqrt{25+4x}$
Swap the domain and range of the original function too that you get the domain and range of the inverse function(s):
4. Finally you have:
$f^{-1}(x)=\left\{\begin{array}{l}\dfrac54+\ \dfrac14 \sqrt{25+4x}\ ,\ x\geq -\dfrac{25}4\ ,\ y \geq \dfrac54 \\ \dfrac54-\ \dfrac14 \sqrt{25+4x}\ ,\ x\geq -\dfrac{25}4\ ,\ y < \dfrac54\end
5. The graphs of the original function and its inverse functions must be reflections over the line y = x (firste median?). See attachment. | {"url":"http://mathhelpforum.com/pre-calculus/64607-inverse-print.html","timestamp":"2014-04-20T01:49:39Z","content_type":null,"content_length":"8381","record_id":"<urn:uuid:eabbcea0-8b3f-42f0-9539-cc10d54f009b>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00525-ip-10-147-4-33.ec2.internal.warc.gz"} |
Expected revenue of all-pay auctions and first-price sealed-bid auctions with budget constraints
Results 1 - 10 of 14
- JOURNAL OF ECONOMIC SURVEYS , 1999
"... This paper provides an elementary, non-technical, survey of auction theory, by introducing and describing some of the critical papers in the subject. (The most important of these are reproduced
in a companion book, The Economic Theory of Auctions, Paul Klemperer (ed.), Edward Elgar (pub.), forthco ..."
Cited by 349 (2 self)
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This paper provides an elementary, non-technical, survey of auction theory, by introducing and describing some of the critical papers in the subject. (The most important of these are reproduced in a
companion book, The Economic Theory of Auctions, Paul Klemperer (ed.), Edward Elgar (pub.), forthcoming.) We begin with the most fundamental concepts, and then introduce the basic analysis of optimal
auctions, the revenue equivalence theorem, and marginal revenues. Subsequent sections address risk-aversion, affiliation, asymmetries, entry, collusion, multi-unit auctions, double auctions,
royalties, incentive contracts, and other topics. Appendices contain technical details, some simple worked examples, and a bibliography for each section.
- In Proceedings of the 7th ACM Conference on Electronic Commerce , 2005
"... We study a multi-unit auction with multiple bidders, each of whom has a private valuation and a budget. The truthful mechanisms of such an auction are characterized, in the sense that, under
standard assumptions, we prove that it is impossible to design a non-trivial truthful auction which allocates ..."
Cited by 74 (10 self)
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We study a multi-unit auction with multiple bidders, each of whom has a private valuation and a budget. The truthful mechanisms of such an auction are characterized, in the sense that, under standard
assumptions, we prove that it is impossible to design a non-trivial truthful auction which allocates all units, while we provide the design of an asymptotically revenue-maximizing truthful mechanism
which may allocate only some of the units. Our asymptotic parameter is a budget dominance parameter which measures the size of the budget of a single agent relative to the maximum revenue. We discuss
the relevance of these results for the design of Internet ad auctions.
- In Proceedings 42nd ACM Symposium on Theory of Computing , 2010
"... In this paper, we present the first approximation algorithms for the celebrated problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple
(heterogeneous) items and when bidders can have arbitrary demand and budget constraints. Our mechanisms are surprisingly ..."
Cited by 19 (1 self)
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In this paper, we present the first approximation algorithms for the celebrated problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous)
items and when bidders can have arbitrary demand and budget constraints. Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the
optimal ex-interim truthful mechanism with discrete correlated type space for each bidder. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal
ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard
rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our
results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles. We believe this approach will be useful in other Bayesian
mechanism design contexts.
"... Motivated by sponsored search auctions with hard budget constraints given by the advertisers, we study multi-unit auctions of a single item. An important example is a sponsored result slot for a
keyword, with many units representing its inventory in a month, say. In this single-item multi-unit aucti ..."
Cited by 4 (0 self)
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Motivated by sponsored search auctions with hard budget constraints given by the advertisers, we study multi-unit auctions of a single item. An important example is a sponsored result slot for a
keyword, with many units representing its inventory in a month, say. In this single-item multi-unit auction, each bidder has a private value for each unit, and a private budget which is the total
amount of money she can spend in the auction. A recent impossibility result [Dobzinski et al., FOCS’08] precludes the existence of a truthful mechanism with Pareto-optimal allocations in this
important setting. We propose Sort-Cut, a mechanism which does the next best thing from the auctioneer’s point of view, that we term semitruthful. In our mechanism, it is a weakly dominant strategy
for all agents to state their true budgets and to not understate their values.
, 2005
"... This dissertation studies mechanism design for various combinatorial problems in the presence of strategic agents. A mechanism is an algorithm for allocating a resource among a group of
participants, each of which has a privately-known value for any particular allocation. A mechanism is truthful if ..."
Cited by 3 (2 self)
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This dissertation studies mechanism design for various combinatorial problems in the presence of strategic agents. A mechanism is an algorithm for allocating a resource among a group of participants,
each of which has a privately-known value for any particular allocation. A mechanism is truthful if it is in each participant’s best interest to reveal his private information truthfully regardless
of the strategies of the other participants. First, we explore a competitive auction framework for truthful mechanism design in the setting of multi-unit auctions, or auctions which sell multiple
identical copies of a good. In this framework, the goal is to design a truthful auction whose revenue approximates that of an omniscient auction for any set of bids. We focus on two natural settings
— the limited demand setting where bidders desire at most a fixed number of copies and the limited budget setting where bidders can spend at most a fixed amount of money. In the limit demand setting,
all prior auctions employed the use of randomization in the computation of the allocation and prices. Randomization
, 909
"... Abstract. Inspired by Internet ad auction applications, we study the problem of allocating a single item via an auction when bidders place very different values on the item. We formulate this as
the problem of prior-free auction and focus on designing a simple mechanism that always allocates the ite ..."
Cited by 2 (0 self)
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Abstract. Inspired by Internet ad auction applications, we study the problem of allocating a single item via an auction when bidders place very different values on the item. We formulate this as the
problem of prior-free auction and focus on designing a simple mechanism that always allocates the item. Rather than designing sophisticated pricing methods like prior literature, we design better
allocation methods. In particular, we propose quasi-proportional allocation methods in which the probability that an item is allocated to a bidder depends (quasi-proportionally) on the bids. We prove
that corresponding games for both all-pay and winners-pay quasi-proportional mechanisms admit pure Nash equilibria and this equilibrium is unique. We also give an algorithm to compute this
equilibrium in polynomial time. Further, we show that the revenue of the auctioneer is promisingly high compared to the ultimate, i.e., the highest value of any of the bidders, and show bounds on the
revenue of equilibria both analytically, as well as using experiments for specific quasi-proportional functions. This is the first known revenue analysis for these natural mechanisms (including the
special case of proportional mechanism which is common in network resource allocation problems).
, 2008
"... A number of heterogeneous items are to be sold to a group of potential bidders. Every bidder knows his own values over the items and his own budget privately. Due to budget constraint, bidders
may not be able to pay up to their values. In such a market, a Walrasian equilibrium usually fails to exist ..."
Cited by 1 (0 self)
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A number of heterogeneous items are to be sold to a group of potential bidders. Every bidder knows his own values over the items and his own budget privately. Due to budget constraint, bidders may
not be able to pay up to their values. In such a market, a Walrasian equilibrium usually fails to exist and also the existing auctions might fail to allocate the items among the bidders. In this
paper we first introduce a rationed equilibrium for a market situation with financially constrained bidders. Succeedingly we propose an ascending auction mechanism that always results in an
equilibrium allocation and price system. By starting with the reservation price of each item, the auctioneer announces the current prices of the items in each step and the bidders respond with their
demand sets at these prices. As long as there is overdemand, the auctioneer adjusts prices upwards for overdemanded items until a price system is reached at which either there is an underdemanded
set, or there is neither overdemand nor underdemand anymore. In the latter case the auction stops. In the former case, precisely one item will be sold, the bidder buying the item leaves the auction
and the auction continues with the remaining items and the remaining bidders. We prove that the auction finds a rationed equilibrium in a finite number of steps. In addition, we derive various
properties of the allocation and price system obtained by the auction.
, 2011
"... This paper studies the impact of financial constraints on the persistency of high markups in a class of markets, including public procurement, known by practitioners as bidding markets. We
develop an infinite horizon model in which two firms optimally reinvest working capital and bid for a procureme ..."
Cited by 1 (1 self)
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This paper studies the impact of financial constraints on the persistency of high markups in a class of markets, including public procurement, known by practitioners as bidding markets. We develop an
infinite horizon model in which two firms optimally reinvest working capital and bid for a procurement contract each period. Working capital is constrained by the firm’s cash from previous period and
some exogenous cash flow, it is costly and it increases the set of acceptable bids. We argue that the latter is a natural consequence of the presence of progress payments or the existence of moral
hazard. We say that the firm is (severely) financially constrained if its working capital is such that only bids (substantially) above production cost are acceptable. We show that markups are
positive (high) if and only if one firm is (severely) financially constrained. Our main result is that markups are persistently high because one firm is severely financially constrained most of the
"... We study the problem of maximizing revenue for auctions with multiple units of a good where bidders have hard budget constraints, first considered in [2]. The revenue obtained by an auction is
compared with the optimal omniscient auction had the auctioneer known the private information of all the bi ..."
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We study the problem of maximizing revenue for auctions with multiple units of a good where bidders have hard budget constraints, first considered in [2]. The revenue obtained by an auction is
compared with the optimal omniscient auction had the auctioneer known the private information of all the bidders, as in competitive analysis [7]. We show that the revenue of the optimal omniscient
auction that sells items at many di#erent prices is within a factor of 2 of the optimal omniscient auction that sells all the items at a single price, implying that our results will carry over to
multiple price auctions. We give the first auction for this problem, to the best of our knowledge, that is known to obtain a constant fraction of the optimal revenue when the bidder dominance (the
ratio between the maximum contribution of a single bidder in the optimal solution and the revenue of that optimal solution) is large (as high as 2 ). Our auction is also shown to remain truthful if
canceled upon not meeting certain criteria. On the negative side, we show that no auction can achieve a guarantee of 2-# the revenue of the optimal omniscient multi-price auction. Finally, if the
bidder dominance is known in advance and is less than 5.828 , we give an auction mechanism that raises a large constant fraction of the optimal revenue when the bidder dominance is large and is
asymptotically close to the optimal omniscient auction as the bidder dominance decreases. We discuss the relevance of these results for related applications.
, 2013
"... We develop a model of bidding markets with financial constraints a la Che and Gale (1998b) in which two firms optimally choose their budgets. First, we provide an alternative explanation for the
dispersion of markups and “money left on the table” across procurement auctions. Interestingly, this expl ..."
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We develop a model of bidding markets with financial constraints a la Che and Gale (1998b) in which two firms optimally choose their budgets. First, we provide an alternative explanation for the
dispersion of markups and “money left on the table” across procurement auctions. Interestingly, this explanation does not hinge on significant private information but on differences, both endogenous
and exogenous, in the availability of financial resources. Second, we explain why the empirical analysis of the size of markups may be biased downwards or upwards with a bias positively correlated
with the availability of financial resources when the researcher assumes that the data are generated by the standard auction model. Third, we show that large concentration and persistent asymmetries
in market shares together with occasional leadership reversals can arise as a consequence of the firms internal financial decisions even in the absence of exogenous shocks. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=824763","timestamp":"2014-04-16T09:20:34Z","content_type":null,"content_length":"40629","record_id":"<urn:uuid:b8825109-9e46-4b77-96ce-6dee691eb407>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00353-ip-10-147-4-33.ec2.internal.warc.gz"} |
Information Primitives and
Information Primitives and Laws of Nature
A workshop 13-17 May 2008 at ETH Zürich.
This workshop saw theoretical computer science meet quantum physics to address two interrelated questions of overarching importance:
1. Why are the laws of nature quantum?
2. What are the fundamental possibilities and impossibilities of information processing?
Computer science and physics give two different ways of understanding the world. Consider for example a computation. From the perspective of computer science a particular protocol is carried out.
This can be broken up into a set of primitive operations and the possibility of performing a given protocol hinges on whether each primitive protocol is possible.
The physics perspective would instead view the computer as a physical system and therefore governed by quantum theory. Thus from the perspective of physics, the fundamental possibilities and
impossibilities of information processing derive from the laws of nature. Understanding this derivation is crucial for understanding the potential of quantum information science. Going in the other
direction one can attempt to formulate our theory of nature in terms of information primitives. This may provide a derivation of quantum physics from operational restrictions such as no cloning. It
is hoped that the physical content of such an axiomatization would be much clearer than the current abstract quantum axioms.
This workshop revolved around these questions, including addressing recent work on:
-Finding information theoretic considerations that explain why quantum correlations have their exact form.
-Fully axiomatizing quantum theory in terms of information primitives (c.f. the Fuchs-Brassard conjecture).
-Deriving given information primitives, such as key distribution, from a limited set of physical assumptions, rather than all of quantum theory.
-Considering more general notions of information.
-Studying mathematical frameworks for tackling these questions, such as generalized probabilistic theories.
-Information processing in operational foil theories such as Barrett's GNST.
Howard Barnum (Los Alamos)
Jonathan Barrett (Cambridge)
Cyril Branciard (Geneve)
Caslav Brukner (Wien)
Nicolas Brunner (Geneve) Ernst Specker (ETH Zurich)
Matthias Christandl (Cambridge/München) Antoine Suarez (Cent. Quant. Phil., Zurich)
Maurio D'Ariano (Pavia) Alain Tapp (Montreal)
Torsten Franz (Braunschweig) Ben Toner (CWI Amsterdam)
Chris Fuchs (Perimeter Institute) Alexander Wilce (Susquehanna)
Nicolas Gisin (Geneve) Stephanie Wehner (Caltech)
Jonathan Oppenheim (Cambridge) Reinhard Werner (Braunschweig)
Terry Rudolph (Imperial) Hans Westman (Perimeter)
Valerio Scarani (Singapore) Jurg Wullschleger (Bristol)
Ruediger Schack (Royal Holloway)
Volkher Scholz (Braunschweig)
Tony Short (Cambridge)
Rob Spekkens (Cambridge)
Howard Barnum
Non-classicality without
entanglement enables bit
Jonathan Barrett
Is quantum theory special?
Matthias Christandl
The Classical and Quantum
Finite de Finetti Theorems
Terry Rudolph
Data Tables and Ontological
We show that any number of parties can coherently exchange any one pure quantum state for another, without communication, given prior shared entanglement. Two
applications of this fact to the study of multi-prover quantum interactive proof systems are given. First, we prove that there exists a one-round two-prover quantum
Ben Toner interactive proof system for which no finite amount of shared entanglement allows the provers to implement an optimal strategy. More specifically, for every fixed input
string, there exists a sequence of strategies for the provers, with each strategy requiring more entanglement than the last, for which the probability for the provers
Coherent state exchange in to convince the verifier to accept approaches 1. It is not possible, however, for the provers to convince the verifier to accept with certainty with a finite amount of
multi-prover quantum shared entanglement. The second application is a simple proof that multi-prover quantum interactive proofs can be transformed to have near-perfect completeness by the
interactive proof systems addition of one round of communication.
Joint work with Debbie Leung and John Watrous.
Caslav Brukner
Physics of systems with
limited information resources
Jonathan Oppenheim
Generalized evolution laws
Slides (.pdf)
In a well-known generalization of classical probability theory, arbitrary compact convex sets serve as abstract state spaces for physical systems (with classical
Alex Wilce systems corresponding to simplices and quantum systems, to state spaces of $C^{\ast}$ algebras). One can define tensor products of such abstract state spaces, modeling
composite systems subject to a no-signaling condition. Once this is done, many things one might have been inclined to regard as characteristically {\em quantum}
Teleportation Protocols for phenomena -- notably, the general phenomenon of entanglement, as well as no-cloning and no-broadcasting theorems -- turn out to be quite generic features of all
Abstract State Spaces non-classical probabilistic theories. On the other hand, the existence of a teleportation protocol is a strong constraint, moving us closer to quantum theory. In this
talk, after briefly summarizing the framework of abstract state spaces, I'll outline what we currently understand about teleportation in this setting.
Joint work with Howard Barnum, Jon Barrett and Matt Leifer.
Ernst Specker
Infuturabilia and Quantum
Rob Spekkens
The power of epistemic Slides (.ppt)
restrictions in axiomatizing
quantum theory: from trits to
Chris Fuchs
SICkening axioms for quantum
Reinhard Werner
Three Old Schools of Axiomatic
Quantum Mechanics
Mauro D'Ariano Paper (.pdf)
Matrix-algebra representation In all probabilistic operational theories the
of probabilistic theories: the "transformations" have a C*-algebra representation over the linear space spanned by "effects". The Choi-Jamiolkowski isomorphism is sufficient to restrict the
special case of Quantum operational algebra to the quantum one, allowing to derive a composition law for effects, and leading to a systematic routine to derive a quantum operator
Mechanics. representation of probabilities. Possible candidates for operational principles for the Choi-Jamiolkowski isomorphism are discussed.
Ruediger Schack
Slides (.pdf)
Quantum Bayesianism,
tomography, and random numbers
Nicolas Gisin
Slides .pdf
Towards understanding Quantum
Jurg Wullschleger
Single-Serving Quantum
Slides (.pdf)
The relevant references:
Stephanie Wehner 1. The quantum moment problem and bounds on entangled multi-prover games
A. Doherty, Y.-C. Liang, B. Toner and S. Wehner, to appear in IEEE Conference on Computational Complexity '08, arXiv:0803.4373
The (quantum) moment problem:
deriving bounds on (quantum) 2. Higher entropic uncertainty relations for anti-commuting observables
correlations S. Wehner and A. Winter, To appear in Journal of Mathematical Physics, arXiv:0710.1185
3. Generalized non-local theories from relaxed uncertainty relations
W. van Dam, G. Ver Steeg and S. Wehner, In preparation.
Hans Westman In deBroglie-Bohm theory the quantum state plays the role of a guiding field. We investigate whether this is a universal feature shared by all hidden variable theories,
or if it is merely a peculiarity of deBroglie-Bohm theory. We exhibit a class of theories for which the quantum state plays the role, not as a guiding agent, but only
A Construction Method for as information about an underlying ontic state. We show that such theories are consistent with the Schrodinger equation if and only if there are positive solutions to a
Candidates of \psi-Epistemic gigantic system of linear equations. We end by modeling quantum measurements of a qubit within such theories and discuss possible ramifications of Hardy's ontological
Theories. excess baggage theorem.
Slides (.ppt)
The pioneering work by Barrett, Hardy and Kent stimulated the study of the link between the possibility of distributing non-local correlations and the possibility of
establishing secure communication.
Valerio Scarani Two approaches have been explored:
Cryptography and non-locality (1) Eve is not constrained by quantum physics, only by no-signaling. The latest results imply that security cannot be guaranteed in the most general possible scenario;
the minimal assumptions that have to be made are under study. The interest of this approach resides in the exploration of the ultimate requirements for unconditional
composable security of communication.
(2) Eve is constrained by quantum physics but (contrary to usual security proofs of QKD) Alice and Bob have to work in a black-box scenario: they can only choose their
input and receive an outcome. The bound for this "device-independent security" has been found against collective attacks. This approach relies on the exactness of
quantum physics to derive the bound, but does not require the end users to have any knowledge of quantum (or even classical) physics.
Volkher Scholz
Tensor norms and Tsirelson's
Slides (.ppt)
Bell proved that quantum entanglement enables two spacelike separated parties to exhibit classically impossible correlations. Even though these correlations are
Alain Tapp stronger than anything classically achievable, they cannot be harnessed to make instantaneous (faster than light) communication possible. Yet, Popescu and Rohrlich have
shown that even stronger correlations can be defined, under which instantaneous communication remains impossible. This raises the question: Why are the correlations
A limit on nonlocality in any achievable by quantum mechanics not maximal among those that preserve causality? We give a partial answer to this question by showing that slightly stronger
world in which communication correlations would result in a world in which communication complexity becomes trivial.
complexity is not trivial
G. Brassard, H. Buhrman, N. Linden, A. Méthot, A. Tapp et F. Unger, A limit on non-local correlations in any world where communication complexity is not trivial,
Physical Review Letter, vol. 96, 250401, 2006.
Slides (.ppt)
Tony Short
I will show that two copies of a noisy entangled state cannot be purified into a single less noisy, and hence more entangled, state using local operations and classical
No purification for two copies communication. This result is quite general, requiring only that there exists some local `twirling'
of a noisy entangled state. operations that map all states onto the relevant family, and hence applies to Werner states in quantum theory, noisy PR-boxes in `box-world' (Generalised Non-Signalling
Mechanics), and many other families of states in probabilistic theories.
In recognition of this workshop having been inspired by the Foils workshop 2007, the organizers of that workshop all chaired a session. They are: Jonathan Barrett, Tony Short, and Rob Spekkens.
Furthermore ETH members Roger Colbeck, Oscar Dahlsten, Renato Renner, and Stefan Wolf all chaired a session.
We gratefully acknowledge generous support from the Centre for Theoretical Studies and additional support from QSIT. | {"url":"http://www.qit.ethz.ch/workshops/IPLN2008","timestamp":"2014-04-18T10:34:27Z","content_type":null,"content_length":"38874","record_id":"<urn:uuid:d2e88616-54b2-43c3-8e2c-e69749184256>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00337-ip-10-147-4-33.ec2.internal.warc.gz"} |
Shania Twain
By Julie Dwyer | Published December 10, 2012 | Full size is 600 × 391 pixels
Shania Twain performs onstage at The Colosseum at Caesars Palace on Saturday, Dec. 1, 2012, in Las Vegas. The show kicks off a two-year residency at Caesars. (Photo by Eric Jamison/Invision/AP)
Tennessean Music
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A senseful meaning of 'approximation of manifolds'?
up vote 7 down vote favorite
Any continuous function can be uniformly approximated by smooth functions.
I would like to have something similar - in what-ever sense - for continuous manifolds.
For example, by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\mathbb R^{2n}$. You can construct a
continuous function $f$ with image $[-1,1]$ on $\mathbb R^{2n}$, whose zero level set is exactly (the image of) $M$.
A meaning of "approximating a manifold" would be to approximate such a level set function by smooth functions. However, Whitney's theorem is non constructive, you need a metric on the manifold for
the question to make sense, and there are likely to appear difficulties.
Do you where to find a elaboration on questions like the above? (Of course, different approaches are of interest as well.). Thank you.
approximation-theory manifolds reference-request
3 Gromov-Hausdorff metric would be an obvious one if you allow metrics. See the wikipedia page for details. IMO this is perhaps one of the most sensible approaches -- even in your function space
example, one way or another a metric lurks in the background. If you remove metrics, do you consider diffeomorphic manifolds to be identical for these purposes, or do you want to allow something
else? – Ryan Budney Dec 13 '10 at 9:45
3 There's also the "space of manifolds", things like the quotient $Emb(M, \mathbb R^\infty)/Diff(M)$, this is the space of submanifolds of $\mathbb R^\infty$ which are diffeomorphic to a given
manifold $M$, with the Whitney/weak topology. So "approximation of a manifold" here would be described by a neighbourhood base of a given point in this embedding space. – Ryan Budney Dec 13 '10 at
AWESOME question,+1 for Martin. – Andrew L Dec 14 '10 at 11:01
add comment
4 Answers
active oldest votes
A senseful meaning of ‘approximation of manifolds’?
A simply-looking meaning is the one of Gromov-Hausdorff, as mentioned by Ryan Budney. It has many cool applications, but none that I'm aware of are to topological manifolds.
The kind of approximation that one normally uses to prove something about topological manifolds is representing your manifold as an inverse limit of polyhedra (rather than smooth
manifolds). Examples of such use are incidentally given below.
I don't think that
by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\Bbb R^{2n}$.
up vote 8
down vote Whitney's embedding theorem applies only to smooth manifolds. To embed a non-triangulable topological manifold in $\Bbb R^{2n+1}$ (note the dimension shift) one "approximates" the
accepted manifold by polyhedra and embeds those first (using PL general position); no simpler way is known. This is called the Menger-Nöbeling-Pontryagin embedding theorem (or after various
subsets of the 3 authors); a clear proof appears in J. R. Isbell, Embeddings of inverse limits, Ann. of Math. 70 (1959), 73-84 and in Isbell's book "Uniform spaces". (Many other books
give a less explicit proof based on the Baire category theorem.)
If you really want to embed a non-triangulable topological $n$-manifold $M$ in $\Bbb R^{2n}$ (and not just in $\Bbb R^{2n+1}$), this is harder. Earliest results that seem to imply this
are in Bryant-Mio and Johnston's 1999 papers in Topology, available from Ranicki's website. A more direct (and I think much easier) proof is in this paper (based again on approximation
by polyhedra). In wondering about other possible approaches, I don't see how Kirby-Siebenmann could help: they show that $M\times\Bbb R$ is still not a PL manifold, if $n>4$; I don't
know if it could be triangulable.
add comment
I doubt that this is what you are looking for, but I cannot resist mentioning this kind of beautiful approximation. One way to define a (riemannian) manifold is via a commutative
version of Connes's spectral triples.
up vote 2 As shown by Roggenkamp and Wendland in this beautiful paper, there exist abstract conformal field theories which in a certain limit (typically a large volume limit) give rise to such
down vote commutative spectral triples. A expository version of their results can be found in this more recent preprint.
add comment
Here's an example where a sequence of manifolds approximates one given manifold in a definite sense.
For $i\in {\Bbb N}$, let $M_i$ be an orientable surface of genus $i$.
Let $T$ be the boundary of a tubular neighborhood in ${\Bbb R}^3$ of $$\{ (x,0,0)_{x\ge 0} \} \cup \{ (x,1,0)_{x\ge 0}\}\cup \{ (i,y,0)_{i\in{\Bbb N},y\in[0,1]}\}\ .$$
up vote 1 Now one can have maps from $T$ to each $M_i$, each map a homeomorphism when restricted to an ever larger open set, in such a way that these open sets exhaust $T$.
down vote
In general I suppose you'd want a net of manifolds of a fixed dimension and a corresponding directed set of open sets that exhaust the manifold you mean to approximate. You wouldn't even
have to have the limit manifold in advance if you had a suitable family of compatible maps connecting the various manifolds in your net.
I doubt you can, in any reasonable way, approximate the topology of a compact topological manifold by distinct compact topological manifolds of the same dimension.
add comment
You may also be interested in the geometric measure theory perspective on what is a nonsmooth surface supporting a "kind of differential geometry" (formulation from Frank Morgan's book
Geometric measure theory: a beginner's guide). The relevant notion is that of "rectifiable currents."
up vote 0
down vote Specifically regarding convergence of manifolds, have a look e.g. at the paper How Riemannian manifolds converge: A survey by Christina Sormani (arXiv:1006.0411) and references therein.
add comment
Not the answer you're looking for? Browse other questions tagged approximation-theory manifolds reference-request or ask your own question. | {"url":"http://mathoverflow.net/questions/49236/a-senseful-meaning-of-approximation-of-manifolds/49316","timestamp":"2014-04-21T10:31:16Z","content_type":null,"content_length":"69318","record_id":"<urn:uuid:91bf7956-cce5-47d8-939c-5808e2b94b32>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00137-ip-10-147-4-33.ec2.internal.warc.gz"} |
A note on the exact completion of a regular category, and its infinitary generalizations
- University of Edinburgh , 2000
"... Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be
constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
Cited by 13 (4 self)
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Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed.
Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a
simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the
realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with
finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to
- Theory Appl. Categ , 2001
"... A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D
, we describe a universal functor D # D ex preserving finite products and finite coproducts, for wh ..."
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A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D , we
describe a universal functor D # D ex preserving finite products and finite coproducts, for which D ex is extensive; that is, for all objects A and B the functor D ex /A D ex /B # D ex /(A + B) is an
equivalence of categories. As an application, we show that a distributive category D has a full distributive embedding into the product of an extensive category with products and a distributive
preorder. 1.
"... Abstract. We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κ-ary exact categories ” are a reflective sub-2-category of “κ-ary
sites”, for any regular cardinal κ. A κ-ary exact category is an exact category with disjoint and universal ..."
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Abstract. We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κ-ary exact categories ” are a reflective sub-2-category of “κ-ary
sites”, for any regular cardinal κ. A κ-ary exact category is an exact category with disjoint and universal κ-small coproducts, and a κ-ary site is a site whose covering sieves are generated by
κ-small families and which satisfies a solution-set condition for finite limits relative to κ. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2046385","timestamp":"2014-04-18T21:39:16Z","content_type":null,"content_length":"17486","record_id":"<urn:uuid:ac10eecb-7222-45f8-ab1d-668f2e2fea3f>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00577-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bits (Money)
Date: 02/26/2001 at 17:58:32
From: privers
Subject: Money
In the cheer: Two bits, four bits, six bits, a dollar
All for (school)
Stand up and holler!
what are bits?
Date: 02/27/2001 at 10:15:44
From: Doctor Roy
Subject: Re: Money
Thanks for writing to Dr. Math.
In the old days, a Spanish doubloon was worth enough so that people
would often split it up into 8 pieces, called bits or pieces of 8.
Similarly, the cheer refers to dividing a dollar up into 8 pieces, or
bits. So, 2 bits would be 25 cents, and 4 bits would be 50 cents.
I hope this helps.
- Doctor Roy, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/58446.html","timestamp":"2014-04-20T14:29:40Z","content_type":null,"content_length":"5819","record_id":"<urn:uuid:f8a3a805-031e-4ee0-a46e-3b0531f7dfad>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00180-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
Determine which, if any, of the three statements are equivalent. Give a reason for your conclusion. I) If the cat does not have claws, then the cat cannot scratch the furniture. II) If the cat can
scratch the furniture, then the cat has claws. III) If the cat has claws, then the cat can scratch the furniture. a. I and II are equivalent b. I and III are equivalent c. II and III are equivalent
d. I, II, and III are equivalent e. None are equivalent
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this is hard.
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The first thing you need to do is transform these statements into statements using propositional logic: \[p = \text{cat has claws}\]\[q = \text{cat can scratch furniture}\] 1) \(\neg p \implies \
neg q\) 2) \(q \implies p\) 3) \(p \implies q\) Since 1 & 2 are contrapositions of each other, therefore they are logically equivalent. However 3 is not equivalent to 1 or 2. Therefore I would
say the answer is \(a\)
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To be more specific the converse of a statement is not logically equivalent to the statement.
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So to be clear p->q does not imply q->p
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Do you understand?
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heck no!!!
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saifoo not funny!
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Here, I will explain. Lets look at these two statements: \[p=\text{"it is raining"}\]\[q=\text{"i am getting wet"}\] Suppose \(p \implies q\) If it is raining, then I am getting wet. Is that the
same as \(q \implies p\)? If I am getting wet then it is raining. The answer is clearly no. Do you understand so far?
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You could be getting wet, even when it is not raining (shower for instance).
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saifoo not funny!
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okay i get that part!
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So we've established that \(p \implies q\) does not necessarily mean \(q \implies p\). However, as I have stated before. I have made the claim that if \(p \implies q\) then \(\neg q \implies \neg
p\). So \(p \implies q\) is If it is raining then I am getting wet. Is that the same as \(\neg q \implies \neg p\) If I am not getting wet, then it is not raining. Well we have stated that if it
is raining then I am getting wet. So if I am not getting wet then it is not raining, or the first implication would not hold. Do you see how these two are logically equivalent?
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yes i understand now!
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So now apply this to the original problem. Do you see how only the first two statements about the cat are logically equivalent?
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saifoo not funny!
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so what is my awser
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Well look at the choices. What do you think is the correct answer?
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A sound reasonable!
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Yes, A is the correct answer.
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so how wouldi right this!
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I suggest you read about contrapositive and converse statements.
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Once you fully understand those two concepts, it will be immediately clear that only 1 & 2 are equivalent.
Your question is ready. Sign up for free to start getting answers.
is replying to Can someone tell me what button the professor is hitting...
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Two port negative resistance element - diyAudio
The four op-amp circuit is simple to understand.
Let R1,2,3,4,5 be 100 ohm.
Let R6,7,8,9,10,11,12,13 be 100k ohm.
Let a voltage difference of dv appear between in1 and in2.
Then by the op-amp action of U1 and U2, dv occurs across R1.
This creates a current through R1 which is the same current
through R4 and R5.
Since R1=R4=R5 then dv also occurs across R4 and R5.
The op-amp circuit of U3,R6,R7,R10,R11 and R2 form a voltage to
current converter, i.e. the voltage across R4 occurs across R2.
Thus a current flows through R2 equal to dv/R2 and
since R1=R2 then this current is also equal to Dv/R1.
The same reasoning also applies to the lower op-amp U4 circuit
as a voltage to current converter.
Thus the currents at the in1 and in2 nodes will be equal and
opposite in sign and equal to +/- dv/R1.
Then for a positive dv, current flows out of in1 and into in2.
Thus the input Z is -R1.
The circuit is an op-amp version of two second generation current
conveyors forming a negative impedance converter or nic.
See the attachment. | {"url":"http://www.diyaudio.com/forums/parts/75925-two-port-negative-resistance-element.html","timestamp":"2014-04-21T11:03:57Z","content_type":null,"content_length":"76842","record_id":"<urn:uuid:f8e5f5e3-d520-48f2-ab53-a553581ef923>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00595-ip-10-147-4-33.ec2.internal.warc.gz"} |
A. The coupled mode space approach
B. The uncoupled mode space approach
C. A fast uncoupled mode space approach
A. Benchmarking of the FUMS approach
B. Device physics and characteristics
A. Theory
B. Results | {"url":"http://scitation.aip.org/content/aip/journal/jap/96/4/10.1063/1.1769089","timestamp":"2014-04-16T13:39:43Z","content_type":null,"content_length":"113088","record_id":"<urn:uuid:377cbb0e-c76f-4181-8c2a-d7efd7160bc8>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00147-ip-10-147-4-33.ec2.internal.warc.gz"} |
This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis.
It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators,
inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of
The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the
mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory
of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal
answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.
"The nature of the writing is impressive, and any library owning this volume, and other volumes of he series, will be a rich library indeed. This book can work out well as a text for further study at
higher graduate level and beyond. For many a mathematician, it works well as a collection of enjoyable chapters; and most importantly, it can comfortably serve well as a reference resource and study
material. They will be grateful to the publishers and the authors, for the volume includes a wealth of interesting and useful information on many important topics in the subject. . . . In short, a
scintillating volume, full of detailed and thought-provoking contributions. Readers who bring to this book a reasonably strong background of the topics treated in the volume and an open mind will be
well rewarded."--Current Engineering Practice
Subject Area:
• Mathematics
Shopping Cart: | {"url":"http://press.princeton.edu/titles/8958.html","timestamp":"2014-04-18T10:36:25Z","content_type":null,"content_length":"17038","record_id":"<urn:uuid:59845714-91da-47cd-95e3-704fd05073aa>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00474-ip-10-147-4-33.ec2.internal.warc.gz"} |
More discrete mathematics questions
1) suppose you have 70 books (35 novels, 20 history books and 15 math books). Assume that all 70 books are different
A) in how many different ways can you put 70 books in a row on a shelf?
B) in how many different ways can you choose a set of 12 books to give to a friend?
C) in how many different ways can you choose a set of 4 history books and 8 novels to give to a friend?
D) in how many different ways can you put the 70 books in a row on a shelf if the novels are on the left, the math books are in the middle and the history books are on the right?
2) what is the coefficient of x^40 and y^10 in the expansion of (2x+y)^50?
3) give a simple expression for the value of the following sum (as a function of n):
C(n,0) + 3 * C(n,1) + 3^2 * C(n,2) +...+ 3^n-1 * C(n,n-1) + 3^n * C(n,n).
Your formula should not involve sums or combinatorial symbols like P(n,r) or C(n,r). Using the binomial theorem briefly justify why your answer is correct.
4) you have 20 pennies, 60 nickels and 40 dimes. Assume that the pennies, dimes and nickels are identical. In how many different ways can you put all the coins in a row?
5) assume we are using LaPlace's probability model, where all outcomes are equally likely. An urn contains 70 balls, of which 10 are red, 20 are blue and 40 are green. Let b be a randomly chosen
A) what is p(b is not green)?
B) what is p(b is blue | b is not red)?
Please help with these questions! Thanks! | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=242590","timestamp":"2014-04-18T19:14:20Z","content_type":null,"content_length":"30000","record_id":"<urn:uuid:84e69dcb-d7ae-4a0d-9ede-1eb27228d007>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00073-ip-10-147-4-33.ec2.internal.warc.gz"} |
How do I preallocate memory in MATLAB...
1. The problem statement, all variables and given/known data
I get the error that the variable changes size with each loop iteration then it suggests that I preallocate memory to overcome this. How would I do that?
2. Relevant equations
3. The attempt at a solution
here is my code
for i = 1:10
y(i+1) = y(i)+ (t(i)^2*0.5);
thanks for the help | {"url":"http://www.physicsforums.com/showthread.php?t=569719","timestamp":"2014-04-18T21:19:23Z","content_type":null,"content_length":"19960","record_id":"<urn:uuid:70a39162-e24b-43d3-a190-d2565b695e5b>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00019-ip-10-147-4-33.ec2.internal.warc.gz"} |
the PHP ternary operator equivalent on Python
bonono at gmail.com bonono at gmail.com
Sat Nov 19 08:19:47 CET 2005
Steven D'Aprano wrote:
> Why do you assume that everything you need for your list comprehension has
> to go into a single line? Chances are your list comp already calls
> functions, so just create one more for it to use.
> py> def describe(cond):
> ... if cond:
> ... return "odd"
> ... else:
> ... return "even"
> ...
> py> L = [describe(n % 2) for n in range(8)]
> py> L
> ['even', 'odd', 'even', 'odd', 'even', 'odd', 'even', 'odd']
this makes me "jump" from the list comprehension to the function,
harder to follow than 'in place', especially when the function is
simple, it is not necessary. What is the reason for introducing list
comprehension and then later, generator expression when :
for x in range(8):
return a
for x in range(8):
yield describe(x%2)
can do the same thing ?
Doesn't it violate the general idiom that it better has one and only
one way to do certain thing ? I believe part of the reason is that one
doesn't need to "jump" up and down.
> One major advantage is that this makes it easier to test your function
> describe() in isolation, always a good thing.
> Another advantage is that the idiom "call a function" is extensible to
> more complex problems:
> def describe(n):
> if n < 0:
> return "negative " + describe(-n)
> elif n == 0:
> return "zero"
> elif n % 2:
> return "odd"
> else:
> return "even"
> L = [describe(n) for n in range(8)]
> if much easier to understand and follow than using ternary expressions:
> # obviously untested
> L = ["zero" if n == 0 else \
> "negative " + ("odd" if n % 2 else "even") if n < 0 else \
> "odd" if n % 2 else "even" for n in range(8)]
> Yes, I've seen ternary expressions nested three and even four deep.
Now that is obsession for(and bad use of) one-liner. But picking a bad
example of one liner to rule out its effectiveness and readability is a
stretch I would say.
More information about the Python-list mailing list | {"url":"https://mail.python.org/pipermail/python-list/2005-November/296876.html","timestamp":"2014-04-18T04:29:29Z","content_type":null,"content_length":"5167","record_id":"<urn:uuid:e4049ffa-20da-4aef-ab64-909cf9894b79>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00644-ip-10-147-4-33.ec2.internal.warc.gz"} |
Absolute min and max help.
April 17th 2010, 07:31 PM #1
Dec 2009
Absolute min and max help.
The instructions say sketch graph by hand and use sketch to find absolute and locx and min for values of f.
$f(x)= ln x, 0 < x \leq 2$
The answer is abs max is f(2) = ln 2
I can see that there is no local max or min (i cheated though and didn't draw the graph, just that f'(x) can't equal to 0)
But I can see how 2 would be the absolute max because it's the biggest number in the interval, and no absolute min because the function will always try to reach 0 but doesn't reach it.
But the Extreme value theorem in the book says that
If F is continous on a closed interval [a,b], then f attains an absolute maximum value of f(c) and an absolute min at f(d) at some number c and d in [a,b]
doesn't that mean to have absolute max or min, the function has to be on a closed interval?
But the Extreme value theorem in the book says that
If F is continous on a closed interval [a,b], then f attains an absolute maximum value of f(c) and an absolute min at f(d) at some number c and d in [a,b]
doesn't that mean to have absolute max or min, the function has to be on a closed interval?
Note that:
$x \in [a,b] = a \leq x \leq b$
$x \in (a,b] = a < x \leq b$
$x \in [a,b) = a \leq x < b$
$x \in (a,b) = a <x<b$
So the definition in your book says the max lies within [a,b]. This includes when x = a or b too.
April 17th 2010, 08:34 PM #2
Super Member
Jan 2008 | {"url":"http://mathhelpforum.com/calculus/139762-absolute-min-max-help.html","timestamp":"2014-04-18T17:26:51Z","content_type":null,"content_length":"33468","record_id":"<urn:uuid:1ca6e36d-9349-4087-923c-b4c7bb0f24ad>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00007-ip-10-147-4-33.ec2.internal.warc.gz"} |
Is this tower of blocks going to fall?
Given a stacking configuration for a series of square blocks, your function should return true if they are at equilibrium and false otherwise.
The block configuration for N blocks is provided as a input vector x with N elements listing the x-coordinates of the left-side of each block. The blocks are square with side equal to 1 (so the i-th
block left side is at x(i) and its right side is at x(i)+1). The y-coordinates of each block are determined implicitly by the order of the blocks, which are dropped "tetris-style" until they hit the
floor or another block.
All blocks are identical (same dimensions and mass) and perfectly smooth (friction is to be disregarded).
Intermediate positions may be unstable. You are only required to determine whether the final configuration is stable.
Example (1)
x = [0 0.4];
The first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.4 1). This configuration is stable so your function should return true.
Example (2)
x = [0 0.6];
The first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.6 1). This configuration is unstable (the second block will fall) so your
function should return false.
Example (3)
x = [0 1.5 0.6];
The three block bottom-left corner coordinates are (0,0) (1.5,0) and (0.6,1). This configuration is stable so your function should return true.
Example (4)
x = [0 .9 -.9 zeros(1,5)];
This configuration is unstable, but note that if instead of five we add a few more blocks on top of this at the 0 position that will keep the tower from falling!
Example (5)
x = cumsum(fliplr(1./(1:8))/2);
This configuration is stable (see the classic optimal stacking solution) so your function should return true.
If you wish, you may display any given block configuration x using the code below:
for n=1:numel(x), y(n)=max([0 y(abs(x(1:n-1)-x(n))<1)+1]); end
Visit Block canvas for a related Cody problem. | {"url":"http://www.mathworks.com/matlabcentral/cody/problems/1908-equilibrium","timestamp":"2014-04-16T16:47:57Z","content_type":null,"content_length":"26214","record_id":"<urn:uuid:848d8616-35bb-4c9f-9c1c-1af47afd775e>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00285-ip-10-147-4-33.ec2.internal.warc.gz"} |
How to Solve Nonlinear Systems
In a nonlinear system, at least one equation has a graph that isn’t a straight line — that is, at least one of the equations has to be nonlinear. Your pre-calculus instructor will tell you that you
can always write a linear equation in the form Ax + By = C (where A, B, and C are real numbers); a nonlinear system is represented by any other form. Examples of nonlinear equations include, but are
not limited to, any conic section, polynomial of degree at least 2, rational function, exponential, or logarithm.
How to solve a nonlinear system when one equation in the system is nonlinear
If one equation in a system is nonlinear, you can use substitution. In this situation, you can solve for one variable in the linear equation and substitute this expression into the nonlinear
equation, because solving for a variable in a linear equation is a piece of cake! And any time you can solve for one variable easily, you can substitute that expression into the other equation to
solve for the other one.
For example, follow these steps to solve this system:
1. Solve the linear equation for one variable.
In this example, the top equation is linear. If you solve for x, you get x = 3 + 4y.
2. Substitute the value of the variable into the nonlinear equation.
When you plug 3 + 4y into the second equation for x, you get (3 + 4y)y = 6.
3. Solve the nonlinear equation for the variable.
When you distribute the y, you get 4y^2 + 3y = 6. Because this equation is quadratic, you must get 0 on one side, so subtract the 6 from both sides to get 4y^2 + 3y – 6 = 0. You have to use the
quadratic formula to solve this equation for y:
4. Substitute the solution(s) into either equation to solve for the other variable.
Because you found two solutions for y, you have to substitute them both to get two different coordinate pairs. Here’s what happens when you do:
Therefore, you get the solutions to the system:
These solutions represent the intersection of the line x – 4y = 3 and the rational function xy = 6.
How to solve a nonlinear system when both system equations are nonlinear
If both of the equations in a system are nonlinear, well, you just have to get more creative to find the solutions. Unless one variable is raised to the same power in both equations, elimination is
out of the question. Solving for one of the variables in either equation isn’t necessarily easy, but it can usually be done. After you solve for a variable, plug this expression into the other
equation and solve for the other variable just as you did before. Unlike linear systems, many operations may be involved in the simplification or solving of these equations. Just remember to keep
your order of operations in mind at each step of the way.
When both equations in a system are conic sections, you’ll never find more than four solutions (unless the two equations describe the same conic section, in which case the system has an infinite
number of solutions — and therefore is a dependent system). Four is the limit because conic sections are all very smooth curves with no sharp corners or crazy bends, so two different conic sections
can’t intersect more than four times.
For example, suppose a problem asks you to solve the following system:
Doesn’t that problem just make your skin crawl? Don’t break out the calamine lotion just yet, though. Follow these steps to find the solutions:
1. Solve for x^2 or y^2 in one of the given equations.
The second equation is attractive because all you have to do is add 9 to both sides to get y + 9 = x^2.
2. Substitute the value from Step 1 into the other equation.
You now have y + 9 + y^2 = 9 — a quadratic equation.
3. Solve the quadratic equation.
Subtract 9 from both sides to get y + y^2 = 0.
Remember that you’re not allowed, ever, to divide by a variable.
You must factor out the greatest common factor (GCF) instead to get y(1 + y) = 0. Use the zero product property to solve for y = 0 and y = –1.
4. Substitute the value(s) from Step 3 into either equation to solve for the other variable.
This example uses the equation solved for in Step 1. When y is 0, 9 = x^2, so
When y is –1, 8 = x^2, so
Be sure to keep track of which solution goes with which variable, because you have to express these solutions as points on a coordinate pair. Your answers are
This solution set represents the intersections of the circle and the parabola given by the equations in the system. | {"url":"http://www.dummies.com/how-to/content/how-to-solve-nonlinear-systems0.html","timestamp":"2014-04-18T19:10:27Z","content_type":null,"content_length":"58964","record_id":"<urn:uuid:452d57a5-0e59-4537-b9c7-574c2794fccc>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00286-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: [Haskell-cafe] Re: (flawed?) benchmark : sort
Dan Weston wrote:
6.3.2 (The Ord Class):
"The Ord class is used for totally ordered datatypes."
This *requires* that it be absolutely impossible in valid code to distinguish equivalent (in the EQ sense, not the == sense) things via the functions of Ord. The intended interpretation of these
functions is clear and can be taken as normative:
forall f . (compare x y == EQ and (f x or f y is defined))
==> f x == f y)
Thanks Dan. I didn't grasp the significance of this at first, but
I believe you are correct. But maybe it should be "=" not "=="
in the last line?
forall f . (compare x y == EQ and (f x or f y is defined))
==> f x = f y)
So assuming your (and my) logic is correct, the existing report text
does indeed settle the original dispute that sparked this thread.
Essentially you can't have 2 distinct values that compare equal,
so if they do then they must be indistinguishable? Is that right?
So there is no need for the sort on a list of elements whose type
is an instance of Ord to be "stable" as the difference between
the results of a stable and unstable sort cannot be observable
for any (correct) Ord instance (assuming the the instances compare
method was used to perform the sort).
So if we have a compare method on this type we can establish the
== method:
x == y = case compare x y of
EQ -> True
_ -> False
and from this it follows that (x == y) = True implies x and y are
So I believe for types that are instances of both Eq and Ord, this
settles the question of what (x == y) = True implies.
So now I'm wondering what about types that are instances of Eq
but not of Ord? Well from para. 6.3.1
"The Eq class provides equality (==) and inequality (/=) methods."
Well I guess assuming that saying two values are "equal" is another
way of saying they are indistinguishable then I think it's pretty
clear what the report is saying. This interpretation also ensures
consistency between Eq/Ord instances and Eq only instances.
Assuming this is all correct, I think I can sleep easier now I can
forget about all this "things being equal and not equal at the same
time" craziness, at least for Eq/Ord instances that are compliant
with the standard (which are the only ones I care about).
I think anyone wanting standard classes with different mathematical
properties should define them, stick them in Hackage and propose
them for Haskell-prime (if that's still happening?)
Adrian Hey
Haskell-Cafe mailing list | {"url":"http://www.archivum.info/haskell-cafe@haskell.org/2008-03/00456/Re-(Haskell-cafe)-Re-(flawed-)-benchmark--sort.html","timestamp":"2014-04-19T09:33:41Z","content_type":null,"content_length":"16456","record_id":"<urn:uuid:bf818f20-6f07-4634-b2cf-cf056b7b8069>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00214-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Simple loop
02-25-2009 #1
Registered User
Join Date
Feb 2009
Simple loop
Hi, I need some help on the following problem:
An number of students have taken two exams, exam 1 and exam 2. (We don't know exactly how many) The marks obtained are to be typed into the program. (Using scanf() I presume)
The first two numbers typed in are the scores obtained by the first student. The next two numbers are scores for the second student and so on. Write a program that calculates and then print the
average score for exam1 and for exam 2.
The scores entered must be within the 0-100 range. Otherwise the marks should be ignored.
Here is what I have so far and it looks very messy:
#include <stdio.h>
int main()
int e1, e2, sum1, sum2;
sum1 = 0;
sum2 = 0;
scanf("%d %d", e1, e2);
if (e1 >= 0)
if (e1 <= 100)
sum1 = sum1 + e1;
scanf("%d %d", e1, e2);}
if (e2 >= 0)
if (e2 <= 100)
sum2 = sum2 + e2;
scanf("%d %d", e1, e2);
scanf("%d %d", e1, e2);
if (e2 >= 0)
if (e2 <= 100)
sum2 = sum2 + e2;
scanf("%d %d", e1, e2);
scanf("%d %d", e1, e2);
scanf("%d %d", e1, e2);
I know that we must somehow use the 'while' command but I don't exactly know how. How do I even start this problem?
Okay, how do you get the average of something?
For starters I would suggest spending time in coming up with a sound algorithm.
Start by entering the scores for exam 1 and 2 (s1 and s2). The average (a1 and a2) for exam1 will be the sum (sum1, sum2) of all the scores in exam1, likewise for exam 2. Then divide by the
number (n) of students who had taken the test. Since n will increase by one everytime I enter a set of two scores, n++ could be used. But we start off with one student, so I initialized n=1. Same
thing with sum1 and sum2
int s1, s2, sum1, sum2, n;
float average1, average2;
sum1 = 0;
sum2 = 0;
n= 1;
I will enter the two scores by scanf. Then we can start on the sum. Since for the first student, the initial sums are 0. I think I should put
scanf("%d %d", &s1, &s2);
sum1 += s1;
sum2 += s2;
02-25-2009 #2
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Partial Orders
May 2nd 2009, 05:38 PM #1
Junior Member
Mar 2009
Partial Orders
A = {1,3,5,7,9,10,12,15,8,21}
R = {(x,y) | y divided by x}
Im trying to prove that R is a partial order (reflexsive, antisymmetric and transitive).
Reflexsive: I said that it is true, because for all x, xRx. That is, every number in A is divided into itself once.
Antisymmetric: is false. Because if we have (5, 10) and (10, 5) then x is not equal to y. Is that correct? Because we get 2 and 0.5 and thats not equal?
Transitive:: if we have (5, 10, 15) then thats true but (15, 10, 5) false? Hence not transitive?
Can somebody help me out on this question?
I drew a hasse diagram for this but i dont know how to draw one on this forum. If someone can draw me one on the forum so i can check mine it would be of great help?
My maximal was 12 and 15 and minimal element was 1. Correct?
Thats what confusing me alittle.
All is said that "y is divided by x". I need prove that its a partial order.
that is what it says in words? that makes no sense. it should be y "divides" x, or maybe something like "y divided by x is an integer". just saying y divided by x can mean many things. divides
exactly? has a remainder of a certain amount?...etc
if there was a symbol used, say so. do not type the english if you are not sure how the notation is read
A = {1,3,5,7,9,10,12,15,8,21} R = {(x,y) | y divided by x}
Im trying to prove that R is a partial order (reflexive, anti-symmetric and transitive).
Reflexsive: I said that it is true, because for all x, xRx. That is, every number in A is divided into itself once.
Antisymmetric: is false. Because if we have (5, 10) and (10, 5) then x is not equal to y. Is that correct? Because we get 2 and 0.5 and thats not equal?
Transitive:: if we have (5, 10, 15) then thats true but (15, 10, 5) false? Hence not transitive?
Reflexive: I said that it is true, because for all x, xRx. That is, every number in A is divided into itself once.
Anti-symmetric: is false. Wrong
Transitive:: if we have (5, 10, 15) then thats true but (15, 10, 5) false? Wrong again.
You seem to have a most odd understanding of this relation.
2 divides 4 but 4 does not divide 2.
So if x divides y and y divides x then x=y: anti-symmetric.
So if x divides y and y divides z then clearly x divides z: transitive.
Plato, so your saying my relation definition will always only apply too numbers where x and y are the same; for e.g (2,2) (3,3). Hence x and y will divide and y and x will divide?
I tried drawing a hasse diagram of the partial order R to try get better understanding of it...
Can someone who knows how to draw these diagrams check if 12 and 15 is the maximal and 1 the minimal?
Thanks for the help, im not very good at maths but im trying my best!
May 2nd 2009, 05:45 PM #2
May 2nd 2009, 05:48 PM #3
Junior Member
Mar 2009
May 2nd 2009, 05:59 PM #4
May 2nd 2009, 06:07 PM #5
May 2nd 2009, 08:28 PM #6
Junior Member
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